attachment i. lesson plan lesson plan school : smp negeri 18 tangerang subject : mathematics grade/ semester : vii / 1 year school : 2019 / 2020 time allocation : 2 teaching and learning hours (2 x 40 minutes) a. basic competencies 3.6 explaining the linear equations and linear inequalities in one variable and its solution 4.6 solving the problem related to linear equations and linear inequalities in one variable b. competence achievement indicators 3.6.1 modelling the algebra expressions using algebra tiles 3.6.2 solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division using algebra tiles 3.6.3 solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division without using algebra tiles c. learning objectives by the end of this learning, students will be able to: 1. model the algebra expressions using algebra tiles 2. solve the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division using algebra tiles 3. solve the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division without using algebra tiles d. materials linear equations in one variable: 1. algebra expressions 2. solving the linear equations in one variable by applying mathematical operations of addition, subtraction 3. solving the linear equations in one variable by applying mathematical operations of multiplication or division e. teaching and learning method · approach / strategy: student centered learning · model: cooperative learning · method: group discussion, ask and answer, collaboration work, presentation f. teaching and learning materials / media · algebra tiles · worksheet · stationeries (pencil, paper, eraser, ruler, marker, glue, sellotape) g. teaching and learning resources · textbook: mathematika smp/ mts kelas vii semester 1 kurikulum 2013, edisi revisi 2016 (mathematics textbook for 7 grade semester 1 of junior high school, curriculum 2013, revision edition) · textbook: buku guru mathematika smp/ mts kelas vii semester 1 kurikulum 2013, edisi revisi 2017 (mathematics teacher textbook for 7 grade semester 1 of junior high school, curriculum 2013, revision edition) · internet: lesson plan 2: cups and chips – solving linear equations using manipulatives; https://www.learner.org/series/insights-into-algebra-1-teaching-for-learning-2/variables-and-patterns-of-change/lesson-plan-2-cups-and-chips/ h. instructional procedures teacher’s activity student’s activity time opening greets to the students greets back to the teacher 10 minutes ask one of the students to lead pray pray together · teacher explains the learning objectives and teaching and learning method students pay attention and get ready to learn the students are divided into six groups students already seated according to their grup main activity · teacher ask the prior knowledge related solving the linear equations in one variable, that are the integer and operations of addition, subtraction, multiplication or division · teacher ensures that students understand about mathematics operations in the integer by question and answer method students give answer 7 minutes reviewing the previous lesson · teacher informs that the learning about linear equations in one variable implemented by using algebra tiles · by discussing, teacher introduces the algebra expressions and linear equations in one variable · using the question and answer method, teacher explains how to solve the linear equations in one variable using the algebra tiles · teacher make sure that students understand the use of algebra tiles well · teacher gives the worksheet and the algebra tiles to all groups · students pay attention about the material introduced by teacher · students participate actively in asking and answering questions · students understand about the use of algebra tiles well 13 minutes presenting the material and the use of algebra tiles teacher moves around and guides the students in each group: · observe the students as they work · give suggestions or help students who are having difficulties. · look for ‘good’ ideas with the intention of calling them in a certain order during discussion · encourage alternative method to solve the the linear equations in one variable · students discuss in group to do the task based on the worksheet · students model the algebra expressions using algebra tiles · students solve the linear equations in one variable using the algebra tiles · students solve the linear equations in one variable without use the algebra tiles 20 minutes students working on their own teacher guides and lets students discuss among themselves · each group present their work on the board. · other groups give response by asking question or giving other solution. 20 minutes whole-class discussion teacher asks the students about their answer in solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division make the conclusion · students explain the strategy of solving the linear equations in one variable. · students make conclusion about the algebra expressions and how to to solve the the linear equations in one variable 5 minutes highlighting and summarising the main point closing · teacher leads students to reflect on what they have learned. · teacher gives exercise. · students reflect on what they have learned 5 minutes exercises i. assessment assessment technique : test question form : essay instrument : attached ii. student worksheet student worksheet subject : mathematics grade / semester : vii / 1 material : algebra student activities: 1. modelling the algebraic expressions using algebra tiles 2. solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division using algebra tiles 3. solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division without use algebra tiles 1. modelling the algebraic expressions using algebra tiles model the following algebraic expressions using algebra tiles! 1. 4x + 3 2. 5a – 10 3. x2 – 3x + 5 4. m + 7 = –6 5. 3z + 5 = 14 2. solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division using algebra tiles solve the following linear equations in one variable using algebra tiles! 1. x + 3 = 8 5. 3m = 24 2. 9 = m +1 6. 2x + 6 = 12 3. 10 = n – 4 7. 3z + 5 = –16 4. –5 + x = –1 8. 5x + 8 = 2x – 7 3. solving the linear equations in one variable by applying mathematical operations of addition, subtraction, multiplication or division without use algebra tiles solve the linear equations in one variable below! 1. 4 + x = 10 5. 4x = 12 2. 11 = n – 2 6. 2y + 4 = 0 3. a – 7 = 20 7. 14 = 2a + 4 4. –2 = – m + 5 8. 6n – 1 = 2n + 19 iii. assessment assessment subject : mathematics grade / semester : vii / 1 material : algebra solve the linear equations in one variable below! 1. x + 6 = 10 2. 8 = a – 1 3. 4x + 4 = 0 4. 3 + 2m = 15 iv. presence v. schedule classroom implementation has been implemented based on the schedule of the mathematics learning in class vii.7 smpn 18 tangerang. the classroom implementation was carried out on tuesday, 26 november 2019, from 12.20 to 1.40 pm at classroom of vii.7 smpn 18 tangerang. lesson plan school : smp n 1 petanahan subject : mathematics grade/semester : 7th / i subject matter : integer learning goal : students can doing arithmetic operation in integer student can find the properties of operation in integer students can solve the problems related to arithmetic operation in integer duration : 80 minutes based competencies : 3.2 explaining and doing arithmetic operation in integer and fraction using various properties of operation 4.2 solving problems related with arithmetic operation in integer and fraction. mode : online 1. preliminary activity main activity mathematical learning goal conjectured of students’ thinking teacher’s reaction teacher opens the lesson by greeting the students and checking the students’ readiness teacher explores the prior knowledge by asking the student about number students have the prior knowledge of number and numbers operation student know about number and represent the number of object with integer teacher motivates the students and asking how about positif and negative numbers students don’t know about integer teacher recall students ability in daily life case like money hook: teacher and students review prior lessons which are important for the new topic. integer consist of positive number, zero, and negative numbers. operation in numbers are addition, students recall their understanding to find out the concept of integer and operation of integer students know the concept teacher asks the student to tell to other what they have known about integer and operation of integer in real life problem substraction , multiplication and division students don’t know the concept teacher gives some questions, so the student would be found the concept of operation in integer 2. main activity going to geogebra classroom students access and explore activities in the geogebra classroom. student can do some activities in geogebra classroom teacher motivates the students and asks the students to share their activities student difficult to do some activities in geogebra classroom teacher helps students and motivates the students, make longer time to them and check whether there is a technical problem such as internet connection or device problem. understanding the problems 1. student doing the exercise in activity 1 and 2. teacher tells the student about the problem, activities 1: student can do addition with positive integer activities 2: student can do addition with negative number students ask the teacher about how to move the slider to change the number the teacher motivates the student and tell how to change the number by sliding the red point and blue point student can explore and move the slider into the number in the activities and get the right answer teacher motivates the student and ask them to share how they can do the activities to others doing the problem activities 3 : student can make generalization of addition two positive number and two negative number students make a wrong answer teacher motivates student and ask student to check the number line and check the answer of activities 1 and 2 students get the correct answer teacher gives positive feedback and let them share their idea activities 4 and 5: student can do addition negative number with positive number students can find the same result of activities 4 and 5 teacher motivates student and let them share the relation between activities 4 and 5 students cannot find the same result of activities 4 and 5 teacher motivates the student and recall the result in activities 4 and 5 activities 6 : student can make generalization of addition two positive number and two negative number students get the point of commutative properties in addition teacher give positive feedback and let them share their insight with their answer students do not get the point of commutative properties in addition teacher motivates students and asks students to make a correlation between activities 4 and 5 activities 7,8,9,10 and 11 : students can do substraction in integer and find the properties of substraction students get the right answer teacher gives positive feedback and let them share their idea that the propertis od substraction is not commutative, the sunstraction of number with negative number can make the substraction becoming addition. students get the wrong answer teacher motivates the students and recall their result in activities 7 and 8, do they get the same result ? in activities 8,9,10 and 11 they can make other simulation to get the point of substraction properties activities 12,13,14,15 and 16 : students can do multiplication in integer and find the properties of multiplication students make a wrong answer and did not find the concept of commutative in multiplication teacher motivates student and ask student to check the number line and check the answer of activities 12,13,14, and 15 students get the correct answer and find the concept of commutative in multiplication teacher gives positive feedback and let them share their idea activities 17,18,19 and 20 : students can do division in integer and find the properties of division in positive and negative numbers students make a wrong answer teacher motivates student and ask student to check the number line and check the answer of activities 17,18, teacher gives feedback of wrong conclusion in activities 19 and 20 and motivates them to recall their understanding students get the correct answer teacher gives positive feedback and let them share their idea 3.closing activity making conclusion teacher ask the student to share their insight from all activities of the properties operation in integer students get the right conclusion teacher gives positive feedback and let them share their idea to others students get the wrong conclusion teacher motivates the students to recall their understanding about comutative in addition and multiplication. mengetahui, kepala smp n 1 petanahan drs. supriyatin nur widayat nip. 196812071999031003 petanahan, 26 juli 2020 guru mapel sanni merdekawati,s.pd nip. rubric assesment assesment technique : student worksheet instrument : essay and multiple choice base competencies : 3.2 explaining and doing arithmetic operation in integer and fraction using various properties of operation 4.2 solving problems related with arithmetic operation in integer and fraction. no assesment aspect score assessment rubric 1 knowing concept of addition in line number 0 students don’t join the activity in geogebra classroom 2 1 students can’t do the activity in geogebra classroom. their answer doesnot related to the concept of addition in line number 3 2 students able to do the activities but get the wrong answer 4 3 students able to do the activities but can’t explain their idea 5 4 students able to do the activities , answer the task correctly and explain their idea about the concept of addition 6 generalizing / making conclusion about the concept of addition and properties of commutative 0 students don’t join the activity in geogebra classroom 1 students can’t do the activity in geogebra classroom. their answer doesnot related to the concept of addition in line number 2 students able to make summary of activities before 3 students able to share their idea about the properties of addition 4 students able to share their idea about the commutative properties in addition 7 knowing concept of substraction in line number 0 students don’t join the activity in geogebra classroom 8 1 students can’t do the activity in geogebra classroom. their answer doesnot related to the concept of substraction in line number 9 2 students able to do the activities but get the wrong answer 3 students able to do the activities but can’t explain their idea 4 students able to do the activities , answer the task correctly and explain their idea about the concept of substraction in line number 10 generalizing / making conclusion about the concept of substraction in line number 0 students don’t join the activity in geogebra classroom 11 1 students choice the wrong answer 2 students able to share their idea about their answer but still make wrong choices 3 students can do the activity in geogebra classroom. their answer is right 4 students able to share their idea about their choices and get the right answer 12 knowing concept of multiplication in line number 0 students don’t join the activity in geogebra classroom 13 1 students can’t do the activity in geogebra classroom. their answer doesnot related to the concept of multiplication in line number 14 2 students able to do the activities but get the wrong answer 15 3 students able to do the activities but can’t explain their idea 4 students able to do the activities , answer the task correctly and explain their idea about the concept of multiplication 16 generalizing / making conclusion about the concept of multiplication in line number 0 students don’t join the activity in geogebra classroom 1 students choice the wrong answer 2 students able to share their idea about their answer but still make wrong choices 3 students can do the activity in geogebra classroom. their answer is right 4 students able to share their idea about their choices and get the right answer 17 generalizing / making conclusion about the concept of multiplication in line number 0 students don’t join the activity in geogebra classroom 18 1 students choice the wrong answer 19 2 students able to share their idea about their answer but still make wrong choices 20 3 students can do the activity in geogebra classroom. their answer is right 4 students able to share their idea about their choices and get the right answer student’s worksheet “operation of integer” learning goal : • students can doing arithmetic operation in integer • student can find the properties of operation in integer • students can solve the problems related to arithmetic operation in integer this link to geogebra classroom: https://www.geogebra.org/classroom/tw82hm7z screenshot of activities https://www.geogebra.org/classroom/tw82hm7z 69 southeast asian mathematics education journal, volume 10, no 2 (2020) rural vs urban: teachers' obstacles and strategies in mathematics learning during covid-19 pandemic 1 fadhil zil ikram & 2 rosidah 1 universitas negeri malang, malang, indonesia 2 universitas negeri makassar, makassar, indonesia 1 dhilikram@gmail.com 2 rosidah.unesa@gmail.com abstract during the pandemic of covid-19, numerous obstacles arose in education, and some strategies were necessary to tackle them. this study adopted a qualitative approach to investigate teachers' difficulties in mathematics learning in rural and urban areas and how they solved them. the findings revealed that motivation, productivity, and activity are the main problems of the teachers. accessibility also poses a difficulty for teachers in rural areas. several strategies were implemented to solve these obstacles, including using technology, maintaining communication with families, providing flexibility, reducing the number of tasks and quizzes, and peer teaching. keywords: rural, urban, strategies, obstacles, mathematics learning. introduction covid-19 has been trending all around the world due to its impact on various fields, including education. teachers and students must adapt to the new learning situation, especially virtual learning (cheng, 2020). this type of instruction allows students who have limited access to get an education (febrianto, mas’udahdah, & megasari, 2020; radha, mahalakshmi, kumar, & saravanakumar, 2020). moreover, febrianto et al. (2020) said that distance learning also enables teachers to create a learning space for discussion. however, due to sudden adaption to the current situation, various virtual learning problems surely arise. a preliminary study conducted by observing social media revealed that many people complain about distance learning. the most common one was accessibility, with limited internet access students could not fully participate in the learning process. therefore, devising plans for learning activities is crucial for teachers. there is an exponential increase in the number of studies regarding mathematics education during the pandemic in indonesia. for example, there is a study examining the effectiveness of "edutainment" as a learning media (pratama, lestari, & astutik, 2020) and the use of whatsapp and zoom, which facilitate mathematics learning in terms of students’ mathematics achievement (kusuma & hamidah, 2020; yensy, 2020). additionally, there is one investigating the mathematics learning process during the pandemic (irfan, kusumaningrum, yulia, & widodo, 2020; wiryanto, 2020), and a study that explored the difficulties of undergraduate students of mathematics education during their distance learning (annur & hermansyah, 2020). a study by mailizar et al. (2020) investigated mathematics teachers’ views on distance learning implementation. the difference in learning obstacles most likely to occur due to the difference in residential areas during the pandemic, but a study comparing teachers' strategies in rural and urban areas is a rarity. therefore, the researcher 70 southeast asian mathematics education journal, volume 10, no 2 (2020) found it necessary to conduct a study investigating mathematics teachers' obstacles and strategies from rural and urban areas to implement mathematics learning during the pandemic. methods this study adopted a qualitative approach to investigate mathematics teachers' obstacles and strategies during covid-19 pandemic. ten mathematics teachers, consisting of four teachers from urban areas and six teachers from rural areas, participated in an in-depth interview regarding their mathematics teaching problems and how they tackled them. telephone or chatting application was employed to collect data. the interview conducted followed a protocol that contains two main questions: the obstacles that the teachers encountered in teaching mathematics during the pandemic, and the strategies they employed tackle the issues. the interview results were then analysed to compare obstacles and strategies in mathematics learning during pandemic from the perspective of teachers in rural and urban areas. the analysis methods used were data condensation, data display, verification, or concluding (miles, huberman, & saldana, 2014). in data condensation, the researcher thoroughly scanned the interview transcripts to look for relevant information regarding teachers’ struggles and strategies and highlighted them. the other information was left to be considered later to look for related and unique information or findings. the data were displayed in a table to look for patterns and consistencies (data display). then the researcher drew a conclusion based on the data (verification or concluding). results and discussion after conducting the interview, asking the teachers about the obstacles they encountered in distance mathematics' learning during a pandemic situation and the strategies to solve them, the results were analysed as follows. table 1 result of the study obstacles teachers strategies less motivated less productive less active rural use of various media such as whatsapp, zoom quizizz, youtube, and google classroom. utilising many educational platforms. asking students to create a product related to the topic taught. maintaining communication with students’ families. reducing the number of assignment and quizzes. urban students’ access to learning rural homeroom teachers visiting and giving the students teaching materials to students without access to the internet. giving flexibility for students with accessibility problems. asking students to help each other by borrowing or lending their phone or laptop or borrowing the gadgets from their relatives or neighbours. peer-teaching 71 southeast asian mathematics education journal, volume 10, no 2 (2020) table 1 provides information about mathematics teachers' obstacles in rural and urban schools during the pandemic. commitment to learning becomes one of the issues during the pandemic (engelbrecht, llinares, & borba, 2020), and in this study, they are in the forms of motivation, activity, and productivity. interviews with the teachers revealed the source of these difficulties. table 2 interview excerpt teachers interviewer answer t1 why did the students become less motivated, less productive, and less active? they prefer face-to-face learning. they want to experience the classroom's physical environment. t2 why did the problem with students’ commitment to learning happen? they did not like distance learning. they have a better understanding when the teachers deliver the material directly. for mathematics, all of them even said that face-to-face learning is a must. t3 do you know why the students have a commitment problem? yes, they prefer meeting directly in the classroom. table 2 shows that although many students were already capable of studying through internet resources, they still prefer face-to-face learning. several studies also reported that many students still prefer offline learning (febrianto et al., 2020; radha et al., 2020). one of the strategies used was decreasing the number of assignments. according to the mathematics teachers interviewed, the number of assignments contributes to students' anxiety and stress, leading to a decrease in their commitment to learning. nonetheless, because of virtual learning, nguyen et al. (2020) said that teachers should increase the number of tasks. they suggested that this strategy prevents students from cheating. however, in this study, the teachers prioritise their students' mental health to maintain their immunity to the virus. it follows kamsurya (2020) findings, stating that people should support and increase the immune system to avoid the coronavirus. another strategy employed was the optimisation of the use of learning media. the teachers interviewed utilised social media and many learning platforms to implement their learning. george (2020) also proposed some of the strategies used: online teaching platforms, learning applications, and youtube videos. in this study, some teachers even asked the students to create a video presenting a mathematical concept. although both teachers’ strategies were similar, in-depth interview showed a slight distinction. teachers from rural areas revealed that homeroom teachers visited their students with accessibility problems once every two or three weeks to maintain communication with family. the family greatly influences students' learning process during the pandemic (febrianto et al., 2020). many students neglected their assignments because they must help their parents’ jobs. the teachers then discussed it with the parents about the ideal learning for their children. both teachers from rural and urban areas prefer using virtual conference applications to teach. not only did it become the trends (pratama, azman, kassymova, & duisenbayeva, 72 southeast asian mathematics education journal, volume 10, no 2 (2020) 2020), but some studies (kusuma & hamidah, 2020; yao, rao, jiang, & xiong, 2020) also suggested them. however, teachers from rural schools have to consider their students' internet quota by focusing more on the whatsapp group or providing a recorded video that still has effectiveness (morgan, 2020; yensy, 2020). applications such as zoom were used only several times to minimise students' internet data usage. the next obstacle was accessibility to the internet. many students worldwide do not have access to distance learning or even just to read reading resources (bakker & wagner, 2020; engelbrecht et al., 2020; morgan, 2020; sá & serpa, 2020). in indonesia, such a problem is also present (bahasoan, ayuandiani, mukhram, & rahmat, 2020; febrianto et al., 2020; hebebci, bertiz, & alan, 2020; kamsurya, 2020; mailizar et al., 2020). however, in this study, only the teachers from rural areas reported the problem. the teachers have to present equity and flexibility to the students. in rural schools, they gave homeroom teachers teaching materials to visit their students with accessibility problems. such an effort was similar to the strategies employed by china's government, where they asked the social worker to ensure students participate in learning (cheng, 2020). moreover, the interview revealed that the mathematics teacher from rural schools even implemented peer-teaching or asked students to borrow gadgets (phone or laptop) from relatives, friends, and neighbours to ensure learning participation. interview results showed that the teachers have sufficient skill to utilise learning media in delivering mathematics instruction during the pandemic. similar findings also reported that teachers become more innovative and creative in optimising their teaching (febrianto et al., 2020; lestari & gunawan, 2020). in this study, the teachers provide a solution for students without internet access and those with unlimited access. conclusion the study results implied that teachers' obstacles in rural and urban areas are quite the same: students were less motivated, less productive, and less active. the only difference was the internet accessibility problems. regarding teachers’ strategies, they include the use of various media; peer teaching and cooperation among students (rural schools); doing projects; visiting (rural schools), consulting, and maintaining communication with students’ families; reducing the number of quizzes and assignments; and giving flexibility to students with limited internet access. references annur, m. f., & hermansyah, h. 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(2020). minimise online cheating for online assessments during covid-19 pandemic. journal of chemical education, 97(9), 3429–3435. https://doi.org/10.1021/acs.jchemed.0c00790 pratama, h., azman, m. n. a., kassymova, g. k., & duisenbayeva, s. s. (2020). the trend in using online meeting applications for learning during the period of pandemic covid-19: a literature review. journal of innovation in educational and cultural research, 1(2), 58–68. https://doi.org/10.46843/jiecr.v1i2.15 pratama, l. d., lestari, w., & astutik, i. (2020). efektifitas penggunaan media edutainment di tengah pandemi covid-19. aksioma: jurnal program studi pendidikan matematika, 9(2), 413–423. radha, r., mahalakshmi, k., kumar, v. s., & saravanakumar, a. r. (2020). e-learning during lockdown of covid-19 pandemic: a global perspective. international journal of control and automation, 13(4), 1088–1099. sá, m. j., & serpa, s. (2020). the global crisis brought about by sars-cov-2 and its impacts on education: an overview of the portuguese panorama. sci insight edu front, 5(2), 525–530. https://doi.org/10.15354/sief.20.ar039 wiryanto, w. (2020). proses pembelajaran matematika di sekolah dasar di tengah pandemi covid-19. jurnal review pendidikan dasar: jurnal kajian pendidikan dan hasil penelitian, 6(2), 125–132. yao, j., rao, j., jiang, t., & xiong, c. (2020). what role should teachers play in online teaching during the covid-19 pandemic? evidence from china. sci insight edu front, 5(2), 517–524. https://doi.org/10.15354/sief.20.ar035 yensy, n. a. (2020). efektifitas pembelajaran statistika matematika melalui media whatsapp group ditinjau dari hasil belajar mahasiswa (masa pandemik covid 19). jurnal pendidikan matematika raflesia, 05(02), 65–74. 97 southeast asian mathematics education journal, volume 10, no 2 (2020) case study: developing computational thinking skill during pandemic situation 1 mharta adji wardana & 2 iwan pranoto 1institut teknologi bandung, bandung, indonesia 2institut teknologi bandung, bandung, indonesia 1 mhartawardana@gmail.com 2 iwanpranoto@ymail.com abstract in 1980, seymour papert mentioned that computational thinking is the idea where the interaction between a student and computer can become a mental model, assisting the learning process. this idea becomes well known as constructionism. thus, papert believes that the computer presence or even the thinking of computer interaction may help the student to think and learn better. it is more general than the present widely accepted perception of computational thinking, where it focuses on the utilization of computers in problem-solving only. this case study aims to describe how students can learn computational thinking through the traditional curriculum and unplugged setting in this pandemic. three middle school students participated in a concrete mathematics lesson design in a middle school lesson on linear function topic. the result indicates that clear instructions and gradual examples will help students understand the series of operations that are part of computational thinking. keywords: computational thinking, lesson design, linear function, mathematics content, mental model. introduction this study recalls the existing definitions of computational thinking (ct) proposed by previous researchers. cansu and cansu (2019) lists and examines various definitions of ct from 2009 to 2014. some researchers also stated that ct is about formulating problems and creating solutions that a computer can apply effectively (wing, 2006; yadav et al., 2014; denning, 2009; hemmedinger, 2010). there are various existing definitions of computational thinking proposed by previous researchers. cansu and cansu (2019) lists those definitions that emphasize ct's utilization as a problem-solving tool. we try to evaluate those definitions by examining papert’s original idea about constructionism theory as follows. table 1 some definitions of ct source definition evaluation wing (2014) computational thinking is the thought processes used to formulate a problem and express its solution or solutions in terms a computer can apply effectively. from these existing definitions, we learn that previous researchers consider ct as a tool that can be used to solve problems. in contrast, we do not always need problems to hone ct. we yadav et al. (2014) the mental process for the abstraction of problems and the creation of automatable solutions. 98 southeast asian mathematics education journal, volume 10, no 2 (2020) denning (2009) computational thinking has a long history in computer science. known in the 1950s and 1960s as "algorithmic thinking," it means a mental orientation to formulating problems as conversions of some input to output and looking for algorithms to perform the conversions. today the term has been expanded to include thinking with many levels of abstractions, using mathematics to develop algorithms, and examining how well a solution scales across different sizes of problems. try to bring ct to a wider scope. students can think and learn better through ct. computers in mind as a mental model to examine mathematics concepts to study. hemmendinger (2010) computational thinking is to teach them how to think like an economist, a physicist, an artist, and to understand how to use computation to solve their problems, to create, and to discover new questions that can fruitfully be explored. even though papert (1980) did not define ct explicitly in mindstorm, he gave a hint that through ct, students can think and learn better. for instance, through ct, the linear function concept as a series of operations can be represented as a real model. furthermore, it will become a mental model. in traditional approaches to teaching linear function, teachers perhaps do not focus enough that linear functions such as f(x)=2x-3 are algorithms. a linear function consists of a step-bystep series of instructions, namely, when someone gives a number x, to find the value f(x), she or he must do the step-by-step computations. first, x is multiplied by 2; then, it is subtracted by 3. students should understand that the order of the steps is significant. doing subtraction first will not result in the same. to comprehend this algorithm idea of function and linear function was fundamental before and is much more relevant now. to function effectively in contemporary life, people must have the skills to work collaboratively side-by-side with machines. people should develop the consciousness that they now live side-by-side with computers. therefore, they must be able to communicate and utilize computers properly. so, modern people must be able to talk to not only other human beings but also computers. this computer culture should change mathematics teaching. in this pandemic situation, the teaching and learning process is in a distance learning setting. the communication’s between students and teachers is limited. the notion of minimally invasive education (mie) proposed by mitra (2000) can be an alternative way for students to learn independently with the teacher’s minimum intervention. further, mie is a learning approach with minimum or no teacher’s interventions (mitra, 2000). previous research about mie also stated that applying mie can foster mathematics achievement (inamdar & kulkarni, 2007). so, it is a potential strategy to use distance learning with limited communication facilities between students and teachers. this research takes a different notion of ct. this research reports the implementation of papert's original idea (1980) in middle school mathematics teaching. in addition to the concept 99 southeast asian mathematics education journal, volume 10, no 2 (2020) of ct as a problem-solving tool, it will be shown qualitatively that through ct, students can think and learn the function concept better. mathematics teaching with the above understanding in mind raises the awareness of computer’s roles in the present and future life. the approach could also help the students to improve their comprehension of linear function concepts. these are the two aims of the work. methods this research utilized a case study approach since it fits with this research as it involves limited participants. we firstly develop a series of modules, and we examine them. however, most schools are now closed, and it is limited to conduct this research in face-to-face interaction. then, we sent the modules through social media applications (whatsapp.) students were selected randomly. after that, we collect the results through the student’s work (image files) sent through whatsapp. finally, we analysed the data through content analysis. results and discussion mathematics lesson design in an unplugged setting in this pandemic situation, students and teachers do the learning process in a distance mode. to facilitate student’s learning with minimum intervention by the teachers, we develop several self-explained modules on a linear function. moreover, the modules are more visual, and they do not use lengthy and wordy sentences. furthermore, visual mathematics learning can help students imagine the topic, check the truth in their way, and increase student self-confidence (montenegro, 2003). to reach an effective result, we adapt the essential concept of function proposed by nctm (2020) to represent function into a formula, table, and graphic. thus, we focus on three modules, knowing the linear function, convert the table of a linear function to a formula, and convert the formula to a graphic. the following are the modules developed in this study. module 1: knowing the linear function through this module, we try to create a mental model of a linear function. one can illustrate a linear function as a step-by-step series of operations. this model will be a mental model brought by students to understand function or other mathematics topics. figure 1. sample of the task on module 2. 100 southeast asian mathematics education journal, volume 10, no 2 (2020) in simple terms, those step-by-step series of operations is the central part of computational thinking; it is an algorithm. through this visualisation, we invite the students to envisage how meaningful each step of the operation. multiplying by 2 may not be a difficult step if x is a small number. if we input the value of x with the 9-digit specific number, students will realise that they need a computer to calculate it quickly and precisely. it has raised a student’s awareness of computer ability. at the end of the modules, students input the value of x and try to find the gradient value (𝑎) and intercept (𝑏) according to the above steps. module 2: convert the table of linear function to formula we try to represent the straightforward conversion steps to determine the value of 𝑎 and 𝑏. instead of using delta ∆ and 𝑦𝑖 notation, we prefer to present it with an illustration. we try to create this module that is self-explained. figure 2. sample of the task of module 2. goyal (2012) said that mie gives more opportunity for students to explore the ideas that lie in the lesson content. students analyse that the difference between 𝑦 can also be a negative value. here, we provide opportunities for students to uncover ideas about changes in the value of 𝑦. again, each step of the whole algorithm is meaningful for students. through ct, the students could comprehend the linear function concept better. module 3: convert function formula to graphic this module helps students learn that each input 𝑥 will pair with a 𝑦 value as a collection of points. then, these points are plotted on the coordinate plane. 101 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 3. sample of task on module 3. we provided an example of how to sketch a graphic. first, we give a complete illustration; then, we reduce the instructions gradually. students see this process as a routine procedure suitable for computers to do. to enhance the student’s understanding, we provide modules 1b, 2b, and 3b with reverse instructions. implementing mathematics learning to develop computational thinking in an unplugged setting the implementation of ct modules is as follows. the teacher shared the modules in pdf. via whatsapp and asked students to copy all modules to their paper. due to the distance learning setting applied, a teacher cannot meet the students in-person. through mathematics instruction, we try to raise student’s awareness of computer’s roles in life and make students comprehend the linear function concept easier. we expect students to discover each meaning of step, such as imagine what if the input 𝑥 is a large number, so we need the machines help to calculate. to confirm that students think and learn about the linear function concept better, we ask them to do regular tasks. the students do the tasks before and after doing the modules. after that, we 102 southeast asian mathematics education journal, volume 10, no 2 (2020) arrange some interviews through whatsapp to discuss what the students have learned. the following table summarises the student’s results. table 2 students results on the given modules lesson instructions students 1 2 3 pre-test module 1a misconception misconception misconception module 1b misconception inaccurate accomplished module 2a accomplished accomplished inaccurate module 2b inaccurate inaccurate accomplished module 3a inaccurate inaccurate module 3b misconception misconception post-test inaccurate inaccurate the validity of student’s works determines the above classification. in that classification, the label misconception means they do not understand the basic concept, and the label inaccurate means the student's calculation contains some errors. from table 2, two students accomplished module 2a, one student accomplished module 1b, and one student completed module 2b. on the other modules, students could not execute yet. the following pictures show the answer of some students in module 1a. figure 4. the answer of student 1. based on the answer of student 1 on problem 2, it shows he could follow the instruction and fill the missing number correctly, but he missed the minus sign on the answer (it must be -5 instead of 5). meanwhile, on problem 3, he put the incorrect number on the coefficient box. 103 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 5. the answer of student 2. based on student 2’s answer, he also entered the incorrect number in the coefficient box, even though he was correct in filling in the intercept box. he had a misconception about the basic linear function concept. we need to add more examples where the coefficients have been installed gradually, for instance, from 4x, 3x, 2x, to x. figure 6. the answer of student 3. based on students 3’s answer, it is clear that we need even more gradual examples. she did not learn from the first example well, so she put the incorrect number on the intercept box. besides, students could not answer the problems; the awareness of the use of information processing machines has not yet emerged. they did not realise that they do not need to compute large numbers instead of the computer. we need to add more examples on the modules, showing large number inserted and ask them to calculate. hopefully, they will realise that the computer does the calculation of large numbers better. from the results, it seems that it needs extra work to make students understand the instructions. the results might be different if their regular teacher carried out this lesson. this last reflection may be the vital message that our present education system is not yet ready selfteaching environment. we also cannot entirely depend on student’s ability to understand instructions without the teacher's help. 104 southeast asian mathematics education journal, volume 10, no 2 (2020) conclusion this study has provided insight into nurturing ct skills through an unplugged environment. the effectiveness of the modules and the implementation during pandemics need to be addressed in future research. as this study involved limited participants, one must use the results cautiously. references cansu, s. k., & cansu, f. k. (2019). an overview of computational thinking. international journal of computer science education in schools, 3(1), 1-11. https://doi.org/10.21585/ijcses.v3i1.53. denning, p. j. (2009). the profession of it beyond computational thinking. communications of the acm, 52(6), 28-30. https://doi.org/10.1145/1516046.1516054. hemmendinger, d. (2010). a plea for modesty. acm inroads, 1(2), 4-7. https://doi.org/10.1145/1805724.1805725. inamdar, p., & kulkarni, a. (2007). ‘hole-in-the-wall’ computer kiosks foster mathematics achievement-a comparative study. journal of educational technology & society, 10(2), 170-179. retrieved from https://www.jstor.org/stable/10.2307/jeductechsoci.10.2.170. mitra, s. (2000). minimally invasive education for mass computer literacy. proceeding in conference on research in distance and adult learning in asia 2000, pp. 21-25. hongkong: the open university of hongkong. montenegro, j. j. (2003). visual mathematics. proceeding in 2nd international conference on multimedia and icts in education. spain: badajoz. national council of teachers of mathematics (nctm). (2018). catalysing change in high school mathematics: initiating critical conversations. reston, va: nctm. national council of teachers of mathematics (nctm). (2020). standards for the preparation of secondary mathematics teachers. reston, va: nctm. papert, s. a. (1980). mindstorms: children, computers, and powerful ideas. new york: basic books. yadav, a., mayfield, c., zhou, n., hambrusch, s., & korb, j. t. (2014). computational thinking in elementary and secondary teacher education. acm transactions on computing education (toce), 14(1), 1-16. https://doi.org/10.1145/2576872. wing, j. m. (2006). computational thinking. communications of the acm, 49(3), 33-35. retrieved from http://www.cs.cmu.edu/~15110-s13/wing06-ct.pdf. https://doi.org/10.21585/ijcses.v3i1.53 105 southeast asian mathematics education journal, volume 10, no 2 (2020) learning mathematics through mathematical modelling processes within sports day activity 1 sakon tangkawsakul, 2 nuttapat mookda, & 3 weerawat thaikam 1 faculty of education, kasetsart university, bangkok, thailand 2 directorate of education and training, rtaf, thailand 3 faculty of education, nakhonsawan rajabhat university, nakhonsawan, thailand 1 s.tangkawsakul@gmail.com abstract applying mathematics in real-life situations is an important objective of mathematics education. yet in thailand, the students' ability to connect mathematics to real-life situations is insufficient. in this study, we adapted the school sports day to provide opportunities to relate real-life situations with mathematical knowledge and skills. this study aims to describe the students' responses and the teacher's interaction during a modelling task. the designing of the modelling task, inspired by the realistic fermi problems, involves collaboration between mathematics teachers and educators and the participation of 10th grade students. the task is set in the context of bleacher in the school sports day. each experiment's modelling task lasted for 45 minutes and was conducted in the one-day camp with 45 students. data were collected through observation, interview, and written task before being analysed through content analysis. the results showed that the students who had no previous mathematical modelling experience engaged in mathematical modelling processes with their friends under the guidance and support of the teacher. most of them were able to think, make assumptions, collect data, observe, make conjectures and create mathematical models to understand and solve the modelling task. keywords: mathematical modelling, realistic fermi problems, sports day, real-world problems. introduction in several countries, the promotion of science, technology, engineering, and mathematics (stem) education is an essential educational topic that enables students for a scientific and technological society. one of the important teaching and learning approaches for the transition to stem education and interdisciplinary mathematics education is mathematical modelling (borromeo ferri & mousoulides, 2017; tezer, 2019). besides, mathematical modelling can be considered good examples of stem integration (kertil & gurel, 2016). actually, mathematical modelling supports mathematical learning and enables students to deal with real-world problems (blum & borromeo ferri, 2009). although the ability to apply mathematical knowledge and solve real-world problems has been recognised in the basic education core curriculum in thailand, both students and teachers have little experience in mathematical modelling. moreover, about 50 per cent of 15-year-old thai students did not achieve the international basic proficiency level (level 2) at mathematical literacy in pisa 2009 and pisa 2012. these results show that thai students lack the ability to connect realworld problems with mathematics (klainin, 2015). hence, we were interested in observing and describing how the students respond to modelling activities. to achieve that purpose, we started by designing tasks to encourage students to connect inside and outside classroom mathematics and foster their ability to solve 106 southeast asian mathematics education journal, volume 10, no 2 (2020) real-world problems with mathematical modelling processes. we adapted the idea of using realistic fermi problems about the bleacher in the school sports day, closely related to thai students’ experiences, to introduce mathematical modelling in upper secondary mathematics following ärlebäck and bergsten (2013). the questions we aim to answer in this paper are as follows: 1. how does the teacher interact with students during the mathematical modelling task? 2. how do the students respond to mathematical modelling task? this study aims at describing the students' responses to the modelling task and the teacher's interaction with their students during mathematical modelling processes. theoretical framework mathematical modelling processes mathematical modelling is an essential educational topic that fosters the students' ability to deal with real-world problems (blum & ferri 2009). mathematical modelling processes showed how the process connects real-life contexts and mathematics content. it might look different and highlight different perspectives depending on the research's purpose and focus (blum, galbraith, & niss, 2007). in this study, we adapted an ideal modelling process that includes six phases allowing cognitive activities to solve the modelling tasks described by ferri (2006) as shown in figure1. figure 1. modelling cycle under a cognitive perspective described by ferri (2006). based on ferri's model, modelling processes consisted of two components: phases and transitions that intertwined between two domains, reality and mathematics. the six phases comprise a real situation, mental representation of the situation, real model, mathematical model, mathematical result and real result. simultaneously, the transitions include six activities: understanding the task, simplifying/structuring the task, mathematising, working mathematically, interpreting and validating. 