Microsoft Word - Final Waddell Vol 3 No 2.doc Journal of Urban Mathematics Education December 2010, Vol. 3, No. 2, pp. 116–154 ©JUME. http://education.gsu.edu/JUME LANETTE R. WADDELL is an assistant professor of the practice in the Teaching and Learning – Mathematics Education Department in Peabody College at Vanderbilt University, 230 Appleton Drive, 261 Wyatt, Nashville, TN, 37203; email: lanette.r.waddell@vanderbilt.edu. Her work and research interests include implementation and evaluation of professional development models in teaching mathematics with understanding for K–8 pre- and in-service teachers; using representa- tions, tools, and technology to enhance mathematics teaching and learning; and the effects of cul- turally relevant pedagogies on teaching practices in mathematics. How Do We Learn? African American Elementary Students Learning Reform Mathematics in Urban Classrooms Lanette R. Waddell Vanderbilt University In this article, the author uses qualitative methodology to investigate how African American elementary students in an urban school engaged with a National Coun- cil of Teachers of Mathematics standards-oriented mathematics curriculum and how their engagement converged with or diverged from the offered patterns of teaching practices in classrooms. The findings suggest that student practices con- verged with teaching practices that reflected the African American cultural di- mension of social/affective interactions such as focused collaboration and active participation and diverged when students enacted practices that reflected expres- sive creativity and nonverbal interactions as with dramatic expression and im- provisation. Rather than looking at the divergent behaviors as social problems or behaviors needing remediation or punishment, considering what can be learned from these behaviors could enhance the mathematical identity and academic achievement of African American students. KEYWORDS: African American children, elementary mathematics, student learn- ing, reform mathematics, urban education n the United States, the mathematical under-achievement of many African Americans has been a source of concern for numerous reasons. Studies have shown that mathematics achievement is affected more by the school environment (e.g., curriculum, teacher qualifications and expectations, materials) than by the home environment (Lee, 1998; Roscigno, 1998); that mathematics is critical in advancing to higher education due to its filtering effect (Schoenfeld, 2002); and that mathematics is a civil rights issue in that, if children are not mathematically literate, they will be relegated to second-class economic status (Moses & Cobb, 2001). These studies demonstrate the importance mathematics holds in the arena of academic advancement and success. Researchers such as Gloria Ladson- Billings (1998), William Tate (1995), Na’ilah Nasir (2002), Danny Bernard Mar- tin (2000) and others have studied African American students and mathematics I Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 117 teaching and learning in an effort to understand many African American students’ persistent low levels of achievement on standardized assessments and in schools (The College Board, 1999; Langland & Emeno, 2003; National Center for Educa- tional Statistics, 2001). Many solutions have been suggested in the effort to provide better educa- tional opportunities for African American and low-income students: higher cur- riculum standards, testing regiments and accountability measurements, supple- mental education programs, improved teacher–student relationships, and more qualified teachers (see, e.g., The College Board, 1999; Ferguson, 2001; National Center for Educational Statistics, 2001; Ogbu, 2003). These solutions focus closely on improving schools, curricula, and the work of teachers to create better and more specific opportunities for African American students to excel. What is missing are studies that look closely at how students respond and interact with mathematics in light of these efforts by teachers and other school personnel. In this article, I describe a study about how a group of African American elementary students in an urban school negotiated and learned mathematics in Na- tional Council of Teachers of Mathematics (NCTM) standards-oriented class- rooms. By studying their patterns of interaction and engagement, I consider how their actions connected to and supported theories of African American cultural dimensions. Four questions guided the study: 1. What kinds of individual mathematical interactions occur when African American students engage in standards-oriented activities, tasks, and events? 2. What patterns of practice emerge across these African American students in their mathematical interactions? 3. In what ways do student practices converge with or diverge from the classroom practices and teacher anticipated norms? 4. To what extent do these student patterns of practice connect with research on Afri- can American cultural dimensions? What is Standards-Oriented Teaching? The NCTM (2000) Standards consist of two main parts: content standards and process standards. They are “inextricably linked” (p. 4); the foundation of the reform vision is built upon the fusion of these parts. The content standards are what NCTM has selected as the most important mathematical topics for each grade level. They are comprised of related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning as defined by NCTM. The process standards describe the five teaching and learning processes that NCTM promotes as necessary elements in the development of an investiga- tory mathematics classroom: problem solving, communication, connections, rea- soning and proof, and representation. These processes are crucial as the Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 118 framework of a classroom that strives to develop students who are mathematical inquirers and critical thinkers. The process standards support what is called mathematizing, “developing mathematical understandings from initial instruc- tional activities that focus on material objects, actions, and events, through the process of coming to see concrete situations in mathematical terms” (Lampert & Cobb, 2003, p. 240). A standards-oriented mathematics classroom is supported by a social con- structivist view of learning. Ernest (1996) described social constructivism in terms of the interconnectedness between the individual and the social. He stated, “Human subjects are formed through their interactions with each other as well as by their individual processes” (p. 342). In his work on social interactionism, Voigt (1996) also explained that, in learning, the focus is on the interaction between the individual, or subject, sense making and the social processes in which the individ- ual participates. Students bring their prior knowledge and cultural understandings to bear on mathematical activities or tasks. They then negotiate meanings within these tasks through individual cognitive dissonance, through their work with oth- ers, or through teacher feedback (explicit or implicit). This negotiation of mean- ing occurs mainly through discourse, which can be rigidly directed by teacher actions or more fluidly built through numerous interactions between students, teachers, and tools. As students have more and more opportunities to interact with others around particular meanings and concepts, “taken-as-shared” (p. 33) mean- ings develop. Only through the interaction of the participants in the development of routines and obligations can the stabilization of meanings and the creation of mathematical themes occur (p. 41). Wood’s (1994) concept of mutual orienting also describes the actions and interactions of classroom community members. She explains, “Patterns of interaction are seen as emerging from the individual’s inter- pretation of another’s actions and from the mutual orienting that occurs between the teacher and students…they build up negotiated expectations and obligations over the course of the school year” (p. 151). Cobb (2000) called this negotiation of classroom actions and interactions “social norms”; these norms are established through the interactions between the teacher and the students and amongst stu- dents themselves. From the view of these researchers, the vision of standards-oriented mathe- matics learning can be described as the negotiated mathematical meanings that develop through participation with provided tasks and tools; meanings that be- come shared by the group or community through interactions leading to the estab- lishment of classroom norms. To enhance this process of negotiating mathematical meanings within an activity or task, teachers provide students with opportunities to interact with other students and tools. Students should have op- portunities, when working on mathematical tasks, to reflect on their own proc- esses and the processes of others; this opportunity provides time to reorganize Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 119 information and to be taught to engage in the critique of and inquiry into mathe- matical concepts. These opportunities to reflect, critique, and inquire also provide the teacher with more information about the nature of students’ prior knowledge, backgrounds, worldviews, and interests, thereby allowing more connections to the students’ lives and understandings. Research on the Implementation of NCTM Standard-Oriented Curricula In the current political landscape, it is important to consider if standards- oriented practices for teaching mathematics will help engage all students in learn- ing mathematics deeply and with understanding and increase levels of achieve- ment in course grades and testing for all students. Numerous studies have analyzed and considered the effects NCTM standards-oriented curricula have had on student test score achievement and how African American students have fared in these classrooms and schools. One such study that focused on overall test score achievement was con- ducted by the Alternatives for Rebuilding Curriculum (ARC) Center (2002), a part of the Consortium for Mathematics and its Application in Massachusetts. They conducted a study that focused on the three National Science Foundation (NSF, 2002) funded standards-oriented curricula to establish what effect these types of curricula had on student test score achievement. The three curricula— Investigations in Data, Number, and Space; Everyday Mathematics; and Math Trailblazers—were implemented in three states: Massachusetts, Illinois, and Washington.1 Schools and grades chosen to participate in the study were selected by their usage of the NSF curricula and the length of implementation (at least 2 years) determined through telephone surveys and standardized test score record availability. It is important to note, however, that this data collection process did not provide details into particular teaching practices, nor did it examine the mean- ing or level of curricula implementation beyond the requirement that the curricula be in place for 2 years. Nevertheless, it yielded 742 classrooms and more than 100,000 students from Grades 3 to 5 in the three states. Each school was then matched to a comparison school based on reading test scores, income measures, and percentage of White students in the school. Overall, they found standards- oriented schools outperformed their matched counterparts on all test measures and mathematics strands, with measurement and computation showing the largest 1 Everyday Mathematics is a comprehensive pre-kindergarten through sixth-grade mathematics curriculum developed by the University of Chicago School Mathematics Project, published by Wright Group McGraw-Hill. Math Trailblazers is a full mathematics curriculum for grades K–5 that was developed by the Teaching Integrated Math and Science (TIMS) Project at the University of Illinois at Chicago, published by Kendall/Hunt Publishing Company. A complete description of the Investigation curriculum is described later in the methods section of this article. Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 120 gains. When the results for all three states were disaggregated by race/ethnicity and SES, all reform schools again outperformed the comparison schools. When examining data on African American students as a racial group, students in the standards-oriented classrooms performed approximately 4 percentile points better on all subtests and total scaled score than comparison students. What was discon- certing, however, was that African American students, as a group, scored far be- low all other race/ethnic categories of students regardless of the type of curriculum implemented in the school. A study by Briars and Resnick (2000) of the Pittsburgh, Pennsylvania School District analyzed achievement test scores and curriculum implementation levels to determine how well students achieved when the school district adopted the standards-oriented curriculum Everyday Mathematics. They grouped schools by two levels of teacher curriculum implementation, strong and weak, and matched these schools by demographics (e.g., free and reduced-priced lunch, fam- ily structure, mobility rate, and percentage of African American students). For their study, they defined strong implementing schools as having a majority of teachers who used all of the components of the curriculum and provided students with opportunities to engage in mathematics according to the NCTM process and content standards. Weak implementing schools were schools that had only one or two teachers using the curriculum as prescribed by the curriculum developers. They found that students in the strong implementing schools outperformed those in weak implementing schools on three strands of the standardized test used to measure achievement: computation skills, conceptual understanding, and problem solving. The scores of African American students in the strong implementing schools also rose, and, although the test score gap between them and White stu- dents narrowed, it was still large except on computational skills where the African American students outscored the White students (see also McCormick, 2005; Se- cada, 1992). In a similar study, Riordan and Noyce (2001) also found a positive correla- tion between the length of time (in years) that a teacher used the Everyday Mathematics curriculum and the achievement of the students, though as in the ARC study, they did not have a detailed way of analyzing the level of implemen- tation other than the length of time the curriculum was in use. Their study, using regression analysis on test score data, compared three groups: early implementers (more than 4 years of implementation), late implementers (less than 4 years), and non-implementer schools. The data were collected from the statewide Massachu- setts standardized test scores from 1999. They found that the student scores of the early implementer schools were the highest, followed by the late implementer schools. The non-implementer schools had the lowest average scores, though just slightly lower than late implementing schools. Thus, the longer a school has been using the standards-oriented curriculum, the higher the students scored on the Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 121 standardized test. An important note for this study, however, was the improved achievement of the African American students. The African American students in early implementing schools had average scores 9 points higher than African American students in the non-implementing schools. This point difference be- tween test schools was larger than all other racial groups except Hispanics. How- ever, in all schools, the overall scores of the African American students were lower than those of other student groups. Overall, these studies suggest that standards-oriented curricula have a posi- tive impact on the achievement of African American students after at least 2 years of implementation. However, the test score achievement gap continues. This points to a need to consider the implementation of curricula and teaching practices found in standards-oriented classrooms to have a more detailed view of the impact standards-based teaching practices have on students. Interactions with standards- oriented teaching practices, teacher beliefs and knowledge, response of the stu- dents to these practices; each of these can create markedly different patterns of practice in classrooms generally regarded as standards-oriented. Being able to document the practices, established norms, and cultures in standards-oriented classrooms can assist in creating a sharper picture of implementation and what standards-oriented classrooms might “look like.” African American Cultural Dimensions and Learning For several decades, social scientists have studied and described how (some) African Americans interact with and interpret the world and how this interpreta- tion affects learning (see, e.g., Boykin, 1983, 2001; Boykin & Bailey, 2000; Jones, 2003; Ladson-Billings, 1994; Shade, 1982, 1992). It is important to note, however, that this growing body of research does not intend to essentialize Afri- can Americans into some monolithic group of sameness but rather attempts to highlight shared beliefs, values, and customs (i.e., culture) that might be uniquely shared historically among Black Americans. For instance, Boykin (1983) identi- fied nine cultural dimensions that might be considered unique among African Americans: spirituality, harmony, movement, verve, affect, communalism, ex- pressive individualism, orality, and social time. He offered these dimensions as “motifs, patterns of behavior, and predilections” (p. 348) that distinguish African Americans from other groups of people in our country. More recently, Boykin and Bailey (2000) researched three of the African American learning dimensions that stemmed from Boykin’s earlier work on dimensions of African American cultural themes: communalism, an acculturation toward social and family relations; movement, rhythmic and expressive orientation toward life; and verve, heightened appreciation toward physical stimulation. Their study, conducted with 163 low- income African American students in grades 2 through 5 in a large urban city, Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 122 was designed to understand how these three cultural dimensions (might) affect learning preferences and orientations. They administered six surveys to see how cultural practices related to movement, communalism, and verve in the students’ homes correlated to the same areas in the students’ learning preferences in school. Their findings suggested that students most strongly oriented toward movement factors both at home and in school, which were expressed by items related to how music is incorporated into and improved everyday well-being. The communalism theme was shown through student preferences focused on a sense of common duty and support within the family (and extended family) on the home scale and a similar preference for working within a group and valuing friends on the school scale. Verve, while having a higher correlation between home and individual learning preferences than communalism, showed average scaled scores that were at or below the Likert mid-point range for those same items in school, which in- cluded student preferences in active oriented activities in the classroom and when playing. Though Boykin and Bailey’s (2000) work did not focus specifically on the mathematics learning of students, it could be said that most U.S. mathematics classrooms have social and participation norms that are at odds with these three themes of African American cultural dimensions. Vervistic and movement oppor- tunities, such as activities that require high levels of energy and participation, do not regularly show up in classrooms (other than physical education). Berry (2002), in his study of successful African American middle school boys, found that boys who displayed vervistic behaviors were kept out of higher-level mathe- matics classrooms, as the teachers and administrators did not consider the stu- dents’ academic abilities, only their social ones. According to Boykin (1983), this mismatch in cultural styles is a challenge that needs to be addressed as we move toward improving the academic achievement of African American students. Similarly, Shade (1992) studied an expanded view of cognitive style— perceptual, intellectual, and social domains—to examine the possibility of a unique African American learning style. She defined cognitive style as “a cultur- ally induced way in which individuals organize and comprehend their world” (p. 256). In her research, 178 grade 9 students were stratified by achievement levels and sex, and consisted of 92 African Americans and 86 European Americans with similar socioeconomic backgrounds. The students were administered three tasks that aligned with three areas of cognitive style under consideration. Her results found a significant difference between the African American and European American students on the perceptual processing task (p<0.0001), which aligned with the domains of field sensitive/field dependent, in which African American students showed preference for field sensitive tasks. This finding suggested that African American students focused on ideas holistically, in context, and in rela- tion to the environment rather than in a field independent view—using analytic Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 123 and systemic approaches to situations found in many mathematics classrooms (Stiff & Harvey, 1988). On the social interaction style instrument, a small, but sta- tistically significant difference (p<0.025) was found between African American and European American students in the area of orientation toward the world around us, where African American students tended to be more adaptive and spontaneous to occurrences in the world, as opposed to making plans and attempt- ing to organize the outside world. These findings led Shade to suggest that Afri- can Americans demonstrated a preference for considering the world in a more spontaneous, flexible, and less structured way than European Americans. In other research on African American cultural orientation, Jones (2003) de- veloped TRIOS, a theory that represents the attitudes, beliefs, and values of Afri- can American culture that arose out of African Americans’ need “for self- protection and self-enhancement in a universal context of racism” (p. 239). TRIOS is defined by five main components: Time, Rhythm, Improvisation, Oral- ity, and Spirituality. Through these components, Jones noted a collaborative, rather than oppositional, duality of individualism and collectiveness in African American culture: a sense of independence, “be yourself at all times” through im- provisation and orality; along with interdependence, “harmony in my group” through a sense of collectiveness with other African Americans. This duality means that there is a sense of a collective group identity that supports a need for individuality, rather than conformity. Jones’s (2003) research on TRIOS, which in many respects mirrored Boykin’s (1983) work, was conducted with 1415 respondents of different races and ethnicities ranging in age from 14 to 62 from across the country, with most respondents being college-aged students. They responded to a 77-item survey in which they were asked to respond to statements concerning different aspects of the TRIOS dimensions. In his study, he found that four dimensions of TRIOS were captured well in the statements on the survey: spirituality, belief in a higher power that influences all living things; improvisation, goal directed individualistic and creative problem solving in a distinctive style; orality, preference to face-to- face communication and personal expression; and time (also called present orien- tation), personal perspectives are in the present and time is derived from tasks and not prescribed by them. In an analysis of racial and gender differences in orienta- tion to the different dimensions of TRIOS, the study showed that African Ameri- cans tended to exhibit more dimensions of TRIOS than non-African Americans. In particular, spirituality and time provided the strongest African American corre- lations and greatest Black–White differences in his study. Although this study fo- cused on adolescent and adult respondents, the findings can also be relevant to younger African Americans. Willis’ (1989) review of the research on African American learning styles allowed her to categorize the research on cultural dimensions into four areas. In Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 124 this study, those categories were used to frame an overarching organization to the cultural themes and orientations found in the work of Boykin (1983), Shade (1992) and Jones (2003) (see Table 1): social/affective response, African Ameri- cans are people-oriented, emphasize the affective domain, interaction and learning in social groups is important; harmonious, interdependence and communal as- pects of people and environments are respected, knowledge is to be relevant and thought about holistically; expressive creativity, African Americans tend to prefer environments that offer adaptive, variable, and novel situations, and stylistic si- multaneous stimulation is preferred with verve and oral expression; and nonverbal response, use of intonation and body language are important ways of communi- cating, movement and rhythm are vital. Here, I added a fifth category, spirituality, to capture the belief that a greater power than humans influences all things and ac- tions, as reflective of the work by both Boykin (1983) and Jones (2003). When considered in school settings, these cultural dimensions act as filters for the stu- dents’ understanding of and participation in classroom interactions. Table 1 Categories of African American Cultural Dimensions (Adapted from Willis, 1989) African American Cultural Dimension Categories Related Cultural Dimensions as Described by Researchers Boykin (1983) Jones (2003) Shade (1982) Social/affective – affective response to stimuli, values social interactions above object interactions Communalism Affect Orality Social time Time orality Social cognition Harmonious – interdependence, holistic approaches to the world, purposeful uses of information Harmony Communalism Rhythm Field sensitive Worldview Expressive creativity – creative, adap- tive, spontaneous, vervistic, dramatic and enthusiastic oral expression, multiple stimuli preferred Orality Verve Expressive individualism Improvisation orality Stimulus variety Conceptual tempo Nonverbal – use of intonation in oral lan- guage, body language, movement and rhythmic expression Movement Verve Affect Harmony Rhythm Stimulus variety Field sensitive Spirituality – belief that a greater power than humans is at work and influences all things and actions Spirituality Spirituality Conceptual tempo In more recent work, Boykin and Jones (2004) discussed how African American cultural dimensions and themes can be used to analyze the interactions between the learner and the learning environment provided by teachers and schools. Using the previous research on African American cultural and TRIOS dimensions, they developed a psychosocial integrity approach to teaching that can Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 125 enhance opportunity for learning and achievement not only for African American students but also for all students. The notion of psychosocial integrity considers that people experience life through: complexity, experiences are understood in multiple facets, variations, and depths; coherence, making sense of life through our frame of reference; and texture, experiences can be examined through many points of view. Boykin and Jones believe that schooling practices that draw on psychosocial integrity ideas as assets of the students could lead to enhanced out- comes for all students. Such an approach incorporates five practices that should be the focus of school activities: promoting meaning making through multiple modalities; teaching thinking and learning strategies while fostering critical think- ing processes; building a learning community; utilizing cultural resources of stu- dents, families, and their communities; and providing a supportive yet demanding learning environment. This approach is supported by the work on social construc- tivism; individual sense making is considered and built upon in the process of de- veloping and negotiating community norms and practices in the classroom (Ernest, 1996). Psychosocial integrity also closely connects with the research on culturally relevant pedagogy as both approaches draw heavily on understanding and incorporating students’ lives into classroom practices (Ladson-Billings, 1994). Table 2 shows how psychosocial integrity practices relate to research on African American cultural and learning dimensions as well as culturally relevant pedagogy. Promoting meaning making through multiple modalities is guided by cul- tural dimensions that encourage students to draw on their knowledge and the knowledge of others to make sense of tasks. Students draw on ideas and beliefs of the world around them and on feelings and thoughts within themselves to under- stand the world. The dimensions of harmony, social/affective responses, and spiri- tuality are reflected in this theme. By teaching thinking and learning strategies, teachers are providing students with opportunities to focus deeply on tasks and topics in many forms and in many ways and use whatever tools are necessary to accomplish those tasks. Allowing movement and physical and oral expression when working, along with paying close attention to task understanding and com- pletion, supports the dimensions of harmony, nonverbal response and expressive creativity. To have the collaboration skills necessary to engage with others on a topic and use others as learning resources, teachers build learning communities in their classrooms. Working communally, speaking with others while working, and interpreting others’ ideas encourage students to turn to others as learning partners. Understanding that students come from a neighborhood or community that sup- ports particular cultural and social habits and these habits have a large impact on their lives in school is an important reason for considering and using students’ cultural resources in teaching. Using individual differences as a teaching tool and making clear connections to students’ lives outside of school can provide students Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 126 with a sense of belonging. The African American cultural dimensions of har- mony, expressive creativity, and nonverbal response support these ideas. Finally, creating a supportive but demanding environment can be used to develop a sense of harmony among the students as they rally together in support of success for all (Ware, 2006). Learning about each student and their individual levels of understanding can help teachers focus on student improvement and create attainable goals and ex- pectations. These psychosocial integrity practices could allow school personnel and teachers to develop pedagogical practices for African American students that draw on their often-preferred modalities and cultural and psychological schemas. Table 2 Psychosocial Integrity Practices and Related African American Cultural Dimensions Psychosocial Integrity Teaching Practices Psychosocial Integrity Practices Defined Supporting Cultural Dimension Categories (Willis, 1989) Culturally Relevant Pedagogy (Ladson-Billings, 1994) Promoting meaning making Making connections among topics, to students experiences, and prior and common knowledge Harmonious Social/Affective Spirituality Drawing on prior knowledge, cultural themes, individual differences Teaching thinking and learning strategies Providing tools for deep, constructive, active engagement with academic content; multiple stimuli and a variety of activities are used in teaching Harmonious Nonverbal Expressive creativity Considering knowledge as something constructed and critiqued Building a learning community Learning is interdependent, collaboration is fostered through personal interactions Social/Affective Harmonious Development of learning community and collective empowerment through collaboration Using cultural resources Drawing on students’ every day practices, cultural habits, and worldviews in every day practices Harmonious Expressive creativity Nonverbal Community connections out- side of the school walls Creating a supportive and demanding learning environment Maintaining high expectations, focus on effort and improvement Social/Affective Academic skill development and achievement is paramount Standards-oriented practices in mathematics most strongly correlate to the psychosocial integrity practices of promoting meaning making, teaching thinking and learning strategies, and building a learning community. These practices offer opportunities for teachers to develop mathematical understandings in their stu- dents, draw on students’ prior mathematical knowledge, help students use a vari- ety of tools and modalities to communicate mathematically, and support student collaboration and interdependence. Through the explicit development of mathe- matical ideas in discussions and in tasks, and supported by continuous classroom discussions on expected behaviors and norms, standards-oriented practices en- Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 127 courage the development of a supportive and demanding learning environment as well. Understanding how to incorporate these psychosocial practices into the eve- ryday work of schooling will require long-term investment into teachers’ profes- sional development. African American Student Identity, Engagement, and Agency in Mathematics Nasir (2002) and Nasir and Hand (2008) studied how student identity and engagement might create opportunities for students to learn. Studying African American students in mathematics-based activities that occurred outside of school as well as in-school mathematics activities raised the question of how these stu- dents could be successful with mathematics in out-of-school activities but unsuc- cessful with mathematics in school, and the impact of student identity in each these settings. Nasir framed identity as “being constructed by individuals as they actively participate in cultural activities.…[It] both shapes and is shaped by the social context” (p. 219). The concern was how the learning communities or set- tings influenced the development of identity. In particular, Nasir and Hand fo- cused on practice-linked identities, “identities that people come to take on, construct, and embrace that are linked to participation in particular social and cul- tural practices” (p. 147). Drawing on Wenger’s (1998) idea of how identities are formed within a community of practice through engagement, alignment, and imagination, Nasir analyzed how students developed identities and goals when playing dominoes and basketball. She focused on the practice-linked goals the students created for themselves that allowed them to participate in the games. Embedded in these were mathematical goals, such as calculating shot percentages or making number combinations, that allowed the students to reach their practice- linked or learner’s goal. She noticed that students would develop goals that per- mitted them to be a part of the play and part of the player community and this, in turn, helped them identify with the game. This desire to be a part of the commu- nity of dominoes or basketball players motivated the students to create personal goals that meshed with the goals and rules of the game. In her view, learning was intertwined with student goals and their developing identity within the community of practice, in this case the community of dominoes or basketball. Students en- gaged and developed goals that allow them to participate in the community, which in turn helped them imagine and create new identities relative to the com- munity. This engagement led to a desire to learn more about the activity/game and generate and align new goals to continue participation in that community. In this case, the ability of the students to imagine themselves as part of the basketball or dominoes community provided a motivation to engage with the community, when given the opportunity. In another study that focused on African American students and their Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 128 mathematics learning, Martin (2000) studied the mathematical socialization of Af- rican American middle school students through a framework that allowed him to analyze how community, school, and individual themes and beliefs affected the success of African American students in school mathematics. Students in his study were interviewed and observed over a 1-year period, along with interviews of their guardians and teachers. He found that successful students were able to overcome a great number of negative forces, such as poor past mathematical ex- perience or pressure to underperform, through student agency, which, for these students, was defined as “actively constructing meanings for mathematics learn- ing and mathematics knowledge and acting on those meanings accordingly” (p. 170). The negative forces came from different areas of the students’ lives—some from family, some from school, most strongly from peers—leading Martin to suggest that the students were motivated to succeed “by an inner drive and self- determination to succeed” (p. 183). He was less certain about how these students developed a strong sense of identity and motivation toward success in school be- cause the students had widely varied lives and family backgrounds. Martin wrote, “although community and school forces do have the potential to affect their mathematics socializations and identities, these forces are not deterministic” (p. 185). How these students developed and used their identity and agency to manage negative forces and make beneficial academic choices determined their level of success in mathematics. Why might students decide to engage with particular communities of prac- tice? In these examples, students chose to participate in the community of domino or basketball players; they invoked their agency to make choices about what they needed to learn and what actions they needed to take in order to participate fully. Basketball and dominoes, for these students, were part of the social and historical culture of African Americans; these students saw other African Americans engage in these activities and that allowed them to imagine that they could participate in these activities as well. African American cultural dimensions such as improvisa- tion could occur when making moves during these activities, the communalism among players that emerged when participating in these activities, or the verve with which these games can be played made these activities attractive to young African Americans. Standards-oriented mathematics has