107 southeast asian mathematics education journal, volume 10, no 2 (2020) the realistic fermi problems there are several important principles in model-eliciting activities (lesh, hoover, hole, kelly, & post, 2000). two of them are: 1) model construction principle, as in the problem must evoke the need to mathematise and model meaningful situation to solve a problem, and 2) reality principle, as in the problem need to be relevant and meaningful to the students; the kind that they might encounter in real life. effective modelling activities should also be open, in a sense that there are no predetermined right answers. the students have the freedom to choose the most suitable mathematical concepts they will use to solve the problem. one of the notions that fit these principles is the fermi problem. the term fermi problem originates from the 1938 italian nobel prize winner in physics enrico fermi (1901-1954). he had posed and solved problems like how many piano tuners are there in the us. he demonstrated that using a few reasonable assumptions and estimates could give accurate and reasonable answers (efthimiou & llewellyn, 2007). the fermi problem was always answered by simplifying, making assumptions, estimating, and doing rounded calculations while the exact answer is often not available. it is the main important feature of the fermi problem-solving process (sowder, 1992). according to ärlebäck and bergsten (2013), the characters of realistic fermi problems was described as follow: 1. the realistic fermi problem does not necessarily demand any specific pre-mathematical knowledge. all individual students or groups of students can access and solve the problem; 2. the realistic fermi problem is more than just an intellectual exercise. the context is realistic and presents clear real-world connection; 3. the realistic fermi problem is open. the specifying and structuring of the relevant information and relationships are needed to tackle the problem; 4. the realistic fermi problem does not show the numerical data. the problem solver needs to make reasonable estimates of relevant quantities, and 5. the realistic fermi problem is to promote group discussion. design of the study the question was posited as the students' responses to the modelling task and teachers' interaction with their students during each mathematical modelling processes. participants the study participants were divided into two groups. the first comprised three mathematics teachers and two mathematics educators interested in mathematical modelling, while the second consisted of 45 tenth grade students. these students had no previous experience in mathematical modelling in the classroom based on the basic education core curriculum of thailand. methods in designing the bleacher task, the task's objectives were to encourage students to connect inside and outside classroom mathematics and enhance their ability to solve real-world problems with mathematical modelling. we designed and validated the task by collaborating 108 southeast asian mathematics education journal, volume 10, no 2 (2020) with three mathematics teachers and two mathematics educators interested in mathematical modelling. the task validation is based on four notions below. 1. the characters of realistic fermi problems described by ärlebäck and bergsten (2013). 2. the information on the students’ experiences and prior knowledge relevant to the task. this information was collected by conducting interviews with mathematics teachers who had experience teaching these students. the interview was important to confirm that the bleacher in the school sports day problem is both encouraging and engaging context for the students. 3. the framework for designing mathematical modelling learning experience described by ang (2018). 4. the modelling cycle under a cognitive perspective described by ferri (2006). the bleacher task is as follows: the bleacher task: in the school sports day, all students were organised into several groups with wide-ranging sports and varied performance abilities. each group was labelled by colour to participate in sports, cheerleading and bleacher cheer show performance. you and your team members are given the relevant materials. draw your own conclusions and answer these following questions: 1. how many students that can reasonably be arranged in the bleacher for cheer show performance? 2. what is the relationship between the sizes of bleacher and the numbers of students? in implementing the bleacher task, a qualitative approach was adopted. the teacher's role (one of the authors) was as a facilitator, encouraging and guiding a small group of students to deal with the bleacher task. the 45 students were divided into eight groups of 4 – 5 students with wide-ranging abilities in mathematics. each experiment lasted for 45 minutes and was conducted with two groups of students in the one-day camp. the students' responses and teachers' interactions during mathematical modelling process were gathered through observations, written work, and interviews. the data were analysed by content analysis according to the mathematical modelling framework by ferri (2006). the result is presented in a narrative description. results and discussion this study showed the students' responses to the bleacher task and the way that teacher interact with their students during each mathematical modelling process as follows. initially, the teacher and students met at bleacher in the school playground (figure 2). the teacher asked the student about their experience with the bleacher on the school sports day. all of the students participated in the conversation about the bleacher. the students were given the bleacher task. the teacher then gave the students time to understand the task and discuss what the problem wanted to know with their friends (figure 3). 109 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 2. meeting at bleacher in the school playground. figure 3. understanding the task. next, the teacher discussed the factors related to determining the number of students sitting on the bleacher to guide students to simplify the task and make assumptions. most of the students' assumptions were as follows: 1. each student must be seated according to the size of the bleacher, fit, and not overcrowded; 2. the sitting position must look neat and straight; and 3. the sitting posture must be the same. besides, one group assumed the distance between each student must be appropriate for movement in performance. then, the teacher asked the students what the relevant and necessary data is, which we have to search for solving the problem. some groups of students identified some unnecessary data (width and height of the bleacher). in these cases, the teacher asked them how the bleacher's width and height were relevant and necessary for solving the problem. they discussed and found that the bleacher's width and height were not necessary (figure 4). figure 4. identifying some unnecessary data. 110 southeast asian mathematics education journal, volume 10, no 2 (2020) on the other hand, most students were available to identify the relevant and necessary data (the size of the student who sat on a bleacher or area for sitting per person) as shown in figure 5. during their search for the data, two groups have interesting reasons for determining the size of the student which connects between real-world and mathematics such as knowing the height and weight of the student to choose the right seated student, fit and balance to the bleacher, and sitting in a straight line. figure 5. identifying the relevant and necessary data. after that, the teacher guided the students to create a real model which simplified and structured students' mental picture (figure 6). they used drawings and diagrams to represent the bleacher's size, sitting position, and distance between each student. some student groups identified conditions about sitting posture and performance on the bleacher such as space for placing cheer prop and students' size and height. figure 6. a real model which simplified and structured students’ mental picture. the teacher then allowed the students time to discuss and think about the appropriate mathematical knowledge and concepts related to solving the problem. moreover, they tried to sit (figure 7) and measure by using the measuring tape (figure 8) to collect data, observe, make conjectures, and create mathematical models to solve the problem under the teacher's guidance and support. figure 7. sitting experiment. 111 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 8. measuring using the measuring tape. the students used the bleacher sizing and the sitting experiments and obtained measurement from the different width of the body (the lap and shoulder). some groups used either the actual measurements (2 decimal places) or approximate values from their measurements which round it to integers to make calculations easier (figure 9). finally, some groups used only one value from one measurement, while others choose to use the mean of values from their repeated measurements. next step, the students worked mathematically based on unit rate ideas to represent the relation of seat length per person, then created the relation between seat length and the number of the students in general term. the teacher asked the students to interpret and check the mathematical results. figure 9. working mathematically. furthermore, students validated the result by comparing the solution with their experience. finally, they have to share their problem-solving processes and explain the meaning of all variables in the equation based on the real-world situation to others, as shown in figure 10. in this step, we found that some groups have mistaken in interpreting. for example, the students used the equation (y = 25x) to model the general relation between seat length (y) and the number of students (x). figure 10. sharing their problem-solving processes. 112 southeast asian mathematics education journal, volume 10, no 2 (2020) in this case, the teacher asked students to think about applying the student's model in a normally real-world context because actually we knew the size of the bleacher first then found the number of students. so, the teacher tried to lead students to concern about the independent variable (input) and the dependent variable (output) based on the real-world situation. it was evident from the result that the use of open and complex problems for a diverse solution, like fermi problems, can be an effective modelling task. the problem can elicit the cognitive processes essential to mathematical modelling (ferri, 2006) from the students. this is in line with the studies by peter-koop (2004) and ärlebäck (2009), which demonstrate fermi problems' potential to engage the students and encourage them through multiple modelling cognitive stages. conclusion the description of the students' response and the teacher's interaction shows that typical fermi problem, which is open, ill-structured, and possesses high-complexity, can effectively engage the students throughout the modelling task and elicit an appropriate cognitive response. the real-world context that is closed and related to students' experience also enhances students' engagement in the mathematical modelling processes with their friends under the teacher's guidance and support. it is recommended for the teacher to plan to deal with students in the modelling activity to know about the phases and their transitions in each modelling cycle because it is essential to guide and support them in dealing with a real-world problem. references ang, k. c. (2018). mathematical modelling for teachers: resources, pedagogy and practice. london, england: routledge. ärlebäck, j. b., & bergsten, c. (2013). on the use of realistic fermi problems in introducing mathematical modelling in upper secondary mathematics. in lesh r., galbraith p., haines c., hurford a. (eds) modeling students' mathematical modeling competencies (pp. 597-609). dordrecht, netherlands: springer. doi: 10.1007/978-14419-0561-1_52 blum, w., & borromeo ferri, r. (2009). mathematical modelling: can it be taught and learnt? journal of mathematical modelling and application, 1(1), 45-58. borromeo ferri, r. (2018). learning how to teach mathematical modelling in school and teacher education. springer international publishing. doi: 10.1007/978-3-319-68072-9 borromeo ferri, r., & mousoulides, n. (2017). mathematical modelling as a prototype for interdisciplinary mathematics education? theoretical reflections. in t. dooley & g. gueudet (eds.), proceedings of cerme 10 (pp. 900-907). dublin, ireland: erme. 113 southeast asian mathematics education journal, volume 10, no 2 (2020) efthimiou, c. j., & llewellyn, r. a. (2007). cinema, fermi problems and general education. physics education, 42(3), 253-261. doi: 10.1088/0031-9120/42/3/003 kertil, m., & gurel, c. (2016). mathematical modelling: a bridge to stem education. international journal of education in mathematics, science and technology, 4(1), 44-55. doi: 10.18404/ijemst.95761 sowder, j. t. (1992). estimation and number sense. in d. a. grouws (ed.), handbook of research on mathematics teaching and learning: a project of the national council of teachers of mathematics (p. 371–389). reston, va: the national council of teachers of mathematics, inc. sunee, klainin. (2015). mathematics education at school level in thailand the development – the impact the dilemmas. retrieved from https://library.ipst.ac.th/handle/ipst/959 tezer, m. (2019). the role of mathematical modeling in stem integration and education. in k.g. fomunyam (ed.) theorizing stem education in the 21st century. intechopen. doi: 10.5772/intechopen.88615 https://library.ipst.ac.th/handle/ipst/959 114 southeast asian mathematics education journal, volume 10, no 2 (2020) 87 southeast asia mathematics education journal, volume 10, no 2 (2020) how does guided inquiry enhancing students’ mathematical literacy? an experimental study for mathematics learning 1 luthfi nur azizah & 2 ali mahmudi 1 department of mathematics education, graduate school, universitas negeri yogyakarta 2 faculty of mathematics and science, universitas negeri yogyakarta 1 luthfinurazizah.2017@student.uny.ac.id abstract this paper reports a quasi-experimental study with one group pre-test post-test design on promoting mathematical literacy using guided inquiry method. as one of the important education goals globally, mathematical literacy focuses on a concrete dimension with context and content to be studied. a total of 32 secondary school students of class viii participated as a sample and 192 students as a population in the study. students' mathematical literacy was measured by using an essay test. the students' responses were analysed and compared using paired sample t-test. it was found that the guided inquiry method successfully promotes students' mathematical literacy as indicated by the result of paired sample t-test values of significance (2-tailed) of 0.000 < 0.05. thus, the guided inquiry can be used as an alternative learning method to improve students' mathematical literacy. keywords: guided inquiry, mathematical literacy, quasi-experimental. introduction one of the goals of today's education is to prepare students to apply the knowledge they have in everyday life. this is in line with moretti and frandell's (2013) opinion, which stated that education has roles in preventing risks and a tool that can improve the quality of human life. mathematical literacy helps someone recognise the role of mathematics and assess it to make decisions that a constructive, active, and reflective citizen needs (oecd, 2017). in addition, the results of research conducted by higo, wada, and sato (2012) showed that students who have good mathematical achievement have good mathematical literacy skills. in fact, students' mathematical literacy skills in indonesia are still low and far below the average. this can be seen from the results of pisa 2009, pisa 2012, and pisa 2018 which revealed that more than 75 % of indonesian students were only able to work up to level 2 (oecd, 2010; 2014; 2019) according to khoirudin, setyawati, and nursyahida (2017), low mathematical literacy skills are influenced by learning process at school, classroom environment, support for the family environment, test readiness and implementation, and abilities of each student. school mathematics learning needs to be designed by teachers in situations where students are active in every learning activity. therefore, traditional learning is not the right alternative for designing learning activities. there are several learning methods and strategies that can be applied to make students more active in learning activities. as the opinion of khan (2012, p.1) which states that "educationists suggest various student-centred strategies for making students more autonomous in the learning process". therefore, various student-centred 88 southeast asia mathematics education journal, volume 10, no 2 (2020) learning strategies are highly recommended to make students explore mathematics independently during the learning process. there is one of the learning methods that can be used in mathematics learning, namely inquiry-based learning. the inquiry comes from the word to inquire, which means participating or being directly involved in asking questions, seeking information, and conducting investigations (kuhlthau, maniotes, & caspari, 2007). kuhlthau et al. (2007) define inquiry-based learning as a learning approach in which students find and use various sources of information and ideas to increase their understanding of a particular problem or topic. this approach is not only about answering a question or getting the right answer, but also requires investigation, exploration, search, research, pursuit, and study. this is also emphasised by duran & dökme (2016) that describe inquiry-based learning as a way of asking questions, seeking information and finding new ideas related to an event. furthermore, inquiry-based learning supports students in applying students' knowledge, understanding realworld situations and reinforcing the discovery process. there are two types of inquiry-based learning: guided inquiry and open inquiry (jiang & mc comas, 2015). furthermore, colburn (2000) classified inquiry into structured inquiry, guided inquiry, and open inquiry. the structured inquiry is a learning activity in which the teacher gives students direct problems to investigate, procedures, and materials, but does not tell them about the expected results. students must find relationships between variables or generalise from the data collected. this inquiry type usually includes more directions than structured inquiry activities into what students should observe and what data they should collect. guided inquiry is a learning activity where the teacher only provides material and problems to be investigated. students design their own procedures for solving problems. open inquiry is likely a guided inquiry, with the addition that students also formulate their own problems to be investigated. in essence, the types of inquiry that have been previously disclosed are not much different. the difference between each type is only whether assistance is provided in the form of questions or problems (hypotheses), completion procedures or instructions, and solutions at each of these types. then, for the steps in inquiry, learning is not different at each type. this study uses guided inquiry in mathematics learning in schools. the steps in guided inquiry are formulating problems, submitting hypotheses, collecting data, testing hypotheses, drawing conclusions, implementing, and following up. methods this research is a quasi-experimental research with one group pre-test post-test design. this research used one group of students without any control group with type one group pretest post-test design (privitera & ahlgrim-delzell, 2018). this study's population was the entire class viii in smpn 2 sleman, which consists of 6 classes. random sampling technique was used and resulted in class viii d as a sample with 32 students. the instrument used in measuring mathematical literacy skills in this study is an essay test. 89 southeast asia mathematics education journal, volume 10, no 2 (2020) results and discussion the implementation of learning with the guided inquiry method was assessed based on observations from two professional observers in mathematics learning during four meetings. the learning implementation observation sheet contains a set of statements that refer to the guided inquiry method steps. the scale used is the guttman scale with two options, yes or no. the learning results with the guided inquiry method showed the average percentage when viewed from teacher activities was 97%, and student activities were 92%. this shows that mathematics learning with the guided inquiry method has been carried out well. as for the pre-test and post-test results, students' mathematical literacy abilities can be seen in table 1 below. table 1 results of the mathematical literacy test process components indicator type of test pre-test post-test formulate 1. selecting relevant information 52 92,7 2. creating a mathematical model 31 81,3 employ 3. using mathematical formulas or concepts 54 80,5 4. determining a mathematical solution 37 65,2 interpret 5. interpreting mathematical solutions 9,4 54,7 6. giving reasons 9,4 54,2 overall average 39,1 74,8 based on table 1, the students’ pre-test score of mathematical literacy skills in that class has not yet reached the minimum score of 70. however, after being given treatment, there was an improvement in the students' mathematical literacy skills. the class's average score using the guided inquiry method learning increased by 35.7, from an initial score of 39.1 to 74.8. it can also be seen that the indicator related to the use of mathematical formulas or concepts have the highest average score compared to other indicators. this means that students know the right mathematical formula or concept to solve the problem, but still make mistakes in performing calculations or determining mathematical solutions. furthermore, for the interpretive process component, which is divided into two indicators, namely interpreting the mathematical solution into the context of the problem and providing the right reasons based on the mathematical solution, both of the average scores are 9.4. this score is the lowest compared to the other indicators. most students made mistakes in interpretation and did not provide reasons based on the answers obtained correctly. inquiry-based learning supports students in applying students’ knowledge, understanding real-world situations and supports the discovery process (toth, ludvico, & morrow, 2014). the research results conducted by wang, rodr, maxwell, and algar (2016) found that guided inquiry learning can improve students' understanding of the concept and increase students' research skills. the learning step using the guided inquiry begins with presenting the problems and asks students to formulate the problems. after that, students are asked to submit hypotheses to solve these problems. this activity asks students to provide a provisional guess on how to solve the problems presented. 90 southeast asia mathematics education journal, volume 10, no 2 (2020) furthermore, students are asked to collect the necessary data to solve the problem based on predetermined assumptions through various sources. this activity of formulating problems, proposing hypotheses, and collecting data can help students develop the formulation process skills in mathematical literacy. this is because there is a process of formulating and recognising or identifying the problems given. the next activity is to test hypotheses using data that has been previously collected, which in this case are concepts, facts, and mathematical procedures that have been adjusted to the previous expectations. this activity allows students to develop skills in the process of employing process components in mathematical literacy. then, students are asked to conclude the results of the hypothesis that have been tested. after that, students are allowed to apply and follow up on the concepts that have been found previously. in this final step, students are expected to explain their understanding of the concepts and processes that have been obtained, which are then allowed to apply these concepts in the new contexts or situations to develop a deeper understanding. this is possible to develop the ability to interpret mathematical literacy processes. this interpretive process refers to interpreting and reflecting mathematical solutions or conclusions obtained into the context of the problem. the sample about mathematical literacy problems is as follows. context: scientific (revolving door) figure 1. sample of mathematical literacy problems. the problem in figure 1. belongs to the process component, namely formulate and employ. in this problem, students are asked to select relevant information to solve problems, create mathematical models, use mathematical formulas or concepts appropriately, and determine mathematical solutions. the following is a sample of student's pre-test response. 91 southeast asia mathematics education journal, volume 10, no 2 (2020) figure 2. sample of student’s pre-test response. figure 2 shows that the student has used mathematical concepts to solve the problem in number 1a and do the calculations correctly. for problem number 1b, the student still has not chosen the relevant information yet, because he thought that four-door turns only contain a maximum of 2 people. besides, the student has not tried to make a mathematical model based on the problem, so that the score obtained was not optimal. the post-test results of students' mathematical literacy abilities showed a significant increase in scores for each component of the mathematical literacy process. even though there were still some students' errors in answering the problems. most students are good at writing down relevant information to solve problems, but they still make mistakes when making mathematical models. besides that, students could use the mathematical formulas or concepts correctly, but some students still make mistakes in calculations or determine the mathematical solutions. then, in terms of interpreting the solution, some students answered correctly, but some still had difficulties interpreting the mathematical solution correctly. the result of miscalculation and interpretation of mathematical results or solutions is that students were not precise in providing reasons for the answers so that their scores are not optimal. the following sample of mathematical literacy problem for components of the employ and interpret process can be seen below. context: occupational (fostering a garden) figure 3. the sample of mathematical literacy problem for components of the employ and interpret process. the problem asks students to use the right mathematical formula or concept to solve the problem and determine the mathematical solution. the mathematical concept that students need to solve this problem is the concept of a circular area. then, from the predetermined concept, students are asked to determine the solution. then, the solutions are interpreted into 92 southeast asia mathematics education journal, volume 10, no 2 (2020) the context of the problem being asked, namely determining the adequacy of fertilisers that must be used. furthermore, from the student's interpretation, questions are given about the reasons for the student's answer. if students do this series of steps correctly, they will get the maximum score. the snippets of student' post-test responses can be seen in figure 4 below. figure 4. sample of student’s post-test response. based on the results of the student's response in figure 4, it can be seen that the student has used the right mathematical formula or concept to solve problems and determine mathematical solutions. then the student determines the amount of fertiliser needed by interpreting the answers obtained in the context of the problem. next, he gave the exact reasons for the questions given. thus, the student gets the maximum score. furthermore, based on the results of descriptive analysis, it can be seen that the average score of students' mathematical literacy skills has increased, this is because in the guided inquiry method there are several learning steps such as formulating problems, proposing hypotheses, collecting data, testing hypotheses, drawing conclusions, and applying and following up. after the teacher conveys the learning objectives, it is followed by giving students motivation to be actively involved in the problem-solving process. the guided inquiry learning method is also closely related to providing real-world problems as a source of student learning. this certainly can develop a component of the process, namely the formulate in mathematical literacy skills. when viewed in inferential statistics, the post-test univariate normality assumption test of students' mathematical literacy abilities shows a kolmogorov-smirnov significance value of 0.68 > 0.05 can be concluded that the data comes from a normally distributed population. based on the test of homogeneity of variances' results, the significance value of the post93 southeast asia mathematics education journal, volume 10, no 2 (2020) test of mathematics literacy is 0.282 > 0.05, so it can be concluded that the two-sample data come from populations that have the same variance. furthermore, the inferential analysis was carried out to determine the average difference before and after treatment was given. the results of the average difference test analysis according to paired sample t-test with the help of ibm spss statistics 21 software shows that significance (2-tailed) of 0.000 < 0.05. it means that guided inquiry method successfully promotes students’ mathematical literacy. this result is in line with sulistyowaty and prafianti (2017) and matthew & kenneth (2013) which reveal that this method can improve student achievement. one of the students' achievements in learning mathematics is mathematical literacy ability. these results also align with the opinion from kuster, johnson, keene, and andrews-larson (2017) that inquiry learning allows students to be actively involved, and students have a commitment to active learning in class. this is also emphasised by ural (2016) that in an inquiry-based learning environment, students are more active, and they guide their own learning process. besides, the research results from hassi and laursen (2015) support that learning mathematics in a classroom situation that uses student activity, deep involvement, and collaboration can improve not only students’ thinking skills but also their abilities in problem-solving. furthermore, research conducted by duran and dökme (2016) also reveals that mathematical literacy related to providing real-world problems can help students build their critical thinking skills. conclusion based on the research, we found that the guided inquiry method effectively enhances students' mathematical literacy skills. therefore, to improve mathematical literacy skills, teachers should provide students with clear guidance in the activities. teachers also need to remind students of the initial purpose of the inquiring carried out because the objectives are closely related to the conclusions. references colburn, a. (2000). an inquiry primer. science scope, 23(6), 42–44. retrieved from https://eric.ed.gov/?id=ej612058 duran, m., & dökme, i. (2016). the effect of the inquiry-based learning approach on student’s critical-thinking skills. eurasia journal of mathematics, science and technology education, 12(12), 2887–2908. https://doi.org/10.12973/eurasia.2016.02311a hassi, m., & laursen, s. l. (2015). transformative learning: personal empowerment in learning mathematics. journal of transformative education, 13(4), 316–340. https://doi.org/10.1177/1541344615587111 higo, a., wada, y., & sato, y. (2012). exploring the relationship between iranian students’ mathematical literacy and mathematical performance. journal of american science, 59(11), 562–571. retrieved from https://www.jstage.jst.go.jp/article/nskkk/59/11/59_562/_pdf 94 southeast asia mathematics education journal, volume 10, no 2 (2020) jiang, f., & mc comas, w. f. (2015). international journal of science the effects of inquiry teaching on student science achievement and attitudes: evidence from propensity score analysis of pisa data. international journal of science education, 37–41. https://doi.org/10.1080/09500693.2014.1000426 khan, a. w. (2012). inquiry-based teaching in mathematics classroom in a lower secondary school of karachi, pakistan. international journal of academic research in progressive education and development, 1(2), 1–7. retrieved from http://ecommons.aku.edu/pakistan_ied_pdcc/7 khoirudin, a., setyawati, r. d., & nursyahida, f. (2017). profil kemampuan literasi matematika siswa berkemampuan matematis rendah dalam menyelesaikan soal berbentuk pisa. aksioma, 8(2), 33–42. https://doi.org/10.26877/aks.v8i21839 kuhlthau, c. c., maniotes, l. k., & caspari, a. k. (2007). guided inquiry: learning in the 21th century. london: libraries unlimited. kuster, g., johnson, e., keene, k., & andrews-larson, c. (2017). inquiry-oriented instruction: a conceptualisation of the instructional principles. primus, 1970(august). https://doi.org/10.1080/10511970.2017.1338807 matthew, b., & kenneth, i. o. (2013). a study on the effects of guided inquiry teaching method on students achievement in logic. international researcher, 2(1), 135–140. moretti, g. a. s., & frandell, t. (2013). literacy from a right to education perspective. oecd. (2019). pisa 2018 (volume 1): what students know and can do. retrieved from https://www.oecd-ilibrary.org/education/pisa-2018-results-volume-i_5f07c754-en. oecd. (2017). pisa for development assessment and analytical framework: reading, mathematics and science, preliminary version (preliminary). paris: oecd publishing. oecd. (2014). pisa 2012 result in focus: what 15-year-olds know and what they can do with what they know. retrieved from http://www.oecd.org/pisa/keyfindings/pisa-2012results.htm. oecd. (2010). pisa 2009 results: vols. i – v. retrieved from https://www.oecd.org/pisa/pisaproducts/pisa2009keyfindings.htm privitera, g. j., & ahlgrim-delzell, l. (2018). research methods for education. new york, ny: sage publishing. sulistyowaty, r. k., & prafianti, r. a. (2017). implementation of inquiry strategy on exponent, roots and logarithm implementation of inquiry strategy on exponent, roots and logarithm. journal of physics: conference series paper, 1–5. https://doi.org/doi :10.1088/1742-6596/895/1/012078 toth, e. e., ludvico, l. r., & morrow, b. l. (2014). blended inquiry with hands-on and virtual laboratories: the role of perceptual features during knowledge construction. interactive learning environments, 22(5), 614–630. https://doi.org/10.1080/10494820.2012.693102 95 southeast asia mathematics education journal, volume 10, no 2 (2020) ural, e. (2016). the effect of guided-inquiry laboratory experiments on science education students’ chemistry laboratory attitudes, anxiety and achievement. journal of education and training studies, 4(4), 217–227. https://doi.org/10.11114/jets.v4i4.1395 wang, j. j., rodr, j. r., maxwell, e. j., & algar, w. r. (2016). build your own photometer: a guided-inquiry experiment to introduce analytical instrumentation. journal of chemical education, 93, 166–171. https://doi.org/10.1021/acs.jchemed.5b00426 96 southeast asia mathematics education journal, volume 10, no 2 (2020) southeast asian mathematics education journal volume 12, no. 1 (2022) 1 students’ creativity profiles in constructing independent gates learning activity using 4dframe 1 agus setio & 2 gusnandar yoga utama 1 smpn 1 tirtoyudo, indonesia 2 seameo qitep in mathematics, indonesia 1 agus.ulfa@gmail.com 2 gusnandaryogautama@gmail.com abstract creativity is one of the 21 st century skills that students require for the future. the implementation of steam education in teaching and learning, whether in content or learning activities, has contributed to the creativity development. the objective of the study is to investigate the students’ creativity profiles by employing steam approach and 4dframe to construct a miniature independence-day memorial gate. this activity was selected because indonesian independence day, organized in august, is very special. many activities are frequently conducted to celebrate this annual event. one of which is by building a gate in front of the village. this research utilized a qualitative research method with a narrative design. there was twelve 8 th -grade students who were recruited as respondents with various abilities grouped into high, medium, and low. the student’s products were evaluated by cpam (creative product analysis matrix) instrument. the cpam assessment score presented 88% in novelty, 72% in resolution, and 88% in elaboration. overall, their creativity profile is 82%, included in the high category. the conclusion is that student creativity profile through the steam approach assisted by 4dframe is in the high category and can be applied in learning. keywords: 4dframe, creativity, cpam, steam introduction as a huge multicultural country, indonesian owns a diverse way in celebrating the big days, including in the independence day commemoration. during the entire month of august, numerous activities are conducted to celebrate it as raising indonesian flag, red and white decoration, carnivals, as well as local community contests. the independence day festivities are frequently welcomed by indonesian citizen with excitement and enthusiasm (laeli, maulana, & hamid, 2020). people also generally build independence memorial gate, called “gapura kemerdekaan” in their communities’ entrance as a tribal representation of cultural characteristic, tradition, or belief (purnegsih & kholisya, 2019). the gates will look more beautiful during the night as they are equipped with lamp decoration. the familiarity and festivities of independence memorial gate make it a potential context for hands-on learning activity. however, since real memorial gate is not easy to build, the students prefer to build a miniature. designing and building an independent gate miniature can be a platform for the students in learning to be an engineer and employ the knowledge they gained in science, mathematics, and technology to create a real product. mathematics, for instance, can be utilized for precision and measurements. for the final touch, it is expected that they do not ignore the aspects of art and beauty in the design. such an mailto:agus.ulfa@gmail.comorg students creativity profiles in constructing independent gates learning activity using 4dframe 2 educational approach, combining or consists of science, technology, engineering, art, and math, is understood as steam. maeda (2013) explained that steam is adding art to stem. he asserted that “art and science were once inextricably incorporated, both dedicated to identifying that truth and beauty are better together.” eger (2013) elaborated that steam ensures the whole brain’s development in stem through art. hence, it can be implied that steam education is a learning approach which utilizes science, technology, engineering, by employing art and mathematics as the access points for directing student inquiry, dialogue, and critical thinking (the institute for arts integration and steam, 2021). the objective of steam approach is to create students who are able to solve the problem issues through innovation, creativity, critical thinking, effective communication, collaboration, and ultimately latest knowledge (quigley & herro, 2016). particularly in increasing creativity, implementing steam education can encourage and stimulate creativity, as learners produce better and more creative packaging designs (sakon & petsangsri, 2021). in a research conducted by wandari, wijaya, and agustin (2018), it was discovered that the students’ creativity through the steam project is calculated based on cpss rubric concerning on tree dimension which are novelty, resolution, and elaboration synthesis. the study encompassed that upon the end of the project, the students’ creativity on novelty gains 76%, on the resolution 78%, on the elaboration and synthesis 69%, with a conclusion that the overall creativity is categorized as good. from the studies above, it is evident that steam is significantly suitable to assist the students in obtaining 21st century life skills understood as 4c skills, which are critical thinking, creativity, communication, and collaboration. in enhancing those goals, creativity in particular, teachers can opt for physical manipulatives, either naturally acquired from the students’ environment or produced commercially. the related example of manipulatives is 4dframe. a set of 4dframe consists of hollow tubes (similar to drinking straws) and combining stars (bridges). one prior benefit of the set is its flexibility, as the tubes can be cut to preferred sizes and connected with specific elements to lengthen them. furthermore, the connecting stars are freely bendable (park, 2013). the essential aim of 4d frame is to elevate the students’ familiarity with geometric structures, within the context of problem-solving. this approach is based upon the creative exploration of the structures, attained through the stepby-step, scientific analysis of each stage in the construction process (lavicza et al., 2018). 4dframe can be employed for geometric modelling and creative real-world problem-, particularly concerning engineering, architecture, and applied mathematics. 