the potential to provide a framework in which the identity and engagement needs of African American learners could be met. Teachers can create mathematical tasks and lessons that consider students’ prior knowledge and ways of knowing, which can encourage students to develop personal goals in conjunction with the tasks. The whole class/community can share ideas, which can spark student participation in mathematics lessons. With the increase in participation, students begin to see mathematics as something they can engage in and, in turn, can increase their identification and alignment with Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 129 mathematics. Developing such a mathematics community can provide a space for students to interact with mathematical ideas through many modalities and pre- ferred learning styles. In doing so, opportunities can emerge that entice students to engage with mathematics and align their practices and interactions with the mutu- ally established meanings, routines, and obligations of the classroom. Methods The work in this study was part of a larger ethnographic study, entitled Mathematics PLUS (MathPLUS), which examined the relationship between teacher learning and student performance in two urban elementary schools. The study reported here focuses on student performance and interaction within one of the schools. This school enrolled approximately 275 students from a range of eth- nic and social class backgrounds during the data collection period. Sixty-five per- cent of the students were eligible for free or reduced-priced lunch. The school drew many of its students from the surrounding neighborhood, which was pre- dominately African American, and included many low-income families along with some families from middle class backgrounds. As designated by the school district, the school was a desegregation school and, as such, drew a small number of White students who were admitted from outside the immediate neighborhood. From this small school of 11 classrooms, nine classroom teachers are repre- sented; the teachers’ classroom experience ranged from 6 years to over 30 years by the end of the study. Because this study focused on the learning and interac- tions of African American students over a period of at least three years, seven Af- rican American students from these classrooms had sufficient data to be included (see Table 3; names of all students and teachers are pseudonyms). Table 3 Students with Years and Grades During Study Student Name Years in Study Grades Kiana Ali 1999–2003 1st through 4th Candace Brown 1999–2003 1st through 4th Maya Connor 1999–2002 2nd through 4th Jordan Jones 2000–2003 1st through 3rd Samuel Quon 1999–2003 1st through 4th Felix Robinson 1999–2002 2nd through 4th Royce Rush 1999–2003 1st through 4th The Investigations Curriculum The school in the study adopted the mathematics program, Investigations in Number, Data, and Space (TERC, 2004), and this program was used throughout Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 130 the 4 years of the study. Investigations, a standards-oriented curriculum, focuses on developing conceptual understanding and mathematical thinking through vis- ual models, strategy development, inquiry, collaboration, and communication. The goals of the curriculum, as stated by the developers, are to offer students meaningful mathematical problems, to emphasize depth in mathematical thinking rather than superficial exposure to a series of fragmented topics, to communicate mathematics content and pedagogy to teachers, and to substantially expand the pool of mathematically literate students. During lessons, students spend the ma- jority of the mathematics period working on mathematical tasks, activities, or games that are designed to promote mathematical communication, questioning, and problem-solving skills. Teachers attended monthly professional development sessions throughout the study, facilitated by the principal investigator, to support their use of the program and discuss issues related to implementing a standards- oriented curriculum. Data Sources and Analysis Researchers visited the study classrooms informally once per week and took field notes on the classroom activities. Once a month, more formal observations were conducted in which classroom actions were audiotaped and transcribed. During each audiotaped observation, one researcher would be in the classroom observing the students in the room during the lesson, while another would observe the teachers. The teachers and students were also interviewed three times per year, in the fall, winter, and spring. The teacher interviews were audiotaped and tran- scribed, while the student interviews were videotaped and transcribed. Data ana- lyzed for this study included teacher and student observations and the clinical student interviews. To discuss how the African American children of this study interact with standards-oriented teaching practices, there was a need to define what those prac- tices were and whether the teachers in this study enacted practices that could be regarded as standards oriented. The teachers’ practices were drawn from either a 1 or 2-year period of observations, depending on the years the students in the study were members of each teacher’s class. In order for a practice to be considered a pattern for an individual teacher, the teacher must have engaged in that practice in more than half the observations for the years being analyzed. And at least six teachers had to enact a particular practice for it to be included as a pattern of prac- tice for this study. In all, there were nine standards-oriented practices enacted by the teachers that became common experiences for the students throughout the study: whole group discussion, encouraging active listening, questioning and probing, small group work, teacher support, sharing and explaining ideas, activat- ing prior knowledge, modeling strategies and thinking, and modeling means of Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 131 communication (see Table 4). The patterns that emerged from the coding process for each teacher were analyzed to determine how they might fit with and reflect the themes reflected in the theories of African American cultural dimensions and psychosocial integrity themes as described earlier (see Tables 1 and 2). A compi- lation of all the teachers was then created to determine what learning opportuni- ties were common across all the teachers, and thus, all the students (see Table 5). Table 4 Nine Patterns of Practice Common Among Teachers Teacher Practice Description of Practice Leading whole group discussions Whole group discussion about mathematics topic of the day Encouraging active listening Encouraging students to listen to other students and focus on topic at hand during discussion Questioning and probing student thinking Teacher questioning used to solicit student ideas or information; probe student thinking Creating opportunities for small group/partner work Small group work offers opportunities to actively engage with task or activity Facilitating student independent work The teacher supports student learning during independent or small group mathematical tasks Encouraging the sharing and explaining of student ideas Allowing opportunities for sharing and explaining ideas and solutions Activating prior math knowledge Teachers consider and use prior mathematical knowledge as a mathe- matical tool Modeling thinking and solution strategies Modeling games and activities to emphasize strategies and thinking Modeling multiple means of communication Students are encouraged to use a variety of means and media to com- municate in the classroom (e.g., models, drawings, graphs, materials, manipulatives) The analysis of the research data continued with the individual coding of the student observations. Each transcribed observation was broken into events, de- fined as a teacher or student initiated segment of a lesson in which interactions be- tween teachers, students, and their ideas are focused around the same mathematical goal. Each event was described using a number of categories: whole group participation, preparing for independent work, doing independent work, small group/partner interactions, choice and use of mathematics tools, communi- cating mathematical ideas, mathematical errors, mathematical understandings, and other. These categories reflected the different interactions and classroom practices that are expected in standards-oriented classrooms (NCTM, 2000). For the student data, codes were used that captured how students interpreted and responded to mathematical tasks, what they did in the face of confusion or lack of knowledge, Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 132 how they engaged with their work, and how they engaged with peers, teachers, and the classroom environments or routines. Each category had between four and eight codes to describe the students’ interactions. These codes were drawn in part from the research on African American cultural dimensions and provided multiple ways to interpret what was happening for the students beyond a more typical in- terpretation, such as on-task or off-task. After the coding of all students was com- pleted, counts of each code were made to provide initial information on patterns in students’ interactions. These counts were tabulated for each student by grade and then again over the entire study. Patterns were determined by the percentage of times a particular practice was enacted by each student and then whether the practice was enacted over the number of years the student was in the study. The individual student practices were compared; those practices that were consistently found among at least five students were considered patterns (see Appendix A). Some patterns occurred among smaller student subgroups (e.g., just the girls) and this was noted and discussed as well. Table 5 Psychosocial Integrity Practices and African American Cultural Dimensions Related to Teacher Patterns of Practices Psychosocial Integrity Themes Themes Defined Supporting Cultural Dimension Categories (Willis, 1987) Teachers’ Patterns of Practice Promoting meaning making Making connections among topics, to students experiences, and prior and common knowledge Harmonious Social/Affective Spirituality Sharing and explaining ideas, leading whole group discussions, activating prior mathematics knowledge Teaching thinking and learning strategies Providing tools for deep, constructive, active engagement with academic content; multiple stimuli and a variety of activities are used in teaching Harmonious Nonverbal Expressive creativity Leading whole group discussion, encouraging active listening, questioning and probing, facilitating student independent work, modeling multiple means of communication, modeling thinking and solutions strategies, activating prior math knowledge Building a learning community Learning is interdependent, collaboration is fostered through personal interactions Social/Affective Harmonious Creating opportunities for small group/partner work, leading whole group discussions, facilitating student independent work Using cultural resources Drawing on students’ every day practices, cultural habits, and worldviews as a tool in everyday practices Harmonious Expressive creativity Nonverbal Modeling multiple means of communication Creating a supportive and demanding learning environment Maintaining high expectations, focus on effort and improvement Social/Affective Facilitating student independent work, modeling multiple means of communication Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 133 How Have We Chosen to Learn? Coherence Between Student and Teacher Patterns of Practice Candace, Felix, Jordan, Kiana, Maya, Royce, and Samuel—these seven African American students each had a story to tell. Each student’s pattern of practice was connected to three areas: which classroom opportunities would allow or encourage the student pattern to emerge, which African American cultural dimension(s) were reflected in the practice, and which of the enacted teacher practices supported or allowed the engagement of a particular student practice. Once the student patterns of practice were connected to the teacher patterns of practice, the coherence, or lack thereof, was defined to be convergent, episodic convergence, supported divergence, or divergent. For a practice to be considered convergent, the students would have consistently engaged with that practice in about 80% of the events noted for that practice. That is to say, the classroom practices provided space and opportunity for the students to respond to mathematical tasks as the teachers intended. Episodic convergence practices occurred when students, for the most part, engaged with a particular practice in a way that was anticipated by the teacher and classroom norms, but occasionally interacted with activities and tasks in ways that are not expected or anticipated. These practices occurred in more than 50% but less than 80% of the noted events for that