4dframe also provides opportunities for children to experiment with creative methods associated with mathematical art which stimulates inquiry, problem solving, as well as inter and transdisciplinary cooperation in the classroom (fenyvesi et al., 2016). based on the introduction above, we are concerned on scrutinising how students implement their creativity in constructing “gapura kemerdekaan” miniature in the context of indonesia independence celebration by applying 4dframe set. the knowledge of students’ creativity profiles is useful as preliminary information on the students’ creativity and is able to reinforce teachers to enhance their future teaching and learning process to better support the students’ creativity. agus setio & gusnandar yoga utama 3 methods this research employed a qualitative approach, in which the researcher portrayed the things happening during class activities, then collected and explained stories about students’ lives and experience in the form of narratives (creswell, 2011) with a quantitative descriptive approach. according to the characteristics of qualitative research elaborated by sugiyono (2013), this study illustrated the meaning of the research data obtained, and the findings were not generalized. in collecting data, the researcher owns a role as a participant in the study. the researcher involved in the classroom activities and also observed situation. this research was conducted in a state junior high school in malang regency, indonesia. the participants of this study were twelve 8 th grade students, of which four possess high cognitive ability, four medium, and four low. the number of participants is limited due to the availability or practical tools and the healthy protocol that we had to conform in the pandemic situation. the participants were selected through purposive sampling, in which researcher employed their personal judgment, to select participants who would likely provide the data they require (fraenkel, wallen & hyun, 2011). in this case, the author considered the participants based on their learning outcomes. data was collected through observation and student worksheets and analysed by employing descriptive approach. additionally, we also utilized creativity product analysis matrix (cpam) rubric developed by besemer and trefinger (1981). besemer and trefinger (1981) explained that analysing creative products is tremendously crucial in the assessment and study of creativity. as this research applied a project-based learning using steam approach, the researcher administered cpam rubric to profile students’ creativity. the students’ creativity data is based on a product that they have designed during a steam project-based learning activity in the classroom. the students’ creativity was scored from 1 to 3 scale for each criterion as presented in table 1. table 1 cpam rubric by besemer and trefinger (1981) creative dimension criterion score 1 2 3 novelty germinal the lower level of germinal: the product inspires others with the creation medium level of germinal: the product inspires others to try something new high level of germinal: the product inspires others to try something new by directly provide ideas to develop more product design original the lower level of originality: students mostly employ the previous finding as their product idea medium level of originality: students employ the previous finding as their product idea, but they create a modification of the product high level of originality: the product idea origins from their own understanding students creativity profiles in constructing independent gates learning activity using 4dframe 4 creative dimension criterion score 1 2 3 resolution valuable the lower level of valuable: the product is not compatible with the purpose and does not relate to the concept medium level of valuable: the product is compatible with the purpose and does not relate to the concept high level of valuable: the product is compatible with the purpose and relates to the concept useful the lower level of usefulness: the product can be utilized once medium level of usefulness: the product can be utilized continuously with a certain requirement high level of usefulness: the product can be utilized continuously without any requirement elaboration well crafted the lower level of well crafted: the product is completed well medium level of well crafted: the product is completed well with the good-looking design high level of well crafted: students attempt to provide interesting product design by applying some materials expressive the lower level of expressive: the product is displayed with lacking body language and require controlling tone, not understandable medium level of expressive: the product is displayed with lacking body language and require controlling tone, but understandable high level of expressive: the product is displayed in a communicative (by employing effective body language and clear voice) and understandable manner the research stages encompass identifying a phenomenon to identify an educational problem; designing and preparing learning experience, incorporating the tools and materials; selecting participants to study; observing the participants and analysing their product; describing the participants’ learning experience in writing (adapted from creswell, 2011). this study requires two meetings to complete all stages of steam project-based learning. first meeting, researchers conducted a preparation stage leading students to understand the theme and scope. second meeting, researchers conducted an implementation stage which let students create the product based on their drawing design. the learning activities of each stage are illustrated in a table 2 below. table 2 research stage meeting stage activity 1 st designing students recognize the project theme and scope students identify the information from the internet regarding the basic concept in designing the project students discuss tools and materials which will be employed students produce design drawing agus setio & gusnandar yoga utama 5 meeting stage activity 2 nd implementation students create the project based on the design drawing students conduct an actual test of their product before the first meeting, the researcher obtained permits, prepared proposals, purchased tools and materials, and the revised the proposals with the guidance of the advisory teams from one of southeast asia ministry of education organization concerning on math teacher development. the researcher also validated the instruments (lesson plan and student worksheet) to two validators, comprising of the head of the mathematics teachers group and a colleague math teacher at the school of implementation. the validation aspect was about the component of lesson plan, steam approach, and students’ worksheet. the instrument validation results revealed that the research instrument was employed with revisions. the things that need to be revised encompasses learning objectives which should be clearer, and the reinforcement for closing activities that must be provided. after the preparation is complete and the research permit is received, the research was conducted on september 4 and 5, 2021, taking place in schools with the implemented health protocols. results and discussion on the first day, through a power point slide show, the teacher provided an initial overview of the lesson which would be performed, and the activities that students would conduct (refer to figure 1). the students were tremendously curious and paid attention to the information provided, as presented in the following photo. figure 1. apperception activities. the next activity was that the teacher was distributing student worksheet 1 containing the problems which had to be solved by each group. the problem was that they had to help their society to design a gate for celebrating indonesian independence. the success criteria of the gates were the strength and sturdiness against shocks and wind, the beauty and patriotism portrayed by the design, minimal use of material, and the inclusion of 2d geometrical shape(s). in completing worksheet 1, students were required to employ the edp (engineering design process) principle as they were expected to act as an engineer (refer to figure 2). they began discussing what a strong and sturdy gate might look like and the way to create it (ask), then students creativity profiles in constructing independent gates learning activity using 4dframe 6 identified example in the internet (research). next, they discussed in groups which design they preferred, made plan for their own gate, and created it by drawing on their worksheet. figure 2. edp stage. after about 60 minutes, all groups were finally able to complete their designs and drew them in the worksheet 1. the gate designs that they produced were displayed in the figure 3. figure 3. students’ gate design. the next step was as instructed on the student worksheet. each group had to discuss which design was the best and appropriate the success criteria, then drew it on flip chart paper and presented it to the other to obtain criticism and input from other groups. their selected designs are shown in figure 4. agus setio & gusnandar yoga utama 7 figure 4. students selected gate design. the second implementation day was the create stage. in this stage, the students were demanded to build their own independence gate design by utilizing 4dframe. aside from the success criteria mentioned previously, the students also had to consider the constraint, that was having to employ as little material as possible. in this case, they were only allowed to apply maximum 50 pieces of 4dframe and 10 cm of red-and-white ribbon. figure 5. students’ activities with 4dframe. at this meeting, the teacher also provided reinforcement about what they had to perform in qualifying the success criteria. each group was tremendously enthusiastic and excited to establish their gate using 4dframe (figure 5), because it was something new and fun for them. their excitement is illustrated in the following photo. figure 6. students’ activities build and test their gates students creativity profiles in constructing independent gates learning activity using 4dframe 8 after working for about 90 minutes, they finally completed it. to examine whether the design fulfilled the success criteria, each group had to assess the resistance of their gates to shocks and wind. the students were placing their designs on top of a specifically designed platform alongside a gadget with vibrometer application, shaking the platform with the increasing vigorousness, then taking note of the degree displayed on the gadget and the state of the gate on worksheet 2 (figure 6). some examples of the students’ products are presented in figure 7. group 1 group 2 group 3 figure 7. students’ products after they assessed their gates through shock and wind resistance, their products (independence gate) were observed for creativity with the help of cpam rubric. the result is as follows (see table 3). table 3 student product creativity profiles creativity dimension criteria group score average i ii iii novelty germinal 2 3 2 2.7 original 3 3 3 resolution valuable 2 3 2 2.2 useful 2 2 2 elaboration well crafted 3 3 3 2.7 expressive 2 3 2 total average 2.3 2.8 2.3 2.46 group 1 and group 3 acquired score 2.5 for novelty because their designs were almost similar with the gate designs discovered on the internet, while the novelty aspect of group 2 attained score 3 because its new design and concept. on average, the three groups received 2.7 for the creativity. for the resolution aspect, group 1 and group 3 scored 2 because their product almost fit or meet the problematic situation, while group 2 scored 2.5 because their product fit the problematic situation, and on average, the students received scores 2.2 for the resolution aspect. for the elaboration aspect, group 1 and group 3 also acquired score 2.5 because their product possessed a sense of wholeness or completeness, while group 2 obtained score 3 because their product worked with care to develop to its highest possible agus setio & gusnandar yoga utama 9 level, and on average students gained scores 2.7. overall, the product creativity score of the three groups was 2.46 or 82%, which based on the criteria incorporated in the high category. it is in accordance with the study of aguilera and ortiz-revilla (2021) which unveiled that stem and steam education both possess robust potential to foster students’ creativity. conclusion based on the results of the research, it was revealed that the profile of students’ creativity in building the independence gate by employing 4dframe was in the high category (82%). as the objective of this study, this research result can be utilized as initial information about students’ creativity, particularly in steam learning assisted by 4dframe. it has been evident to enhance elevating the students’ creativity, hence, it can be implemented in mathematics learning. this research is merely preliminary research, thus, next research concerning on how to foster creativity by utilizing steam approach is highly recommended. references aguilera, d., & ortiz-revilla, j. (2021). stem vs. steam education and student creativity: a systematic literature review. education sciences, 11(7), 331. https://doi.org/10.3390/educsci11070331. besemer, s. p., & trefinger, d. j. (1981). analysis of creative products: review and synthesis. the journal of creative behavior, 15(3), 158–178. https://doi.org/10.1002/j.2162-6057.1981.tb00287.x creswell, j. w. (2011). educational research: planning, conducting, and evaluating quantitative and qualitative research (4th ed.). prentice hall upper saddle river, nj. creswell, j. w. (2012). educational research (4th ed.). nebraska: university of nebraskalincoln eger, j. (2013). steam... now. the steam journal, 1(1), 1–7. https://doi.org/10.5642/steam.201301.08. fenyvesi, k., park, h. g., choi, t., song, k., & ahn, s. (2016). modelling environmental problem-solving through steam activities: 4dframes warka water workshop. proceedings of bridges finland conference, pp. 601–608. retrieved from http://archive.bridgesmathart.org/2016/bridges2016-601.pdf fraenkel, j. r., wallen, n. e., & hyun, h. h. (2011). how to design and evaluate research in education (8th ed.). new york: mc graw hill. laeli, s., maulana, a., & hamid, m. s. (2020). penyadaran dan pengelolaan semangat indonesia melalui perayaan ulang tahun republik indonesia ke 74. educivilia. jurnal pengabdian pada masyarakat, 1(1), 71–77. lavicza, z., fenyvesi, k., lieban, d., park, h., hohenwarter, m., mantecon, j. d., & prodromou, t. (2018). mathematics learning through arts, technology and robotics: multi-and transdisciplinary steam approaches. east asia regional conference on mathematics education, pp. 110–122. national taiwan normal university. students creativity profiles in constructing independent gates learning activity using 4dframe 10 maeda, j. (2013). stem + art = steam. the steam journal. 1(1), 1–3. https://doi.org/10.5642/steam.201301.34 park, h.-g. (2013). a workshop on n-regular polygon torus using 4d frame. proceedings of bridges 2013: mathematics, music, art, architecture, culture, pp. 597–600. purnegsih, i. i., & kholisya, u. k (2019). representasi kosmologi jawa pada gapura kontemporer di desa-desa kabupaten karanganyar. cakrawala-jurnal humaniora, 19(1), 113–120. quigley, c. f., & herro, d. (2016). “finding the joy in the unknown”: implementation of steam teaching practices in middle school science and math classrooms. journal of science education and technology, 25(3), 410–426. https://doi.org/10.1007/s10956016-9602-z sakon, t., & petsangsri, s. (2021). steam education for enhancing creativity in packaging design. archives of design research, 34(1), 21–31. https://doi.org/10.15187/adr.2021.02.34.1.21 sugiyono. (2013). metode penelitian kuantitatif, kualitatif, dan tindakan. bandung: alfabeta. the institute for arts integration and steam. (2021). what is steam education? retrieved from: https://artsintegration.com/what-is-steam-education-in-k-12-schools/ wandari, g. a., wijaya, a. f. c., & agustin, r. r. (2018). the effect of steam-based learning on students’ concept mastery and creativity in learning light and optics. journal of science learning, 2(1), 26–32. https://doi.org/10.17509/jsl.v2i1.12878. southeast asia mathematics education journal, volume 10, no 1 (2020) 1 development of comic learning media for buying and selling practices on social arithmetic husnuz zaimah mtsn 2 mojokerto, east java abstract this study was conducted to develop comic learning media which was valid, practical and effective to help students understand the concept of social arithmetic. this study is a developmental study which adapts stages of borg and gall. the stages are 1) initial study and gathering information, (2) planning, (3) developing the initial product, (4) limited field trial, (5) revising the initial product, (6) field trials, and (7) revision of the final product. the result of this study was a comic learning media for buying and selling practices on social arithmetic which has met the valid, practical, and effective criteria. based on the validity results involving media and educational experts showed that this comic learning media was valid, with the percentage of validity respectively 93.6% (very feasible) and 96% (very feasible). based on the implementation of this media to vii grade students of mts n (junior high school) 2 mojokerto, an assessment by students as the media users showed that this media was practical with the percentage of 82.3% (very good) during the limited field trial, and 92.31% (very good) during field trials. the effectiveness of this media based on students’ achievement result that 91% students were in good and excellent category. it means that this comic learning media is effective. based on these results, it can be concluded that this media was appropriate to be used as learning medium on the topic of social arithmetic. keywords: development, comic media, buying and selling practices. introduction formulation of mathematical discussion is not limited to determining relations, patterns, shapes and assemblies as targets. these four goals only emphasize that mathematics involves abstract ideas. hudoyo (2013) stated that learning mathematics is a high mental activity. thus, in the learning process of mathematics, the teacher must be able to provide good activities so that abstract mathematical concepts can be understood. in mathematics learning, it should also be noted that the activities or things students learn must be something interesting and challenging. as stated as one of the principles in school mathematics in the national council of mathematics teachers, when learning is challenging, students will be confident to solve math problems and to understand new knowledge (nctm, 2000). learning media is one thing that can challenge and attract students' attention in learning mathematics. there are so many types of media. according to bretz (1971), in general, types of media can be divided into visual media, audio media and audiovisual media. as one of the visual media, comics are two-dimensional media in the form of sheets and contain several panels of textual images. wright (1979) stated that comics as a medium of learning are something that can still attract children's attention. the use of comics as a medium will provide an opportunity for readers to form their own understanding after being interested in the visual mailto:%20zaimah335@gmail.com development of comic learning media for buying and selling practices on social arithmetic 2 appearance (mcvicker, 2007). specifically, comic as a learning media in mathematics learning can help students to prepare their competencies in 21st century learning (toh, cheng, ho, jiang, & lim, 2017). comics as learning media as a packaged of picture and colored books that contain interactive dialogue pictures seems to has potential possibility to help students understand the concept of mathematics during the classroom activities. some researches showed that comic media can improve students’ learning achievement (indaryati & jailani, 2015; adi widodo, turmudi, afgani dahlan, istiqomah, & saputro, 2018; rasiman, & agnita, 2014; sulistyani, & retnawati, 2015). it means that comic seems to be a promising media that can help students understand mathematical concepts. in social arithmetic material there are subjects of selling prices, buying prices, profits, loss, percentage and discounts. based on the researcher experiences, many students got difficulties on this topic. it is supported by the research of fajriah, salasi, suryawati, & fatimah (2019) that students may have some difficulties on it. on the formal textbook, the concept of social arithmetic is also presented in a mechanistic way (fauzan, armiati, & ceria, 2018). hence, teacher needs to find a strategy to teach social arithmetic in order to help students understand the concept. based on the mentioned fact, then researcher developed a comic as a learning medium that can effectively help students to understand the concept of social arithmetic. methods this study is a type of research and development (r&d). borg and gall (1983) stated the notion of research development is a process used to develop and validate educational products. the development procedure based on borg and gall's (1983) consist of 10 stages, namely (1) conducting research and information gathering, (2) planning, (3) developing initial product formats, (4) design validation, (5) making validation revisions design, (6) first trial (7) revising the product, (8) conducting a second trial (9) conducting an effectiveness test on the product, and (10) revising the final product. then, adapted from borg and gall, the stages on this study are (1) initial study and gathering information, (2) planning, (3) developing the initial product, (4) limited field trial, (5) revising the initial product, (6) field trials, and (7) revision of the final product. the implementation of this comic media was conducted at mts n 2 mojokerto in march 2018 may 2019 with research subjects in class vii d, even semester, for the academic year of 2018/2019 and 2019/2020. the assessment of the product in this development research involves experts (media experts and mathematics subject experts) to measure its validity. furthermore, this study also involved students as users to measure its practicality. the effectiveness of the media is determined by carrying out implementation activities on 34 students and testing student learning outcomes after using this comic media to see the potential effect of the developed comic on learning achievements. the research data consisted of qualitative and quantitative data. the qualitative data were obtained from the validator's suggestions and the students when determining the validity and practicality of the media. quantitative data were obtained from validator assessment scores, student assessment questionnaires and student learning outcomes test scores. husnuz zaimah 3 the instruments in this study include: (1) instruments for measuring validity in the form of an assessment sheet by a media expert and an assessment sheet by a material expert; (2) instruments to measure practicality, namely student practicality assessment sheets; (3) instruments to measure practicality, namely test questions for student learning outcomes. the data obtained from each category were then analysed to obtain criteria for its validity, practicality, and effectiveness. data validation scores from media experts and material experts, and media practicality scores from student assessments are then converted using a formula adapted from akbar (2013) as follows: v ah = 𝑇𝑠𝑒 𝑇𝑠ℎ × 100% (1) v au = 𝑇𝑠𝑒 𝑇𝑠ℎ × 100% (2) v-ah = percentage of validity v-au = percentage of practicality tse = total empirical score achieved tsh = total expected score 100 = constant the level of validity of the comic learning media is defined using the criteria on the following table. the media is stated as valid if the percentage of the expert’s judgement at least meets “valid” criteria. further, the comic media is said practical if the result of students’ response is at least in “good” criteria. table 1 criteria of media's validity and practicality percentage (%) level of validity level of practicality 85.01 – 100 very valid very good 70.01 – 85 valid good 50.01 – 70 less valid fair 01.00 – 50 invalid poor adapted from kristanto (2019) the effectiveness of the media is defined based on the students result of learning achievement test. adapted from zainil, prahmana, helsa, and hendri (2018), the media is said to be effective if most of students at least got “good” criteria of the learning result. development of comic learning media for buying and selling practices on social arithmetic 4 table 2 criteria of students' learning achievement score interval (x) criteria 80 ≤ 𝑥 < 100 excellent 66 ≤ 𝑥 < 80 good 56 ≤ 𝑥 < 66 fair 40 ≤ 𝑥 < 56 poor 0 ≤ 𝑥 < 40 failed adapted from zainil, prahmana, helsa, and hendri (2018) results and discussion there are 7 stages performed in this study namely: (1) initial study and gathering information, (2) planning, (3) developing the initial product, (4) limited field trial, (5) revising the initial product, (6) field trials, and (7) revision of the final product. the results on each stage are described as follow. initial study and gathering information this stage included preliminary study to see the students’ needs for learning media as well as gathering information (curriculum analysis for the preparation of the developed media). researcher used questionnaire for students at mts n 2 mojokerto. based on the results of the questionnaire, 85.92% of students showed high enthusiasm for comic-based learning media. analysing the curriculum were done to select and to adjust the materials of buying and selling in social arithmetic in order to make the media appropriately with the indonesian curriculum. further, the analysis of the basic curriculum that can be achieved through this comics media were done as well as its learning objectives. planning in the planning stage, the researcher arranged the flow of the developed comic. in addition, at this stage the researcher determines the application used, namely using the comic life 3 application. the following is an example of how the application looks. husnuz zaimah 5 figure 1. interface of comic life 3 application. at this planning stage, several materials needed in the comic development process were also collected. because the comics were developed using real photos of students instead of pictures or animation, the researchers collected the photos needed to compile the comic story. developing the initial product the researcher defined the specifications of the comic media. the product developed in this research specifications are as follows: (1) this product is based on multimedia using an android-based and pc-based comic life 3 application ; (2) each page contains several panels featuring some scenes; (3) pictures used in the comic are non-fictional photographs; (4) contains balloon text in the form of a daily dialogue about the process of buying and selling practices on social arithmetic material; (4) inserting honest entrepreneurial principles into stories in comics. the prototype of the comic as the initial product can be seen on the figure below. figure 2. prototype of the comic. part of the scene which contains photos of practice in class there are 4 panels each featuring a scene adding stickers to show students' expressions input dialog for text balloon development of comic learning media for buying and selling practices on social arithmetic 6 in this comic media, pictures and stories are presented in an interesting plot so that it can attract students' interest in reading and capturing the messages in comics. this is an advantage that can be obtained when comics are used as a medium in learning. this is in accordance with the opinion of recine (2013) that comics which contain images integrated into words will engage students more than the isolated images that fill textbooks. in addition, besides the objectives of learning mathematics about social arithmetic, developing character values about honesty in entrepreneurship is the main message to be conveyed through this comic. so, it is hoped that, through this comic, students can indirectly learn mathematics as well as train their character. on this developing stage, the prototype was then evaluated in order to validate its feasibility. two experts were involved to determine the validity of the product. the following table depicted the result of validity assessments. table 3 validation results from material expert aspect percentage criteria content 86% very feasible presentation and interface 93% very feasible language 90% very feasible total 89% very feasible based on the above table, the result from each aspect is in the “very feasible” criteria. in total, 89% came from the media expert. it means that based on the media expert the comic media has met the validity criteria. table 4 validation results from media expert aspect percentage criteria dimension of the comic paper 88% very feasible cover design 97% very feasible comic content 91% very feasible total 93% very feasible material expert were asked to validate the comic from the aspect related to content but not limited to the aspect of dimension of the comic and cover design as well. the result for aspect of dimension, cover design, and content are 88% (very feasible), 97% (very feasible), and 93% (very feasible) respectively. it means that the comic media has met the criteria of husnuz zaimah 7 validity. in total, based on material expert can be said that the comic is valid, with very feasible criteria (93%). in conclusion, based on the expert’s judgement, this comic media was valid. limited field trial after getting the valid comic media, to know its practicality it is implemented in limited students. the limited field trial was conducted to 17 students. students used learning comic media, then were given a questionnaire. the results of the questionnaire assessment (questionnaire) are intended to improve and perfect the media to be tested on operational field tests. the result of students’ questionnaire showed the result of 82.30% (very good). based on table 1, this comic media has met the practicality criteria. revising the initial product based on experts (media and materials) and limited field trial, some suggestions and recommendation were gathered. revision of the initial product was done. the following table are some points of revisions for the initial comic media. table 5 revisions of the initial design comment/recommendation revision decision subject matter expert subject matter expert a. the initial instructions should be explained as there was a picture of the teacher giving the order to arrange a bench for the booth. b. the character should be named. a. slides are given by the teacher to give directions when students have to set up a booth. b. b. the character's name is raised. media expert media expert a. the colour is too bold. b. it is better to have a frame to make it interesting. c. there should be colour variations. a. the colour was revised to be bright b. each scene is framed. c. colour variations are implemented. the revised version of the comic was then implemented to know the effectiveness of the media. field trials comic learning media was implemented to 34 students in this field trial stage. this stage aims to know the potential effect of the media for students understanding of social arithmetic concept. in this implementation, students were given the opportunities to demonstrate the story in the comics, so that they need to understand the content of the comics. on the demonstration part, there were two groups, namely sellers and buyers. a total of 34 students were divided into 6 groups and each group consisted of 3 groups of buyers and 3 development of comic learning media for buying and selling practices on social arithmetic 8 groups of sellers. the instructions given to the seller group are carrying used goods packaging, price tags, toy money, sales memorandum and preparing the name of the store, while the seller group brought in the toy money as capital, and book purchase tables. activities were started from giving directions, structuring the sales stall, and other preparations. the core buying and selling activities were limited to 60 minutes with a whistle mark from the teacher. finally, the presentation of each group reports the results of sales and purchases. then, they were documented in photos and videos. initial media products were in jpeg format photos and mp4 format videos that contained the buying and selling process. the results of students' work groups of buyers are shown in figure 3. table inventory and price group experienced a 12.26% gain evidence memorandum of sale of goods group experienced a 25.70% loss obstacles during practice figure 3. students' work (buyer group) based on the demonstrations of the buyer group, students experienced how to solve the problem related to social arithmetic such as price calculations, gain, and loss. students who belonged to seller group also experienced making some mathematics calculations and solving problems related to price, gain, loss. indirectly, through this simulation, students actually solve the real-life problems related to social arithmetic. students learn and understand the case from the comic, then it is emphasized through demonstrations. husnuz zaimah 9 based on students questionnaire of this the field trials, it resulted 92.31% (very good). it means that this comic is practical based on field trials with 34 students. the evaluation of the comic effectiveness is defined by students test on social arithmetic topics after using the comics on their learning process as shown below. table 6 students' learning achievement using comic score interval (x) percentage of students criteria 80 ≤ 𝑥 < 100 23% excellent 66 ≤ 𝑥 < 80 68% good 56 ≤ 𝑥 < 66 9% fair 40 ≤ 𝑥 < 56 poor 0 ≤ 𝑥 < 40 failed from the above table, it is shown that there are 68% students are in “good” criteria and 23% students are in “excellent” criteria for their learning achievements. it means that 91% students achieve the learning results. it leads to the conclusion that this comic is effective to help students understand the concept of social and arithmetic. revision of the final product in the final improvement of the comics media for buying and selling for social arithmetic subject, the researcher made some revisions. the final design of the comic was printed as shown in figure 4. figure 4. final version of the comic. development of comic learning media for buying and selling practices on social arithmetic 10 conclusion based on the research result, a valid, practical and effective comic as a learning media for social and arithmetic has been successfully developed. the validity results involving media and educational experts showed that this comic learning media was valid, with the percentage of validity 93.6% are (very feasible) and 96% (very feasible) respectively. based on the implementation of this media to vii grade students of mts n 2 mojokerto, an assessment by students as the media users showed that this media was practical with the percentage of 82.3% (very good) during the limited field trial, and 92.31% (very good) during field trials. the effectiveness of this media based on students’ achievement result shows that 91% students were in “good” and “excellent” categories. it means that this comic learning media is effective. based on these results, it can be concluded that this media was appropriate to be used as learning media on social arithmetic. references adi widodo, s., turmudi, t., afgani dahlan, j., istiqomah, i., & saputro, h. (2018). mathematical comic media for problem solving skills. in proceedings of the joint workshop ko2pi and the 1st international conference on advance & scientific innovation (pp. 101-108). icst (institute for computer sciences, social-informatics and telecommunications engineering). akbar, s. (2013). instrumen perangkat pembelajaran. bandung: pt remaja rosdakarya bretz, r. (1971). a taxonomy of communication media. educational technology. borg, w. r and gill, m d. (1978). educational research: an introduction. new york: longman fajriah, f., salasi, r., suryawati, s., & fatimah, s. (2019). analysis of problem solving ability in social arithmetics. in journal of physics: conference series (vol. 1157, no. 4, p. 042102). iop publishing. fauzan, a., armiati, a., & ceria, c. (2018). a learning trajectory for teaching social arithmetic using rme approach. in iop conference series: materials science and engineering (vol. 335, no. 1, p. 012121). hudoyo, h. (1990). teaching mathematics, ministry of education and culture, directorate general of higher education project development institute for education. indaryati, i., & jailani, j. (2015). pengembangan media komik pembelajaran matematika meningkatkan motivasi dan prestasi belajar siswa kelas v. jurnal prima edukasia, 3(1), 84-96. kristanto, a., & mariono, a. (2019, november). development of education game media for xii multimedia class students in vocational school. in journal of physics: conference series (vol. 1387, no. 1, p. 012117). iop publishing. mcvicker, c. j. (2007). comic strips as a text structure for learning to read. the reading teacher, 61(1), 85-88. rasiman, agnita siska pramasdyahsari. (2014). development of mathematics learning media e-comic based on flip book maker to increase the critical thinking skills and character of junior high school students. international journal of education, vol.2, nov 11, 2014. recine, d. (2013). comics aren't just for fun anymore: the practical use of comics by tesol professionals (doctoral dissertation). husnuz zaimah 11 sulistyani, n., & retnawati, h. (2015). pengembangan perangkat pembelajaran bangun ruang di smp dengan pendekatan problem-based learning. jurnal riset pendidikan matematika, 2(2), 197-210. toh, t. l., cheng, l. p., ho, s. y., jiang, h., & lim, k. m. (2017). use of comics to enhance students’ learning for the development of the twenty-first century competencies in the mathematics classroom. asia pacific journal of education, 37(4), 437-452. wright, g. (1979). the comic book: a forgotten medium in the classroom. the reading teacher, 33(2), 158-161. microsoft word seamej.journal.vol1.draft 5 edited by wahyudi jan 2012.docx recent decades have seen growing concern over the lowering levels of engagement with mathematics in australia and internationally. this paper reports on a longitudinal study on engagement with mathematics and explores the influences of teachers on the students’ engagement with mathematics. findings reveal that the development of positive pedagogical relationships between students and their teachers forms a critical foundation from which positive engagement can be promoted. introduction in recent years there has been growing concern over the lowering levels of engagement with mathematics in australia (commonwealth of australia, 2008; state of victoria department of education and training, 2004; sullivan & mcdonough, 2007; sullivan, mcdonough, & harrison, 2004) and internationally (boaler, 2009; douglas willms, friesen, & milton, 2009; mcgee, ward, gibbons, & harlow, 2003). the issue of lowered engagement levels in mathematics during the middle years of schooling (years 5 to 8 in nsw) has the potential to cause wide-reaching consequences beyond the obvious need to fill occupations that require the use of mathematics. lowered engagement with mathematics can lead to reducing the range of higher education courses available to students through exclusion from courses that require specific levels of mathematics. students who discontinue studying mathematics can potentially limit their capacity to understand life experiences through a mathematical perspective (sullivan, mousley, & zevenbergen, 2005). one of the most significant influences impacting on engagement in mathematics is the teacher and teaching practices, or pedagogy (hayes, mills, christie, & lingard, 2006; nsw department of education and training, 2003). this paper is derived from a longitudinal case study on engagement with mathematics during the middle years of schooling. in this study a group of 20 students experienced a range of mathematics teachers and pedagogical practices in their final year of primary school and the first two years of secondary school. data was collected from the group across the three school years through individual interviews and focus group discussions. this paper is an investigation of the influences of teachers and their practices on the participants’ engagement with mathematics. the theoretical framework underpinning this paper is based on current theories and definitions of engagement, and literature defining ‘good’ teaching of mathematics. a brief overview of the literature is now provided. catherine attard <c.attard@uws.edu.au> university of western sydney catherine attard engagement with mathematics: the influence of teachers catherine attard engagement seminal australian research into student engagement, the fair go project (fair go team, nsw department of education and training, 2006) focussed on understanding engagement “as a deeper student relationship with classroom work” (p. 9). the fair go team found students need to become ‘insiders’ within their classroom, feeling they have a place and a say in the operation of their classroom and the learning they are involved with. students have a need to identify themselves as ‘insiders’ as well as to be identified as ‘insiders’ by their teachers, students and all stakeholders. there are other definitions of engagement that should also be considered. some view engagement only at a behavioural level (hickey, 2003), where others view it as a multidimensional construct (fredricks, blumenfeld, & paris, 2004). fredricks et al. (2004), define engagement as multi-faceted and operating at operative, affective, and cognitive levels. operative engagement involves the idea of active participation and involvement in academic and social activities, and is considered vital for the achievement of positive academic outcomes. affective engagement includes students’ reactions to school, teachers, peers and academics, influencing willingness to become involved in school work. cognitive engagement includes the idea of investment, recognition of the value of learning and a willingness to go beyond the minimum requirements. in terms of engagement with mathematics, engagement occurs when students are procedurally engaged within the classroom, participating in tasks and ‘doing’ the mathematics, and hold the view that learning mathematics is worthwhile, valuable and useful both within and beyond the classroom. in an investigation into the reasons students are choosing not to pursue higher-level mathematics courses, mcphan, moroney, pegg, cooksey and lynch (2008), claim “curriculum and teaching strategies in the early years which engage students in investigative activities and which provide them with a sense of competence are central to increasing participation rates in mathematics” (p. 22), yet attempts to investigate the lack of engagement with mathematics have failed to find good reasons for students’ difficulties. it is claimed students who are engaged with school are more likely to learn, find the experience rewarding and continue with higher education (marks, 2000). engagement with mathematics: the influence of teachers ‘good’ teaching and mathematics the teaching practices employed within mathematics classrooms cover a wide spectrum ranging from ‘traditional’, text book based lessons, to contemporary or ‘reform’ approaches of problem solving and investigation based lessons, or a combination of both. when recalling a typical mathematics lesson, many students cite a traditional, teacher-centred approach in which a routine of teacher demonstration, student practice using multiple examples from a text book and then further multiple, text book generated questions are provided for homework (even & tirosh, 2008; goos, 2004; ricks, 2009). an alternate approach to teaching mathematics reflects a constructivist perspective where students are given opportunities to construct their own knowledge with a focus on conceptual understanding rather than instrumental understanding. such an approach promotes problem solving and reasoning and is consistent with australian frameworks for quality teaching (newmann, marks, & gamoran, 1996; nsw department of education and training, 2003). although there are arguments for using either or both approaches, there is strong support for an investigational, contemporary approach to teaching and learning mathematics (anthony & walshaw, 2009; boaler, 2009; clarke, 2003; lovitt, 2000). open-ended, rich tasks transform students’ beliefs about problem solving and alter the culture of mathematical engagement. evidence suggests that providing students with engaging mathematical tasks supported by appropriate teaching strategies leads to sustained improvement in learning outcomes (callingham, 2003). much research has been conducted on effective teaching of numeracy and mathematics, with a particular emphasis on the pedagogical content knowledge (pck) required for effective teaching of mathematics (askew, brown, rhodes, johnson, & wiliam, 1997a; delaney, ball, hill, schilling, & zopf, 2008; hill, ball, & schilling, 2008; schulman, 1986). in support of the need for strong pck it can be argued that teachers with higher mathematical qualifications do not necessarily produce strong learning outcomes in their students as a result of weak understandings of how students learn and the pedagogies that are appropriate for particular mathematics content (askew, brown, rhodes, wiliam, & johnson, 1997b). in recent years the national mathematics teaching professional association, the australian association of mathematics teachers (aamt) (2006), developed a set of standards that reflects current literature on effective teaching of mathematics and represents national agreement of teachers and stakeholders on the required knowledge, skills and attribute of quality teachers of mathematics. data informing this paper were analysed against the backdrop of the above literature on engagement, effective teaching and current teaching standards. the following is a brief description of the methodology used in the study. catherine attardcatherine attard methodology the participants in this case study were originated from a year 6 (the final year of primary schooling in new south wales) cohort in a western sydney catholic primary school. the students were identified through martin’s motivation and engagement scale (high school) (2008), as having strong levels of engagement with mathematics. the instrument consisted of a 44 item likert scale requiring students to rate themselves on a scale of 1 (strongly disagree) to 7 (strongly agree) and was adapted to be specific to mathematics. all students in the group of 20 made the transition together to the local catholic secondary college which had been in operation for only two years prior to the group’s arrival. the participants had a diverse range of mathematical abilities and came from a range of cultural backgrounds, and most came from families with two working parents. during the study the students participated in individual interviews during year 6 and again in year 8, and a series of focus group discussions at five points across the duration of the study. teachers identified by the students as ‘good’ mathematics teachers were interviewed and observed during several mathematics lessons. the students formed three focus groups, a boys group, girls group and mixed gender group. each interview and focus group discussion was based on the following set of discussion points/questions: (a) tell me about school; (b) let’s talk about maths; (c) tell me about a fun maths lesson that you remember well; (d) when it was fun, what was the teacher doing?; and (e) what do people you know say about maths? the data gathered were transcribed and coded into themes. in terms of the students’ perceptions of mathematics teaching, two major themes emerged as being influential on their engagement with mathematics: teachers’ pedagogical practices, those day-to-day routines that teachers implement in their teaching of mathematics, and the pedagogical relationships formed between teachers and students. results and discussion during year 6 the participants experienced pedagogies that included an emphasis on cooperative learning. the opportunities for interaction and discussion that this provided had a positive impact on the students’ engagement with mathematics, with one student saying: “you’ve got like more options to choose from rather than if you’re by yourself” and another: “working with partners is fun because you could find different strategies and you have fun and it’s easier.” it can be argued that the classroom practice of cooperative learning has positive results in terms of providing a safe environment in which the students are able to learn within a positive classroom culture. the ability to associate learning in mathematics with fun appeared to be a powerful influence on engagement, and the following quote engagement with mathematics: the influence of teachers summarised the collective feeling of most of the participants: “the group can work it out together to try and solve the problem and you’ve like learned something new or how to work out something.” one year 6 teacher, mrs. l, who was identified by the students as the ‘best’ mathematics teacher, was described by several students as someone who enjoyed teaching and had a passion for mathematics. alison believed this quality to increasing her own engagement: “she just puts a lot of enthusiasm in maths and makes it really fun for us. she gets all these different maths activities. she just makes it really fun for us and i quite enjoy maths now because of that.” it appeared the mrs. l’s enthusiasm for mathematics promoted positive attitudes and excitement towards mathematics, reflecting the findings from research (askew et al., 1997b) and recommendations by the aamt (2006). in addition to her passion for mathematics, the students witnessed mrs. l. as appearing to have fun teaching. tenille said: “it’s fun when the teacher, like, while you’re doing the work she also has fun teaching the maths as well.” when the students moved on to their first year of secondary school, year 7, they experienced a new set of pedagogies and a new group of mathematics teachers. in contrast to the teaching approaches used during their primary years, the students were expected to work on an individual basis, using computer-based interactive tutorials and mathematics textbooks. this caused a reduction in classroom interaction and discussion, and rather than having a single mathematics teacher, the students were provided with a rotation of four different teachers. although the provision of computer technology provided the opportunity for teachers to deliver a new and relevant way of teaching and learning (collins & halverson, 2009), they instead appeared to be used as replacements for teachers. alison commented on this emerging idea among the students: … it's probably not the best way of learning because last year at least if you missed the day that they taught you, you still had groups so your group could tell you what was happening. where now, we’ve got the computers and it’s alright because there is, um, left side of the screen does give you examples and stuff, um, but if you don’t understand it, it’s really, hard to understand. it is reasonable to suggest that the website and textbook were not necessarily inferior resources. however, the data was showing that it was the way they were used in isolation from other resources that meant the students began to disengage from mathematics. during term 2 of year 7 the students were provided with the opportunity to participate in tasks that were more interactive and hands-on, consistent with recommendations from research (boaler, 2002; callingham, 2003; lowrie, 2004). several of the students commented on this change, with fred saying: “it’s more interesting”. the students found the incorporation of concrete materials made their mathematics lessons more interesting, and the opportunity to work in groups during one particular activity made those lessons memorable, with rhiannon giving catherine attardcatherine attard this reason: “because we got to create the shape by using straws, in groups. not by ourselves.” in addition to the benefits of being able to work collaboratively, george felt he and his group made more of an effort than usual: “it was good because we could make it ourselves and we could like put effort into it.” when the students reached their second year of secondary school, year 8, the school’s structure had been reviewed and during term 2, the students were allocated one regular mathematics teacher per group. the newly formed mathematics classes appeared to increase the students’ engagement, allowing stronger teacher/student and peer relationships to develop. in terms of the resources that were used in the year 8 lessons, there was less reliance on the students’ laptops and more emphasis on using text books. kristie described a typical routine: well, we just got our text book and the laptops don’t come out in maths as much or at all, unless you’ve forgotten your text book or something like that. and, um, maths is good, we separated into groups and the teacher’s out the front and he’ll tell us what to do and you pretty much put your hand up if you need help, and he’ll help you and then you have the text book out and you answer the questions in your maths book. although it has been found that a traditional approach to teaching mathematics may have a negative influence on student engagement, in this particular case the students saw it as an improvement on previous pedagogies and appeared to experience higher levels of engagement. one aspect of the teachers’ pedagogies that had a positive effect on the students’ engagement was the students’ perceptions of an improvement in teacher explanations. george made this comment which reflected the feelings of many of the students: “i think maths has improved because the teachers go through it with you more, whereas last year they would just set you a task and leave you with it.” billy, a student who had difficulty maintaining his engagement with mathematics added: “sir just writes stuff on the board and then he explains it really good and we learn about stem and leaf graphs. he teaches it really good and other teachers just write it down and say ‘go do that’.” during the final focus group discussions, alison made a comment that was reflective of the group’s feelings once they were assigned their regular teachers and were able to begin building positive pedagogical relationships: “the teachers know where we’re coming from and what we need to learn and they learn, not what the group needs, but what we need.” the data shows that the students appeared have begun to re-engage with mathematics because they felt the teachers knew them in terms of their mathematics learning needs. the opportunity to establish positive pedagogical relationships with teachers appeared to provide students with a sense of belonging, an important aspect of an effective mathematics classroom (boaler, 2009). engagement with mathematics: the influence of teachers implications and conclusion the biggest influence on engagement with mathematics for these students appeared to be that of their teachers. this influence can be viewed at two interconnected levels. the first level includes the pedagogical practices employed by the teacher, and the second, the pedagogical relationships that occur between the teachers and students. that is, the connections made between the teachers and students, and the teachers’ recognition of and response to the learning needs of his or her students. although this study has limitations in terms of the selective nature of the sample, it is suggested that the development of positive pedagogical relationships forms a critical foundation from which positive engagement can be promoted and this may be applicable to a wider student population. the findings discussed in this paper imply many students in the lower secondary years of schooling are still dependent on high levels of interaction within the mathematics classroom. repetition of the current study within different school contexts would be of benefit in further exploring the concept of engagement with mathematics. further studies on engagement with mathematics during the later years of schooling and beyond into tertiary education would be beneficial in terms of investigating whether pedagogical relationships remain as important for older students. although student achievement and its relationship to engagement levels was not a focus of this study, such an exploration would also be worthwhile for future research. catherine attardcatherine attard references anthony, g., & walshaw, m. 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(2005). increasing access to mathematical thinking. the australian mathematical society gazette, 32(2), 105–109. catherine attardcatherine attard 75 southeast asian mathematics education journal, volume 10, no 2 (2020) using 'paddy field' graphic organiser to enhance students' ability in expanding and simplifying algebraic expressions lim yi wei smk sanzac, kota kinabalu, malaysia abstract the purpose of this action research was to enhance students' abilities in expanding and simplifying algebraic expressions. a total of 26 students from 3 anggerik (9th grade) smk sanzac, malaysia, were involved in this action research. based on the analysis of their marks from the previous test and mid-year examination, they were unable to solve algebraic expressions, even though they had learnt them since form 1(7 th grade). however, expanding and simplifying algebraic expressions are fundamental to solving mathematics problems, with approximately 46% of lower form mathematic topics requiring these skills. hence, the "paddy field" method was introduced to help students expand and simplify algebraic expressions. post-test results showed a significant improvement in students' abilities to expand and simplify algebraic expressions. keywords: algebra, learning difficulties, expanding and simplifying algebraic expressions. introduction the problem occurred in my classroom initiated this research. one of the challenges that the researcher encountered in my mathematics class was making students understand the expansion and simplification of algebraic expressions. their low achievement results in this topic have proven that i failed to make them understand how to expand and simplify algebraic expressions with my conventional teaching method as a teacher. this situation seems to persist across the grades. many of them are still struggling with basic algebraic expression questions ever since form 1 (7 th grade). they claimed that algebra was the toughest topic in mathematics. i felt worried, especially for my form 3 (9 th grade) students because they will sit for the year-end form 3 assessment. this has motivated the researcher to look into the problem to understand their learning difficulties in this topic. from students‟ answers, it is found out that the main difficulties encountered while expanding and simplifying algebraic expressions were the lack of conceptual understanding and poor computational procedures. these common mistakes included equal sign errors, negative sign errors, and bracket expansion errors, as shown in table 1. this is similar to what has been identified by jupri et al. (2014). in school, algebra expression is the macro concept that has a great influence in developing other micro mathematics concepts like functions, quadratics, arithmetic sequences, and more. it is also an important foundation to prepare students for mastering other stem subjects. if these common mistakes persist without intervention, they will hinder students' ability to solve even simple calculations. that is why enhancing students' abilities to expand and simplify algebraic expressions are so important. 76 southeast asian mathematics education journal, volume 10, no 2 (2020) table 1 sample of students works with conception and procedural errors type of error examples description conclusion equal sign errors example 1 the correct answer is . this student combined the unlike terms and obtained as the final answer. this has shown that the student misinterpreted the equal sign solely as a signal to get a single 'answer.' example 2 the correct answer is . however, this student combined the unlike terms and obtained as the final answer. negative sign errors example 3 the correct answer is . student failed to simplify the last term with a negative sign attached to the integer before the bracket. students had problems working with integers and operation signs. example 4 the correct answer is . this student multiplied with each term inside the bracket without considering the product of two negative terms is positive. example 5 the correct answer is . this student failed to simplify the last term with two negative signs and obtained . bracket expansion errors example 6 the correct answer is . this student only multiplied the first term by 4 and ignored the second term. this has shown that students failed to retrieve the correct distributive law in an algebraic expression. example 7 the correct answer is . this student multiplied the first term by 4 but added up the second term with 4. example 8 the correct answer is . this student added up the terms instead of multiplying them. 77 southeast asian mathematics education journal, volume 10, no 2 (2020) previous studies have identified that students‟ difficulties with early algebra are most prone to happen during the transition from arithmetic to algebra (cai, ng, & moyer, 2011; welder, 2012). one of these difficulties is bracket usage. in arithmetic, brackets are used to show students which operation to perform first (welder, 2012). for instance, the bracket in 18 – (9 -10) means to carry out the operation 9-10 inside the bracket first. however, students need to have much more flexible thinking in algebra because the bracket indicates the priority of operation and multiplication (welder, 2012). for instance, the bracket in 4(a+5) means multiplying both terms in the bracket by 4. oftentimes, students are confused and misapply the first rule into expanding and simplifying the algebraic expression. another difficulty in algebra is the misconception in the equal sign (hewitt, 2012). according to strand & mills (2014), there are different meanings of the equal sign. an equal sign can be viewed as an operation symbol, which means „to do something‟ or „find the total‟ or „the answer comes next‟ (powell, 2012; strand & mills, 2014). however, many students misinterpreted the equal sign solely as a signal to get a single 'answer.' this happened because, throughout primary school arithmetic learning, students are repeatedly exposed to standard equations with one answer 8+4=12 or 8-4=4 (powell, 2012). hence, when an algebraic expression such as 6+a is given to them, they tend to compress it down into a single entity with no operation shows, such as 6a (hewitt, 2012). the equal sign can also be viewed as a relational symbol which keeps two sides of an equation balanced and equivalent (welder, 2012). this relational understanding is critical in learning algebra. however, most of the teacher manuals and student textbooks do not provide enough opportunities to foster a relational understanding of the equal sign (mcneil et al., 2006). this research focused on improving students‟ ability in expanding and simplifying algebraic expressions. expanding and simplifying algebraic expressions are the fundamental parts of solving mathematics problems, yet so many students failed to do so and lost interest in mathematics. previous studies (booth, barbieri, eyer, & paré-blagoev, 2014; lim, 2010; ottmar & landy, 2017) had shown that the concept of expanding and simplifying an algebraic expression was very abstract for students to visualise concretely. students who have reached the middle grades and are still experiencing difficulty with mathematical concepts are often provided with rote approaches only, keeping them from gaining the conceptual frameworks necessary for success with algebraic tasks(dacey & gartland, 2019). one of the interventions to deal with these students‟ difficulty is by supporting it with digital manipulative and helping students expand brackets (jupri, drijvers, & van den heuvelpanhuizen, 2015). in this study, the researcher did not use ict based activity instead of hands-on activity. to help students visualise and organise the computational procedures of expanding and simplifying algebraic expressions, a graphic organiser, "the paddy field" is introduced to them. ives & hoy (2003) have pointed out that when students are allowed to expand and simplify the algebraic expression visually and systematically, they are more likely to succeed in algebraic tasks. this study aims to enhance students' ability in expanding and simplifying algebraic expressions. this study tried to support the student to expand and simplify one bracket 78 southeast asian mathematics education journal, volume 10, no 2 (2020) algebraic expressions, expand and simplify two brackets algebraic expressions, and ensure hands-on learning activities in expanding and simplifying algebraic expressions. methods this is a classroom action research study. a pre-test was given to students before the intervention lesson to determine the target group's baseline knowledge in expanding and simplifying algebraic expressions. it consisted of 10 questions on these mathematics skills. each question was designed to measure various operations of simplifying and expanding in algebraic expressions. for instance, item one was used to see if students could expand and simplify one bracket algebraic expression with unlike terms. while item three was used to see if students could expand and simplify one bracket algebraic expression with a negative premultiplier. table 2 displays the questions which students were asked in the pre-test. table 2 pre-test questions question number question question rationale 1 ( ) to test students understanding of multiplying one bracket algebraic expression with unlike terms and positive sign 2 ( ) to test students understanding of multiplying one bracket algebraic expression with unlike terms and a negative sign 3 ( ) to test students understanding of multiplying one bracket algebraic expression with negative pre-multiplier and a positive integer 4 ( ) to test students understanding of multiplying one bracket algebraic expression with negative pre-multiplier and negative integer 5 ( )( ) to test students understanding of multiplying two brackets algebraic expression with only positive integers 6 ( )( ) to test students understanding of multiplying two brackets algebraic expression with positive and negative integers 7 ( )( ) to test students understanding of multiplying two brackets algebraic expression with two negative integers 8 ( ) to test students understanding of multiplying single bracket algebraic expression with a square 9 ( ) to test students understanding of solving one bracket complex algebraic expression 10 ( ) to test students understanding of solving two brackets bracket complex algebraic expression the rationale for using these questions was to fulfil the requirement of the secondary school mathematics curriculum and assessment standard offered by the ministry of malaysia education (2016). besides, these items were discussed with other mathematics teachers to fulfil the instrument's validity. similar questions were reshuffled and were used as post-test at the end of the research study for comparison and analysis purposes. students have to complete each test within thirty minutes. the target group consists of 26 students from 3 anggerik. the selection of respondents was based on the items‟ analysis of their mid-year mathematics exam paper. i discovered that this group of students have difficulties in expanding and simplifying algebraic expressions. based on a 2019 mid-year mathematics examination paper, it was obvious that most of my students struggled with algebraic expressions. this was reflected in table 3 below, with the 79 southeast asian mathematics education journal, volume 10, no 2 (2020) second column from the left showing the percentage of students who did not get the correct answer. the third column shows the percentage of students who left the question blank without answering. looking at their results, i could conclude that they failed because i failed to make them understand how to expand and simplify algebraic expressions with my conventional teaching method. table 3 analysis items of 2019 form 3 anggerik mid-year mathematics exam paper question incorrect answer (%) did not answer (%) 66.7 0.0 33.3 0.0 56.7 6.7 86.7 13.3 the data was analysed using descriptive statistics. my limitation was that the post-test was only conducted once due to time factors. it can be further enhanced with another post-testing after a period of time to see whether students can still apply this method. results and discussion implementation of “paddy field” the “paddy field” is adopted from the grid method to clear students‟ confusion in applying the conventional foil (first, outer, inner, last). it consists of 4 square grids which look like the chinese word “田” (tian) which means paddy field in english. the "paddy field" helps students to find the product of algebraic expressions in a tidier and more organised way. 80 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 1. an example with one binomial. figure 2. an example of two binomials. firstly, each of the students is given one set of “paddy field”. there are two types of the set, algebraic expression with one binomial as in figure 1 and algebraic expression with two binomials as in figure 2. they are asked to put the algebraic expression question in the blue box in the right upper corner. figure 3. an example with one binomial. figure 4. an example with two binomials then they have to write each term of the algebraic expression in the grey boxes beside "田". if the question consists of only 1 term outside the bracket, they have to write that term in the first vertical grey box and the other two terms inside the bracket by the two horizontal grey boxes (figure 3). if the question consists of two brackets, they have to write each term of the first bracket in the vertical grey boxes and write each term of the second bracket in the horizontal grey boxes (figure 4). however, students are reminded it does not matter whether one puts the terms of the first bracket in the vertical grey boxes first or horizontal grey boxes first. 81 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 5. an example with one binomial. figure 6. an example of two binomials. after that, they have to write down the product of the numbers in its row and column. for instance, in figure 5, in its column and in its row. the product is multiplied by r which is . students have to be reminded that they will get the area of each partition of "the paddy field". they have to multiply the terms and not to sum up the terms from each side. figure 7. an example with one binomial. figure 8. an example of two binomials. students are told to move the colourful cards from “田" and arrange them from the term with the largest exponent to the constant on the row of boxes provided as illustrated in figure 7 and figure 8. x x 82 southeast asian mathematics education journal, volume 10, no 2 (2020) figure 9. an example of two binomials. lastly, students have to simplify the algebraic expression by combining the like terms. then they have to write the algebraic expression in the simplest form. observation during the activity, students seemed to be more confident in solving the algebraic expression questions. when i asked them how they felt about this method, they told me this method was easier to use than the foil method. teacher : how do you feel about this method? student : it's easy. i like it! teacher : which one do you prefer? the foil method or the paddy field method? student : definitely this paddy field method. teacher : why not foil method? student : it's very confusing for me, especially the arrows. i get lost most of the time and get the wrong answer. teacher : what is the thing that you like about the paddy field method? student : i can see the steps more clearly with the boxes provided in the paddy field method. figure 10. an example of the confusion of a student a using the foil method. x 83 southeast asian mathematics education journal, volume 10, no 2 (2020) they claimed that the foil method was confusing. most of the time, they get lost while halfway through expanding the algebraic expressions. they claimed that this method actually guided them well to expand every term in the boxes provided. through observation, i found that student a, who did not answer most of the pre-test questions, was busy solving the given algebraic expressions questions using "the paddy field" graphic organiser. i was glad to see he could expand and simplify algebraic expressions. meanwhile, he still made some negative sign errors and operation errors while working on those given questions. i believe this can be overcome with more practice and guidance. other students seemed to immerse themselves in the activity too. i noticed that they could do well in expanding algebraic expressions. their problems were similar to student a, which were negative sign errors. they seemed to forget to change the sign while multiplying two negative terms. this is in-line with the study by booth, barbieri, eyer & blagoev (2014). it turned out that in the classroom, it becomes a common mistake. another common error that students made was forgetting to put the variable in the final answer. this mistake also can be found in lim (2010)‟s research. these struggles can be improved with more practice and guidance, as they were not familiar with writing the final answer with variables. post-test during the post-test, students used "the paddy field" graphic organiser in answering posttest questions. in comparison between the pre and post-test results in table 4, we can see a great improvement in every respondent's ability to expand and simplify algebraic expressions. three students from no correct answer in the pre-test managed to score full mark in the post-test. table 4 comparison of students’ scores in the pre-test and post-test min improvement 3 max improvement 10 average improvement 5,33 from table 4, it is observable that all students improved the numbers of correct answers. none of the students received less correct answers than their pre-test scores. table 5 comparison of students’ correct responses in the pre-test and post-test algebraic expressions skills pre-test post-test changes in the average % (+/-) items no. of items average (%) of students with correct responses items no. of items average (%) of students with correct responses expand one bracket algebraic 1,2,3,4,9 5 70.77 2,4,7,8,9 5 90.00 +19.23 84 southeast asian mathematics education journal, volume 10, no 2 (2020) expressions simplify one bracket algebraic expressions 1,2,3,4,9 5 55.38 2,4,7,8,9 5 84.62 +29.24 expand two brackets algebraic expressions 5,6,7,8,10 5 31.54 1,3,5,6,10 5 92.30 +60.76 simplify two brackets algebraic expressions 5,6,7,8,10 5 10.77 1,3,5,6,10 5 80.77 +70.00 results from table 5 indicated that students' skill in simplifying two brackets algebraic expressions improved the most, from 10.77% to 70.00% out of four algebraic expression skills. followed by expanding two brackets algebraic expressions, from 31.54% to 92.30%. most students did not have a problem expanding one bracket algebraic expressions before and after the intervention, as reflected in high scores in both pre-test and post-test scores. students exposed to the "paddy field" graphic organiser could expand and simplify algebraic expressions more confidently. compared to the conventional foil method, this grid method, which was integrated with a hands-on activity, is not only fun to learn, but it helped students to expand and simplify the algebraic expression in a tidier and more systematic way. furthermore, "paddy field" graphic organiser managed to attract students' attention and increased their involvement in its graphic design that closely resembles daily life. when i saw that my weak students could solve algebraic expressions questions with "the paddy field" graphic organiser, i realised how important an effective graphic organiser is for improving algebra knowledge in students. future research should look into graphic organisers to help students solve algebra-related topics such as linear equations and inequalities, quadratics and polynomials, graphing lines and slope, functions, and many more. before doing so, i will conduct another test in the next cycle to see whether students can apply this grid method without the "paddy field" graphic organiser in answering the algebraic expression questions. i will also upgrade the present "paddy field" graphic organiser to a magnetic version so that the four coloured cards will glide smoothly and firmly on the graphic organiser. conclusion the overall findings of this action research have shown a positive response to its objectives. it has shown that using an appropriately designed graphic organiser to teach mathematical skills truly helps students be more successful in mathematics. however, it occurred in my classroom only, and its generalisability is limited. furthermore, teachers play an important role in thinking of ways suitable for their students' learning. finding the proper and suitable for students is crucial. it needs extra work for teachers to develop competencies to design/ creates suitable activities for students. 85 southeast asian mathematics education journal, volume 10, no 2 (2020) acknowledgement i would like to express my sincere gratitude to previous excellent principal jusa c of smk sanzac, dr shirley tay siew hong, for her patient guidance, valuable comments and constant encouragement throughout this action research. i would also like to thank mr wahid yunianto, mathematics specialist, and his team from seameo regional centre for qitep in mathematics for the valuable comments, proofreading and layout editing. references booth, j. l., barbieri, c., eyer, f., & paré-blagoev, e. j. (2014). persistent and pernicious errors in algebraic problem-solving. journal of problem solving, 7(1). https://doi.org/10.7771/1932-6246.1161 cai, j., ng, s. f., & moyer, j. c. (2011). developing students’ algebraic thinking in earlier grades: lessons from china and singapore. https://doi.org/10.1007/978-3-642-177354_3 dacey, l., & gartland, k. (2019). math for all: differentiating instruction, grades 6-8 (j. ann cross, ed.). sausalito: california: scholastic inc. hewitt, d. (2012). young students learning formal algebraic notation and solving linear equations: are commonly experienced difficulties avoidable? educational studies in mathematics, 81(2). https://doi.org/10.1007/s10649-012-9394-x ives, b., & hoy, c. (2003). graphic organisers applied to higher-level secondary mathematics. learning disabilities research and practice, 18(1). https://doi.org/10.1111/1540-5826.00056 jupri, a., drijvers, p., & van den heuvel-panhuizen, m. (2014). difficulties in initial algebra learning in indonesia. mathematics education research journal, 26(4). https://doi.org/10.1007/s13394-013-0097-0 jupri, a., drijvers, p., & van den heuvel-panhuizen, m. (2015). improving grade 7 students‟ achievement in initial algebra through a technology-based intervention. digital experiences in mathematics education, 1(1). https://doi.org/10.1007/s40751-015-00042 lim, k. s. (2010). an error analysis of form 2 (grade 7) students in simplifying algebraic expressions: a descriptive study. electronic journal of research in educational psychology, 8(1). mcneil, n. m., grandau, l., knuth, e. j., alibali, m. w., stephens, a. c., hattikudur, s., & krill, d. e. (2006). middle-school students‟ understanding of the equal sign: the books they read can‟t help. cognition and instruction, 24(3). https://doi.org/10.1207/s1532690xci2403_3 86 southeast asian mathematics education journal, volume 10, no 2 (2020) ministry of education malaysia. (2016). kurikulum standard sekolah menengah: standard kurikulum dan pentaksiran matematik tingkatan 2. putrajaya, malaysia. ottmar, e., & landy, d. (2017). concreteness fading of algebraic instruction: effects on learning. journal of the learning sciences, 26(1). https://doi.org/10.1080/10508406.2016.1250212 powell, s. r. (2012). equations and the equal sign in elementary mathematics textbooks. elementary school journal, 112(4). https://doi.org/10.1086/665009 strand, k., & mills, b. (2014). mathematical content knowledge for teaching elementary mathematics: a focus on algebra. mathematics enthusiast, 11(2). welder, r. m. (2012). improving algebra preparation: implications from research on student misconceptions and difficulties. school science and mathematics, 112(4). https://doi.org/10.1111/j.1949-8594.2012.00136.x microsoft word seamej.journal.vol1.draft 5 edited by wahyudi jan 2012 br.docx ng kit ee dawn towards mathematical literacy in the 21st century: perspectives from singapore ng kit ee dawn nanyang technological university, national institute of education, singapore <dawn.ng@nie.edu.sg> abstract the organization for economic cooperation and development (oecd) postulates that a major focus in education is to promote the ability of young people to use their knowledge and skills to meet real-life challenges (oecd, 2006). pisa, an international standardised assessment of students’ (aged 15) performance in the literacies of mathematics, science, and reading, was developed by the oecd in 1997 to evaluate the achievement of students who are about to finish their key stages of education (anderson, chiu, & yore, 2010). the concept of mathematical literacy has been defined and interpreted in various ways as recorded in the curriculum documents around the world. this paper will share perspectives from singapore on how mathematical literacy is interpreted in the mathematics curriculum through the use of three tasks: interdisciplinary project work, applications, and modelling. it will surface challenges to improving the mathematical literacy of students when using such tasks. introduction a major focus in education is to promote the ability of young people to use their knowledge and skills to meet real-life challenges (organisation for economic co-operation and development [oecd], 2006). in other words, equipping learners with literacy relevant to day-to-day real-world competencies is perceived to be an important current goal in education. pisa, which refers to the “programme for international student assessment” from the oecd, assesses and compares students’ reading, scientific, and mathematical literacy. since 2000, pisa tests are run every three years to elicit the knowledge and skills of 15 year-old students because they are nearing completion of their compulsory schooling. students’ familial and institutional backgrounds are factors considered in providing explanations for differences in performance during pisa among countries. pisa defines literacy to include various “competencies relevant to coping with adult life” (anderson, chui, & yore, 2010, p. 374). in particular, mathematical literacy is: the capacity of an individual to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen. (oecd, 2006, p. 21) hence, teaching for mathematical literacy is discussed within a broader, utilitarian view of mathematics whereby students are provided opportunities to engage with different types of mathematical problems, especially those relating to real-world contexts. indeed, 19 towards mathematical literacy in the 21st century: perspectives from singapore proponents for the incorporation of problem solving tasks involving real-world contexts in the mathematics curricula (e.g., gravemeijer, 1994; ng & stillman, 2009; van den heuvelpanhuizen, 1999; zevenbergen & zevenbergen, 2009) have long argued for the importance of cultivating students’ ability to work with tasks relating to real life. in relation, other educators (e.g., english, 2008; galbraith, 1998) espoused the need for students to draw upon their repertoire of interdisciplinary learning rather than tapping upon subject-specific knowledge and skills for more holistic mathematical learning through making connections between school mathematics and real-world problems which are often interdisciplinary in nature. according to stacey (2009), the current concept of mathematical literacy proposed by pisa is related to several other concepts ingrained in mathematics education. among them is mathematical modelling, a process of representing real world problems in mathematical terms in an attempt to understand and find solutions to the problems (ang, 2010). the singapore mathematics curriculum framework (curriculum planning and development division [cpdd], 2006) shown in figure 1 highlights mathematical problem solving as central to mathematics learning. teachers are encouraged to use a wide range of problem-solving situations, including non-routine, open-ended, and real-world contexts in their mathematics classrooms. some of these tasks can also be interdisciplinary where subject-specialist teachers work in collaboration during task implementation within a class. although mathematical concepts form the foundation of this pentagonal framework, skills and processes are perceived to be the pillars of this framework. real-world problem solving tasks involving applications and modelling are one of the latest infusions in the processes component of the framework which recognises the importance of mathematical reasoning, communication, and connections. it is postulated that such tasks provide platforms for the analysis of mathematical situations, construction of logical arguments, as well as links between mathematical ideas, between school-based subjects, and between school mathematics and everyday life (english, 2008). nonetheless, the syllabus documents did not set out to distinguish between applications and modelling tasks, perhaps contributing to the limited use of modelling activities in singapore schools (ng, 2011a). though both involve the use of real-world contexts, stillman, brown, and galbraith (2008) articulated the differences between applications and modelling tasks in the following ways. application tasks are commonly evident in situations where the teacher looks for real-world contexts to match specific taught mathematical knowledge and skills for use. in contrast, each modelling task starts with the real-world context where any of a variety of mathematical knowledge and skills can surface during mathematisation (de lange, 2006) of the context for model development in problem solving. 20 ng kit ee dawn figure 1. framework of the singapore mathematics curriculum (cpdd, 2006) the purpose of this paper is to share selected perspectives from singapore on how mathematical literacy is interpreted in the mathematics curriculum through the use of three tasks: interdisciplinary project work, applications, and modelling. implications will be drawn from these interpretations to suggest future developments in teacher education for mathematical literacy as there are still challenges to be overcome. mathematical literacy in interdisciplinary project work interdisciplinary project work (pw) has been implemented in singapore primary, secondary, and pre-university institutions since 2000 (cpdd, 1999). as from 2005, students’ performance in pw has been part of entry requirements to universities in singapore (moe, 2001). pw is an applications task embedded in real-world context which draws upon the integrated use of at least two areas of discipline-based knowledge and skills for problem solving. there were two main impetuses for widespread introduction of pw in singapore schools. firstly, each pw makes explicit connections between content knowledge and skills of its anchoring school-based subjects. pw encourages more holistic learning in preparation of students for work in a knowledge-based economy (tharman, 2005) where work-related real-world problems are often interdisciplinary in nature (sawyer, 2008). secondly, pw promotes student-centred learning (quek, divaharan, liu, peer, williams, wong et al., 2006). figure 2 shows an example of a pw task for year 7-8 students (aged 13-14) comprising mathematics, science, and geography as anchoring subjects (ng, 2009). 21 towards mathematical literacy in the 21st century: perspectives from singapore figure 2. an example of pw involving mathematics (ng, 2009) the task required students to work in groups to design an environmentally friendly building in a selected location within singapore and then construct a physical scale model of their building using recycled materials. it draws upon mathematical concepts and skills such as scale drawings, proportional reasoning, arithmetic, and measurement. students were encouraged to make decisions on their building design and location based on mathematical calculations after considering scientific and geographic real-world constraints. here, mathematical literacy is explored when students provided appropriate mathematical reasoning and arguments for their choices using their interpretations from their day-to-day experiences. nonetheless, research into the mathematical thinking of the students in casestudy groups reported in ng (2011b) revealed that some students face challenges in mathematical literacy during real-world mathematical applications and decision making. one such example came from a year 8 group where each member drew independent scale drawings of the front, top, and side views of their eco-friendly house but using different scales and dimensions, not considering that all drawings of the house should come together to form a coherent image of the house. mathematical literacy in mathematical applications tasks mathematical applications tasks have been commonly used in singapore classrooms. an example is shown in appendix a (adapted from foo, 2007) where year 7 students (aged 12-13) worked on calculating the budget for painting and waxing an office room, given a floor plan of it. the task was designed to elicit mathematical concepts and skills such as measurement, arithmetic and area, drawing upon students’ real-world understanding of 22 ng kit ee dawn painting and waxing of a room as well as the sale packaging of paint and wax which are usually sold by the litre. again, a challenge in mathematical literacy is detected in a student’s work shown in figure 3. although the student has successfully worked out the floor area and the actual amount of wax needed to cover the floor, he had assumed that he could purchase 1.15 litres of wax without much consideration about real-world constraints. figure 3. an example of a student’s work on the painting task indeed, as much as challenges exist in student’s display of mathematical literacy cited above, it was found that pre-service teachers at times faced the same challenges. echoing the findings from verschaffel, decorte, and borghart (1997), some postgraduate pre-service teachers who were undergoing teacher education for teaching of middle school mathematics (years 12-14) in singapore were unable to critically examine the fallacy in realistic mathematical meaning-making in the given context as shown in appendix b. whilst many of the 15 respondents could detect that it was almost impossible to solve the problem, most were unable to present sound mathematical reasoning and arguments as to why this was the case. only two of the respondents mentioned a lack of information on the dimensions of the flask and how its shape changed at different parts, hence removing the possibility of direct proportion reasoning. mathematical literacy in mathematical modelling tasks ng (2010) investigated the initial experiences of primary school teachers in their attempt at a mathematical modelling task (appendix c) and found that many teachers in her sample size of 48 were constrained by their beliefs and conceptions of what mathematics was during their modelling process. they embarked on immediate translation of information provided in the context of the task into mathematical expressions or known algorithms in order to obtain unique answers to the problem. the teachers needed some time to accept that mathematical representations can also take the forms of tabulation of data, graphical, and written statements containing mathematical reasoning based on data, along with assumptions floor area calculated first coat of wax remaining amount of wax needed 23 towards mathematical literacy in the 21st century: perspectives from singapore made and conditions set during their chosen approach. it appeared that the teachers’ interpretation of mathematical literacy during contextualised tasks was still confined to abstract mathematical representations. implications and future directions for teacher education this paper set out to present perspectives of mathematical literacy as interpreted in the singapore mathematics curriculum through exploring the use of three tasks types: interdisciplinary project work (pw), applications, and modelling. although proponents of mathematical tasks embedded within meaningful real-world experiences espoused that such tasks provide platforms for enhancing the mathematical literacy of students, there are challenges to overcome in teacher education so that the potentials of these tasks could be harnessed for stated educational goal. one of the challenges is that of producing quality mathematical outcomes from the tasks which should incorporate a reasonable degree of mathematical accuracy within the appropriate choice of approaches used bound by real-world constraints. another challenge relates to the preparation of students for mathematical arguments tapping upon various forms of mathematical representations. this is because pw and modelling tasks are deliberately open-ended to encourage multiple interpretations and solution pathways. last but not least, a third challenge involves changing the mindsets of teachers towards a more encompassing view of what mathematics for the purpose of promoting mathematical literacy in students. teachers can be encouraged to take a less prescriptive pedagogical approach which can limit the nature and variety of mathematical interpretations and representations in contextualised tasks. references anderson, j. o., chiu, m. h., & yore, l. d. (2010). first cycle of pisa (2000-2006) international perspectives on successes and challenges: research and policy directions. international journal of science and mathematics education, 8(3), 373-388. ang, k. c. (2010, december 17-21). teaching and learning mathematical modelling with technology. paper presented at the linking applications with mathematics and technology: the 15th asian technology conference in mathematics, le meridien kuala lumpur. curriculum planning and development division [cpdd]. (1999). project work: guidelines. singapore, ministry of education: author. curriculum planning and development division [cpdd]. (2006). mathematics syllabus. singapore, ministry of education: author. de lange, j. (2006). mathematical literacy for living from oecd-pisa perspective. tsukuba journal of educational study in mathematics, 25(1), 13-35. 24 ng kit ee dawn english, l. d. (2008). interdisciplinary problem solving: a focus on engineering experiences. in m. goos, r. brown & k. makar (eds.), proceedings of the 31st annual conference of the mathematics education research group of australasia (vol. 1, pp. 187-193). brisbane: merga. foo, k. f. (2007). integrating performance tasks in the secondary mathematics classroom: an empirical study. unpublished masters dissertation, nanyang technological university, singapore galbraith, p. (1998). cross-curriculum applications of mathematics. zentralblatt für didaktik der mathematik, 30(4), 107-109. gravemeijer, k. p. e. (1994). developing realistic mathematics education. utrecht, the netherlands: freudenthal institute. ministry of education [moe]. (2001, december 2). press release: project work to be included for university admission in 2005. retrieved december 13, 2002, from http://www1.moe.edu.sg/press/2001/pr20062001.htm ng, k. e. d. (2009). thinking, small group interactions, and interdisciplinary project work. unpublished doctoral dissertation, the university of melbourne, australia. ng, k. e. d. (2010). initial experiences of primary school teachers with mathematical modelling. in b. kaur & j. dindyal (eds.), mathematical modelling and applications: yearbook of association of mathematics educators (pp. 129-144). singapore: world scientific. ng, k. e. d. (2011a). facilitation and scaffolding: symposium on teacher professional development on mathematical modelling initial perspectives from singapore. paper presented at the connecting to practice teaching practice and the practice of applied mathematicians: the 15th international conference on the teaching of mathematical modelling and applications. ng, k. e. d. (2011b). mathematical knowledge application and student difficulties in a design-based interdisciplinary project. in g. kaiser, w. blum, r. borromeo ferri & g. stillman (eds.), international perspectives on the teaching and learning of mathematical modelling: trends in the teaching and learning of mathematical modelling (vol. 1, pp. 107-116). new york: springer. ng, k. e. d., & stillman, g. a. (2009). applying mathematical knowledge in a design-based interdisciplinary project. in r. hunter, b. bicknell & t. burgess (eds.), crossing, divides: proceedings of the 32nd annual conference of the mathematics education research group of australasia (vol. 2, pp. 411-418). wellington, new zealand: adelaide: merga. organisation for economic co-operation and development [oecd]. (2006). pisa 2006: science competencies for tomorrow's world (vol. 1). paris: oecd. quek, c. l., divaharan, s., liu, w. c., peer, j., williams, m. d., wong, a. f. l., et al. (2006). engaging in project work. singapore: mcgraw hill. sawyer, a. (2008). making connections: promoting connectedness in early mathematics education. in m. goos, r. brown & k. makar (eds.), proceedings of the 31st annual conference of the mathematics education research group of australasia (vol. 2, pp. 429-435). brisbane: merga. 25 towards mathematical literacy in the 21st century: perspectives from singapore stacey, k. (2009). mathematics and scientific literacy around the world. in u. h. cheah, r. p. wahyudi, k. t. devadason, w. ng, preechaporn & j. aligaen (eds.), improving science and mathematics literacy theory, innovation and practice: proceedings of the third international conference on science and mathematics education (cosmed) (pp. 1-7). malaysia, penang: seameo recsam. stillman, g., brown, j., & galbraith, p. l. (2008). research into the teaching and learning of applications and modelling in australasia. in h. forgasz, a. barkatsas, a. bishop, b. clarke, s. keast, w. t. seah & p. sullivan (eds.), research in mathematics education in australasia: new directions in mathematics and science education (pp. 141-164). rotterdam: sense publishers. tharman, s. (2005). speech by mr tharman shanmugaratnam, minister for education, at the opening of the conference on 'redesigning pedagogy: research, policy and practice' on 30 may, at the national institute of education, singapore. retrieved may 27, 2008, from http://www.moe.gov.sg/media/speeches/2005/sp20050530.htm van den heuvel-panhuizen, m. (1999). context problems and assessment: ideas from the netherlands. in i. thompson (ed.), issues in teaching numeracy in primary schools (pp. 130-142). buckingham, uk: open university press. verschaffel, l., decorte, e., & borghart, i. (1997). pre-service teachers' conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems. learning and instruction, 7(4), 339-359. zevenbergen, r., & zevenbergen, k. (2009). the numeracies of boatbuilding: new numeracies shaped by workplace technologies. international journal of science and mathematics education, 7(1), 183-206. 26 ng kit ee dawn appendix a painting task suppose you are a painter and have been asked to give a quote for painting an office room. it is requested that only ceiling and wall should be painted in silken blue while the floor in the room need to be waxed. a floor plan of the room is shown below: � � � � � � � �� � window� 2. 1� m � 3. 6� m � 0.5�m� 0.4�m 5�m 2.1�m� 2.5�m door 0.8�m room�height��������������2.4�m door�height����������������2.1�m� window�height����������0.9�m� 0.1�m additional information: 1 litre of paint covers 16 m2; 1 litre of paint costs $19.90; 1 litre of wax covers 15 m2 for first coat; 1 litre of wax covers 30 m2 for subsequent coats; a litre of wax costs $34.80. please help to provide the budget for both paint and wax: wax (3 coats are needed): paint (2 coats are needed): [adapted from foo, k. f. (2007). integrating performance tasks in the secondary mathematics classroom: an empirical study. unpublished masters dissertation, nanyang technological university, singapore.] 27 towards mathematical literacy in the 21st century: perspectives from singapore flask t some st li wei flask is seconds here ar solution 3 x 3.5 after 30 solution 3 x 3.5 after 30 solution 3.5 cm after 30 solution i can’t g your ta think th [task ad beliefs ab and instr task tudents are is having a being filled s, how deep http://upload. re some of th n 1: cm = 11.5 c 0 seconds, t n 2: cm = 10.5 c 0 seconds, t n 3: + 20 cm = 2 0 seconds, t n 4: get a precise ask is to dec he other solu dapted from v bout the role ruction, 7(4), 3 given the fo chemistry d at a consta p will the wa .wikimedia.or heir solutio cm the depth of cm the depth of 23.5 cm the depth of e answer! cide which s utions are in verschaffel, l of real-world 339-359.] a ollowing tas class. he h ant rate. if th ater be in th [p rg/wikipedia/c erlenm ns: f the water i f the water i f water in th solution is c ncorrect. ple ., decorte, e. d knowledge i appendix b sk to do: has to fill a c he depth of he flask afte icture taken fr commons/thum meyer_flask_ in the flask in the flask he flask will correct. exp ease write y ., & borghart n mathematic b conical flas f the water i r 30 second from mb/0/00/erlen _hg.jpg] will be 11.5 will be 10.5 l be 23.5 cm plain your ch your explan t, i. (1997). p cal modelling sk with wate s 3.5 centim ds? nmeyer_flask_ 5 cm. 5 cm. m. hoice. also ations on th re-service tea of school wo er from a tap metres after _hg.jpg/450px explain wh he next page achers' concep ord problems. p. the 10 xhy you e. ptions and learning 28 ng kit ee dawn appendix c youth olympic games problem singapore will be hosting the first youth olympic games (yog) from 14 to 26 august 2010. it will receive some 3,600 athletes and 800 officials from 205 national olympic committees, along with estimated 800 media representatives, 20,000 local and international volunteers, and more than 500,000 spectators. young athletes between 14 and 18 years of age will compete in 26 sports and take part in culture and education programme. the singapore 2010 youth olympic games will create a lasting sports, culture and education legacy for singapore and youths from around the world, as well as enhance and elevate the sporting culture locally and regionally. there are altogether 201 events featuring 26 different kinds of sports such as swimming, badminton, cycling, fencing, table tennis, volleyball and weightlifting. singapore is well-known in the region for grooming young swimmers. the singapore sports school is having difficulty selecting the most suited swimmers for competing in the women’s 100m freestyle event. they have collected data on the top five young female swimmers over the last 10 competitions. to be fair, codes are used to represent the swimmers until the selection process is over. the list of swimmers and their codes are only known to you and your group. as part of the singapore sports school yog committee, you need to use these data to develop a method to select the two most suited women for this event. study the data presented below collected over two years (2007-2008). records for competition 1 are the most recent. records for competition 10 are the oldest. women’s 100m freestyle results recorded (seconds)* competition no. time in 100 m freestyle (minutes and seconds) swimmer a swimmer b swimmer c swimmer d swimmer e 1 00:56 00:49 01:02 00:57 00:55 2 00:55 dnc 00:59 01:05 dnc 3 dnc 00:56 dnc 00:59 00:55 4 00:58 01:01 00:57 dnc 01:04 5 dnc 00:57 dnc 00:58 00:49 6 00:59 dnc dnc 00:56 00:57 7 01:00 dnc 00:59 00:56 00:57 8 00:59 00:56 00:55 00:58 00:57 9 00:59 00:56 dnc dnc 00:58 10 00:59 00:57 00:57 dnc 00:58 dnc: did not compete. *best time across heats, semi-finals and finals. (1) decide on which two female swimmers should be selected. (2) write a report to the yog organizing committee in singapore to recommend your choices. you need to explain the method you used to select your swimmers. the selectors will then be able to use your method to select the most suitable swimmers for all other swimming events. � adapted from english, l. (2007). modeling with complex data in the primary school [electronic version]. thirteenth international conference on the teaching of mathematical modelling and applications. retrieved july 22-26, from http://site.educ.indiana.edu/papers/tabid/5320/default.aspx. 29 southeast asian mathematics education journal volume 11, no 2 (2021) 95 comparison on the effectiveness of modular learning in general mathematics among the senior high school strands imee borinaga-gutierrez valencia national high school ormoc city, leyte, philippines imee.borinaga029@deped.gov.ph abstract this study aimed to compare the effectiveness of modular learning among senior high school strands and investigate the influence of students’ demographic profiles on their academic performances. using one way anova, results showed a significant difference in the test scores obtained from the different strands. results with tukey hsd revealed that abm performed best under modular learning, followed by stem, gas, ats, and eim, respectively. pearson’s r revealed a significant correlation between student’s sex, monthly income, and parents’ employment status to students’ academic performance. furthermore, the z test for two means determined that there was a significant difference in the scores of males and females. from these results, the researcher suggests that teachers handling tvl tracks must exert extra effort in delivering their mathematics lessons to close the gap in academic performance with the students from academic tracks. keywords: modular, general mathematics, strands, senior high school, academic performance introduction the philippines education system has implemented the use of modules for instruction along with other modalities such as online learning, use of radio, television, and blended learning. 