teacher practice. Supported divergence practices were practices enacted by the students that did not cohere with a pattern of practice of the classrooms in this study. These patterns, however, were often unnoticed, overlooked or ignored by teachers, possibly leading to the continued use of the pattern. Divergent practices were patterns of practice enacted by the students that were not supported by teacher practices or classroom norms. Even in the face of redirection or reprimand, students continued to engage with these divergent practices throughout their years in the study (see Appendix A). Focused Collaboration The students most often displayed direct and focused collaboration on the assigned tasks or activities when working in small groups with their peers. They were mathematically engaged during this time, with their collaborative focus be- coming stronger in grades 3 and 4. Teachers consistently provided small group independent work time for students. Work on learning styles by Dunn et al. (1990) found that African American students preferred to work with others more often than other ethnic groups (e.g., Mexican American, Greek American, Chi- nese American), supporting the finding that the students look for opportunities to interact with each other when working. This finding is also supported by the work of Nasir (2002). She found that, when African American children were engaged Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 134 with others during activities, they would work toward learning more about that activity in order to align their practices with the activity’s expectations. The stu- dents in this study made the effort to be a part of the classroom mathematics community and worked toward aligning their practices with the expectations of the teacher, thus establishing a norm about the importance of working together with others when learning mathematics. Considering African American cultural dimensions, the strong collaborative focus during independent time allowed the students to enjoy social/affective interactions with others while learning. In this example from grade 3, Candace was using the traditional regrouping algorithm to subtract. The students were given place value mats and base ten blocks (see Fig- ure 1) to help them solve the problems. She was then assigned to work with Ki- ana, and together they worked through solving the problem 95 minus 56. Figure 1. Illustration of base ten blocks. OBSERVATION: They carried a supply of base ten blocks with them and a place value mat. Kiana instructed Candace: “Show me 95.” Candace built 95 without any problem. Kiana asked her if she could subtract six from five and Candace said she needed more ones. Kiana indicated the tens section and said, “Take one away from here and add ten more.” Candace did that correctly. Kiana said, “Make sure you have 95 still.” She then took Candace through this same problem on the paper. “You would have to turn that 5 into a 10. So what do you have up here? You have a 15.” Candace crossed out the five in 95 and writes 15. Kiana continued her coaching, “So the nine… if you are taking one away from nine what do you have?” Candace didn’t seem to follow. “What comes before nine?” Kiana hinted. Candace said, “You cross that out and make an eight.” Then they worked the subtraction: “15 minus 6.” Candace counted on her fingers and Kiana did too in order to confirm Candace’s an- swer of nine: “8 minus 5.” They got the answer 39. We see here the collaborative nature between these two students; they were fo- cused on the task and worked through the problem without interruption. Although to some, this interaction may appear to be more directed by Kiana in that she is telling Candace what to do and is not focused completely on understanding the mathematics. The interaction, however, demonstrates the efforts of one student to Tens Ones Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 135 support the work of another, with Kiana acting as the more knowledgeable other. In many “traditional” classrooms, this type of student interaction is often discour- aged. Kiana took on the role of coach and supported Candace’s effort to make sense of the subtraction by asking Candace questions and allow her to participate in the problem solving—moving the blocks and writing all the steps. Active Participation Students actively participated in whole group lessons, by listening, watch- ing, and engaging in mathematics discussions without engaging in other activities unrelated to the mathematics at hand. They followed the lesson or activity, and many students eagerly supply responses and ideas. By creating almost daily op- portunities for the students to come together for whole group discussions and us- ing questioning and probing techniques to encourage participation, the teachers provided a space in which the students interacted with mathematical ideas. In the following fourth-grade observation, Maya is actively engaged with the lesson, even though her engagement pushed against the norm of quiet listening and turn taking. OBSERVATION: Teacher Laura began working on the poster board at the front of the rug and wrote: “How many hundreds” and below that question wrote 200+400. She asked the students to show with their fingers how many hundreds there were. Maya held up six fingers. Teacher Laura then wrote 201+403, and again asked how many hundreds. Maya whispered “604,” and then raised her hand. Maya said, “It would be six hundred and something, these are easy.” Teacher Laura wrote the next one 199+404, and then asks what they should do. Maya whispers, “It’s 603”. Teacher Laura then asked, “What if we had 199 pennies and something cost $2.00….” Maya commented softly, “If they gave her $2.00 then she can give one penny back.” Teacher Laura asked, “How much is 404, how many hundreds?” Maya said softly, “Can I blurt out the answer? I want to so badly.” The class decided there are four hundreds in 404. As the class prepared to move on to the next activity, Maya raises her hand and says, “Teacher Laura, can somebody say the answer now?” Teacher Laura seemed confused. “Can I say the answer?” Teacher Laura said, “We said the answer.” Maya responded, “That’s not the answer, the answer is 603.” This observation illustrates that within most of the study classrooms, there were limits on how students were allowed or encouraged to interact during whole group lessons. Students were expected to listen closely to others, respond to the given question or prompt, and stay focused and attentive to what was happening in the discussion. When students called out a response or idea without acknowledgment from the teacher, chose to attend to other things during the lesson (active inatten- tion), or did not pay attention to what was going on (passive inattention), they were most often reprimanded or redirected toward the expected behavior. Here, Maya maintained a steady stream of comments while continuing to focus on the Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 136 lesson, even though she was not interacting in the discussion as Teacher Laura had intended. Looking back to African American cultural dimensions, calling out by students or engaging in other related activities during whole group time could be construed as a student’s need for orality or movement expressiveness (Boykin & Cunningham, 2001; Foster, 1992). Physical Tool Use The next convergent pattern of practice that focused on the students’ interac- tions with mathematical tasks and activities was the practice of using physical tools to solve problems and accomplish tasks. Physical tools were the concrete mathematics materials for modeling and counting (blocks, cubes, etc.), drawings, pictures, or written algorithms, but not mental tools or models. The students were adept at using physical tools in many different ways to support their problem solving. The teachers supported tool use as they supplied the students with access to tools, modeled tool use often, and encouraged students to share how they used tools in their work. Thus, heavy use of tools was expected and the data showed that the percentage of physical tool use in both the classroom and interview events ranged from a high of 95% of the time by Felix to a low of 51% by Samuel. As an example, when solving problems, Felix was adept in using tools and models to support his efforts. In grade 4, he was able to use snap cubes to solve the problem 4 times 8: Interviewer: What is 4 times 8? Felix: (counting the cubes by 4, taking sticks of 4 cubes from the long stick, lining up the short sticks on the desk): These are 4s and then add them up. Add 4 eight times. And then…4, 8…(counting the cubes of the third stick one by one) 12, (count- ing the cubes of the fourth stick) 16, (counting the cubes of the fifth stick) 20, (counting the cubes of the sixth stick) 24, (counting the seventh) 28, (counting the eighth) 32. Interviewer: Great. So just how you explained to me. Can you explain to me one more time? Felix: I put all these 4s and put them into one group (piling up the sticks, holding them together, separating the sticks and placing them one by one on the desk, murmuring). Then you add them together (putting the sticks back together). You’ll come up to 32. He also used physical tool modeling for addition and subtraction problems, as well for solving story problems for all 3 years in the study. Having access to many tools connects to the African American cultural di- mension of nonverbal interactions and expressive creativity; tools allowed stu- dents physical activity while working with mathematics, drawing on the ideas of Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 137 rhythm/patterns and movement as integral parts of the African American life ex- perience. Tool use, as it was enacted in these classrooms, also allowed personal interactions with others in that students generally shared tools and worked to- gether on mathematical tasks requiring tools. Although using physical tools was well supported by the teachers in this study, the reliance on tools was expected to lessen, as the students got older. By the end of third grade and all of fourth grade, students were expected to rely on mental models to support solving problems more than was expected of them in the earlier grades, particularly with basic facts and to support work on more com- plex problems. However, only two of the students, Samuel and Maya, used mental models efficiently and effectively. The other five students had some basic strate- gies for using mental models, but only for simpler problems. They instead contin- ued to rely on physical tools to support their problem solving efforts. Fuson (2003) found that some students using a reform curriculum continued to use lower-level counting strategies to solve computation problems without moving on to higher-level strategies up into the fourth grade. Similarly, Siegler’s (2003) re- view of literature on individual differences in students’ mathematics cognition found that students with mathematics difficulties had limited strategies for retriev- ing correct answers from memory and would rely on elementary counting strate- gies. He contended some of these mathematics difficulties arose from the children’s “limited exposure to numbers before entering school” and “from poor families with little formal education” (p. 295). Although I do not agree with Siegler’s deficit family view, he did suggest that greater practice and instruction in how to execute strategies as well as addressing limited background knowledge and conceptual understanding would allow students with mathematics difficulties to learn with reasonably high levels of proficiency. Addressing the issue of bridg- ing concrete and physical representations to more abstract mathematical symbols and language is an important topic and provides for a possible extension of this research study. Direct Communication Students in the study made attempts to clearly and directly respond to ques- tions and prompts by the teacher during whole group discussions and, most times, their responses stayed on the topic under discussion. The teachers encouraged strong communication through the practices of encouraging active listening and modeling clear mathematical communication; the teachers’ practices also sup- ported students’ communication efforts by allowing the students to share their ideas during whole group discussions and independent work time, which also cre- ated opportunities for peer modeling. Considering African American cultural di- mensions that would support these practices, social affective opportunities entail Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 138 encouraging and valuing interaction among classroom members. By engaging in whole group discussions orally and through the use of tools, students were given the opportunity to engage in personal interactions with other classroom members and to develop a sense of community in the classroom. Lampert and Cobb (2003) discussed using communication to learn as a goal in standard-oriented classrooms; as