39.6% or 8.8 million students who answered deped’s national learner enrolment and survey forms (lesfs) prefer modular learning over other modalities (manlangit, 2020). modular distance learning is a form of instruction which makes use of self-learning materials such as modules, textbooks, hand-outs, or learning activity sheets presented in digital or printed form. in this form of instruction, teachers deliver lesson content without facing the students personally. parents, guardians, or para teachers act as more knowledgeable ones (mkos) helping the student learn in the comfort of their home. this means that mkos will have to guide the flow of instruction while teachers teac the lesson content. learning materials will be received by students through email, messenger, cd/dvd, usb storage, designated stations set by each school, or delivery by teachers. in modular learning, teachers monitor students’ progress through email, messenger, text message, call, or home visitation. generally, it implies self-directed learning. while this means difficulty in teaching various subjects, there is no more significant challenge than teaching mathematics. the republic of the philippines ranks lower than the other countries in terms of mathematics. the philippines joined the programme for international student assessment (pisa) of the organization for economic co-operation and development (oecd) in the year 2018. this is a step undertaken to increase the quality of the philippines basic education. as discussed in the pisa 2018 philippine national report, filipino students achieved an average of 353 points, which is lower than the 489 points oecd average in mathematical literacy and is classified as below level 1 proficiency (deped, 2019). https://twitter.com/bnzmagsambol/status/1288718985232388096 https://twitter.com/bnzmagsambol/status/1288718985232388096 comparison on the effectiveness of modular learning in general mathematics among the senior high school strands 96 delivery of lesson content in mathematics poses a challenge, not just to primary or junior high school teachers, but also to the senior high school teachers. the philippine senior high school offers four (4) tracks for students to choose from academic, technical vocational livelihood (tvl), arts & design, and sports. there are four (4) strands under the academic track, which include science, technology, engineering & mathematics (stem), accountancy & business management (abm), general academic strand (gas), and humanities & social sciences (humss). under the tvl track, there are four (4) strands including, information and communication technology, home economics, industrial arts and agri-fishery arts (senior high school in the philippines: curriculum breakdown). valencia national high school offers automotive servicing (ats) and electrical, installation, and maintenance (eim) for this track. general mathematics is one of the core subjects in senior high school, which means that every strand shall take this subject. the first quarter involves topics on the key concepts of functions, key concepts of rational functions, and key concepts of an inverse function. by this, the modular learning efficacy among the shs strands is measured through their academic performance in the subject. the operational definition of academic performance used in this study is the students’ test scores for the summative test covering the first quarter topics. modular teaching in general mathematics poses a question on its effectiveness as implemented in the different strands of senior high school. with this, the researchers find the need to determine if there is a significant difference in the students’ test scores in the different strands with the use of modular instruction. the influence of students’ demographic profiles on their academic performance is another concern that needs to be addressed. methods the study was conducted at valencia national high school, located at brgy. valencia, ormoc city, leyte, the philippines. the respondents for the study were the 213 grade 11 students at valencia national high school comprising the five strands offered in the school, namely science, technology, engineering & mathematics (stem), accountancy & business management (abm), general academic strand (gas), automotive servicing (ats), and electrical, installation, and maintenance (eim). in calculating the sample size, slovin’s formula was used with a confidence level of 95%. after determining the number of needed respondents for the study, stratified random sampling was used to highlight the differences in the academic performance between the strands, which will be a relevant result in comparing the effectiveness of modular learning. the number of respondents in each stratum was determined and was drawn randomly for the researchers to identify the respondents belonging to each stratum. this enables the researcher to be free from bias and obtain a sample population from the entire population, ensuring that each subgroup is well represented. a structured survey was carried out for this study. it comprised of two parts: personal information and questions about the respondents’ demographic profile. following was the 30-item multiple-choice test about the first quarter topics in general mathematics. data gathered was presented using a table and percentage to highlight the demographic profile of students in the different strands. one way analysis of variance (anova) was used to determine a significant difference between students' test scores in the other various strands. tukey hsd followed this to decide which strands have differed significantly. pearson’s correlation was then used to determine a imee borinaga-gutierrez 97 correlation between students’ academic performance and their demographic profile. students drawn randomly from each stratum were classified as male or female and determine if this has a significant difference with their academic performance. the z test for two means was used to determine if there is a significant difference between the scores of males and females under modular learning. pearson product moment correlation was used to test the validity of the test questions used. cronbach’s alpha was used in evaluating the consistency of the instrument. kolmogorov-smirnov test was used to determine the normality of the data. levene’s test was used to test the equality of variances. from the data gathered, the researchers generated necessary conclusions and recommendations for the study. results and discussion table 1 shows that there is a greater number of females than the number of male respondents. table 1 number of male and female respondents table 2 monthly income of parents note. adapted from the philippine institute for development studies the majority of the respondents, 82.2%, came from a family classified as poor, which has a monthly income of less than php 10,481. the data was from the philippine institute for development studies. 30 respondents were classified as a low-income class of those with a monthly income of between php 10,481 and php 20,962. 6 of the 213 respondents were lowermiddle-income, and two were classified under middle-income class. sex frequency valid percent female 107 50.2 male 106 49.8 total 213 100.0 classifications frequency valid percent poor (below php 10,481) 175 82.2 low income class (php 10,481 php 20,962) 30 14.1 lower middle-income class (php 20,962 php 41,924) 6 2.8 middle income class (php 41,924 php 73,367) 2 .9 upper middle-income class (php 73,367 php 125,772) 0 0 upper income class (php 125,772 php 209,620) 0 0 rich (php 209,620 and above) 0 0 total 213 100.0 comparison on the effectiveness of modular learning in general mathematics among the senior high school strands 98 table 3 number of persons in the household table 3 shows that most of the respondents, which is 25.8%, came from a family comprising of 5 members in the household. nine respondents came from a family of greater than or equal to 10. these are 4, 3, 1, and 1, respectively. table 4 employment status of parents the respondents were also asked about their parent's employment status, and the result showed that 115 of the respondents answered that their parents are not working or comprising more than half, which is 54% of the total respondents. this is because the school is situated in a rural area, so most of the respondents’ parents depend on agriculture for a living. there were only 32 respondents with parent/s working full time while others are working part-time (16.4%), self-employed (9.4%), or unemployed because of the pandemic (5.2%). household number frequency valid percent 2 4 1.9 3 15 7.0 4 41 19.2 5 55 25.8 6 36 16.9 7 28 13.1 8 15 7.0 9 10 4.7 10 4 1.9 11 3 1.4 12 1 .5 14 1 .5 total 213 100.0 employment status frequency valid percent full time 32 15.0 part time 35 16.4 self employed 20 9.4 unemployed due to covid 11 5.2 not working 115 54.0 total 213 100.0 imee borinaga-gutierrez 99 table 5 descriptive statistics of the five strands in valencia national high school students under the academic tracks performed better in the summative test than those in the tvl tracks. the result showed that there is greater score gains from students in the academic track than students from the other tracks (guill, lüdtke, & köller, 2017).the highest mean, 18.17 with a standard deviation of 6.436, came from the abm strand, which means that among the five strands, abm performed well in the general mathematics summative test given by the researcher, followed by stem with a mean of 15.97 and ga with a mean of 14.34. these three strands landed on the top three because these are academic strands, while ats and eim, both tvl strands, landed on the bottom two. students taking up tvl strands are more focused on hands-on activities, especially on their major subjects. most of the students enrolled in this strand are working during the pandemic, making it difficult to juggle academics and work. they have less time to study than those belonging to the academic strands. this explains why the two strands landed low while the academic strands landed on the top three spots. ats has a mean of 10.67 and a standard deviation of 5.360. eim with a mean of 10.15 and a standard deviation of 3.913 performed the least among the five strands. with this result, teachers in general mathematics, or even in other subjects, may exert extra effort in teaching the lessons to tvl students, especially under modular learning. additional activities that will enable them to focus on learning the modules and engaging activities are highly encouraged. table 6 one way analysis of variance on students’ academic achievement in general mathematics sum of squares df mean square f sig. between groups 1682.044 4 420.511 17.905* .000000000001194 08 within groups 4885.036 208 23.486 total 6567.080 212 *p<0.001 table 6 shows the one-way analysis of variance (anova) on the academic performance of the respondents in the different strata in general mathematics. the result shows a significant difference in the academic performance of students under modular learning f(4,208)= 17.905, p=0.000. this is supported by a study that also revealed a significant difference in the strands n mean std. deviation minimum score obtained maximum score obtained abm 23 18.17 6.436 6 30 stem 30 15.97 3.489 6 22 ats 52 10.67 5.360 1 24 ga 56 14.34 4.999 5 25 eim 52 10.15 3.913 4 20 total 213 13.07 5.566 1 30 comparison on the effectiveness of modular learning in general mathematics among the senior high school strands 100 proficiency level across shs strands (cerbito, 2020). a post hoc test using the tukey hsd test was conducted to determine which strands show the significant result. table 7 tukey hsd test on the different strands (i) strand (j) strand mean difference (i-j) sig. abm stem 2.207 .472 ats 7.501* .0000000331 ga 3.835* .014 eim 8.020* .000 stem abm -2.207 .472 ats 5.294* .0000347807 ga 1.627 .574 eim 5.813* .000 ats abm -7.501* .000 stem -5.294* .000 ga -3.666* .001 eim .519 .982 ga abm -3.835* .014 stem -1.627 .574 ats 3.666* .001 eim 4.185* .000 eim abm -8.020* .000 stem -5.813* .000 ats -.519 .982 ga -4.185* .000 results in table 7 revealed that abm showed a significant difference to ats, ga, and eim. stem significantly differs from ats and eim. ats showed a significant difference with ga, and ga showed a significant difference with eim. mamolo (2019) revealed that the strands under the academic track are more focused on the academe, while strands under the tvl track focus more on skills development. meanwhile, abm showed no significant difference with stem, stem showed no significant difference with ga, and ats showed no significant difference with eim. this means that respondents under the abm strand scored significantly higher among the five strands than the others. specifically, abm performed well in the test compared to the different strands. since abm students are also exposed to mathematical practices concerning their strand, students have developed mathematical skills making them more knowledgeable on the topics. abm strand offers other branches of mathematics, including those major subjects intended just for their strand, e.g., fundamentals of abm. these major subjects helped them develop their mathematical skills making them performed best among the other strands. most students enrolled in this strand are also girls. female senior high *.the mean difference is significant at the 0.05 level. imee borinaga-gutierrez 101 school students tend to have more favorable study habits than males (casinillo, batidor, may, & casinillo, 2020). this is supported by almerino et al. (2020) who found that abm outperformed stem in terms of mathematical application despite their exposure to problem solving and other mathematical applications. table 8 pearson correlation result on demographic profile of grade 11 students and score in general mathematics table 8 presents the correlation result of the demographic profile of the respondents and the scores in general mathematics. the result showed a correlation between sex, income, and employment to the score of the respondents. there is a significant, moderately, and negative correlation between sex and score (r= -0.330, p<0.001). further, there is a significant, low, and negative correlation between the employment status of parents and score (r= -0.147, p=0.032). this indicates that as parents are working on a full-time basis, time spent on the student will lessen, thus decreasing academic performance. there is little time for parent participation since most of the time will be spent on work. there is a difficulty in monitoring the status of their child’s submission of modules due to heavy workload. working parents lack the time to guide their child than stay-at-home parents, especially mothers as housewives (carbonel, banggawan, & agbisit, 2013). dumont et al. (2012) supports this by claiming that students’ academic performance is affected by the level of parental involvement. however, there is a significant, low, and positive correlation between income and academic performance (r= 0.142, p=0.039). parents’ low income means difficulty providing a conducive environment for studying, gadgets used such as cellphone or computers, and other supplementary materials. higher parents’ income also attributes to parents who are more capable of teaching their children. students whose parents earn higher income graduated from college and landed a stable job, thus have higher salaries. with this, they totally have the capacity to tutor their child or hire one increasing their academic performance. farooq et.al. (2011) strengthens this result that socioeconomic status affects the overall performance of a student in mathematics. the result of this study will imply the need for parents to provide adequate support in their child's schooling. the government must also help alleviate the gap between the high and the low socialeconomic status by providing financial assistance to the less privileged, opening a job fair, or offering scholarship grants to deserving students. meanwhile, there is no significant correlation sex income household employment score sex pearson correlation 1 .422** .530** .580** -.330** sig. (2-tailed) .000 .000 .000 .000 income pearson correlation .422** 1 .751** .352** .142* sig. (2-tailed) .000 .000 .000 .039 household pearson correlation .530** .751** 1 .752** -.001 sig. (2-tailed) .000 .000 .000 .991 employment pearson correlation .580** .352** .752** 1 -.147* sig. (2-tailed) .000 .000 .000 .032 n 213 213 213 213 213 **. correlation is significant at the 0.01 level (2-tailed). *p<0.001 *. correlation is significant at the 0.05 level (2-tailed). comparison on the effectiveness of modular learning in general mathematics among the senior high school strands 102 between the numbers of persons in the household and the score in general mathematics (r= 0.001, p=0.991). it can also be seen from the data presented that sex and income has a significant, moderate, and positive correlation (r= 0.422, p<0.001). this means that higher income correlates to a particular sex. table 9 z-test for the academic performance of grade 11 students based on sex female male mean 14.88785047 11.22641509 known variance 29.81749251 25.64348607 observations 107 106 z 5.074629801* z critical two-tail 1.959963985 *.the mean difference is significant at the 0.05 level. table 9 reveals the result or statistical difference between males and females. the result shows a mean of approximately 14.89 for females with a variance of 29. 82. meanwhile, a mean of 11.23 was computed for males and a variance of 25. 64. this means that the female significantly scored higher than the male. the value of z is 5.07 at a 5% significance level and is greater than z critical, which is approximately 1.96. with this, there is a significant difference in male and female academic performance in modular learning. z test for two means showed that females performed better than males. females are more conscious about their school performance, and they tend to compete with their other classmates. female students are more emotional than males, which means that they are easily affected by increased or decreased scores. al-mutairi (2011) supports this claim that female achievement in academics is higher than male. female senior high school students tend to have more favorable study habits than males (casinillo, batidor, may, & casinillo, 2020). it was also found out by ganley & lubienski (2016) that girls value mathematics achievement more than boys. casinillo et al. (2020) further stated that females tend to be more anxious than males in studying mathematics. contrary to this, goetz et al. (2013) demonstrated that girls performance in mathematics is not affected negatively despite their higher level of anxiety. with this result, the gender gap will lessen. thus, mathematics intervention should start at an early age to eliminate anxiety or other factors intertwined with sex affecting academic performance in mathematics. test of validity was done using pearson product moment correlations using spss. item is valid if 𝑟𝑣𝑎𝑙𝑢𝑒 (𝑖𝑡𝑒𝑚) > 𝑟𝑡𝑎𝑏𝑙𝑒. result revealed that at alpha 0.05, there is only one item invalid, that is, item number 1. there are 29 items valid, and these items are items 2-30. in this study, cronbach’s alpha was used to evaluate the internal consistency of the instrument, which is the most common validation as presented in the literature (paiva, et al., 2014). the result showed the reliability of the 30-item questionnaire administered to the respondents is 0.792. based on descriptors provided by taber (2016), reliability of 0.792 is classified as fairly high. this means that the questionnaire will yield a high probability of a consistent result. the researcher used the kolmogorov-smirnov test to determine whether the distribution of data is normal or not. for the normality test, if the significance level is greater than 0.05, the distribution is normal. based on the test conducted to the data, the significant value of the test scores is 0.102 and for imee borinaga-gutierrez 103 the strands is 0.188. thus, both data were normally distributed. a test of equality of variances was conducted using levene’s test to determine the homogeneity of variances. if the p-value is above 0.05, equality of variance is assumed—otherwise, violation of the assumption. the result verified the equality of variances with a p-value of 0.631, which is greater than the alpha 0.05, concluding equality among the group variances. conclusion this study showed that the accounting, business and management strand (abm) performed the best under the modular learning approach among the senior high school strands of valencia national high school (abm). this was because most students in this strand are girls, and based on the result above, sex and score correlate with one another. evident in the results presented, among the five strands, those belonging to the academic track, namely abm, stem, and ga, performed better than eim and ats strands, which fall under the tvl track. this result shows the need to promote the tvl track in terms of their academy while focusing on developing their skills. teachers designated to teach in this track must exert extra effort in using their teaching methods and strategies, especially in this time of current shift in the educational system. results also showed that females performed better in modular learning than males. this result eliminates the gender gap between the male and the female, erasing the notion that males perform better in mathematics than females. on the other hand, the majority of the respondents’ parents are not working. thus, there is less or no income at all, decreasing students’ scores in the summative test. parents are not capable of teaching nor have the financial capability to provide for their child’s needs. with these, modular learning is effective for students with a strong foundation in mathematics and the essential learning tools that aid the understanding of lessons. the researcher would like to extend the study by focusing on male and female academic performance in each strand. the correlation test among the demographic profile of the respondents and their academic performance in each strand is also a favorable study to conduct. further, the significant and positive correlation shown between sex and income is also an excellent study to perform in the future. acknowledgement i would like to extend my gratitude to the student respondents as well as to their parents for their effort in the participation of this study. this would also not be possible without the approval of our school head and head teacher to conduct the study at valencia national high school. i would also like to extend my gratitude to my family for the constant care and support you have given me (financially, emotionally, physically, or mentally) throughout this research journey. this would not be possible without your words of encouragement. comparison on the effectiveness of modular learning in general mathematics among the senior high school strands 104 references al-mutairi, a. 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(2020). supercharging filipino parents is key for successful modular distance learning. flipscience. paiva, p. c. p., de paiva, h. n., oliveira filho, p. m. d., lamounier, j. a., ferreira, e. f. e., ferreira, r. c., kawachi, i., & zarzar, p. m. (2014). development and validation of a social capital questionnaire for adolescent students (scq-as). plos one, 9(8), e103785. https://doi.org/10.1371/journal.pone.0103785 taber, k s. (2016). the use of cronbach’s alpha when developing and reporting research instruments in science education. research in science education, 48(6), 1273-1296. https://doi.org/10.1007/s11165-016-9602-2 https://doi.org/10.13189/ujer.2019.070913 https://doi.org/10.1371/journal.pone.0103785 https://doi.org/10.1007/s11165-016-9602-2 comparison on the effectiveness of modular learning in general mathematics among the senior high school strands 106 southeast asian mathematics education journal volume 11, no 2 (2021) 107 identifying ethnomathematics in the old mosque of tosora negara mangkubumi k. smp negeri 2 sengkang, wajo, south sulawesi, indonesia negarak55@guru.smp.belajar.id abstract the superficial understanding and application of cultural values in the life of a pluralistic community is the main reason to conduct this research as a way to introduce the importance of national identity to students through mathematics learning. the provision of real-life problems in this study is expected to enhance the relationship between reality and mathematical knowledge. the old mosque of tosora in wajo can be an alternative to introduce mathematical activities based on local wisdom (ethnomathematics). this is qualitative descriptive research using a case study. data were obtained through interviews, observation, and documentation. data triangulation was used to check the validity. furthermore, the data were analysed through spradley model based on domain analysis and taxonomic analysis. the results showed that the old mosque of tosora contains some mathematical concepts, mainly geometry. in addition to the philosophy of the building, the four pillars namely panrita (wise and honest), warani (brave), macca (intelligent), and sugi (rich) become the focal principles of buginese’s life. keywords: ethnomathematics, tosora mosque, architecture, geometry. introduction global change is not always about a massive change of mankind, but more about the changing aspect of human individuality as an asset of human wealth (tilaar, 1997). furthermore, it is obvious that instead of destroying nation’s culture, globalization introduces a new diverse global culture that contributes to the birth of a more widespread international cultural mosaic. in the context of education, tilaar (1997) further explained that globalization is underpinned by the spirit of unlimited information. this results in a cross-cultural process that brings together cultural values, thus making it necessary to reproduce knowledge for future development. indonesia is renowned for its abounding natural resources and cultural diversity that spread across the archipelago and have become a powerful attraction to the foreign tourists. however, this wealth is mainly disrupted by the rapid digitalization and technological development. these developments in all walks of life have gradually eroded the national cultures, which may lead to self-identity and moral crises of indonesians. such disconcerting fact is attributed to the shallow understanding and implementation of cultural values in a plural community (wahyuni et al., 2013). education and culture are inseparable from daily life. however, cultural values are less implemented in the formal and non-formal education sectors in any levels. amir & marzuki (2021) held the view that the cultural inheritance is an important part of a community. it has a fundamental value for the moral development of humans as members of society. furthermore, it was conveyed that cultural inheritance can take place through a learning process, mainly identifying ethnomathematics in the old mosque of tosora 108 through education that transforms cultural values. the inculcation of sublime characters for students at an early level is necessary, one of which is through mathematics learning. an individual’s competency to think and to act depends upon his/her competency to interact and to communicate with his/her cultural context (forbes, 2018). good communication skills as part of the interaction in a student's learning environment will make it easier to convey ideas. in mathematics learning, this ability serves as an important element in higher order thinking skills. to get used to cognitive level of thinking highly correlates with the understanding of mathematical objects. four mathematical objects, namely facts, concepts, principles, and skills are embedded in the human mind (zaenuri & dwidayanti, 2018). mathematical concepts in daily life are seen in human creations, involving many activities like finding, drawing, measuring, and designing. these activities are based on interest and intention, which reflects cultural traditions and values through buildings. the results of preference and intention are developed into a cultural civilization in the form of buildings, such as temples, mosques, churches, houses, and others as human cultural constructions resulting from mathematical activities. mathematical explanation is included in the cultural element through the application of ethnomathematics. it acts as pedagogical action in learning mathematics, returns sense of involvement, and increases creativity in doing mathematics (d’ambrosio, 2007). according to rosa & orey (2016), ethnomathematics is growing as an appropriate alternative to traditional learning in terms of history, philosophy, cognitive, and pedagogics. it becomes an entry point that focuses on legitimizing students' knowledge that comes from experiences built in their own way. in line with this matter, risdiyanti & charitas (2020) articulated that ethnomathematics helps teachers and students to understand mathematics in the context of ideas, ways, and practices used in everyday life which will ultimately encourage the understanding of academic mathematics at school. patrianto & rahiem (2019) held another notion that ethnomathematics has an influence in the formal school through mathematics learning and provides the necessary contextual meaning to a lot of abstract mathematical concepts. this learning process starts from some basic learning activities such as adding, subtracting, counting, and measuring, which can attract students' learning attention. activities that involve locating and designing can stimulate their critical and creative abilities. several types of games in learning activities that are practiced by children, both in the form of spoken language, written symbols, pictures, and physical objects will certainly provide meaningful learning experience. game enables students to learn a new experience to help them comprehend the materials. freudenthal (2012) claimed that mathematics, to be of human value, must relate to reality, remain close to children, and must be relevant to society. the potency of cultural-based local wisdom becomes an alternative that can be integrated in mathematics learning. this is in opposition to mathematics learning that is considered to be less varied, more theoretical, less contextual, and more abstract, which affects students’ psychological aspect. the stiffness, formality, and numerical oriented characteristics make mathematics far from the real-world context. what students get during the learning process shows less of contextual connection to social, cultural, and historical issues. therefore, formal education should take the combination of mathematics learning and culture into account to bridge mathematics and the local wisdom negara mangkubumi k. 109 and culture that is the basis of daily life. suastra et al. (2017) stated that local wisdom is defined as the truth that has become a tradition. in regards with cultural context, tosora is one of the historical areas in majauleng, wajo, south sulawesi. tosora is derived from bugis word to sore, which is roughly translated as people who have just arrived by air travel or have just landed (proyek pemugaran dan pemeliharaan peninggalan sejarah dan purbakala sulawesi selatan, 1984). wajo is well known as the old city, and thus holds various traces of the wajo kingdom, a prolonged legacy of cinnotabi kingdom. latenri bali was noted in the history as the first king, who was granted the title as batara wajo i. it was him who changed the name of the city of cinnotabi to wajo (badja, 2012). upon the arrival of islam in wajo, tosora started to become the centre of islamic propagation and the capital of wajo kingdom. this fact is marked by many building relics in the old mosque, mushalla (an open space for praying), gaddonge (flower buildings), and ancient tombs as the evidence of islamic development (duli, 2012). wajo people highly adopt bugis culture. a widely held notion believed that the behaviour of leaders, systems, and ethos were the determinants of one's life. this principle was clearly marked by the presence of old mosque of tosora in a rectangular shape. the shape adheres to the philosophy of the origin of human life: earth, fire, water, and wind. the material used for the old mosque of tosora consisted of limestone and rocks mixed with shell fossils, sand, and brick (proyek pemugaran dan pemeliharaan peninggalan sejarah dan purbakala sulawesi selatan, 1984). the adhesive materials were composed of sand and brown limestone, which symbolized the colour of sulapa appa (the four principles of buginese life). for buginese, black, red, white, and yellow represent land, fire, water, and wind respectively. to actualize cultural-mathematical integrated learning, it is important to explore the construction of the historical heritage sites or local cultures as a way to relate it with mathematical concept. some studies have tried to relate the exploration of local cultural heritage. huda (2018) contributed to the discovery of geometry elements, such as square planes and solid figures. another study, utami (2018) disclosed the mathematical activity in the lampung traditional house, which contains the mathematical concepts of one-dimensional geometry, two-dimensional geometry, three-dimensional geometry, geometrical transformation, odd numbers, even numbers, and rational numbers. a similar study by putri (2017) revealed the existence of mathematics elements in the traditional tambourine art, such as geometrical concepts and numeration techniques that produce harmonious tone patterns. it is not only cultural context that is needed in learning mathematics, but also character building. as stated in the 2013 indonesian curriculum, character and etiquette are required during the learning. thus, it is important for this study to identify religious heritage sites of tosora old mosque in wajo, south sulawesi through mathematical concepts since it is related to these curriculum requirements. in junior high school, quadrilaterals (grade 7), two dimensional shapes (grade 8), and three-dimensional shapes (grade 9) are relevant to students' daily lives. the application of mathematical concepts such as area, volume, line, angle, and various other concepts is found in community activities. this fact indicates that the cultural elements have mathematical values to be taught to students contextually. tracing certain parts of the historical buildings can serve as a concrete example of learning mathematics for the identifying ethnomathematics in the old mosque of tosora 110 application of mathematical concepts that produce an aesthetic, so that learning in schools becomes more realistic. based on the previous explanation, the researcher was interested in conducting a study involving the identification of ethnomathematics within the heritage site of the old mosque of tosora. the core of the problem discussed in this study is based on a research question: “how can we use ethnomathematics to reveal the cultural values of the heritage site of the old mosque of tosora?” in terms of theoretical benefit, this study aims to compile information regarding the connection between mathematics and the heritage site of the old mosque of tosora. in addition, this study is also expected 1) to improve interest and motivation to learn mathematics outside the classroom; 2) to introduce an innovation for mathematics learning and cultural to the students; and 3) to construct a reference for developing mathematics learning on the realistic potential of mathematics in cultural contexts. methods this research used descriptive research with a case study method. the case study is used to provide a detailed description of the background, characteristics, activities, specificity or uniqueness that characterizes the case or the status of a particular individual, group or community, as a way to generalize the findings into general matters (agasi & wahyuono, 2016). the two steps in this case study were choosing a cultural object and conducting interviews. the scope of this research is the heritage site of the old mosque of tosora. the researcher interviewed local people who have knowledge of the history of the old mosque and collected data from these process. the data were collected from a literature study and observation, documentation, and interview. the description and in-depth analysis about the cultural heritage of the old mosque of tosora was taken from a fieldwork. the researcher served as the only one to collect data and information from the informant based on an interview guidelines and observation checklist. the data were then validated by way of triangulation, mainly by utilizing any other forms of data to compare the existing data (moleong, 2014). the researcher validated the data by comparing interview results from the informants to the previous research findings, , the observations, and documentation. collected data were analysed through the spradley model based on domain and taxonomic analysis (sugiyono, 2013). domain analysis was carried out to get general and overall description of the research object based on the category or domain. in this research, the data related to the concepts of mathematics were categorized based on their ethnomathematics categories or domains that have been formulated, for instance in the form of numbers, algebra, geometry, statistics, and so on. then, of the data were analysed taxonomically to explain the chosen domains in more detail based on the existing mathematical concepts in the old mosque of tosora in wajo. the data were presented in a matrix (table). negara mangkubumi k. 111 results and discussion the results of data triangulation based on the interviews with research informants related to the old mosque of tosora are presented in table 1. table 1 indicates that based on a religious perspective, tosora old mosque serves as a centre for propagating islam. in terms of the building design, langkara building or mushalla supports the main mosque, which indicates that the area used to be densely populated. mushalla served as a place for prayer. in terms of the structural aspect of the building, it is apparent that the building design has applied mathematical concepts, including elements of geometry. table 1 result of study no indicator results of interview results of observations result of documentation 1 the establishment of tosora old mosque it was built around 16th century, when the wajo kingdom was newly established and the wajo citizens started to embrace islam. the building materials were composed of egg white and limestone. the history of the old mosque of tosora is explained according to documentation sources, including traces of relics that are estimated to be hundreds of years old. the graves of great scholars at that time included the tombs of several royal servants. 2 community activities tosora was the capital of the wajo kingdom as well as the main centre of economic activities. democracy has been applied since the presence of the kingdom and the old mosque of tosora. the mosque is not only used for worship but also for economic activity. such function was indicated by the traces of the langkara. at that time, it ssupported the main mosque. this fact illustrates that the area was densely populated. the existence of langkara 3 geometrical aspects the building embodied geometrical shapes, one of which was the mihrab, a place for the imam to lead prayers. parts of the mosque building comprised mathematical concepts. the mihrab shape is slightly curved. there were square windows on its left and right. in addition, there were other parts that concerned with geometric concepts. badja (2012) revealed that the basic plan of the mosque was a square shape. the inside of the mosque was equipped with a curved mihrab. mihrab is a semi-circular niche that juts into the front wall of the mosque and shows the direction of qibla, where the imam leads the prayer. identifying ethnomathematics in the old mosque of tosora 112 figure 1. the tosora old mosque (surur et al., 2017). in addition, the relic of four pillars of the mosque shows that the the building design not only considers function and aesthetics, but also uses mathematical concepts in its design. in line with surur et al. (2017) findings, the architectural form is described as a square as illustrated in figure 1. it is related to the philosophy of the bugis nobleman, sulapa appa, which means the four principles of life. these four elements serve as life-guidelines to bring improvements in the field of leadership, social, and self-excellence (yudono et al., 2018). the roof of the old mosque that was designed in a triangular shape resembled the roof of an old mosque in south sulawesi. it was also supported by four main pillars as the local characteristics. the four pillars illustrate the qualities that should be possessed by buginese. those are panrita (wise and honest), warani (brave), macca (intelligent), and sugi (rich). it was further explained that the qualities must be possessed by buginese to manifest their living conditions as panrita (wise and honest), warani (bold), macca (intelligent) and sugi (rich) (surur et al., 2017). the ethnomathematics categories through observation, documentation, and interviewrevealed that the mosque has mathematics elements. the building, the niche, windows, walls, and pillars are related to geometrical concepts, especially two-dimensional shapes. these elements can be used to connect the mathematics learning at school to the cultural value. below is the detailed analysis of the building related to the mathematical aspects. negara mangkubumi k. 113 the building foundation figure 2. the building foundation of tosora old mosque. figure 2 shows the foundation of the old mosque which is in the square shape. the mathematical activities and aspects are as follows. 1. observing the object, which resembles a square, and studying its elements such as angles, sides, and diagonals. 2. identifying the following characteristics of a square on the object. • all sides are equal in length. • its four angles are right angle. • opposite sides are parallel. • each diagonal bisects one to another at right angles. the point is the diagonals intersect and divides each diagonal into two equal parts. • the two diagonals are in the same length. • it has four axes of symmetry. 3. evaluating the area of a square using the formula: 𝑠𝑖𝑑𝑒 × 𝑠𝑖𝑑𝑒 = 𝑠2. 4. evaluating the perimeter of a square and finding its formula that is four times its side. the niche figure 3. the niche. identifying ethnomathematics in the old mosque of tosora 114 the niche, as displayed in figure 3, becomes the object of observation. from twodimensional perspective, it is a compound shape formed by a rectangle and a half-circle. while according to three-dimensional point of view, it is composed of a cuboid and a cylinder, which was cut such that half of the curve surface appears. the mathematical activities and aspects are as follows. 1. two-dimensional concept to determine the total perimeter of compound of two-dimensional shapes, we need to add the three sides of the rectangle and the circular arc (half of a circle). therefore, we get the total perimeter equals to 𝑤 + (2 × 𝑙) + πr. 2. three-dimensional concept related to this concept, the activity encourages to mention the elements of cuboid; i. e. face, edge, vertex, space diagonal, face diagonal, and diagonal plane and the elements of cylinder, such as the curved surface, height, base, radius, and diameter. to add, we can analyse the compound volume that is (ℎ × 𝑤 × 𝑙) + ½𝑟2ℎ and the compound surface area that is (𝑙 × 𝑤) + 2(𝑤 × ℎ) + 2(𝑙 × ℎ) + 𝑟 (𝑟 + ℎ). the corner figure 4. the mosque’s corner. the part of the mosque’s corner as in figure 4 indicates the mathematical concepts of lines and angles as explained in the following activities. 1. identifying the elements of angles, such as arms, vertex, and ray and pointing out that the right angle is equal to 900. 2. pointing out the concept of lines that is two intersecting lines. two rays, k and l, are said to be intersected if they have common end point, which is known as vertex. two intersecting lines have one vertex. negara mangkubumi k. 115 the ventilation figure 5. the mosque’s ventilation. there is a ventilation on the mosque as shown in figure 5. it contains the mathematical concepts of two-dimensional shapes, particularly a rectangle. 1. observing the elements of a rectangle and pointing out its angles, sides, and diagonals. 2. identifying the following characteristics of a rectangle. • the four angles are right angle. • the opposite sides are equilateral, and they are parallel. • both diagonals are in the same length and intersect in the middle. • it has two axes of symmetry. 3. measuring the perimeter of a rectangle and finding its formula that is twice of the sum of its width and length. the identified mathematical aspect in the mosque indicates that the ancestors of wajo people integrated the building design with the cultural philosophy. it makes the mathematical concept in building parts of tosora old mosque as a significant work clearer; the bond indicates cultural activities of the previous society that are still preserved until now. saputro (2018) explained that concrete and related elements can be used as learning sources to connect the daily experiences of students. it includes students’ development in the mathematics learning innovations based on local cultural. therefore, we can organize mathematics learning outside the classroom. these activities are expected to improve students’ experiences and knowledge on mathematical concepts. by introducing and connecting mathematics with the cultural-based local wisdom, students can expand their knowledge and open their minds to a multidisciplinary course as a way to incorporate mathematics, culture, and history. the learning process can be more fun and more meaningful as it involves contextual problems. it is expected to change the old paradigm of mathematics learning that is only restricted with learning symbols, numbers, or formulas, to the new paradigm of learning mathematics by way of a cultural aspects or civilization. this notion is in line with adam (2004), who articulated that cultural aspect contributes to seeing mathematics as part of the daily life. these cultural aspects can be developed into a mathematical activity which has meaningful connection and competency. the activity is expected to enable students to form a deeper understanding of mathematical concepts. identifying ethnomathematics in the old mosque of tosora 116 conclusion based on the result and discussion, these findings indicate that mathematical knowledge can be obtained from many aspects outside of school. one of the possible methods to integrate mathematics and other culture aspects is ethnomathematics that is applicable in the heritage site in wajo, an old mosque of tosora. the building not only possesses elements of mathematics but also principles of buginese’s life. this is indicated by the four pillars of the mosque that is in line with sulapa appa value. the philosophy inspires wajo people to be panrita (wise and honest), warani (brave), macca (intelligent), and sugi (rich). acknowledgements i would like to express my gratitude to the wajo cultural figure for being the resource person in this research. this study is extensively supported by the wajo regency education and culture office and the principal of smp negeri 2 sengkang for providing permission and opportunities to complete the manuscript. references adam, s. 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(2018). menggali etnomatematika: matematika sebagai produk budaya. prisma, prosiding seminar nasional matematika, pp. 471-476. identifying ethnomathematics in the old mosque of tosora 118 southeast asia mathematics education journal volume 13, no 1 (2023) 31 promoting mathematical literacy using desmos polygraph 1faradillah haryani & 2yola yaneta harso 1department of mathematics education, sampoerna university 2department of mathematics, smas pahoa 1faradillah.haryani@sampoernauniversity.ac.id abstract language plays a fundamental role in the teaching and learning of mathematics. students rely on their literacy skills to comprehend problem-solving situations before applying their mathematical knowledge. unfortunately, there have been numerous reports of illiteracy, particularly in the area of communication, which is one of the most important skills in mathematical literacy. the students' lack of experience connecting formal mathematical language with their everyday language is one factor contributing to their communication illiteracy. to address this issue, the current study uses desmos polygraph as a tool to promote mathematical literacy by encouraging a thorough understanding of formal language. desmos polygraph's effective use encourages students to use formal language and understand the relationship between formal and informal language. this study, on the other hand, is solely concerned with analyzing the language used in the activity. further research could assess the student-teacher interaction during the desmos polygraph activity to determine other potential in enhancing mathematical literacy. keywords: mathematical literacy, formal language, desmos, desmos polygraph introduction the use of language in mathematics instruction and learning is becoming increasingly important. language has enabled access to understanding concepts and providing mathematical instruction. furthermore, students' mathematical performance success can be predicted by their general knowledge of how to use language (jourdain & sharma, 2016; van der walt et al., 2009). the more frequently students understand and utilize mathematical language, the better performance of students in mathematics. national research council (2001) elaborated that there are five strands in mathematical proficiency encompassing (a) conceptual understanding, (b) procedural fluency, (c) strategic competence, (d) adaptive reasoning, and (e) productive disposition. it is critical to foster language comprehension to not only comprehend the concept but also to justify and communicate the reason effectively. all four strands (a-d) are interconnected together with the use of language, because one student can grasp the concept, know how to compute fluently by following the proper procedure, and there is a need for them to justify and communicate the appropriateness of the procedure. additionally, cultivating a productive disposition toward mathematics, which entails believing that mathematics is meaningful and useful, is accomplished through active engagement and continuous communication using mathematics. furthermore, riccomini et al. (2015) defined mathematics proficiency as the integrated connection and combination of concepts, procedures, problem-solving, and language. both the nrc (2001) and riccomini (2015) definitions of mathematical proficiency emphasize the importance of language in learning mathematics. promoting mathematical literacy using desmos polygraph 32 language comprehension plays a crucial role in the process of learning mathematics, especially in problem-solving (doyle, 2005). this connection between language and mathematics is evident in the emphasis on literacy skills, which focuses on reading, writing, speaking, and listening, all with the objective of achieving specific objectives. individuals with strong literacy skills have certain characteristics, such as the ability to persuade others of the accuracy of the information, highlight key points, effectively explain concepts, and transform ideas into different forms of language (gardner, 2011). recognizing the significance of literacy in mathematics, it becomes essential to promote mathematical reading, writing, and discourse as a means of improving problem-solving skills (beaudine, 2018; hillman, 2014). nctm emphasizes the link between literacy and mathematics in their connection and communication standard (altieri, 2009), while the oecd has focused on the relationship between mathematics and literacy through pisa mathematical literacy, (oecd, 2018). both of them, as well as many researchers, believe that by combining mathematics and literacy, not only the way to comprehend the problem but also the way to communicate the strategy to solve the problem, can be enhanced (oecd, 2018; ojose, 2011; sumirattana et al., 2017). individuals who are mathematically literate can estimate, comprehend facts, solve everyday problems, reason in numerical, graphical, and geometric contexts, and communicate effectively using mathematics (oecd, 2018). these abilities and knowledge are critical in today's society. as a result, the oecd established the pisa test to ensure that 15-year-olds who have completed compulsory schooling have these skills, including mathematical literacy. the most recent pisa mathematics 2018 results revealed that indonesia is at level 2 out of 6 levels (schleicher, 2019). indonesia received a score of 379, while the average oecd member score is 489. indonesia's pisa mathematics score remains below average. although the pisa mathematics framework has established six levels of mathematics proficiency, all oecd members can only reach level four. levels 5 and 6 represent the ability to mathematically communicate their actions as follows: “level 5: …………. they begin to reflect on their work and can formulate and communicate their interpretations and reasoning”. level 6: “……...can reflect on their actions, and can formulate and precisely communicate their actions …...”. furthermore, mathematical communication competency is essential because anyone learning or practicing mathematics must engage in receptive or constructive communication about mathematical matters. it can be accomplished by attempting to comprehend written, oral, figurative, or gestural mathematical communication, or by communicating one's mathematical ideas and reasoning to others in similar ways. furthermore, communication is one of the 21st century competencies embedded in the pisa mathematics literacy assessment (oecd, 2018). it will be required of the student for him or her to recognize and comprehend a problem situation. reading, decoding, and interpreting statements, questions, tasks, or objects allows the student to form a mental model of the situation, which is an important step in comprehending, elaborating, and formulating. during the solution process, it may be necessary to summarize and present intermediate results. later, once a solution has been discovered, the problem-solver may need to communicate the solution to others, along with an explanation or justification (oecd, 2018). faradillah haryani, yola yaneta harso 33 mathematical communication competence in mathematical literacy emphasizes the significance of formal and technical language and multiple mathematics representations (stacey & turner, 2015). selecting, interpreting, translating between, and employing a variety of representations, as well as using formal and technical mathematical language to capture a situation, interact with a problem, or present one's work, may be required. several studies also discovered that students frequently make literacy mistakes before attempting to use their mathematical abilities. these literacy errors are mostly related to communication issues, such as describing a situation or presenting a solution (fitriani et al., 2018; schüler-meyer et al., 2019; simpson & cole, 2015; thompson & rubenstein, 2014). because their everyday language differs from mathematical language, students are perplexed when it comes to selecting the appropriate mathematical representation and formal language to describe the situation (simpson & cole, 2015). this problem is associated with the mathematical literacy challenge, which is the multimodal formulation (i.e., language, mathematical symbolism, and images) of mathematical knowledge and the complex linguistic structures found in mathematical discourse (o’halloran, 2015). furthermore, this research's pre-study survey reveals that students with high scores in certain mathematical concepts do not guarantee that they have consistently used precise mathematical formal language in work. to avoid confusion, it recommends the practice of instruction that promotes the consistent use of formal mathematical language by bridging it into everyday language. the objective of this research is to introduce the desmos polygraph activity as a tool for encouraging the consistent use of mathematical formal language and its relationship to everyday language. desmos polygraph is a desmos platform activity that is designed to engage students in mathematical conversation. students are paired up for this activity, with one acting as the picker and the other as the guesser. there are sixteen cards with various mathematical representations related to a specific topic. the picker selects one card, and the guesser attempts to identify it by asking questions. students are encouraged to describe mathematical representations using formal language when using desmos polygraph, allowing them to observe the connection between everyday language and formal mathematical language. this practice also allows students to interact with and comprehend the reasoning of their peers. by consistently implementing these practices, the mathematics classroom transforms into a community of mathematical discourse in which students and teachers collaborate to construct mathematical knowledge and improve mathematical literacy (thompson & rubenstein, 2014) this study acknowledges the significance of using desmos polygraph and its relationship to mathematical literacy, an aspect that previous research has not extensively emphasized (caniglia et al., 2017; chorney, 2022; danielson & meyer, 2016). the studies by caniglia et al., (2017) and danielson and meyer (2016) used desmos polygraph to strengthen the students’ oral language, but the connection to the mathematical literacy process was not discussed when students needed to formulate, employ, and interpret the information. furthermore, the study by chorney (2022) focuses more on how to integrate desmos in the classroom to see the challenges in crafting the knowledge. this study aims to shed light on the importance of incorporating this tool into educational practices by investigating the potential of desmos polygraph in enhancing mathematical understanding and communication, two aspects of mathematical literacy. as a result, the following research questions are proposed: promoting mathematical literacy using desmos polygraph 34 1. how is the development of formal language acquisition when utilizing desmos polygraph activity in the classroom activities? 2. how do desmos polygraph activities help to promote mathematical literacy? methods this study employed a content analysis technique. content analysis is a suitable technique as it allows the researcher to examine the communication that occurred (fraenkel et al., 2018). three classroom meetings were conducted with 11th grade mathematics students who had finished studying limit and continuity. students were divided into three groups based on their level of competence, as determined by their most recent limit and continuity assessment scores: high competence for those who received an a, middle competence for those who received a b, and low competence for those who received a c. students used the desmos polygraph in the following three meetings, which could involve multiple sessions, with 15 minutes of discussion following each session. students were informed prior to the activities that they would be divided into three groups (a, b, and c), but the reason for grouping based on competence was not stated. to ensure pairings among students of the same competence level, three consecutive desmos polygraph codes were utilized. the first research question is concerned with the progression of formal language acquisition when using the desmos polygraph. the recorded responses on the desmos polygraph teacher dashboard will be analyzed to answer this question. the terms in the questions are classified into two categories: formal language and everyday language. a mathematical formal language question is one that contains a mathematical term. during the three-day activities, the frequency of formal language occurrence is then counted. analyzing data by looking at the frequency is a common practice in content analysis techniques (fraenkel et al., 2018). the researcher further examined the trends, such as the most commonly used mathematical formal language on a daily basis by different competence levels. the researcher summarized the terms used, including formal and informal (everyday language), and informed the teacher so that the teacher could emphasize the material that students had learned to bridge the gap between everyday and formal language. finally, the researcher discusses how the development of formal language acquisition differs depending on the three levels of competence. the second research question investigates how the desmos polygraph aids in the promotion of mathematical literacy. the qualitative descriptions of the desmos polygraph activities will be employed to determine which activities align with the pisa mathematics framework, with a focus on the communication component. the activities are evaluated utilizing the framework outlined below: faradillah haryani, yola yaneta harso 35 table 1 content, process, and context of desmos polygraph activity based on pisa mathematical literacy framework (oecd, 2021) content process context a. formulating b. employing c. interpreting change and relationship a1. translating a problem into mathematical language or a representation b.1 using and switching between different representations while finding solutions c1. interpreting information presented in graphical and/or diagram form scientific a2. understanding and explaining the relationships between a problem's contextspecific language and the symbolic and formal language required to mathematically represent it; result and discussion number of success students were divided into three distinct competency levels during the meetings: low, medium, and high. participants in each group were paired to participate in the desmos polygraph activity. one student selected a mathematical image, while the other asked yes/no questions to guess which representation was chosen. the guesser correctly recognizing the chosen mathematical representation determined the activity's success. the desmos teacher dashboard displayed the number of successful pairs as well as a list of questions asked. figure 1 depicts the number of victories from day 1 to day 3. figure 1. the number of success of desmos polygraph from day 1-3. 2 5 7 4 12 10 4 16 14 0 2 4 6 8 10 12 14 16 18 day 1 day 2 day 3 high middle low promoting mathematical literacy using desmos polygraph 36 regardless of competence, the number of successes increases from day 1 to day 3. on day 3, it became clear that the most successful pairs were made up of students with high levels of competency, whereas on day 1, the proportions of success were nearly equal across the three competence levels. while the number of successes indicated the ability of the pairings to identify the chosen image, it does not always indicate progress in formal language acquisition. to further investigate this, we are categorizing the terminology used in the questions to determine the extent of formal language acquisition. formal language acquisition we counted how many formal languages were used during the questioning activity to guess which picture was chosen at each meeting. the result is illustrated in figure 2. figure 2. the trend of formal language acquisition in percentages. the figure demonstrates that on day one, the formal language acquisition used by all three groups is nearly identical. the highly competent group, on the other hand, quickly adopts the use of formal mathematical language. although low competence is slow to adopt formal mathematics vocabulary, there is evidence of a shift in how they select the appropriate words to depict a mathematical representation. table 2 compiles the most formal language used over three days. 0% 10% 20% 30% 40% 50% 60% 70% 80% day 1 day 2 day 3 low middle high faradillah haryani, yola yaneta harso 37 table 2 the most formal language used day level of competence low middle high day 1 passes (0,0) y value approaches a certain value at x=a the line crosses the x-axis at point (a,b) passes (0,0) day 2 function continues function value defined limit exists passes (0,0) passes (0,0) function continues limit exists function discontinues day 3 function continues function value defined limit exists passes (0,0) passes (0,0) function continues limit exists function discontinues y value approaches a certain value at x=a passes (0,0) asymptote function value undefined the development of formal language acquisition observed from table 2 is the result of a three-day activity that combined the use of desmos polygraph with intensive discussions between teacher and student to build a bridge between the everyday language they used to describe the image and the formal mathematical language. table 3 summarizes the relationship between informal and formal language during the teacher-student discussion. table 3 everyday terms vs formal language everyday term formal language “garis menyambung” function continues break apart in the middle of the graph function discontinues the graph has 3 lines function discontinues/left limit≠ right limit/the function value is undefined point outside of the line limit value ≠function value there is hole limit exists, but function value undefined graph has a black dot on the graph the function value is defined this finding is consistent with simpson and cole's (2015) findings that the higher the students' competence, the faster they acquire formal language. furthermore, the greater the acquisition of formal language and exposure to the relationship between formal and informal language, the greater the development of conceptual knowledge (simpson et al., 2014). furthermore, as mathematics is also developed through social and cultural activity, there is an promoting mathematical literacy using desmos polygraph 38 urgency to bridge the context of mathematics with everyday activities (alex et al., 2021; solomon, 2008). it should be highlighted that using desmos activity should be followed by active discussion (haryani & hamidah, 2022). the desmos polygraph activity, followed by discussion, consistently assists students in bridging the gap between everyday language and formal mathematics language. to build more mathematical sense and elaborate the clarification of the concept meaning, it is necessary to engage in a complex process of crossing over or code-switching between formal and informal language and creating a bridge between them (solomon, 2008). the pisa process indicator as demonstrated in table 1, the pisa process indicator used in this study focuses specifically on the communication component. the pisa process indicator will be employed to evaluate the desmos polygraph activities. students are paired in the desmos polygraph activity, with one serving as the picker and the other as the guesser. there were 16 mathematical representations of limit and continuity, and the picker chose one while the guesser posed questions to guess the chosen image. the guesser who formulated the questions by selecting the appropriate formal language to translate the symbol or mathematical figure was successful in achieving the a1 process indicator. students must identify distinguishing characteristics of the representations and use them in their questions in the desmos polygraph activity. students who recognize the opportunity to apply mathematical content by using appropriate mathematical terminology demonstrate proficiency in question formulation (oecd, 2021). students may not realize that the problem can be solved using mathematics if they focus solely on the physical characteristics without connecting them to the mathematical content they have learned. for instance, student a's question demonstrates a lack of understanding that the graph's characteristics can be linked to the concepts of limit and continuity. student b, on the other hand, recognizes that a graph breaking apart can be related to the continuity concept. student a: “does the graph break apart?” student b: “is it a continuous function graph?” both the guesser and the picker meet the a2 indicator after each round of discussion with the teacher. the desmos polygraph activity allows students to review and identify reasons for unsuccessful pairings. more information about this indicator will be presented in a separate scholarly publication. the picker then achieves the b1 indicator by correctly responding to the guesser by translating the question "is the graph continuous?" back to the chosen mathematical representation (see figure where a chosen image is highlighted in a blue box). faradillah haryani, yola yaneta harso 39 figure 3. the situation on desmos polygraph which tells about b1 and c1 indicator. finally, during this polygraph activity, the c1 indicator is attained by the guesser, when they correctly interpret the answer from the picker into the decision on which representation needs to be eliminated. figure 3 showed us how the picker said no, answering the question, “is the graph continuous?”. the guesser then interprets the no-answer to the decision on eliminating the representation of the continuous graph. this activity concentrated on the pisa knowledge areas where students have the most difficulty (pisa, 2018). this practice can help students incorporate everyday language into formal languages, such as "garis menyambung" (connected line) to "fungsi kontinu," (continuous function). desmos polygraph's method provides multiple representations of mathematics and allows students to switch between them, assisting them in developing their formal language. this result is aligned with the study of herbel-eisenmann (2002) which used multiple representations to build the students' formal language. desmos polygraph also improves students' mathematical flexibility when translating mathematical images to formal language or vice versa. when it comes to problem solving, flexibility is essential (haryani, 2020). furthermore, because mathematical literacy emphasizes the importance of problemsolving and communication, this practice encourages students to engage in mathematical discourse (casey & ross, 2022; oecd, 2021; ojose, 2011; ripley, 2013; wilkinson, 2019). conclusion this teaching approach with desmos polygraph must be continued since it promotes an increased significance on formal language in mathematics. in addition, it enables pupils to understand the relationship between the formal language of mathematics and their everyday english. it promotes mathematical literacy and enhances the conceptual understanding of mathematics. future research may examine the teacher-student interaction in the discussion after the activity that may promote better mathematical literacy using the desmos polygraph. promoting mathematical literacy using desmos polygraph 40 references altieri, j. l. 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(2019). learning language and mathematics: a perspective from linguistics and education. linguistics and education, 49, 86–95. https://doi.org/10.1016/j.linged.2018.03.005 southeast asia mathematics education journal volume 13, no 1 (2023) 19 analysis of metacognition ability to solve mathematics problem 1fani yunida anggraheni, 2kismiantini & 3ariyadi wijaya 1 magister of mathematics education, universitas negeri yogyakarta, indonesia 2departement of statistics, universitas negeri yogyakarta, indonesia 3departement of mathematics education, universitas negeri yogyakarta, indonesia 1anggrahenify94@gmail.com abstract in this study, two components of metacognition were examined, namely metacognitive knowledge and metacognitive skills. this study aims to analyse the students’ metacognitive abilities based on predetermined indicators, by looking at the relationship between the performance of metacognition knowledge and metacognition skill. the study discovers that the students with low, medium, and high scores perform differently. the conclusion is that students who have metacognition knowledge do not necessarily have metacognition skills or abilities. the conclusion is that students who have metacognition knowledge do not necessarily have metacognition skills or abilities. keywords: metacognition, metacognition knowledge, metacognition skill. introduction learning is a process that helps students achieve goals actively. these goals are divided into three areas: 1) cognitive domains in the form of intellectual knowledge and skills; 2) affective domains in the form of student feelings and assessments; 3) psychomotor domains in the form of driving or perception skills (brookhart & nitko, 2013). the cognitive domain is divided into two dimensions, namely the process of cognition and knowledge. the process of cognition includes remembering, understanding, applying, analysing, evaluating, and creating. on the other hand, the dimensions of knowledge include factual, conceptual, procedural, and metacognition (anderson et al., 2021). metacognition is one aspect that influences the learning process both directly and indirectly, as well as in mathematics. metacognition is a combination of ‘meta’ and ‘cognition’. meta is a prefix that means after, together with, or outside (national research council, 2005). cognition is a subconscious, intuitive, and affective experience and feeling based on information processing, emotions, awareness, and behaviour (rickheit & strohner, 1998). metacognition refers to one’s awareness of the process and the ability to control it (ovan et al., 2018). metacognition is a skill that can be developed by learning, practicing, and applying a successful approach (conley, 2014). metacognition is defined as ‘thinking about thinking’ or ‘cognition about cognition’ which is the ability to self-reflect from ongoing cognitive processes, something unique to individuals, and which plays an important role in human consciousness (amin & sukestiyarno, 2015). the component of metacognition can be divided into metacognitive knowledge, metacognitive experiences, and metacognitive skills (efklides, 2009). metacognitive knowledge is a part of the knowledge that is stored in memory as a cognitive process with various cognitive tasks, goals, actions, and experiences (flavell, 1979). metacognitive analysis of metacognition ability to solve mathematics problem 20 knowledge can also be interpreted when we store memories and then retrieve memories. it is a process of a task, in which we think about when, why, and what strategies can be used to complete the tasks that are given so that information can be sorted according to the needs (efklides, 2009). metacognitive knowledge stages consist of awareness, regulation (hacker et al., 2009), and planning (veenman et al., 2004). awareness is an activity to receive information given from questions, regulation is an activity to choose and write information needed to solve a problem, and planning is an activity to give an idea or writing in the form of a plan to complete a task. metacognitive experience is an intentional cognitive or the form of affective experience that accompanies and alludes to intellectuals (flavell, 1979), which is defined as a form of cognitive monitoring when completing information related to tasks or processes. metacognitive experience consists of metacognitive feelings, metacognitive judgment/estimates, and special knowledge of online assignments (efklides, 2009). the metacognitive experience stages consist of monitoring and self-control (baker & brown, 2001). monitoring can estimate the difficulty of a particular problem, whereas self-control is being able to determine the value of the completion done. metacognitive skill is defined as the ability to control actions and use the right strategy when applying the strategy consciously and automatically by ensuring that the thinking conforms to what is desired and results in line with its objectives (efklides, 2009). the metacognitive skill stages consist of strategies (flavell, 1979), processes (hacker et al., 2009), evaluations (purnomo et al., 2017), and goals (flavell, 1979). strategies are activities that determine the formula or strategy used to solve the problem, the process is an activity that logically solves problems following the chosen strategy, and evaluation is an activity that draws conclusions according to the problem and reflects whether it can be solved in different ways, and the goal is to achieve objectives following the plan. solving mathematical problems will grow the ability of metacognition knowledge and skills. performing metacognition involves generating strategies to solve problems, implementing strategies, and checking whether the answers obtained correspond logically to the problems identified (walle et al., 2019). by doing mathematics it means discovering patterns and relationships, thinking ways, or defining a mathematical sentence (rahmah, 2018). this requires an analysis of metacognitive abilities, namely metacognitive knowledge and skills, to solve mathematical problems to find out how many metacognition abilities students can explore in solving mathematical problems. however, currently, the education system in indonesia does not give sufficient attention to the metacognition process, especially in terms of student assessment. the assessment only measures work steps and results, while the process of overview and recheck is rarely done. therefore, this study aims to analyse the ability of students to see work outcomes based on predetermined indicators and examine the relationship between the performance of metacognition knowledge and metacognition skill. metacognitive knowledge includes regulation and planning, while metacognitive skill consists of strategies, processes, evaluating, and goals. regulation is an activity in the form of observation of metacognition activities to control the process, for example, looking for and determining information related to the topic (purnomo et al., 2017). planning is an activity that involves thinking about the tasks, looking at experience, and thinking about what will happen next, for example, summarizing notes about fani yunida anggraheni, kismiantini, ariyadi wijaya 21 the steps to be taken (larkin, 2006). strategy, processes, evaluating, and goals are activities to determine patterns or related formulas, implement process structures, evaluate the process and examine targets (flavell, 1979). methods this research is descriptive analysis research using qualitative methods. the aim of qualitative research is data collection, analysis, and creation of a representation that can be shared with others. research design, site, and participants the research subjects are mathematics students on “mathematics power” subjects that are aimed at determining the ability of metacognition possessed in solving mathematical problems. the subjects consisted of 6 students with code names ar, asm, ad, sh, pay, and mr. figure 1. research procedures as shown in figure 1, the initial stage in the research is developing the questions instrument. the questions that were asked consisted of two questions that were used to analyse the students’ metacognition abilities. after compiling the questions, the next step is to validate the research instruments by the validator and then enter the stage of data collection. data collection was done by distributing the questions to students completed within a predetermined time limit. table 1 shows the indicators and assessment scores of metacognition ability. table 1 indicators and assessment scores of metacognition ability components of metacognition ability indicators assessment scores metacognitive knowledge regulation select and write down the information needed to solve the problem 1 planning provide an overview of the completion plan 2 metacognitive skills strategies determine the formula or strategy used 2 process solve the problem logically according to the chosen strategy 3 evaluating make conclusions according to the problem 3 recheck the questions given in a different way 1 goal solve questions in accordance with the goals to be achieved 1 data collection and analysis data analysis is then performed after the test, which aims to describe each student’s metacognition abilities. then from the analysed results, we do the categorization of metacognition abilities. the categorized answers are low, medium, and high levels so it is analysis of metacognition ability to solve mathematics problem 22 necessary to analyze a total of six answers. students are categorized as ‘low’, if in they make errors in the regulation and planning steps in solving questions. students are categorized as ‘medium’ if they are correct in the regulation steps of the process but have not conducted an evaluation so they have not achieved the goal. students are categorized as ‘high’ if they solve the questions correctly from the regulation steps to evaluating that achieves the goals. the conclusions are then drawn based on the result of the analysis. the questions are: 1. it takes a boat 2.5 hours to travel down a river from point a to point b, and 3.5 hours to travel up the river from b to a. how long would it take the same boat to go from a to b in still water (minutes)? 2. the difference between ani’s and budi's money is 7500. if 10% of ani is money given to budi, then budi's money becomes 80% of ani's original money. how much money do they have in total? results and discussion this section provides several examples of student's work and the analysis of their metacognitive knowledge and possessed metacognitive skills. findings from analysis of problem 1 analysis of problem 2 shows the work of students with the lowest, medium, and highest scores. solution: a→b following the current = 2.5 hours a→b following the current = 3.5 hours speed = distance/time time = distance/speed a to b goes with the flow 𝑡𝑖𝑚𝑒 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑠𝑝𝑒𝑒𝑑 2.5 = 𝑑 𝑠 − 𝑐 2.5 + 𝑐 = 𝑑 𝑠 so 𝑑 𝑠 = 2.5 + 0.5 = 3 b to a goes with the flow 𝑡𝑖𝑚𝑒 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑠𝑝𝑒𝑒𝑑 + 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 3.5 = 𝑑 𝑠 + 𝑐 3.5 = 2.5 + 𝑐 + 𝑐 3.5 − 2.5 = 2𝑐 1 2 = 𝐴 when going with the flow the time will decrease by half an hour = 30 minutes. when against the current the time will increase by half an hour = 30 minutes at the same average speed. so that if in calm water (without current), the distance from a to b will be traveled within 3 hours = 180 minutes figure 2. student ar’s solution for problem 1. based on the results of student ar’s solution on the process of metacognitive knowledge, the student did not write the required information in full accordingly and provided an overview. in the metacognitive skill process, a student is not yet correct in writing the formula or strategy used so an error occurs in the process of solving a problem. the ar student did not write other ways on the results of his work so the conclusions obtained show inaccurate results. the ar fani yunida anggraheni, kismiantini, ariyadi wijaya 23 student failed to complete the questions in line with the goals that should be achieved. then the score obtained by students is 5. solution: a→b b→a 2.5 hours = 150 minutes 3.5 hours = 210 minutes s v t  = boat speed v current speed distance p a v s = = = a b → ...(1) 150 p a s v v+ = b a − ...(2) 210 p a s v v− = of 1 and 2 obtained 150 210 p a p a s v v s v v + = − = 185 3150 p v = 1 ...(3) 175 s= 2 150 210 1 2 150 210 p p s s v s s v = +   = +    from equation 3 obtained 175 175 175 minutes 1 p s s v t s t s = = = = figure 3. student asm’s solution for problem 1. he wrote down vab = vp + va, vba = vp va and drew a graphic form as a plan to be completed. in the metacognitive skills process, he wrote the process of solving a problem can be logically solved. however, the asm student did not recheck the results of the work he obtained in other ways, so the answer he gets is a single one. he did not write the conclusions he obtained from a settlement. based on the results of the work obtained, the question was completed by ar following the goals that should be achieved but did not show the evaluation process. the evaluation process that has not been carried out does not re-check the results of work in other ways and does not draw conclusions. then the score obtained by students is 9. analysis of metacognition ability to solve mathematics problem 24 solution: known: t (trip from a to b) = 2.5 hours t (trip from a to b) = 2.5 hours let : boat speed = x boat current = y asked: how long on the same boat going from a to b answer: .s v t= → we already know in physics ( ) ( ) , , 2.5 3.5 2.5 2.5 3.5 3.5 2.5 3.5 3.5 2.5 6 6 6 a b a b b a b a v t v t x y x y x y x y x x y y x y x y x y → → → → = + = − + = − − = − − − = − = = . s s v t t v = → = to find out how long on the same boat going from a to b, we use the formula st v = ( ) 2.5 2.5 6 6 2.5 6 7 2.5 6 7 5 35 11 2 2 hours 55 minutes 6 2 12 12 s t v x y t x x x t x x x x = + =   +    = + =  =  =  = = = figure 4. student ad’s solution for problem 1. based on the results of the work of student ad’s solution to the process of metacognitive knowledge, she wrote the required information in full accordingly and provided an overview of the complete plan. the plan is to write down the speed when assisting with the flow and not the flow. in the metacognitive skill process, she wrote the exact strategies used to obtain a reasonable answer. the ad student re-check the results of the work that she obtained by making statements about the results that she obtained by comparing if influenced by the flow. she also drew conclusions from the settlement that she obtained. based on the results of the work obtained, the ad student completed the questions in accordance with the goals that should be achieved and met the completion criteria in accordance with the ability of metacognition. then the score obtained by students is 13. this is logical when the boat length from a to b is 2 hours 55 minutes because it means the journey from a to b is assisted by the current to 2.5 hours, without being assisted 2 hours 55 minutes fani yunida anggraheni, kismiantini, ariyadi wijaya 25 findings from analysis of problem 2 analysis of problem 2 shows the work of students with the lowest, medium, and highest scores. figure 5. student sh’s solution for problem 2. based on the results of the work of student sh’s solution to the process of metacognitive knowledge, the student did not write the required information in full as per the questions. the sh student made plans by writing the initial clues of b + (10%)a = (80%)a and a – b = 7500. in the metacognitive skill process, the student wrote the formula or strategy used correctly so that the process of solving a problem can be s logically solved. however, the sh student did not recheck the results of the work he obtained and she did not conclude the completion she obtained. based on the results of the work that the sh student obtained, the problem was completed in accordance with the objectives that should be achieved. however, she did take all the completion steps such as not writing the required information, not writing other completion steps as evidence of completion and not concluding the results. the score obtained by students is 9. solution: known: let: anis’s money = x budi’s money = y 7500...(1) 10% 80% 0.1 0.8 ...(2) x y x y x x y x − = + = + = asked: the amount of their money…? the answer: from equations 1 and 2 are obtained 7500 0.1 0.8 x y x y x − = + = 7500 25000 17500 y = + = so the total amount of their money 25000 17500 42500 x y+ = + = so the amount of their money rp42.500 1.1 7500 0.8 1.1 0.8 7500 0.3 7500 7500 25000 0.3 x x x x x x = + − = = = = figure 6. student pay’s solution for problem 2. analysis of metacognition ability to solve mathematics problem 26 based on the results of the work of student pay’s solution on the process of metacognitive knowledge, the student completed the information required and wrote a completion plan in the form of writing equation 1 and equation 2. in the metacognitive skill process, the student wrote the formula or strategy used to solve a problem logically. however, the pay student does not re-check the results of the work she obtained in such a way that the completion conclusion was obtained without re-checking. based on the results of the work obtained, the pay student completed the questions in accordance with the objectives to be achieved but did not check other solutions to the questions in the form. then the score obtained by students is 12. solution: the difference between ani's and budi’s money is 7500. then budi’s money becomes 80% of the original money. how much money? 7500a b− = this means a > b (ani has more money than budi) anis’s money 10%a=  of ani’s money is given to budi and anis’ money 10%a a= − the budi’s money b= because budi gets an additional 10% of anis’s money, budi’s money 10%b a= + there is a statement: budi’s money = 80% of anis’s original money now: 10% 80%b a a+ = obtained: 70% 75000b a a b=  − = 70% 7500 30% 7500 30 7500 100 25000 7500 7500 25000 7500 17500 a a a a a a b a b b b − = = = = − = − = − = = so their money = a+ b = 25000+17500 = 42500 2nd way 70% 70% then 100% 7500 100% 70% 7500 100% 70% 170% 7500 30% 42500 b a b a a b so a b = =  − = + + =  − =  = figure 7. student mr’s solution for problem 2. based on the results of the work of student mr’s solution on the process of metacognitive knowledge, the student wrote down the required information in full according to the questions and drew up a plan for systematically solving it. in the metacognitive skill process, the student wrote a formula or strategy used appropriately so that the process of solving problems can be logically solved. the mr student rechecks his work by using a different way to find the results. thus, it is obtained that the previous answer with the second way is the same. based on the results of the work obtained, the mr student completed the questions in accordance with the goals to be achieved and met the completion criteria according to the metacognition ability. then the score obtained by students is 13. fani yunida anggraheni, kismiantini, ariyadi wijaya 27 based on the above description of students’ solutions, there is a noticeable difference in the students’ metacognitive abilities. moreover, a study also stated that mathematics learning students have different levels (lestari et al., 2019). the level consists of levels students read, wrote, and determined the strategy. medium-level students planned and corrected the mistakes, while high-level students implement the best strategies, analyze, and represent. the research by izzaati and mahmudi (2019) also showed different levels of metacognition that students with medium and low levels did not well aspects of planning, monitoring, and evaluating compared to high levels. blumer and keton’s (2014) research discussed that with high metacognition abilities, it will have high performance. meanwhile, amin and sukestiyarno (2015) showed that students’ metacognition abilities related to cognitive skills. the ability of metacognition will affect metacognition skills, students who have high metacognition abilities will have high metacognition skills. metacognition knowledge and metacognition skills are two aspects that are interrelated and important in the learning process (hartman, 2001). students’ metacognitive skills are influenced by their knowledge. students lacking the ability of metacognitive knowledge, the ability of students is metacognitive skills will be wrong too, and the ability of metacognitive knowledge is thus very influential on the ability of metacognitive skills. conclusion this is evident in student work outcomes where all aspects of metacognition abilities are met by the lowest, medium and highest students. as far as metacognitive knowledge ability is concerned, namely in the regulatory aspect, students can determine the information in the problem but if the strategy used is wrong or wrong then the next process is also wrong. another drawback is that students often do not double-check with different methods or ways to ensure the results are met. the limitations in this study are the relatively small subjects and there are only two problems in the study. further research is suggested to include a sufficient number of subjects and several problem models. despite the limitations, this research is expected to contribute to the studies on metacognition. acknowledgements the researchers would like to thank those who helped in the process of completing the article. references amin, i., & sukestiyarno, y. l. 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(2014). learning strategies as metacognitive factors: a critical review. eugene, or: educational policy improvement center. efklides, a. (2009). the role of metacognitive experiences in the learning process. psicothema, 21(1), 76–82. flavell, j. h. (1979). metacognition and cognitive monitoring: a new area of cognitivedevelopmental inquiry. american psychologist, 34(10), 906–911. https://doi.org/10.1037/0003-066x.34.10.906 hacker, d. j., dunlosky, j., & graesser, a. c. (eds.). (2009). handbook of metacognition in education. routledge. https://doi.org/10.4324/9780203876428 hartman, h. j. (ed.). (2001). metacognition in learning and instruction: theory, research and practice (vol. 19). springer science & business media. izzati, l. r., & mahmudi, a. (2019). analysis of metacognition skills of students in junior high school based on cognitive style. journal of physics: conference series, 1157(3). https://doi.org/10.1088/1742-6596/1157/3/032089 larkin, s. 