students are provided opportunities to participate in mathematical tasks, teach- ers model ways of communicating and make explicit the acceptable forms of classroom communication. As students have additional opportunities to express themselves mathematically, they become more adept in using mathematical lan- guage to express their ideas. In this study, the girls were more vocal than the boys; they averaged over 104 events where they engaged orally with mathematics topics, while the boys averaged only 81 events in comparison. This finding could be related to the fact that the girls used more dramatic expression (discussed in the next section) than the boys and consistently used oral means for expressing their vibrancy when interacting with mathematics. Direct communication is considered an episodic convergence practice be- cause, although the students did communicate mathematically in the way the teachers expected during whole group lessons most of the time, the number of times that they had difficulty being clear and concise when explaining their think- ing and solutions during interviews and small group settings was quite large (33% of all communication events). The students often used imprecise language when attempting to explain a solution or idea—using language that was vague; talking in circles; or starting to talk, stopping, then starting again, but staying on or close to the topic under discussion. This imprecise form of communication was evident in all years of the data analyzed. In the classroom, students were often questioned and guided through imprecise oral explanations by the teacher, which often times interrupted the flow of the speech. In her review of literature on sociolinguistics of African American children, Foster (1992) also found that African American students were interrupted more often by teachers when speaking imprecisely than were White students who spoke in a more factual, linear, lecture style. She noted: Teachers failed to comprehend or appreciate the stories being narrated. Frequently interrupting students with inappropriate questions or attempting to redirect the narra- tive to focus on a particular but often insignificant aspect of the story, the teachers questioned the African American students’ intellectual competence and emotional stability (p. 305). Learning to query, rather than curtail, imprecise speech would honor the students’ cultural funds of knowledge and allow teachers to consider whether a student is using speech to coordinate their thinking or if the student needs support through the thinking and communicating process. Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 139 Understanding how students’ cultural norms impact the communication of their thinking and how to support that communication is important in standards- oriented classrooms, due to the emphasis placed on the use of communication to enhance and solidify learning. Without considering the frameworks students use in their interactions, teachers can get caught up in funneling the student toward a more desired response or action rather than working toward understanding what in the student’s frame of reference produced the particular outcome or solution (Wood, 1994). Moschkovich (2002), in her study on the communication practices of bilingual students in mathematics classroom, concurred: In particular, this perspective can affect how teachers assess a student’s competence in communicating mathematically. For example, if we focus on a student’s failure to use a technical term, we might miss how a student constructs meaning for mathe- matical terms or uses multiple resources, such as gestures, objects, or everyday expe- riences. We might also miss how the student uses important aspects of competent mathematical communication that are beyond a vocabulary list. (p. 193) In this study, although the teachers may have had difficulty understanding the verbal reasoning of students who were attempting to solve a problem, the stu- dents were either able to solve the problem or were working toward solving the problem. In many cases, it appeared as though the students used their oral speech as a thinking tool; speaking their thoughts out loud as a way to solidify their solu- tion. If students were using their speech as a way to organize their thinking, inter- ruptions to that speech could impede the thinking process and make problem solving and explaining difficult for the students. This type of speech was evident in an example of Felix as he tried to explain how to mentally add 10 to 128: Interviewer: What number is 10 more than 128? Felix (murmuring and looking down): I’m not sure. Interviewer: Well, if you started up from 128, and you counted up 10 numbers, what would you get? Felix: I’ve counted 10 numbers, but I made a mistake…so stop counting…kind of get to 38. Interviewer: Okay. What number is 11 more than 128? Felix (looking ahead and touching his chin with his left hand): The second one is 138, and the third one is 139. Felix attempted to solve the problem of adding 10 to 128, but did not readily have a solution. When trying to explain, he sounded as if he was talking to himself about what he was doing in his head, attempting to understand what was being asked of him and how he could go about solving the problem. The interviewer, Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 140 not appearing to understand what Felix was doing to solve the problem, continued on to the next problem. After a few moments, Felix was able to put his explana- tion together and give correct solutions to both problems. Having an opportunity to verbalize his thinking and mentally grapple with the wording and ideas in the problem appeared to be needed communication for Felix to solve the problem. Divergent and Supported Divergence Student Patterns of Practice In Wood’s (1994) research on patterns of interactions, she stated, “teachers have unwittingly undermined their own goals by failing to realize that the conse- quences of their interaction (and inaction) are often quite different from their in- tentions” (p. 149). In supported divergence patterns of practice, the teacher would encourage particular behaviors or beliefs either because they were unaware of the students’ engagement in that practice or belief or they were unaware of how their actions helped create or continue particular patterns or beliefs. Divergent and sup- ported divergence patterns of practice occurred far less frequently than convergent patterns, but appeared, in many ways, to be common practices of interaction and communication for the students. Activating Personal Knowledge When solving problems, students used their personal knowledge as a tool. Students made connections to ideas they held about the world or to the experi- ences they had in their daily lives. These connections not only occur when solving word problems that were designed to draw on student background knowledge but also when students used personal knowledge to solve problems about patterns or to develop solution strategies for computation problems. In most instances, how- ever, the teachers did not solicit personal connections in problems; the students initiated these connections. Nor did teachers explore the personal solutions in an attempt to connect them back to the original problem. Particularly in the inter- views, students, when asked to explain, made connections to their understanding of the world and used that understanding for making sense of the problem. In the following interview from grade 4, Royce was working on problems related to theoretical versus experimental probability. In answering, she considers what is important in trusting someone to tell you the truth rather than considering the mathematics of the situation: Interviewer (after discussing problems of drawing red and yellow cubes from a bag): Suppose someone in your class said, “It’s been yellow five times in a row so I’m due for a red one now,” or, “It doesn’t matter that I got all those yellows. It’s still an equal chance of red or yellow.” What would you say to that person? Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 141 Royce: Disagree. It depends what classmate says it. Interviewer: Would you agree if Brent (a “high level” mathematics student) said it? Royce: No, I wouldn’t agree with him. Interviewer: Would you agree if…who would you agree with then? Royce: I would agree with Jeri because she’s my best friend. She’s my best friend in the classroom. Here, by considering what is important when believing what someone tells you, Royce does not attend to the mathematical concepts of the probability prob- lem. She privileged her personal knowledge over her mathematical knowledge. This action does not mean, however, that she does not possess the mathematics knowledge to solve the problem. Exploring her reasoning and then connecting her response back to the problem would allow her response to be valued, considered, and redirected. Additionally, by allowing her to present her understanding of the task, a teacher can expose misunderstandings or roadblocks to strengthening mathematical ideas. This practice of drawing on personal knowledge when learning resonates with the African American cultural dimension of harmonious interactions: provid- ing students with the opportunities to draw on their personal and previously held knowledge to support their mathematical development through the purposeful use of information. Interestingly, drawing on students’ background knowledge and making connections to the world at large are also practices supported by stan- dards-oriented teaching. Unfortunately, the teachers in this study, as a group, did not consistently make personal connections an explicit part of their teaching prac- tices. It is important to note that although the teachers attended ongoing profes- sional development designed to support the implementation of the curricula, the content of those sessions was not available for analysis to know whether teachers discussed issues of using personal versus school knowledge as a learning tool. Dramatic Expression Another divergent pattern of practice enacted by the students was their use of dramatic expression when working on or talking about mathematics problems. All seven students demonstrated the use of dramatic intonation and movement during interviews and in small group work to demonstrate ideas or punctuate speech. One student, Maya, displayed dramatics in all areas of the classroom set- ting as well as in small groups and interviews. Foster (1992) also noted that Afri- can American students, when sharing their writing, “resembled performed narratives, with stylistic features—gestures, dialogue, sound effects, asides, repe- titions, shifts in verb tenses for emphasis—similar to those in a dramatic stage performance” (p. 304), which supports the findings of this study. In the following Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 142 observation, Royce displayed her dramatic side during a grade 4 interview task: OBSERVATION: Royce (reading the problem aloud): Jeanine has 40 cookies. She wants to share them evenly among 8 friends. How many cookies will each friend get? (sighing loudly) Ok, this is what I hate. (She pulls out blocks and separates them into 8 piles.) Royce (with an exasperated voice): Evenly. Each of them will get 5 cookies. I put these together, each of them equals 10 and ten 4’s equal 40. And then I broke them up by 5s, and 4 plus 4 equals 8. That’s how I got it (she looks at the interviewer with a dramatic pause). Any questions? This pattern of practice is considered divergent because the teachers in the study reprimanded students who used dramatic or excessive movement while working or speaking. Particularly during whole group sessions, students who moved around or talked exuberantly were redirected to sit down, sit still, quiet down, and so forth, because the behavior diverged from the expectation of being a quiet classroom participant and not disturbing other students from learning. Thus, most of the events where students displayed dramatic expression were found out- side of the view of the teacher in a setting where disciplinary action would not be taken. When engaged in dramatic and expressive speech and movement outside of the teacher’s view, they were less likely to be reprimanded and thus were able to engage in this behavior. As an example, Jordan used dramatic action when inter- acting with mathematics tasks during interviews. He used verbal expressiveness— shouting “Timber” when he dumped out a container of cubes—as well as move- ment expression—pretending to be deep in thought while quickly pacing back and forth when working on a difficult problem—throughout his interviews. He also moved around quite a bit during the end of his interviews; he stood up, lay on the floor, and walked around the interview area while working on the tasks. These dramatic examples and constant movements, however, did not deter Jordan from solving the problems he was assigned to do; he still worked on the tasks and fo- cused his attention