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(2018). hakikat pendidikan matematika. al-khwarizmi: jurnal pendidikan matematika dan ilmu pengetahuan alam, 1(2), 1–10. https://doi.org/10.24256/jpmipa.v1i2.88 fani yunida anggraheni, kismiantini, ariyadi wijaya 29 rickheit, g. & strohner, h. (1998). cognitive systems theory. in g. altmann & w. koch (ed.), systems: new paradigms for the human sciences (pp. 404-420). berlin, new york: de gruyter. https://doi.org/10.1515/9783110801194.404 veenman, m. v. j., wilhelm, p., & beishuizen, j. j. (2004). the relation between intellectual and metacognitive skills from a developmental perspective. learning and instruction, 14(1), 89–109. https://doi.org/10.1016/j.learninstruc.2003.10.004 walle, j. a. van de, s.karp, k., bay-williams, j. m., wray, j., & brown, e. t. (2019). elementary and middle school mathematics: teaching developmentally. pearson https://doi.org/10.1515/9783110801194.404 analysis of metacognition ability to solve mathematics problem 30 53 southeast asian mathematics education journal, volume 11, no 1 (2021) convergent and divergent thinking in mathematical creative thinking processes in terms of students’ brain dominance 1bayu sukmaangara & 2 sri tirto madawistama 1 ma sidik jahra, ciamis, indonesia 2siliwangi university, tasikmalaya, indonesia 1bayoosukmaangara@gmail.com 2sritirtomadawistama@unsil.ac.id abstract convergent and divergent thinking play an essential role in a person’s creative thinking process to solve problems, which highlights the significance of this research. aside from that, these two types of thinking are related to the function of the brain’s hemispheres that will affect a person’s perspective in processing information. this research aims to get a view of convergent and divergent thinking in the mathematical creative thinking process in terms of brain dominance. the research was conducted using qualitative method with an exploratory descriptive approach. the instruments used are mathematical creative thinking test, brain dominance tests, and unstructured interviews. the research revealed that left-brain dominant students in the creative thinking process are more prominent in convergent thinking; the balance dominant students in the creative thinking process are balanced in divergent and convergent thinking, while rightbrain dominant students in the creative thinking process are more adept in divergent thinking. keywords: convergent and divergent thinking, mathematical creative thinking process, brain dominance. introduction guilford (cropley, 2006) introduced convergent dan divergent thinking concepts in creativity. divergent thinking is needed in the creative thinking process without eliminating the role of convergent thinking. divergent thinking can create new ideas, while convergent thinking can choose the necessary concepts and relate these ideas to solve problems. in other words, these two types of thinking complement each other in creative thinking to solve problems. brophy (1998) explained that creative thinking is about divergent thinking, but it takes convergent thinking to complete it. hence, convergent and divergent thinking have their own significance in the creative thinking process to solve problems. convergent and divergent thinking are needed in preparation, incubation, illumination, and verification stages during the creative thinking process (cropley, 2006). the preparation stage, incubation stage, illumination stage, and verification stage are the four stages of wallas (savic 2016). indicator’s guidelines of the relationship between convergent and divergent thinking with creative thinking process stages, according to wallas, are combined from two sources namely a thesis written by sukmaangara (2020) and an article by cropley (2006). the indicator guidelines are presented in the following table: 54 southeast asian mathematics education journal, volume 11, no 1 (2021) table 1 indicators of the creative thinking process stages according to wallas no the creative thinking process stages indicators of the creative thinking process stage according to wallas activities process required 1 preparation stage a. students prepare themselves to solve problems in various ways such as the following: 1) students can open books; 2) ask the teacher or other students; 3) students learn from the lessons previously taught. b. students try several ways to solve problems. c. students are able to understand the problem by writing down what is known and asked; the activity to understand the problem and activity to identify problem from general knowledge into specific knowledge and produce the knowledge needed to solve problems convergent thinking 2 incubation stage students seek inspiration by doing various activities such as the following: a. students take a moment to reflect b. students read the questions many times. c. students relate the questions to the material that has been obtained. the activity of combining two different things to produce something in a new way divergent thinking 3 illuminatio n stage a. students get ideas. b. students convey some of their ideas which are used as solutions. the activity of discovering something new divergent thinking 4 verification stage a. students run their ideas to get the right answer by: 1) writing the formula; 2) performing arithmetic operations by assigning known data into formulas. b. students can work on the problem correctly and use many ways. c. students re-examine the answers and look for other ways to solve the problem the activity to show about correct solution of the new configuration divergent thinking and convergent thinking adapted from: (sukmaangara, 2020; cropley, 2006) the convergent and divergent thinking processes are presented in the design of thinking to solve problems. design thinking is based on convergent and divergent thinking to solve problems both for the definition and solution of a problem (androutsos & brinia 2019). according to lindberg et al. (2010), the basic principles of the flow of design thinking are as follows: 55 southeast asian mathematics education journal, volume 11, no 1 (2021) figure 1. basic principles of design thinking flow the importance of convergent and divergent thinking cannot be separated from the human brain function. the brain consists of two hemispheres, namely the left hemisphere and the right hemisphere. both of them play different roles. haryanto (2015) articulated that the source of the left hemisphere function is convergent thinking, and the source of the right hemisphere function is divergent thinking. geske (1992) stated that brain dominance affects a person's perspective in processing information and directly determines learning styles. the different roles of the brain's hemispheres will affect one's perspective, showing the importance of the hemispheres in convergent and divergent thinking. based on the abovementioned description, it is conclusive that convergent and divergent thinking play a vital role in a person's creative thinking process to solve problems. these two types of thinking are related to the function of the cerebral hemispheres, affecting a person's perspective in processing information. thus, the researchers are interested in examining these two types of thinking in the creative thinking process in terms of brain dominance. this study will investigate how convergent and divergent thinking in the creative thinking process are based on the wallas stages during problem solving presented in the design of the creative thinking process. the wallas stage was used because the stages highlight convergent and divergent thinking (cropley, 2006). likewise, with design thinking, convergent and divergent thinking could be seen from a way of design thinking (androutsos & brinia, 2019). this research aims to investigate convergent and divergent thinking in the mathematical creative thinking process in terms of brain dominance. the result of the study is expected to help readers, especially teachers, develop convergent and divergent thinking questions adapted to brain dominance so that the students can be more optimal in solving the problems methods this research used a qualitative method with an exploratory, descriptive approach. researchers deeply explored students' convergent and divergent thinking in the mathematical creative thinking process until they obtained enough data to achieve the research objectives. then, the collected data were described in written words. data were collected using brain dominance tests, mathematical creative thinking questions that met the indicators of fluency and flexibility, and unstructured interviews. the study began by providing 31 students of nine-grade at smpn 1 tasikmalaya for the 2019/2020 academic 56 southeast asian mathematics education journal, volume 11, no 1 (2021) year with a brain dominance test. this study adapted the brain dominance test from tendero's (2000) dissertation. the results of the brain dominance test were grouped into eighteen students with left-brain dominance, six students with balanced brain dominance, and seven students with right-brain dominance. furthermore, only one student with left-brain dominance, one with balanced brain dominance, and one with right-brain dominance were selected as research subjects. subjects were also selected based on their ability to solve problems and ability to provide information orally. the selected subjects were given two more brain dominance tests at different times so that a total of three brain dominance tests were given. this test was carried out to obtain more valid data to make it more credible (sugiyono, 2017). the following are the results of the student's brain dominance test: table 2 student brain dominance test results subject first test second test third test conclusion score category score category score category s1 -5 left brain dominant -6 left brain dominance -4 left brain dominance left brain dominance s2 0 balanced brain dominance 0 balanced brain dominance 0 balanced brain dominance balanced brain dominance s3 3 right brain dominance 4 right brain dominance 4 right brain dominance right brain dominance the selected research subjects did the mathematical creative thinking test. this test aims to obtain data about students' convergent and divergent thinking in the mathematical creative thinking process. unstructured interviews were conducted subsequently after this test to support the data. the given questions for the mathematical creative thinking test is presented as follows: figure 2. mathematical creative thinking problem the data were analyzed using the miles & huberman model which consists of data reduction, data presentation, and drawing conclusions (miles & huberman, 1994). 57 southeast asian mathematics education journal, volume 11, no 1 (2021) results and discussion the convergent and divergent thinking research results in the mathematical creative thinking process will be shown in the design form of students' creative thinking processes. the result of students' thinking process from the mathematical creative thinking test answers in each work step is coded. the design's results of students' thinking processes is presented as follows: convergent and divergent thinking processes in the mathematical creative thinking process of left-brain dominant students figure 3 thinking process design of left-brain dominant students information for figure 3 regarding the thinking process design of left-brain dominance students' is presented in the following table: table 3 description of mathematical creative thinking process design of left-brain dominant students dominance code description code description s question g1 solid figure 3 cubes d looking for ideas l volume = 8cm3 a rib length 2cm d2 volume = 6 x 8cm3 c solid figure volume? v1 volume = 48cm3 e using the cube concept e2 volume = 2 x 8cm3 g using the cuboid concept a2 volume = 16cm3 pn solid figure numbering f2 volume = 3 x 8cm3 v1 solid figure number 1 n1 volume = 24cm3 v6 solid figure number 6 g2 volume = v1 + v2 + v3 + v4 + v5 + v6 + v7 + v8 + v9 + v10 + v11 58 southeast asian mathematics education journal, volume 11, no 1 (2021) v7 solid figure number 7 i2 volume = 48cm3 + 16cm3 + 24cm3 + 16cm3 + 16cm3 + 48cm3 + 24cm3 + 16cm3 + 48cm3 + 48cm3 + 16cm3 v10 solid figure number 10 v volume = p x l x t t1 solid figure 6 cubes u1 volume = 6cm x 4cm x 2cm v2 solid figure number 2 z1 volume = 4cm x 2cm x 2cm v4 solid figure number 4 m1 volume = 6cm x 2cm x 2cm v5 solid figure number 5 h the overall volume = 320cm3 v8 solid figure number 8 cek recheck answer v11 solid figure number 11 ilm illumination stage y1 solid figure 2 cubes 1-19 student workflow number v3 solid figure number 3 divergent thinking process v7 solid figure number 7 convergent thinking process k volume = 2cm x 2cm x 2cm based on figure 3 and table 3, students started looking for ideas by doing various activities to find inspiration (d code) at the incubation stage. looking for ideas is an activity to produce something in a new way that requires a divergent thinking process (cropley, 2006). students did the preparation stage after the incubation stage. students read the questions, memorized the lessons, and wrote down what they got from the questions (a and d codes). activities at the preparation stage aim to identify problems from general knowledge for specific knowledge to produce the required knowledge. these activities require a convergent thinking process (cropley, 2006). subsequently, students obtained ideas and conveyed some ideas that would be used, namely using the concept of a cube (e code) and the concept of a cuboid (g code) to solve problems at the illumination stage. activities at the illumination stage are directed to find something new. these activities require divergent thinking processes (cropley, 2006). the activities carried out are strengthened by the results of interviews as follows: p : what did you think for quite a while before working on the questions? s : i think about materials related to the questions and what the questions need. p : what did you get? s : i got the right solution to solve the problem. p : what solution did you use for the first and second method? s : i used the cube concept for the first method and the cuboid idea for the second method. based on figure 3 and table 3, students gave a number to each solid figure was and divided them into 11 solid figures (v1 – v11 code) at the verification stage. this activity belonged to divergent thinking processes since students used various aspects to find solutions (linberg et al., 2009). students solved the problems with the same solid figure for the first and second methods (v1 – v11 code). all of the spatial structures, which consisted of 11 solid figures, were grouped into three parts, namely solid figure with 6 cubes (t1 code), solid figure with 2 cubes (y1 code), and solid figure with 3 cubes (g1 code). the student's activity of grouping into three parts of solid figures is an activity of uniting different aspects. the activity of combining various elements is a convergent thinking process (linberg et al., 2009). 59 southeast asian mathematics education journal, volume 11, no 1 (2021) the first method used the concept of cube volume by calculating the volume of one cube (k l code). this activity united three different parts of the solid figures using one concept, namely the volume of the cube, which is a convergent thinking process (linberg et al., 2009). the volume of one cube was multiplied by the number of cubes that made up the solid figure. this caused students to calculate the volume of each solid figure with three parts, namely the volume of solid figures of six cubes (d2 code), the volume of solid figures of two cubes (e2 code), and the volume of solid figure of 3 cubes (f2 code). finding various aspects to find solutions is a divergent thinking process (linberg et al., 2009). likewise, in the second method, students calculated volume by dividing three parts, but students did calculations using the volume of the cuboid (v code). students calculated in advance the length, width, and height of each solid figure. next, students calculated using the formula for the volume of the cuboid. this activity resulted in divergent thinking as in the first method shown in figure 3 (code v – u1, v a2, v n1). the results of the three groups of solid figures in both the first and second methods resulted in a volume of 48cm3 (v1 code), 16cm3 (a2 code), and 24cm3 (n1 code). students calculated the total volume with vtotal = v1 + v2 + v3 + v4 + v5 + v6 + v7 + v8 + v9 + v10 + v11 (g2-i2 code). the calculation of the total volume obtained the right result; namely, the size of the wake volume of 320cm3 (h code). this activity presents the right solution that requires a convergent thinking process (cropley, 2006). the activities were reinforced by the results of interviews as follows: p : can you explain your work? s : i calculated the volume of the cube first. i multiplied the cube volume by the number of cubes. the calculation resulted in the solid volume of figure number 1, which consisted of six cubes, with 48cm3 (students explain the same methods from number 1 to number 11). that was for the first method, while for the second method, i used the concept of the cuboid. the solid figures consisted of 6 cubes, and the length was calculated using 2cm + 2cm + 2cm, which resulted in 6cm. the same method was used to calculate the width and height and yielded 4cm and 2cm, respectively. the product of length, width, and height resulted in the volume of the number 1 solid figure, which was 48cm3 (students explained all the same methods from number 1 to number 11) students also carried out the incubation stage during the verification stage by looking for ideas (d code) to produce something new and the illumination stage (ilm code) by finding ideas for something new. these activities require a divergent thinking process (cropley, 2006). students also checked the answers (cek code) which needs a convergent thinking process (cropley, 2006). 60 southeast asian mathematics education journal, volume 11, no 1 (2021) convergent and divergent thinking processes in the mathematical creative thinking process of balanced brain dominant students figure 4. thinking process design of balanced brain dominant students information for figure 2 regarding the thinking process design of balanced brain dominant students' is presented in the following table: table 4 description of thinking process design of balanced brain dominant students code description code description b many cubes 40 f1 volume = 192cm3 j volume = s x s x s k2 volume = 16cm3 x 5 n volume = 8cm3 x 40 l2 volume = 80cm3 u2 volume = 40cm3 m2 volume = 24cm3 x 2 v2 volume = 40cm3 x 4 n2 v = 192cm3 + 80cm3 + 48cm3 w2 volume = 160cm3 h volume = 320cm3 x2 v = 160cm3 + 80cm3 + 48cm3 brt asking hs volume = 288cm3 1-16 student workflow number j2 volume = 48cm3 x 4 based on figure 4 and table 4, students did the preparation stage by reading the questions, writing down what they had learned from the questions that have been understood, and then asking questions to better understand the provided questions (a, brt, b, c codes). activities at the preparation stage are targeted to identify problems from general knowledge into specific knowledge and produce the required knowledge. these activities require a convergent thinking process (cropley, 2006). students looked for ideas (d code) at the incubation stage. looking for ideas is an activity to produce something in a new way that requires a divergent thinking process (cropley, 2006). then, students obtained ideas and conveyed some ideas that will be used, by using the concept of a cube (e code) and the concept of a cuboid (g code) to solve 61 southeast asian mathematics education journal, volume 11, no 1 (2021) problems at the illumination stage. activities at the illumination stage are activities to find something new. these activities require divergent thinking processes (cropley, 2006). the activities carried out were supported by the results of interviews as follows: p : why were you silent for a moment, and after that, you asked the question to me? s : i was looking for a method and also, i convinced myself about the method that i would use. thus, i asked you, sir! (with a smile) p : how would you use the first and second methods? s : i used the concept of the cube for the first method and the idea of the cuboid for the second method. based on figure 4 and table 4, the first method began with students getting an idea (ilm code) after asking a question (brt code). students multiplied the cube volume of 8cm3 by the cube's number to produce a volume of 320cm3 (j – h code). the second method started by asking questions (brt code) to look for ideas (d code), and soon the student got an idea (ilm code). students solved problems by calculating the volume of the solid figure of 6 cubes (u1 – f1 code), the volume of solid figure of 2 cubes (z1 – l2 code), and the volume of solid figure of 3 cubes (m1 – v1 code). this activity is directed to find various aspects to find solutions, which is a divergent thinking process (linberg et al., 2009). in calculating the volume of the solid figure of 6 cubes, the student made a mistake in calculating the multiplication (u2 – x2 code), so that the student re-checked the answer (code checking). after re-checking and knowing the errors, the students started counting again and produced the correct calculations. the total volume of solid figure was calculated by adding the volume calculation of 3 solid figures of different parts, namely v = 192cm3 + 80cm3 + 48cm3 (n2 code) until students produced a total volume of solid figure of 320cm3 (h code). this activity to present the right solution requires a convergent thinking process (cropley, 2009). the activities carried out by the student are supported by the results of interviews as follows: p : can you explain your work? s : the cube formula is s x s x s. the side of the cube is 2cm so the volume of a cube is 2cm x 2cm x 2cm which results in 8cm3. i multiplied the volume of one cube by 40 because there are 40 same cubes and the result was 320cm3. the second method was conducted using the cuboid formula. the arrangement of these cubes formed a cuboid; then, the cuboids were calculated one by one with the cuboid formula and multiplied by the same number of solid figures to produce 320cm3. during the verification stage, students did the questioning activities to identify problems which require a convergent thinking process (cropley, 2006). students also did the incubation stage by looking for ideas (d code) to produce something new and the illumination stage (ilm code) by finding ideas for something new. these activities require a divergent thinking process (cropley, 2006). students also carried out checking activities (check code) which required a convergent thinking process (cropley, 2006). 62 southeast asian mathematics education journal, volume 11, no 1 (2021) convergent and divergent thinking processes in the mathematical creative thinking process of right brain dominant students figure 5 the thinking process of right brain dominant students information for figure 5 regarding the design of the right brain dominant student's thinking process is presented in the following table: table 5 description of thinking process design of right brain dominant students code description code description y2 cube volume? q2 volume = 2 x (6cm x 2cm x 2cm) z2 volume = (2cm)3 x 40 r2 volume = 4 x 24cm3 m volume = volume cube x 40 s2 volume = 5 x (2cm x 2cm x 4cm) o2 volume = 4 x (6cm x 2cm x 4cm) t2 volume = 4 x 16cm3 p2 volume = 4 x 48cm3 1 – 13 student workflow number based on figure 5 and table 5, students carried out the preparation stage by reading the questions, writing down what they had learned from the questions they had understood, and asking questions to better understand the provided questions (a, brt, b, y2 codes). activities at the preparation stage were to identify problems from general knowledge to specific knowledge to produce the required knowledge. these activities require a convergent thinking process (cropley, 2006). students looked for ideas (d code) at the incubation stage. looking for ideas is an activity to produce something in a new way that requires a divergent thinking process (cropley, 2006). afterwards, students obtained ideas and conveyed some ideas they would, namely using the concept of a cube (e code) and the concept of a cuboid (g code) to solve problems at the illumination stage. activities at the illumination stage are directed to find something new. these activities require divergent thinking processes (cropley, 2006). the activities carried out are supported by the results of interviews as follows: 63 southeast asian mathematics education journal, volume 11, no 1 (2021) p : why did you ask me? s : because i thought this was the length of the cube, but it turned out to be the side length of the cube p : have you tried to stay calm for a moment? what did you think? s : i tried to remember and linked the material to figure out how i wanted to use it. p : what was the concept used for the first and second methods? s : i used the cube concept for the first method and the cuboid idea for the second method. based on figure 5 and table 5, the first method began with students reflecting (d code) to look for ideas. after getting the idea (ilm code), students could solve the problem. students multiplied the cube volume of 8cm3 by the cube's number to produce a volume of 320cm3 (z2 – h code). the second way, students solved the problem by calculating the solid figure of 6 cubes volume by multiplying the six cubes number directly with the volume of the cube, which is 4 x (6cm x 2cm x 4cm), resulting in a volume of 192cm3 (o2 – f1 code). students did the same calculation to calculate the volume of two cubes (q2 – f1 code) and the volume of three cubes (s2 – l2 code). this activity is an activity to find various aspects to find solutions. this activity is a divergent thinking process (linberg et al., 2009). students re-checked (check code) to ensure the method used was correct in calculating the solid figure of 6 cubes volume. after reflecting on it (d code), students obtained an idea (ilm code) to solve the problem. the total volume of solid figure was calculated by adding volume calculation results of 3 different parts of solid figure, namely v = 192cm3 + 80cm3 + 48cm3 (n2 code) so as to produce a total volume of solid figure of 320cm3 (h code). this activity aims to present the right solution that requires a convergent thinking process (cropley, 2009). the students’ activities were supported by the results of interviews as follows: p : could you explain your work? s : all right, sir. in the first method, i wrote the information given and the questions. i calculated the volume of a cube of 23 x 40 because there were 40 cubes, resulting in 320cm3. in the second method, i calculated this solid figure (pointing to a solid figure consisting of 6 cubes) 6cm for its length, 2cm in height, and 4cm in width. because there were four solid figures with the same size, i could find 4 x (6cm x 2cm x 4cm) so that it produced 192cm3 (students explained all the same methods to solid figure of other cuboids). the total volume was obtained from the summary of the previous volume until i got the volume of 320cm3. students also did the incubation stage during the verification stage by looking for ideas (d code) to produce something new and the illumination stage (ilm code) by finding ideas for something new. it required a divergent thinking process (cropley, 2006). students also did the answer checking activities (check codes), which require a convergent thinking process (cropley, 2006). 64 southeast asian mathematics education journal, volume 11, no 1 (2021) conclusion based on the description of figure 3, figure 4, and figure 5 and the results of the interview, we can conclude that convergent and divergent thinking in the mathematical creative thinking process were reviewed by students' with the following brain dominance: 1) left-brain dominant students in the creative thinking process were more prominent in convergent thinking; 2) balanced brain dominant students had balanced practice of convergent and divergent thinking in creative thinking process, and 3) right-brain dominant students were more dominant in divergent thinking in creative thinking process. the results showed that students with different brain dominance had different practice of convergent and divergent thinking in the mathematical creative thinking process. it was revealed that students thought divergently or convergently according to the function of the student's brain hemispheres. the results of this study can be used as an illustration for teachers to develop questions related to convergent and divergent thinking in the mathematical creative thinking process by considering the dominance of the student's brain, so that they can solve the problem optimally according to the dominant characteristics of their brain. references androutsos, a. & brinia, v. (2019). developing and piloting a pedagogy for teaching innovation, collaboration, and co-creation in secondary education based on design thinking, digital transformation, and enterpreneurship. education science, 9(2), 1-11. https://doi.org/10.3390/educsci9020113 brophy, d.r. (1998). understanding, measuring, and enhancing collective creative problemsolving efforts. creativity research journal, 11(3): 37-47. https://doi.org/10.1207/s15326934crj1103_2 cropley, a. (2006). in praise of convergent thinking. creativity research journal, 18(3), 391404. https://doi.org/10.1207/s15326934crj1803_13 geske, j. (1992). teaching creativity for right brain and left brain thinkers [presentation]. annual meeting of the association for education in journalism and mass communication, quebec: canada. retrieved from: https://files.eric.ed.gov/fulltext/ed349600.pdf haryanto. (2015). pembelajaran konstruktivistik meningkatkan cara berpikir divergen siswa sd. jurnal penelitian ilmu pendidikan, 8(1), 36-43. https://doi.org/10.21831/jpipfip.v8i1.4927 lindberg, t., gumienny, r., jobst, b., & meinel, c. (2010). is there a need for a design thinking process? dtrs8 interpreting design thinking, 243-254. retrieved from: https://hpi.de/fileadmin/user_upload/fachgebiete/meinel/papers/design_thinking/2010_ lindberg_design.pdf miles, m.b., & huberman, a.m. (1994). qualitative data analysis: an expanded sourcebook (2nd ed.). california, amerika: sage publications. 65 southeast asian mathematics education journal, volume 11, no 1 (2021) savic, m. (2016). mathematical problem solving via wallas’ four stages of creativity: implications for the undergraduate classroom. the mathematics enthusiast, 13 (3), 255278. retrieved from: https://scholarworks.umt.edu/tme/vol13/iss3/6/. sugiyono. (2017). metode penelitian kualitatif (3rd ed.). bandung, indonesia: alfabeta. sukmaangara, b. (2020). analisis proses berpikir kreatif matematis berdasarkan tahapan wallas dan resiliensi matematis ditinjau dari dominasi otak siswa [master thesis]. universitas siliwangi, indonesia. tendero, j.b. (2000). hemispheric dominance and language proficiency levels in the four macro skills of the western mindanao state university college students [dissertation]. western mindanao state university, filipina. 66 southeast asian mathematics education journal, volume 11, no 1 (2021) southeast asian mathematics education journal volume 12, no. 1 (2022) 51 the effect of implementing steam and 4dframe learning in developing students’ computational thinking skills 1 rio mardani suhardi & 2 gusnandar yoga utama 1 smp negeri 8 batam, indonesia 2 seameo qitep in mathematics, indonesia 1 riomardanisuhardi@gmail.com 2 gusnandaryogautama@gmail.com abstract computational thinking skills have been a popular term for teachers worldwide, and pisa 2022 will become the first pisa in evaluating them. computational thinking helps students enhance their potential in contributing to other disciplines. however, students’ computational thinking skills at smp negeri 8 batam were low. in overcoming the problem, the teacher employed steam learning as an alternative approach in stimulating students’ computational thinking skills. a teaching aid, named 4dframe, was utilised to support the steam-based teaching. the objective of this action research study is to illustrate the effect of employing steam approach and the 4dframe as the teaching assistance in developing students' computational thinking skills. the study involved 40 students of 9 th grade in smp negeri 8 batam, indonesia. three steam activities incorporating warka water tower, batam-bintan straw bridge, and planting machine were performed in eight online meetings. in each activity, the students administered decomposition, abstraction, pattern recognition, and algorithm as the cornerstones of computational thinking. the data were gathered through observational forms during the learning and test to evaluate students’ computational thinking skills. the results present that 73% and 88% of students acquired the minimum score for the computational thinking post-tests on the first and second cycle respectively. although sample and methodology limitations prevent any claim to generalisation, this learning strategy could be implemented as an alternative for conducting mathematics learning activities in elevating students’ computational thinking skills with students in similar contexts. keywords: 4dframe, action research, computational thinking skills, steam introduction pisa (2019) presented that, on average, indonesian 15-year-olds scored 379 points in mathematics compared to an average of 489 points in oecd countries which places indonesia in the rank of 70 out of the 77 oecd countries that is indeed significantly low. as a future challenge, for the first time, the pisa 2021 framework incorporated an appreciation of the collaboration between mathematical and computational thinking engendering a similar set of perspectives, thought processes and mental models which learners are necessary succeed in an increasingly technological world (pisa, 2018). computational thinking is an innovative thinking ability in identifying life phenomena to provide various practical solutions toward the investigated problems (fajri, yurniwati, & utomo, 2019). with the rapid flow of technological developments which elevate the economy competitiveness, research planners have concerned their efforts on equipping the younger generation to encounter future challenges through the development of computational thinking in the last 10 years (khine, 2018). mailto:1riomardanisuhardi@gmail.com the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 52 based on the researchers’ observations while teaching grade 9 th in smpn 8 batam, indonesia, students were not used to problems and activities which aim at improving their computational thinking skills. it can be identified from the pre-test results of most students which were still low in solving problems associated with computational thinking. it is corroborated by the results of interviews with mathematics teachers who were teaching 9 th grade in smpn 8 batam in the previous school year. they asserted that students' computational thinking abilities were still low. cuny, snyder and wing (2010) elaborated that computational thinking is a thinking process involved in formulating problems and solutions. hence, it is easily performed solutions by an information processing agent. others explained that computational thinking helps students to enhance their potential in contributing to other disciplines, particularly stem (sands, yadav, & good, 2018). there is a necessity to train student’s computational thinking, one of which is by implementing stem learning. science, technology, engineering, and mathematics (stem) are trending in 21 st century education. stem education is crucial because it accommodates an interdisciplinary approach which plays a pivotal role for the future of a country (gülhan & şahin, 2018). while baines (2019) argued that students possess a perception of stem which tends to be boring and confusing. therefore, the proper and wise utilisation of a (art) can make the steam learning more enjoyable for students. stem learning is the current teaching method that the researcher performs in the classroom. based on several observation results in grade 8 th for the 2020/2021 academic year, in general, students are tremendously interested in stem, but some students seem less active during the activities. however, steam integrates artistic design, expression, reflective and multi-sensory appeal which requires art to associate (daugherty, 2013). therefore, the utilization of art increases the students’ attractiveness in being active during learning activities. this research was formulated to enhance students’ computational thinking skills through steam learning. based on the researcher’s experience in teaching in the classroom, students experience difficulties developing their imagination. there is a need for teaching aids which are able to generate imagination and stimulate student's computational abilities. therefore, the researcher attempts to employ the 4dframe as a steam-based learning media. the 4dframe was initiated by hogul park, a korean engineer and model maker originally inspired by classical korean architecture. the 4dframe produces an advantage on its utilisation which is suitable to be administered as a learning media in schools which integrating science, technology, engineering, arts (including architecture or design), and mathematics (park, 2018). the 4dframe is an educational toolkit which assists students to develop creativity with their imagination. thus, applying stem and the 4dframe learning is expected to enhance students’ computational thinking skills. computational thinking is a cognitive or thinking process incorporating logical reasoning by which problems are solved, and artefacts, procedures and systems are better to comprehend (csizmadia, curzon, dorling, humphreys, ng, selby, & woollard, 2015). it encompasses: (1) the ability to think algorithmically; (2) the ability to think in terms of decomposition; (3) the ability to think in generalisations, identifying and formulating rio mardani suhardi & gusnandar yoga utama 53 patterns; (4) the ability to think in abstractions, selecting appropriate representations; and (5) the ability to think in terms of evaluation. steam education defines a variable as a characteristic which conveys a feature, useful or critical parameters, system elements when identifying a system, or when evaluating its performance, status, and condition. the application of steam educators incorporates three domains of teaching and learning which are pedagogy, assessment, and technology integration (anito, elipane, sarmiento, & butron, 2019). because recognising design is the prior concern to understanding engineering, student engagement in this problem-solving process is crucial. the eie (engineering is elementary) project generated a simple five-step engineering design process for children: ask, imagine, plan, create, and improve (hester & cunningham, 2007). it also produced a series of questions to assist students to get through every step. moving through the engineering design process (edp) involves asking the following questions or making the following decisions as displayed in figure 1. figure 1. moving through the engineering design process (edp) (hester & cunningham, 2007). the 4dframe is suitable for both individual and group activities. in addition to being a complex structural material, the 4dframe has also been associated with various software advanced with the objective of providing visualising attractive geometrical models. when utilising the 4dframe sets to implement geometric modelling in schools, students are able to learn various interdisciplinary topics in an active, meaningful and fun way. these topics accommodates the fields of art, architecture, global or local issues, socio-cultural, or transdisciplinary of all these with the implementation phenomena-based learning methods (fenyvesi, park, choi, song, & ahn, 2016). the question which is formulated in this study is how can the implementation of steam and 4dframe learning improve students’ computational thinking skills? hence, this research the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 54 concentrated on how the implementation of steam and the 4dframe learning can effectively enhance students’ computational thinking skills of 9 th graders at smpn 8 batam. methods this research employed classroom action research by stephen kemmis and robin mctaggart (arikunto et al., 2017) with a qualitative approach. the objective of this study is to elaborate the thinking processes and student behaviour during the learning process, and to determine the effect of applying steam and the 4dframe on student’s development of computational thinking skills. this classroom action research encompasses two cycles, each of which was conducted by employing four online meetings. before the implementation phase, students first administered a pre-test to examine their initial abilities. the steam and the 4dframe learning were performed in three meeting topics incorporating warka water tower, building straw bridge, and the planting machine. in the final meeting in every cycle, a post-test was performed to calculate the improvement of students’ computational thinking skills. in collecting the data, we developed several instruments such as computational thinking skills test obtained from bebras, indonesia challenge and mathematics textbook grade 9. the test was utilised to gather data on students’ computational thinking skills. furthermore, we also formulated questions for the interview to explore things which were not monitored during the observation and to identify the obstacles experienced by students during the phase of implementation. interviews were conducted with several students based on the interview guidelines until the required data were fulfilled. to portray teachers’ actions and students’ responses during classroom activities, we distributed an observation sheet for teachers’ actions and students’ responses. the teachers’ actions refer to the steps of the steam edp during learning activities. meanwhile, students’ response is the response conveyed by students after receiving action from the teacher. the implementation of steam and the 4dframe learning to enhance students’ computational thinking skills is considered successful if the average score of the students’ computational thinking skills has elevated from one cycle to the next cycle and at least, 80% of students pass the minimum score of 71. results and discussion pre-test result the pre-test was conducted on wednesday, july 21 st 2021 in two hours. the test questions were distributed through google classroom along with work instructions. the results presented that the average score of students was 63.00 and the percentage of students who passed the minimum score was 38%. it implies that many students could not answer the questions based on computational thinking skills. then, the researcher organized an orientation to the students by introducing the 4dframe and explaining the application during steam learning activity. rio mardani suhardi & gusnandar yoga utama 55 student activities with steam and 4dframe learning learning with steam and 4dframe was conducted for six meetings. the topics provided to stimulate students’ computational thinking skills encompassed the warka water tower, batam-bintan straw bridge and planting machines. the learning was performed in two cycles and the following is a summary of the learning. warka water tower the warka water tower is a water harvesting system which was designed by arturo vittori and andreas vogler to help ethiopians produce clean water. this topic was selected as an initial form of learning orientation with 4dframe teaching assistance to students. figure 2. students’ warka water tower design process. at the beginning of the teaching and learning activities, the teacher directed students to practice the phases of the edp. from the problem, several students asked questions about how warka tower produces clean water. based on the teacher’s explanation, students were instructed to imagine by making an ideal warka water tower in accordance with the criteria and constrains. in the create phase, students were demanded to generate an ideal warka water tower design plan illustrated by figure 2 (a). then, they employed 4dframe to develop the two models of the water tower as displayed in figure 2 (b) and (c). during this activity, students generated algorithmic thinking in which they establish warka water tower step by step by implementing four frames by adjusting the designs they have formulated, and by generating abstraction thinking skills where students selected the right tubes and connectors of the 4dframe. algorithmic thinking is a method to obtain a solution through clear definition of steps, while the abstraction skill makes it easier to think about making artefacts more understandable by decreasing unnecessary details (csizmadia et al., 2015). in the improve phase, the students performed a trial design, conducting reflection and improvement for the next cycle as presented in figure 2 (c). batam-bintan straw bridge the topic of batam-bintan straw bridge was selected by considering the local issue. kepulauan riau provincial government plans to build a bridge which connects two islands, batam and bintan. the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 56 at the beginning of the lesson, students in groups were provided directions to determine the actual distance between batam and bintan island by utilising google maps. this stage stimulates students’ abstraction skills in which students chose necessary features on google maps app to identify the distance between two points. the learning was continued in accordance with the principles of the edp in steam learning. it began with students observing the problem and asking some questions, such as how to connect the two regions to access links and economic equality. then, in groups, students conducted brainstorming to provide several alternative solutions, until the best solution was selected to bridge the two areas. figure 3. students’ batam-bintan bridge design process. based on figure 3 (a), students were instructed to design a bridge. then, they made a prototype of the first design as displayed from figure 3 (b). during this edp stage, students developed their algorithmic thinking skills in creating ideal bridge. furthermore, in improving the phase, they compared the two design (c), and then evaluated the experimental results, hence, they could be enhanced in the next cycle (d). these steps are tremendously beneficial for training students’ evaluation skills. csizmadia et al. (2015) emphasised that evaluation is a process to ensure that a solution, whether an algorithm, system or process is fit for the purpose. planting machine to inspire sustainable living in agriculture, we required machines which are able to distribute seeds precisely. therefore, the last topic selected was planting machines to make students more active during learning process. rio mardani suhardi & gusnandar yoga utama 57 figure 4. students’ planting machine design process. in the beginning, students were provided the opportunity to ask questions associated with the problems which are what the right solution is, and what the constraints are. students in groups were discussing what the best solution was, until the planting machine was selected as the solution to create an environmental seed planting car as displayed in figure 4 (a). with the contextual problems, students were asked to provide simpler solutions by creating a planting machine which is able to save natural resources as presented in figure 4 (b). this design stage helps students to enhance decomposition skills. on the other hand, students also attempted to sharpen the generalisation skills through identifying problems and associating the solutions provided with the implementation of several disciplines in the fields of science, engineering, mathematics, and art (csizmadia et al., 2015). in the create phase, students made two prototypes suitable with their designs by utilising the 4dframe toolkit. finally, both designs were then examined, compared and improved again in the next cycle as illustrated in figure 4 (c) and (d). reflection of cycle i the objective of the reflection was to determine the success rate of the actions. this research was considered to be successful if the steam and the 4dframe were at least in good category. the second criterion is that at least 80% of students who administered the test passed the minimum test score of 71. based on the observer notes, the application of learning at the first, second, and third meetings was in good category. thus, the implementation of learning has fulfilled the indicators of research success. perceiving the implementation of learning, the results of the post-test in the form of students’ scores became a consideration for the success of the research. the test administered was a computational thinking ability test. the researcher determined a score to the students’ test results with a scale of 0 to 100. students were considered complete if they acquired a minimum score of 71. here, 29 students completed the final test and 11 students did not complete. hence, the percentage of students who obtained minimum score was 73%. the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 58 as less than 80% of students passed the minimum score on the post-test of cycle i, the study did not fulfil the success indicators. therefore, the study was continued to the cycle ii. before continuing the research to cycle ii, researchers and observers evaluated the strengths and weaknesses of the learning in cycle i. strengths in cycle i were maintained, while deficiencies in cycle i were enhanced. in the learning activities, students seemed enthusiastic. it can be identified from students who have attempted to create product designs before the learning started. most of the students in the group also looked quite active during the group discussion. the application of 4dframe was tremendously helpful for students to generate designs and to conduct design trials. by employing 4dframe in implementing geometric modelling in school, students learn various interdisciplinary topics in an active, meaningful, and fun way (fenyvesi et al., 2016). thus, the utilisation of 4dframe was maintained in cycle ii. during group activities, there were some students who did not obtain a role and contribute to the group. it was caused by the large number of group members. each group consisted of 5-6 people with heterogeneous skills. during the discussion, the active and high-achiever students were dominating, while others were passive. in overcoming this problem, the teacher demanded group members to pay attention to the group’s goals and to assign duties or responsibilities to every member. another problem encountered in this cycle was the limited 4dframe toolkit which was distributed to students. they could only create one product design. it was difficult to identify comparisons when a trial was conducted. hence, the teacher provided an alternative solution by providing plastic straws and connectors such as 4dframe made with thick paper. reflection cycle ii the reflection cycle was conducted by observers. researchers examined the results of observations, student interview data, and the results of the post-test. the percentage of students who acquired the minimum score was 88%. it presented an increase of 15% of students who passed minimum score. therefore, the indicators of research success were fulfilled. hence, the research was not continued to the next cycle. in the implementation of learning, there was no problem in implementing edp steps and utilising the 4dframe toolkit. based on the results of observations on the teacher actions and student responses, the implementation of learning at the fifth and sixth meetings were considered very good. it can be observed from all the learning steps provided in the lesson plans which were performed by researchers very well. research findings the implementation of learning in cycle ii was elevated in accordance with findings in cycle i. the results of observations on the application of learning in cycle i for the first, second, and third meetings were in the good category. meanwhile, the observation results of the implementation in the second cycle for the fifth, sixth and seventh meetings were in the very good category. rio mardani suhardi & gusnandar yoga utama 59 the results of the post-test in the cycle ii enhanced compared to the cycle i. in the cycle i, the average score of the students’ computational thinking test was 73%. meanwhile, in the second cycle, the average score of the students’ computational thinking test was 88%. the 4dframe-assisted steam learning had encouraged students to be enthusiastic. furthermore, the learning was effective to encourage students in completing the worksheets. based on interviews with some students, they enjoyed learning by utilising 4dframes. steam learning is a lesson which was time consuming. it requires consideration from the teacher in presenting the number of problems in every meeting. moreover, it was difficult for teachers to manage online classes with 40 students at the same time. students with high abilities were more dominant in group activities and caused students with low abilities to be passive as they were not involved in the learning activities. in overcoming this problem, the teacher explained the groups of students about the importance of working in team and sharing roles in group activities. improving students' computational thinking ability computational thinking skills in this study were students’ abilities in (1) algorithmic thinking, (2) decomposition, (3) providing explanations, (4) generalisation, and (5) evaluation to determine the conjectures they formulated in solving related problems to mathematics material in steam learning of the three topics. students' computational thinking ability was evaluated by providing a final test after students were provided an action in the form of a steam and the 4dframe learning strategy. the objective of the test was to determine the success of the teacher’s actions in enhancing students’ computational thinking skills. the data for the final test of cycle i presented that 73% students or 29 of 40 students obtained a minimum score of 71. the test result of the first cycle increased compared to the results of the initial test conducted before performing the action. the results of the initial test revealed that only 38% of students passed the minimum score. although there was an increase from the initial test to the final test of cycle i, the study was considered unsuccessful because the percentage of minimum passing score did not acquire 88%. in the final test of the second cycle, 88% of students or 35 of 40 students obtained a minimum score of 71. thus, after being provided action in the second cycle, the research fulfilled the established research success indicators. one of the weaknesses in the first cycle is presented in figure 5, which displays students’ answers in the post-test. based on figure 5 (a), student 1 experienced difficulty in abstracting questions. meanwhile, in figure 5 (b), student 2 was quite good at abstracting and generalising the questions provided. student 2 performed basic algorithmic by following instructions by formulating a step-by-step solution. algorithmic thinking is a method to identify a solution through a clear definition of steps (csizmadia et al., 2015). the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 60 (a) (b) figure 5. students’ answers in cycle i. translation problem: robi goes to mira’s birthday. he cannot see colour very well. yellow (c) will be seen as green (a). while blue (d) is seen as red (b). robi holds up a row of balloons to welcome his guests. challenge: select two rows that look the same for robi! answer of the students in figure 4 (a): answer a is the same as answer b answer of the students in figure 4 (b): if all c is replaced with a, you will obtain: a). a d a e d a f a b b). a b a e b a f a d c). a d a e b b f a a d). a b a e a b f a d if all d is replaced with b, you will obtain: a). a b a e b a f a b b). a b a e b a f a b c). a b a e b b f a a d). a b a e a b f a b then, the correct answer is a and b in the first stage illustrated by figure 5 (a), student 2 changed c to a with a series of balloons, then continued by changing d to b. it also presented the recursive strategy (decomposition) that student 2 had performed. students 2’s abstraction ability was identified when she merely concerned on the requested balloon and ignored other balloons. at the end, student 2 formulated generalisations by adjusting the pattern she created with the answer choices. rio mardani suhardi & gusnandar yoga utama 61 another given problem is about cone. given a chocolate shaped cone divided into four parts a, b, c, and d. the height for each part is x. the students will determine (a) the ratio between area surface a to area surface b, (b) the ratio between area surface b to area surface c, and (c) the ratio between area surface c to area surface d. figure 6 illustrates one of answers of the students. figure 6. the answer of student 3 for the first question. based on the problem, student 3 obtained the equation to identify the surface area of a which is the area of the blanket of cone a minus the area of the blanket of cone b (algorithmic thinking) that is other than obtaining the equation, the student was also explaining the work in details as shown by figure 7. figure 7. student 3 elaborated the first question. translation explanation: we understand that: height of cone a:b:c:d = 4x:3x:2x:x or it can be simplified to 4:3:2:1. by employing the principle of congruence, it identified that the ratio between the radius and the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 62 the height of each cone is the same. the ratio of the radius of the cone a:b:c:d = 4:3:2:1 or it could be the same as 4x:3x:2x:x. a = 4, b = 3, c = 2, d = 1. with all this information, it can be implied that the surface areas of a, b, c, and d are the area of the blanket. area a = area of blanket of cone a area of blanket of cone b area b = area of blanket of cone b area of blanket of cone c area c = area of blanket of cone c area of blanket of cone a on the other hand, one of the successful factors of this research is displayed in the figure 8. the first thing that student 3 performed in answering the question was to simplify the equation (decomposition) of the comparison of shapes a, b, c and d: to by utilizing the concept of similarity (abstraction), student 3 obtained the ratio of the radii of the a, b, c, and d planes as well: . figure 8. the answer of student 3 for the second and third question. these solutions were also applied to solve the second and third question by replacing each corresponding variable (generalisation). the ability to generalise can be observed when students are able to adapt a solution, or part of a solution, to be implemented to a whole class of similar problems (csizmadia et al., 2015). conclusion based on the data exposure and discussion, and the methodological limitations due to the size of the sample, the steps of steam and the 4dframe learning were able to enhance students’ thinking and computation abilities by implementing each stage in the edp. based on this research, the following suggestions may be of interest to readers with similar teaching-learning contexts. 