on them. Mathematical communication is an important part of standards-oriented teaching practices. The emphasis is on providing opportunities to verbally com- municate and encourage students to develop a vocabulary that will allow them to engage in reasoning, justifying, conjecturing, and explaining; however, other ways in which students might communicate are not addressed in detail in stan- dards-oriented practices. Consider the African American dimension of orality, which is described as using oral communication to express feelings and emotions with movement and verve. Expressing oneself in dramatic and vervistic ways de- parts from traditional norms of classroom interaction—raising your hand and waiting for your turn to speak, letting one person speak at a time, not embellishing your speech or dramatizing parts of your ideas—as the students in this study have Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 143 done. Boykin and Cunningham’s (2001) research on movement expressiveness of elementary African American children showed that the children had better reten- tion of information when learning in an environment that was movement oriented as well as learning with materials that incorporated high movement themes. Improvisation Another practice that students enacted outside of the attention of the teacher was improvising. As defined, improvising occurred when students followed an ac- tivity or task in an unexpected or un-prescribed manner, but still engaged with the task in a mathematical way. Studies on African American elementary students provided other examples of students creating alternative routes though activities as a way to complete them or make them personal (Dyson, 1999; Gadsden, 2001). In classrooms, improvising most often occurred when students had a limited un- derstanding of the required activity. Teachers modeled activities and held discus- sions about what was expected in an activity prior to sending the students off to work on the activity with their partner or group. Some of the students may not have fully understood all the requirements for the activity but knew enough to create an alternative version of what was expected of them. In grade 1, Kiana was playing a card game with Royce in which the players had to choose, from an ar- rangement of 20 cards, two cards that summed up to 10. OBSERVATION: Once Kiana finished arranging the cards, she said to Royce, “Go, what makes 10?” Royce chose a 0 and a 10 and started to record it. The two had a brief argument over who should record, and finally agreed that the person who drew the cards should record the combination. Royce then wrote 10+0=11 on her paper. When it was her turn, Kiana dug through the pile of unused cards to find a seven. She replaced a six in the original arrangement with this seven card and said, “7 and 2.” She counted the pictures on the cards and realized that it is only nine. She tried 1, 5, and 3, in that order, all coupled with the seven card. Each time she chose a new second card, she counted the pictures again, always starting with the seven card and counting each picture by ones. Eventually, she found that 7+3=10, but she wrote on her paper, “5+5=10.” Next it was Royce’s turn. She also tried to dig through the pile of unused cards to find a specific card, but Kiana got very angry and accused her of cheating. Kiana not only took on the role of activity director but also improvised the rules to suit her needs of finding a combination of 10. Throughout grades 1 and 2, Ki- ana would often change the rules of an activity to match her mathematical under- standing, but did not like others to do the same. As discussed by Forman (2003), research on emergent goals in mathematics learning indicates that “children, especially while working in group settings, will establish their own priorities for problem solving” (p. 339) as they learn what it Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 144 means to work collaboratively on mathematical tasks. Dyson’s (1999) work with African American children in an early literacy study also found students using the limited knowledge they held about a task in order to complete it, creating their own version of the final product. She focused on what students transferred from other tasks—the mechanics of tasks completion such as materials to be used, types of solution displays, and how to “talk” about your work—that helped them fill in missing information as the students then improvised and completed the cur- rent task. Her view was that a negotiation needed to occur between the experi- ences students draw on and the experiences the teacher wished to provide in order to develop a common meaning of the proposed task. Another form of improvisation occurred during the clinical interviews. Stu- dents would be given a task to do by an interviewer and if they knew how to solve the problem, they might solve it in an unexpected way or if they were unsure of how to solve the problem, they might change the problem to a form a problem they could solve. What made the acts of improvisation during interviews intrigu- ing was that although the interviewer was a proxy for the teacher, the students ap- peared to be less driven to hide their improvisation during interview tasks. An example of improvising or answering in an unexpected way was when Jordan was asked to circle three numbers from 1 through 9 that totaled up to 15. Interviewer: Can you circle three numbers on this page that equal 15? Jordan (uses marker, pauses and thinks for 5 to 6 seconds, then circles): I can’t circle four? Interviewer: Just three. (Jordan circles three numbers, 9, 3, 2) How do you know they equal 15? Jordan: Doesn’t. Interviewer: It doesn’t? Jordan: Uh-uh. Interviewer: What does it equal? Jordan: 14. Here, Jordan appeared to understand the task—he asked if he could circle 4 num- bers instead of 3—but still completed the task by creating a sum of 14 rather than 15. These acts of improvising, although infrequent in total number, raised the question: Why would a student engage with a task in an un-prescribed manner? Is it a natural occurrence for young children? Although Forman (2003), drawing on the work of Saxe and colleagues, points out that young children will modify or improvise an activity in an effort to simplify it, there was no mention of why stu- dents would improvise when it was clear they were mathematically able to ac- complish the task as given. If considered from the African American cultural Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 145 dimension of expressive creativity, this practice could be construed as an attempt by the students to express their individualism, particularly in the interview setting where they were less likely to be sanctioned by the interviewer for deviating from the expected path. By creating an alternate, yet mathematically sound response to a problem, students gain control over their work and present themselves as profi- cient and creative problem solvers, which allows a student to “save face” and not be negatively labeled or seen in a stereotypical light (Erikson & Shultz, 1992; Perry, Steele, & Hilliard, 2003). Considering Divergent Student Practices Considering divergent practices forces a closer look at the frameworks stu- dents draw upon when interacting with classroom activity. For many African American students, African American cultural learning dimensions provide stu- dents with a framework that they may use to make sense of classroom activities and tasks, which at times is at odds with the expectations of the teachers and/or school. As teachers attempt to establish particular norms of behavior and interac- tion for students, the disconnect between students’ cultural understandings and the expected responses can, in some cases, make it seem as though the student is be- ing willfully disobedient or is less competent than those students who are adher- ing to the norms. What is important to consider, however, is that students have an implicit knowledge of the meta-discursive rules of school. Sfard (2000) suggested that the ideas of mutual orienting, patterns of interaction, and negotiation of obli- gations all fall under the idea of meta-discursive rules, which regulate the dis- course in mathematics classrooms (and other classrooms) and the ways in which classroom community members speak and interact with each other without overt acknowledgment. In essence, students knew that their dramatic expressiveness and improvisation would be censured by the teachers, but still engaged in these practices. What would cause children at such a young age to engage in behaviors that could possibly get them into trouble? As these students worked on mathematics activities, they attempted to en- gage with mathematics in a way that made sense to them, which allowed them to participate in the activities and show that they are mathematically “sound.” In other words, they engaged in mathematical activities as a way to align their prac- tices with the classroom goal of successful mathematics participation, as might occur with the African American cultural dimension of improvising, or to divert attention away from the fact that they might not be successful, as might occur with the dimension of dramatic expression. This idea is similar to the findings of Nasir (2002) with African American children who aligned their practices to those of domino and basketball players as a way to engage with that community. In her study, even when the children did not know how to participate in parts of the Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 146 domino or basketball activity, they would find a way around their obstacle so they could be seen as successful participants. In this study, the students may have been attempting to create a mathematics identity that would be sufficient to allow them to engage and align with the mathematics practices of the classroom. Nasir (2002) also described how many African American children, when sufficiently motivated in activities outside of school, engaged, imagined, and then aligned themselves as part of a community where they felt they could belong. She claimed that this process has not occurred in schools for African American stu- dents. This study suggests that, in elementary school, students do engage with mathematics, and attempt to align themselves with the practices in a way that makes sense to them. What is missing appears to be the level of imagination. Did the students see themselves as being in the community of strong mathematicians? Only one student in the study consistently stated that he believed himself to be a good mathematics student. The other six students, when asked whom they knew was good at mathematics, often named White boys in their classrooms. The stu- dents gave very similar reasons for why these boys were considered good mathe- matics students: they could answer the teacher’s questions quickly, they studied and practiced at home and over the summer (or so the students thought), and had mathematics tricks that help them get the correct answer. Considering the racial demographics of Carter School, the fact that African American students outnum- bered European American students in every classroom at a ratio of at least 4 to 1, and the presence of some very bright African American students in each class- room (consider Maya and Samuel of this study), it was a surprising finding that an African American student was mentioned only once. The students’ choices, how- ever, reflect the ideas that society holds as a whole regarding who is a good mathematician (Burton, 1994). It would appear that the students held a view of a good mathematician that they did not reflect and thus could not imagine for themselves. This lack of imagination could play an important role in the increasing disidentification noted in many African American students as they progressed through mathematics in school (Erikson & Shultz, 1992; Martin, 2000; Nasir, 2002; Osbourne, 1997). Na- sir (2002) stated: The importance of imagination in this process offers evidence that becoming is more than just what one does as a participant. It also includes the meanings one makes of that participation. Children’s ability to imagine (and the affordances for such imagi- nation in practices) their own learning trajectories and their place in relation to others is critical to the development of new goals and the access to new identities. (p. 241) Although the teachers in this study did work toward breaking down the tra- ditional view of mathematics through their enactment of standard-oriented mathe- matics practices, it is clear that the current societal view of what it means to be Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 147 be successful in mathematics is still a barrier for increasing the successful partici- pation of many African American students. By expanding the definition of who is considered a strong mathematics student to include the ability to explain solutions or create multiple solutions to a problem, more students may begin to imagine themselves as good mathematics students, capable of succeeding in mathematics. This expansion appears to be a difficult task, as the teachers in this study devel- oped classrooms that reflected a less traditional view of what it means to do mathematics but the students still continued to adhere to the beliefs that reflected a narrow view of mathematics success. Although the divergent practices of the students do relate to some of the di- mensions of African American culture, they could also be explained through re- search that explored how lower socioeconomic children interact with the world. Lareau (2003), in her book Unequal Childhoods discussed what she called “the accomplishment of natural growth” (p. 238) to describe the development of chil- dren in working-class and poor (i.e., low-SES) families. She found that these chil- dren were raised with similar parenting practices that led to certain ways of interacting with the world. One characteristic she found was that working-class and poor children tended to spend time creating and recreating their own games and activities for play with other children, unlike middle-class children who were more often engaged with organized, adult-led activities. These independent play- ing opportunities could lead the children to be more comfortable with being in- ventive when engaging in games, similar to improvising that was noted in this study. Moreover, Lareau found that low-income children were more responsible for their lives outside of the home; this led the children to become more self- reliant and believe in their ability to take care of themselves without adult help. Being self-reliant encouraged the children to rely on their own personal knowl- edge as a tool for problem solving and decision-making. These findings from Lareau’s study support the idea that SES can play a role in how students interact in classrooms and, in this case, the behaviors of low-income African American children. However, other studies also show that race also plays a part in student behavior and achievement in school (Lee, 1998; Roscigno, 1998; Wilson, 1998). Considering the work on African American cultural dimensions and how these dimensions take into consideration the historical evidence that African Americans have generally been part of the lower economic station in American society, some coherence between the practices and habits of low-SES families and African American families could be expected. Additionally, SES as well as race tend to explain only some of the differences in achievement and attainment for African Americans in school (Wilson, 1998) and, although more affluent African Ameri- cans score higher on standardized tests and have higher grade point averages than less affluent African Americans, they still score lower than students of other races (Nettles, Millet, & Ready, 2003; Steele, 1992; Vars & Bowen, 1998). Although Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 148 considering SES when discussing the relevance of African American cultural di- mensions in student behavior and achievement confounds the issues, it appears that both factors play a role in these students’ possibilities of achieving, and as such any practices that might support and enhance their success would be war- ranted. Implications and Conclusion In my analysis of the data, there appeared to be alignment with the practices the students enacted, African American cultural dimensions, and the research supporting those dimensions. The students enjoyed social interactions, talking and sharing, and connecting with others. They also used physical tools often and re- lied on them to help solve problems. Dramatic expression and improvising relate to the idea of expressive creativity; students wanted to be individual and unique in their work and using dramatics or creating their own rules allowed for this indi- vidualization and uniqueness. More comparative and in-depth research that fo- cuses closely on the interactions and practices of African American students in standards-oriented classrooms is needed to better understand the influence of Af- rican American cultural dimensions on their learning and achievement and how to use those dimensions to better support African American students’ learning and achievement. Rather than looking at the divergent behaviors as social problems or behav- iors needing remediation or punishment, teachers and schools could look at what can be learned from these behaviors that would enhance the academic achieve- ment of students. As an example, the students desire to use physical tools and oral communication to support their thinking could lead teachers to make more efforts in bridging students’ learning of abstract mathematics concepts. By lengthening the time that students have to use tools to solve problems, and making more ex- plicit connections between how the tool use bridges to symbolic representations, teachers will be drawing on students’ strengths while moving them toward deeper levels of mathematical understanding. Other areas of divergent behavior might be more difficult to incorporate into classrooms, such as dramatic expressions, con- sistent oral communication, or improvisation. However, if teachers understood and used these behaviors as tools to support students’ learning in mathematics, African American students may have more success in learning mathematics and come to perceive themselves as mathematicians. Studies that explore the sur- rounding activities that accompany divergent behaviors can provide clues to un- derstanding how African American students think about the mathematics they are learning while engaged in what may, to some teachers, seem to be unrelated ac- tivities or actions. Most of the students in this study did not appear able to imagine themselves Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 149 as good mathematics students. We are still left with the question: How can Afri- can American students see themselves as part of the mathematical community? Breaking down the traditional view of what it means to do mathematics and who can succeed in it would be a step toward providing African American students with a vision of mathematics success that includes themselves. Teachers need to be encouraged and supported to develop and enact explicit practices that work to directly break down the cultural barriers and traditions about who can be success- ful in mathematics. More in-depth studies of teachers’ practices over time in stan- dards-oriented classrooms could help consider the effects of using cultural connections and resources when teaching African American students mathemat- ics. Candace, Felix, Jordan, Kiana, Maya, Royce, and Samuel represent the vari- ety, complexity, and hope for African American students in our nation’s schools. Although the vision and philosophy of standards-oriented practices in mathemat- ics appears to be sound pedagogy, there is still much to learn about its implemen- tation and effects on student achievement and attainment in mathematics. 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Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 153 Appendix A Enacted Student Patterns of Practice Compared to Teacher Practices and African American Learning and Cultural Themes Enacted Student Patterns, Interactions, Beliefs Opportunity or Space within Classroom that Allow or Encourage Pattern African American Cultural Dimension that Supports Pat- tern Enacted Teacher Pattern Practice that Allow or Encourage Pattern Student–Teacher Pattern Coherence Focused Collaboration – students work on task at hand with little tangential or unrelated activities when working in small groups but occasionally needed teacher assistance in tasks Small group work, teacher en- gaging as a support during in- dependent or small group mathematical tasks, modeling of tasks for students prior to independent work, developing classroom norms of collabora- tive habits Social affective Harmonious Small group/partner work Teacher support Modeling strategies and thinking Convergent – students oc- casionally needed teacher support in understanding the task or the mathematics of the task when working independently Active participation – students listen, watch, and engage in mathematics discussions without a noticeable number of other activities unrelated to mathematics discussion at hand Whole group discussion about math topic of the day, develop- ing norms around participation behaviors, modeling participa- tion behaviors, lively energetic discussions Social affective Whole group discus- sions Active listening Questioning and prob- ing Convergent – students can actively participate but within set norms and limits of classroom Physical Tool Use – stu- dents understood and used blocks, cubes, paper, rul- ers, calculators, shapes, number cards, number lines, pictures, graphs, drawings, and so on to solve problems and ac- complish tasks Small group work to collabo- rate on tool use, modeling the use of tools, access to a variety of tools and ideas Nonverbal Expressive creativity Sharing and explaining ideas Small group work Whole group discus- sion Means of communica- tion Modeling strategies and thinking Convergent – students used tools effectively and regularly; sometimes would rely on concrete tools instead of moving toward more abstract tools Direct Oral Communica- tion – student speaks on the topic/task at hand without moving to a tan- gential or unrelated area when talking about mathematical ideas, tasks, or activities Establishing norms during whole group discussion of lis- tening to other classroom members, allowing opportuni- ties to share ideas, modeling ways of responding and speak- ing Social affective Harmonious Sharing and explaining ideas Whole group discus- sions Encourage active lis- tening Means of communica- tion Modeling strategies and thinking Episodic Convergence – listening and clear com- munication are important elements in classrooms; however, students some- times used mathematical language that was limited or vague, did not always talk in a linear fashion, had false starts, retraced ideas; talked off-topic during speech Activating Personal Knowledge – students used their personal back- ground knowledge to un- derstand and solve problems Whole group discussions about tasks and activities; small group work, encouraging per- sonal solutions to problems; sharing solutions in whole group setting Harmonious Sharing and explaining solutions Small group/partner work Whole group discus- sion means of commu- nication Supported Divergence – students used their prior knowledge and ideas often but sometimes would privilege their ideas over the ideas presented in the problem or task; teachers neither actively engage these connections nor did they sanction them Dramatic Expression – students displayed move- ment while speaking; got up often while working; used body language and hand movements while working; used intonation, excitement, expression when speaking Whole group discussion, small group work, using energetic activities, music and rhyme in activities; allowing movement throughout classroom often Nonverbal Expressive creativity Small group/partner work Whole group discus- sion Divergent – although stu- dents engage in whole group discussion almost daily, they are not encour- aged to be expressive or movement oriented in class; episodes of dramatic expression occur most dur- ing interviews cont. on next page Waddell Reform Elementary Mathematics Journal of Urban Mathematics Education Vol. 3, No. 2 154 Appendix A cont. Improvisation – students worked on a task or activ- ity by making new rules or changing the rules of pre- scribed activity, followed activities in unexpected manner while maintaining mathematical integrity of activity, created mathe- matically unexpected solu- tion paths Small group work time, stu- dent developed solutions; shar- ing of student solutions; allow- ing individualism in work hab- its and products Expressive creativity Harmonious Sharing and explaining ideas Small group/partner work Whole group discus- sions Divergent – classroom ex- pectations were to follow norms and rules, improvis- ing falls outside this realm; students tended to impro- vise out of the sight of teacher, although impro- vising was sometimes an attempt to continue work- ing on an activity or tasks in the face of difficulties Self-Reliance – students believed in relying on one- self to understand and do mathematics Independent and individual work time and space Spirituality Expressive creativity Modeling strategies and thinking Divergent – classrooms practices supported col- laboration and interde- pendence, but students maintained this belief in spite of the classroom practices Boys as Mathematicians – students often professed the belief that boys, and most often white boys, were good at mathematics Using male students examples during whole group discus- sions; using male students as partners for struggling students N/A N/A Divergent – standards- oriented mathematics pos- its all students can be strong and capable mathe- maticians.