4dframe-assisted steam learning is a learning strategy which is able to elevate students’ computational thinking skills and to be employed as an alternative in learning mathematics. learning with the 4dframe-assisted steam strategy is easier to apply if the learning is conducted face-to-face. when it is performed virtually, there are several problems might occur such as limited student access to the internet, and the students were reluctant to discuss in groups. rio mardani suhardi & gusnandar yoga utama 63 references anito, j. j., elipane, l., sarmiento, c., & butron, b. (2019). the philippines steam education mode. philippines: pnu university publication office. arikunto, s., suhardjono, & supardi. (2017). penelitian tindakan kelas. pt. bumi aksara. baines, a. s. (2019). steam education theory and practice. cham: springer. csizmadia, a., curzon, p., dorling, m., humphreys, s., ng, t., selby, c., & woollard, j. (2015). computational thinking a guide for teachers. london: computing at school. cuny, j., snyder, l., & wing, j. (2010). demystifying computational thinking for noncomputer. retrieved from http://www.cs.cmu.edu/~compthink/resources/thelinkwing.pdf daugherty, m. k. (2013). the prospect of an “a” in stem education. journal of stem education: innovations and research, 14(2), 10–15. https://www.jstem.org/jstem/index.php/jstem/article/view/1744/1520 fajri, m., yurniwati, & utomo, e. (2019). computational thinking, mathematical thinking. dinamika matematika sekolah dasar, 1(1), 1–18. fenyvesi, k., park, h.g., choi, t., song, k., & ahn, s. (2016). modelling environmental problem-solving through steam activities: 4dframe’s warka water workshop. bridges finland conference proceedings, pp. 601–605. retrieved from https://jyx.jyu.fi/handle/123456789/50739 gülhan, f., & şahin, f. (2018). activity implementation intended for steam (stem+art) education: mirrors and light. journal of inquiry based activities, 8(2), 111–126. hester, k., & cunningham, c. (2007). engineering is elementary: an engineering and technology curriculum for children. presented at asee annual conference and exposition. retrieved from https://www.eie.org/sites/default/files/research_article/research_file/ac2007full8.pdf khine, m. s. (2018). computational thinking in the stem disciplines. springer. https://doi.org/10.1007/978-3-319-93566-9. park, h. (2018). mathematics learning through arts, technology. 8th icmi-east asia regional conference on mathematics education, p. 111. taiwan. pisa. (2018). pisa 2022 mathematics framework. retrieved from https://pisa2021maths.oecd.org/files/pisa%202021%20mathematics%20framework%20draft.pdf pisa. (2019). pisa 2018 results. retrieved from https://www.oecd.org/pisa/publications/pisa-2018-results.htm sands, p., yadav, a., & good, j. (2018). computational thinking in k-12. in m. s. khine (eds.), computational thinking in the stem disciplines, (pp. 154–164). usa: springer international publishing. the effect of implementing steam and 4dframe learning in developing students' computational thinking skills 64 southeast asian mathematics education journal volume 11, no 2 (2021) 83 growth mindset in mathematics among ninth-grade students via 5ps learning model christian r. repuya bicol state college of applied sciences and technology peñafrancia avenue, naga city crrepuya@astean.biscast.edu.ph abstract this paper determined the effects of implementing the prepare, perform, process, ponder, and practice (5ps) learning model in teaching ninth-grade students’ growth mindset in mathematics. this study employed the quasi-experimental design and mixed-method research approach to answer the research questions with 60 ninth-grade students at a public secondary high school in the philippines. the study administered a growth mindset questionnaire, informal interviews, learning journals, and focus group discussions (fgd) on identifying learning experiences and mindsets of the students. findings presented that employing the 5ps learning model significantly influences students' mindset in mathematics. meanwhile, traditional teaching does not significantly affect students' mindset in mathematics. the implementation of the 5ps learning model has a significant positive effect on students' growth mindset in mathematics. the results of the study are limited merely to the participants included in the study; similar research utilizing the 5ps learning model to other learning areas with a larger sample is recommended for more generalizable results. keywords: growth mindset in mathematics, 5ps learning model introduction mathematics education begins in early education. wherein progressive schools designed play a crucial role as center of curriculum to teach students about mathematical concepts through firsthand experiences and at the same time possess fun while learning. its objective is to develop students’ cognitive ability to identify shapes, patterns, and connections and help them understand the significance of mathematical applications to the real world. however, the study of mazana, montero, and casmir (2019) revealed that students exhibit a positive attitude towards mathematics, but their attitude is less positive when the students move to higher levels of education. it displays that as the students’ grade level increases, their interest and positive attitude towards mathematics decreases, and it becomes a challenging subject to learn and appreciate as students also develop fixed mindsets. the pisa 2018 results in the growth mindset. in the philippines, only 31% of the students possessed a growth mindset, resulting in a score of 353 and ranked 76th, the second lowest ranking among participating countries. the data demonstrates that the students of the countries with higher growth mindsets conducted better than those with a lower growth mindset (oecd, 2018). with a growth mindset, intelligence and “smartness” can be learnt, and that one believes that the brain is able to grow from exercise (boaler, 2013). dweck (2016) further asserted that with a growth mindset, students will be able to work hard, perform more effort, and learn effectively, displaying a desire for challenge and resilience of growth mindset in mathematics among ninth-grade students via 5ps learning model 84 the failure. it frequently occurs in mathematics learning. in contrast, students with a fixed mindset believe that they are intelligent or not wise in mathematics. park et al. (2016) unveiled that student with a growth mindset possessed higher mathematics achievement levels and that the more the teachers are focused on classroom performance outcomes (learning outcomes), the more students develop fixed mindsets over the school year. they emphasized that concentrating on learning outcomes is similar to saying – “you need to know this by the end of the year”. these words orient students toward performance but away from “learning”. it is in accordance with blackwell et al. (2007) who explained that fixed mindsets are associated with lower achievement as supported by many conclusive pieces of evidence. researchers have revealed that mistakes are significant opportunities for learning and growth (heinze & reiss, 2007; moser, schroder, heeter, moran, & lee (2011); sarwadi & shahrill`, 2014), however, students generally regard mistakes as indicators of low ability (fixed mindset). every time students produce mistakes in mathematics; new synapses are generated in their brains. the students and teachers should value mistakes and should not consider them as learning failures but as learning achievements. if students produce pages of correct work, their brains are not growing, and development opportunities are missed. conley (2014) employed a growth mindset to provide feedback to students and discovered that students become more persistent and confident in attempting situations and possess happier outlooks. madden (2015) revealed that the students with low mindset scores definitely own low homework percentages but many other students with higher mindset scores do too. the study concluded that encouraging a growth mindset in students generates improved learning, motivation to learn, and standardized test scores. menanix (2015), in his study, discovered that growth mindset instruction coupled with challenging mathematics and opportunities for students to share authority over their learning was significantly more effective at constructively influencing students' mindsets and performance. the study emphasized that when students were deprived of the opportunity to experience a growth mindset by solely working on procedural routine mathematics, they did not possess authority to develop their ideas. their mindsets did not shift in productive ways for engagement learning. furthermore, the study of amant (2017) uncovered that teacher who scored exhibiting growth mindset characteristics employed activities with higher levels of rigor more frequently, which implies that they plan more rigorous activities or ask more rigorous questions. boaler (2013) also asserted that teachers' practices that contribute to fixed mindset thinking among students should be addressed. the researcher indicates that further research is required to determine the significance of the idea. thus, future studies are more able to examine closely the relationship between student’s academic achievement and teacher’s mindset and practices (sun, 2015). with the significance of developing a growth mindset among students and considering the change of students' learning preferences and instruction modes, to the objective of this study is to investigate the effects of utilizing the 5ps learning model in teaching to the students' growth mindset in mathematics. the 5ps learning model (5ps) was developed based on the learning principles and theories provided in the k-12 curriculum guide of the education department of the philippines, comprising the five key steps (italicized): prepare, perform, christian r. repuya 85 process, ponder, and practice. it was employed as a teaching methodology in designing lesson plans which encompasses three parts: the introduction to the lesson, lesson proper, and closure. the 5ps learning model was explicitly designed for blended learning and possessed the advantage of emphasizing differentiated instruction particularly along with the key steps prepare and process. it also administered the key step ponder to provide students the avenue to develop a positive attitude and growth mindset in mathematics through reflective learning. lastly, it also employed differentiated assessment along practice, which believes that the students should be provided equal opportunity to be assessed for and of learning. the study results will be beneficial to the teachers and students in communicating growth mindsets and as a basis for utilizing the 5ps learning model that suits different learning modalities compared to an existing lesson design because the 5ps learning model supported the students to develop a growth mindset. methods research design this study administered the quasi-experimental design. the intervention (instructional materials) on the use of 5ps lesson design and 5ps-designed activity guide by the teacher and students was employed in the experimental group to determine its effect on students' mindset in mathematics. the controlled group was implemented to compare results without utilizing the intervention; instead, it was taught applying the traditional method following the activity, analysis, abstraction, & application (4as) lesson design with the assistance of the ninthgrade learners manual. the experimental and controlled groups equally consisted of 30 selected ninth-grade students randomly (n=60) from a secondary education institution in the philippines. the quantitative research method was utilized to examine students' scores in pretest and post-test mindset questionnaires employing frequency counts to determine the number of students in mindsets scales, paired t-test cohen’s d to identify the difference between the pre-test and post-test scores of students, and effect size of the intervention to the students' mindset in mathematics. the qualitative research method was employed to demonstrate the students' learning experiences by applying informal interviews, journals, and fgd. during the implementation, the students' learning experiences were identified through triangulation in qualitative educational research exhibiting the students' learning journal, interview of the learners, and fgd while they were under the intervention. research procedure before the study was conducted, the researcher sought written informed consent from school heads, teachers, parents, and the students. ethical considerations encompass as informing the parents and participants about the benefits of the study, duration of the students’ involvement, the collected data, confidentiality, and that they can withdraw anytime. the researcher demanded a cooperating teacher (mathematics teacher) to implement the intervention and teacher-observers (selected teachers) to assist in observing and recording the learning experiences of the students. a week before implementing the intervention, the cooperating teacher, the actual teacher, decided the classes as a controlled group and growth mindset in mathematics among ninth-grade students via 5ps learning model 86 experimental group through toss coin. after the pre-test on the mindset survey, the intervention was performed, comprised of six lessons in the 4th quarter of the ninth-grade curriculum. the same cooperating mathematics teacher was teaching the experimental group and controlled group. the experimental group made use of the developed instructional materials. the cooperating teacher followed the 5ps lesson design in which along prepare (5ps key steps italicized), the students prepared for lessons at home and along perform, the students conducted pre-class learning tasks. along the process, there are three stations – a teacher's station, a collaborative station, and a computer station. the cooperating teacher grouped the students based on their learning needs, readiness, and interest, and guided them in rotating to the three stations. at the teacher's station, the students participated in the teacher's discussion, asked questions, verified concepts, received teacher feedback, and performed individualized assessments. at the collaborative stations, the students participated and worked collaboratively with their classmates to accomplish the assigned tasks. at the computer stations, the students reviewed the lecture videos to comprehend the concepts better (with their group) which were previously watched at home and discussed by the teacher. along ponder, the students wrote their learning journals for reflective learning and proceeded to practice, in which they accomplished individual lesson assessments. in an instance that there were students who completed the lessons quickly or were performed in the stations, the teacher assigned them to help their classmates with their learning needs to avoid congestion in the stations. during the experiment implementation, some notable aspects were noted. one of the aspects is in terms of the student's interest to study the instructional materials, particularly during the key step process. another aspect is the positive behavior in the intervention application reflected by their eagerness to learn the lessons, motivation to participate in the activities, and persistence to solve practice problems by the students, particularly in the process and practice of the 5ps learning model key steps. teacherobservers, teachers invited to observe, were present during the implementations. in the controlled group, the teacher employed the traditional teaching method following the 4as lesson format and administered the department of education ninth-grade learners' manual. the students participated in the activity provided by the teacher at the beginning of each lesson. the teacher presented and discussed the lessons. after the discussions, the teacher equipped students with another activity associated with the topics, then analysis and abstraction, in which the teacher asked questions to solicit higher-order thinking among students. the students summarized the lesson learnt and completed the lesson assessments. after the implementation, the researcher administered a post-test mindset questionnaire, informal interviews, and fgd at each student's home, and the participant applied the minimum health protocol. the unique data gathered in the fgd was the students' statements displaying consensus about their significant learning experiences on the implementation of the 5ps learning model, which became helpful in the thematic analysis. research tool and analysis the effect of the intervention employing the 5ps learning model on the students’ mindset in mathematics was examined by utilizing the interview schedule by dweck (2006) christian r. repuya 87 questionnaire. in an interview schedule, the researcher inquired each student the question and checked their responses in the mindset questionnaire. the students responded, whether with a growth or fixed mindset, to each statement. a student with a growth mindset with some fixed ideas believed that his/her ability could be enhanced in mathematics, but still, he or she still possessed some fixed ideas that some of his or her mathematical ability could not be changed. on the other hand, a student with a fixed mindset believed that his/her ability in math could not be changed or improved; however, he/she had some ideas that his/her math ability could be changed in some ways. it encompassed 20 statements that reflect growth mindset (items 2, 3, 5, 6, 9, 10, 13, 15, 18 and 19) or fixed mindset (1, 4, 7, 8, 11, 12, 14, 16, 17, and 20). the statements were assigned with the following point values: table 1 point values of mindset questionnaire scale fixed mindset growth mindset strongly agree 0 3 agree 1 2 disagree 2 1 strongly disagree 3 0 to verify and obtain students’ ideas not caught in the mindset questionnaire, informal interviews, learning journals, and fgd were conducted. informal interviews were performed because the setting of the interviews was outside the students’ homes, and they were instructed to share their learning experiences. after the interviews, the neighboring students were collected for an fgd, which also aims to corroborate their ideas and responses. learning journals became the source of additional information which also helped in analyzing results. results and discussion table 2 presents the summary of the pre-test and post-test frequency counts of the students’ mindsets in the experimental and controlled group. there were eight students in the experimental group who possessed a fixed mindset with some growth ideas, while 20 students owned a growth mindset with some fixed ideas, and two students who possessed a strong growth mindset in the pre-test. during the post-test, eighteen (18) students owned a growth mindset, and twelve (12) students developed a strong growth mindset. table 2 summary of the pretest and posttest frequency counts of two groups on growth mindset in mathematics mindsets in mathematics pretest posttest experimental group controlled group experimental group controlled group fixed mindset 0 0 0 0 fixed mindset with some growth ideas 8 3 0 6 growth mindset with some fixed ideas 20 24 18 21 robust growth mindset 2 3 12 3 total 30 30 30 30 growth mindset in mathematics among ninth-grade students via 5ps learning model 88 it can be identified that both the students in the experimental and controlled group possessed fixed mindset with some growth ideas in the pre-test, in the post-test, the students of the experimental group developed a growth mindset which increased the numbers of students with robust growth mindset from two to 12 leaving no students with fixed mindsets. however, in the post-test of the controlled group, three students were included in the students with fixed mindsets with some growth ideas, resulting in the increase of students with fixed mindsets. more specifically, 10 students who were placed into group of students with strong growth mindset during post-test in the experimental group, 9 of them were from growth mindset with some fixed ideas, and one of whom originated from fixed mindset with some growth ideas to possessing a strong growth mindset. in contrast, the three students in the controlled who were involved in to possess a fixed mindset with growth ideas in the post-test had a growth mindset with some fixed ideas in the pre-test. the results revealed that students in the experimental group demonstrated a comprehension of growth mindset role in their learning. considering the student’s statement coming from having a fixed mindset with some growth ideas to a robust growth mindset, she stated in an informal interview: “dakula po ang naitabang sakuya kang instructional materials na gamit ang 5ps learning model ta dahil guided na po ako sa gigibuhon ko sa harong sa pagadal ko, mas nakakaprepare ako sa mga lesson saka sa tabang kang mga kaklase ko narealize ko na maiimprove man talaga ang kakayanan ko sa math kapag nagaadal ako tapos naghihigos sa pagparticipate sa mga activities (the instructional materials that feature the implementation of 5ps learning model helped me a lot because i am guided on what i am going to apply at home in studying. i can prepare better for the lessons and with the help of my classmates, i realized that my math ability could really be improved if i am studying and being hardworking in participating the activities)”. based on the student’s statements as previously discussed in their learning experiences, the student's growth mindset development could possibly be attributed to the implementation of the instructional materials, which promotes growth mindset through self-learning that emphasizes efforts in learning mathematics. in the controlled group, the increase of students having fixed mindset could be attributed to the pressure which the students may experience at a provided time and might be having lesser motivation for self-learning because the teacher reluctantly performed with the classroom discussion only, which also resulted to less one-on-one feedbacking. to determine if the implementation of the instructional materials along the experimental group impacts students' pre-test and post-test on mindset in mathematics, the mindset scores for each component were subjected to t-test and cohen's index (table 3). christian r. repuya 89 table 3 pre-test and post-test mindset scores t-test and cohen’s index on mindset in mathematics of the two groups groups t-value p-value interpretation cohen’s index (d) interpretation experimental 5.058 0.000 significant 0.92 large controlled -1.264 0.216 not significant the mindset scores of the pre-test and post-test of the experimental group were highly significant, with a t-value of 5.058 and p-value of 0.000 examined at a 95% confidence level. analysis of the effect employing cohen's index presented a large effect with d=0.92. on the other hand, the pre-test and post-test of the controlled group were not significant, with a tvalue of -1.264 and a p-value of 0.216. it is indicated that the computed effect size of this skill is negative. a negative size implies that the effect decreases the mindset scores. to determine if there is a significant difference in the mindset scores of the students between the experimental and controlled groups, a t-test for independent sample was administered. the results revealed a significant difference between mindset scores of the two groups displayed by the t-value of 4.395 and p-value of 0.000, which was significant at a 95% confidence level. the findings from the study presented that the implementation of intervention significantly affects students' mindset in mathematics, while traditional teaching does not significantly affect students' mindset in mathematics. it implies that the conduct of intervention enhanced students' mindset in mathematics in the experimental group. on the other hand, other factors may affect students' mindset during the implementation of the study due to the delayed posttest of the mindset survey caused by health restrictions. however, based on the data gathered by the study, it was strongly possible that the growth mindset development and robust growth mindset among students in the experimental group could be attributed to the application of the instructional materials. in the 5ps key step, perform, the students were trained not to be afraid to attempt things and produce wrong answers which results the students are reluctant to perform harder in learning. along the process, the station rotation model encouraged students to learn autonomously and that their knowledge will grow through study and engaged in the activities through stations. at the same time, along ponder, the students were expected to reflect on their learning growth and opportunities. finally, along with the practice, the students' patience, perseverance, diligence, eagerness, and other traits were reinforced because they require to strive harder to learn and pass the lesson examination and eventually developed a growth mindset with the assistance of the teacher's encouragement and growth mindset messages. with growth mindsets, students are expected to be more motivated to learn, work harder, are less discouraged by difficulty, and employ more effective strategies for learning. the result in the controlled group was associated with the finding of menanix (2015) which revealed that when students were deprived of the opportunity to experience a growth mindset by only working on procedural routine mathematics in which they did not possess growth mindset in mathematics among ninth-grade students via 5ps learning model 90 authority to present with their ideas, their mindsets did not shift in ways that were productive for engagement or learning. in the study of boaler (2013), she asserted that teachers and schools are better to constantly communicate messages to students about their ability and learning through instructional and classroom practices (i.e., instructional materials). she then argued that if one possesses a true commitment to the communication and teaching of a growth mindset, a thorough examination of all aspects of teaching is necessary. she emphasized that mindset messages were communicated through questions asked, task, grading and feedback, mistakes, grouping, and norm-setting. to identify students' learning experiences and mindsets, informal interviews, learning journals, and focus group discussions were conducted. after thematic analysis, the following themes emerged: students’ attitude and mindset about mathematics affected their learning since the 5ps learning model corroborated mind setting, the students understand the value of attempting and failing to succeed. for this reason, the students comfortably participated during the activities and discussion without being afraid to fail. furthermore, the students become more motivated to try and learn. in an informal interview, one of the students who developed a robust growth mindset at the end of the quarter shared as follows: “in my own belief, the students who get smarter understand the topic with more efforts to get the higher achievement. meaning to say, if you exert an effort to perform things to enhance your knowledge, your life will change, but if you do not do something, you will never find relief in life (student 18)”. in an informal interview, a student (student 20) shared his experience in trying his best to study at home and at school: “dati po sir dae ako nakaka-adal sa harong pero kang naintindihan ko ang importance kang pag-adal lalo na sa math, nagaadal na ako sir. nagngagalas na ngani sakuya si mama ta nag-aadal na ako ngunian. nagcocompute ako sir sa harong, uni ang notebook ko sir oh, si mga pigkorompute ko (before i could not study at home, but when i understood the importance of studying, particularly in math, i am now studying. my mom is wondering because i am now studying. i do computations at home, sir, here is my notebook, the ones that i have been computed)”. the students’ statements corroborated the argument of dweck (2006) that better learning occurs when students comprehend the advantage of trying, failure, and success during the learning process. based on the students' statements, the learning process by implementing the 5ps learning model had upheld students understand the significance of efforts, trying, experiencing failure and success, and untiring “practice” to master the mathematical concepts and possess a full grasp of the learning competences. these lead the students to own a positive attitude on christian r. repuya 91 learning mathematics and a growth mindset affecting their learning performances. this learning principle confirmed one of the learning principles on mathematics framework for philippine basic education, which similarly explained “students’ attitudes and beliefs about mathematics affect their learning” (science education institute department of science and technology [sei-dost] & philippine council of mathematics teacher education, inc. [mathted], 2011). mathematics teaching requires positive attitude and growth mindset to perform more teaching-learning efforts. boaler (2013) explained that mindset involves “knowing that math is a subject of growth and [the math user’s] role is to learn and think about new ideas.” she also stated that “many teach mathematics with their fear of the subject.” a teacher’s mindset manipulates many aspects of his/her teaching and students’ learning, encompassing the learning tasks he provides, how classroom discourse is orchestrated, his response to mistakes, and the assessment practices he employs. when teachers learn about the research behind mathematical mindsets and identify their own current mindsets, they are able to reset their approach to learning and teaching math. they will believe that all students are able to understand and enjoy it and set them all up for success (chapman & mitchell, 2018). with the assistance of instructional materials, the teacher was able to communicate a positive attitude and growth mindset, which was tremendously advantageous in providing an equal learning opportunity for all because the teacher believed that everybody could learn mathematics and that the students just require to possess the right attitude and mindset. it also substantiated the teaching principle of sei-dost and mathted (2011) in mathematics framework for philippine basic education that “mathematics can never be learnt in an instant, but rather requires lots of work and the right attitude.” it indicates that teaching mathematics by communicating a positive attitude and growth mindset in mathematics encourages students to conduct more effort into learning. the students are also able to understand the significance of trying, failure, and success during the learning process that eventually will lead to successful enhanced mathematics learning. overcoming challenges in studying there are challenges that the students experienced during the implementation of the lessons and activities. one of the challenges was on studying and learning autonomously. from an informal interview, student 10 commented on how the instructional materials helped her, and she stated, “kaya ko nang maintindihan ang lesson kahit hindi ito itinuturo dahil nagseselfstudy ako na dati na di ko magawa-gawa (i can now understand the lesson even if not taught because i do self-study that i could never perform before)”. growth mindset in mathematics among ninth-grade students via 5ps learning model 92 student 11 also argued, “i develop myself being always in the mood to whatever task it has, to easily understand.” the statement of student 10 obviously proves that the student’s study habit has been changed for a good while student 11 adjusts her mood (attitude in learning mathematics) so that she is able to understand and learn the lessons. student 30 also experienced difficulties in performing self-study. however, she also managed to continue studying at home. she reflected, “i experienced how hard to study by myself because there’s no one to teach me, so i needed to study it and see that my answer was correct when my teacher checked the prepare.” the students’ journal entries also present evidence that they have overcome challenges in studying while answering the question, which is does the use of the instructional materials helped you learn? the students answered: student 4: “yes because noong gumamit ako ng activity guide, nadagdagan yung kaalaman ko at lalo ko pang tinibayan para marami pa akong maencountered (yes because when i used the activity guide, my knowledge increased and so i conducted more effort so that i might learn more)” student 9: “yes, because it helps students in learning. it has self-assessment to comprehend the student’s knowledge employing their own understanding in the lesson. it helps to study hard.” student 20: “yes, dahil mas madali itong matutunan at dahil nagkukusa kami para matutunan and isang bagay na walang nag-gaguide (yes, because it was easy to learn, and we initiated so we can learn things even nobody is guiding)”. student 22: yes, dahil nalalaman ko kung ano-ano ang pagsunod sunod para malaman mo ang tamang sagot at natulungan ako nito para mapalago ang aking kaisipan. nagustuhan ko ang paggamit ng activity guide dahil mas madali na pag-aaralan kahit nasa bahay lang ako (yes, because i learnt the steps and corrected answer, hence, it helped me to improve my mind. i liked the application of the activity guide because it was easy to be studied, even i was just at home)”. student 30: “yes, because of that, i learnt on how to study by myself.” on a focus group discussion, the students approved that one of their significant learning experiences was overcoming challenges to study. the students’ statements in studying indicate that the students also encountered challenges and were able to study autonomously, resulting in a more meaningful mathematics learning experience. in general, the study results recommended the necessity for instructional materials, learning model, teaching approach, and methods that communicate growth mindset messages. findings of the study propose that the application of the 5ps learning model provided the students with necessary learning experiences that develop a growth mindset in mathematics. however, the study is limited solely to the participants included in the study; a larger sample is required for more generalizable results. christian r. repuya 93 conclusion the implementation of the 5ps learning model in teaching mathematics significantly affects students’ mindset in mathematics, while conventional teaching does not significantly affect students’ mindset in mathematics based on the findings of this study. specifically, the application of the 5ps learning model possesses a significant large effect on enhancing students’ growth mindset in mathematics. the students manifest a growth mindset in mathematics by presenting courage in overcoming challenges in learning. there was an emphasis on the teaching and learning efforts that require the teacher’s positive attitude and growth mindset in mathematics. researchers could examine using the 5ps learning model in teaching other learning areas and with larger samples to develop students’ growth mindset. acknowledgements the study recognized the efforts of the students, parents, observers, teachers, and school heads involved in the study for their participation and cooperation during the implementation and data gathering. it also recognizes the department of science and technology for the financial support to make this study into reality and to the department of education for the approval to conduct the study in the school. references amant, t. r. s. (2017). the effect of teacher mindset to the low-tracked students [doctoral dissertation, university of pittsburgh). blackwell, l., trzesniewski, k., & dweck, c. (2007). implicit theories of intelligence predict achievement across an adolescent transition: a longitudinal study and an intervention. child development, 78(1), 246-263. https://doi.org/10.1111/j.14678624.2007.00995.x boaler, j. (2013). ability and mathematics: the mindset revolution that is reshaping education, forum, 55(1), 143-152. https://doi.org/10.2304/forum.2013.55.1.143 chapman, s., & mitchell, m. (2018). mindset for math. the learning professional, 39(5), 60-63. conley, a. (2014). nurturing intrinsic motivation and growth mindset in writing. edutopia. https://www.edutopia.org/blog/intrinsic-motivation-growth-mindset-writing-amy-conley dweck, c. s. (2016). what having a growth mindset actually means. https://hbr.org/2016/01/what-having-a-growth-mindset-actually-means dweck, c. s. (2006) mindset: the new psychology of success. new york house inc. https://iusd.org/sites/default/files/documents/mindsetquiz_module5.pdf heinze, a., & reiss, k. (2007). mistake-handling activities in the mathematics classroom: effects of an in-service teacher training on students’ performance in geometry. proceedings of the 31st conference of the international group for the psychology of mathematics education, pp. 9-16. seoul: pme about:blank about:blank about:blank https://www.edutopia.org/blog/intrinsic-motivation-growth-mindset-writing-amy-conley about:blank growth mindset in mathematics among ninth-grade students via 5ps learning model 94 madden, j. (2015). mindset and the middle school math student. master’s thesis, university of wisconsin river falls. https://drum.lib.umd.edu/bitstream/handle/1903/11138/ berger_umd_0117n_11694.pdf;sequence=1 mazana, m., montero, c., & casmir, r. (2019). investigating students’ attitude towards learning mathematics. international electronic journal of mathematics education, 14(1), 207-231. https://doi.org/10.29333/iejme/3997 menanix, s. (2015). teaching for a growth mindset: how contexts and professional identity shift decision-making. doctoral dissertation, university of california, berkeley. http://digitalassets.lib.berkeley.edu/etd/ucb/text/menanix_berkeley_002 8e_15164.pdf moser, j., schroder, h., heeter, c., moran, t., & lee, y. (2011). mind your errors: evidence for a neural mechanism linking growth mindset to adaptive post-error adjustments. psychological science, 22, 1484-9 organisation for economic and cooperation development (2018). programme for international student assessment. https://www.oecd.org/pisa/ publications/ park, d., gunderson, e.,. tsukayama, e., beilock, s. (2016). young children’s motivational frameworks and math achievement: relation to teacher-reported instructional practises, but not teacher theory of intelligence. journal of educational psychology, 108, 300-313 sarwadi, h. & shahrill, m. (2014). understanding students’ mathematical errors and misconceptions: the case of year 11 repeating students. mathematics education trends and research 2014(2014), 1-10. https://doi.10.5899/2014/metr-00051 science education institute department of science and technology & philippine council of mathematics teacher education, inc. (2011). mathematics framework for philippine basic education. http://www.sei.dost.gov.ph/images/downloads/publ/sei_mathbasic.pdf sun, k. (2015). there’s no limit: mathematics teaching for a growth mindset. doctoral dissertation, stanford university. https://stacks.stanford.edu/file/druid:xf479cc21 94/sun-dissertation-upload-augmented.pdf about:blank about:blank about:blank about:blank about:blank about:blank about:blank about:blank 13 southeast asian mathematics education journal, volume 11, no 1 (2021) mathematical anxiety as predictor of learning motivation strategies jeovanny a. marticion zamboanga del norte national high school, the philippines jamarticion@up.edu.ph abstract empirical findings showed how the mathematical anxiety predicts the academic performance of learners. as a coping mechanism, learners are left with various choices in dealing with subjects involving mathematical concepts. one way of coping with these subjects is a preference for learning motivation strategies. the motivation strategies were categorized into cognitive, meta-cognitive, non-informational resources management and information resources management. however, there is scarce literature on how anxiety could predict the behaviour of an individual accommodation of these strategies. this led the researcher to investigate the predictive behaviour of mathematical anxiety on utilization of learning motivation strategies among senior high school students enrolled in the science, technology, engineering and mathematics program. the program was crafted for students who are inclined towards sciences and mathematics. results revealed that respondents have a moderate level of anxiety. during the course, anxiety contributes to the level of anxiety of the respondents. the self-regulation strategy was the most commonly utilized learning motivation strategy among respondents, while peer learning was the least utilized among the learning motivation strategies. however, the bivariate analysis showed anxiety was moderately related to rehearsal, organization, effort regulation, time and study environment, peer learning and help-seeking strategies. regression analysis was also applied to reveal how anxiety predicts specific learning motivation strategies. analysis disclosed that anxiety predicts the utilization of effort regulations strategies in learning mathematically inclined subjects. the findings provided a new perspective on how anxiety allows learners to utilize available strategies to understand various concepts. teachers are encouraged to cultivate a culture of regulation, an environment conducive for learning, peer interaction and access to internet-based or digital resources for learning keywords: mathematical anxiety, mathematics anxiety reasoning scale, learning motivation strategies, regression analysis introduction the performance of learners in mathematics mainly relies on the influential factors present within themselves and their environment. acquiring conceptual and procedural knowledge in this subject requires moderate control of these factors. if appropriately addressed, this could enhance or improve their willingness to learn, thus unlocking their full potentials. however, if left unchecked, it might impair the ability of a learner to learn basic concepts. eventually, this will lead to the learner’s decreasing engagement in complex mathematics-related tasks. this manifestation accounts for the mathematical literacy results of the programme for international student assessment (pisa), where only 0.01% of the filipino students could perform on complex problems through the development of techniques and mastery of mathematical symbols, operations, and relationships. this is true for public and private schools (department of education, 2019). despite the various claims on how self-discipline and 14 southeast asian mathematics education journal, volume 11, no 1 (2021) positive outlook determine a learner's success in mathematics, factors related to psychological aspects in approaching mathematics could somehow be linked with their poor performance. studies revealed how learners undergo stress, fear, unfavourable sentiments, and frustration caused by mathematical anxiety (xu, 2004). when this anxiety is combined with lower reports of self-confidence, this increases their levels of distress and tendencies to neglect the significance of mathematics learning (jameson & fusco, 2014; oxford & vordick, 2006). while overcoming anxiety has always been a challenge, learners prefer various strategies in acquiring knowledge and skills. it represents the effort to practice, comprehend and absorb information during the classroom experience providing more meaning to cognitive and affective learning (kafadar, 2013). studies were conducted on investigating and exploring the strategies commonly used in mathematics classes. a study conducted by gurat (2018), revealed that student and teachers of saint mary’s university applied various problem-solving strategies, i.e. cognitive, meta-cognitive, prediction, monitoring and evaluation. in a sample of taiwanese students, liu and lin (2010) found out that students have the weak motivation and less utilization of learning strategies for mathematics. the ability to alter strategy and reasons when introduced to mathematical problems has improved procedural and conceptual knowledge (star et al., 2015). anxiety and achievement the mechanisms of anxiety, performance and achievement of learners in mathematics could be explained by how learners accommodate cognitive sources (processing efficiency theory) (eysenck & calvo, 1992) and how they control (attentional control theory) (eysenck, derakshan, santos, & calvo, 2007). there should be an emphasis among individuals with higher levels of anxiety. since these people are worried that it would threaten their current goal, they will always manipulate ways to employ varying strategies. hence, this reduces the effects of anxiety (derakshan & eysenck, 2009). individual’s anxiety towards mathematics gradually reduces cognitive resources such as working memory. it is the short-term memory that initially handles information. as anxiety increases worry, worry takes significant parts of the working memories, overloads the resources, and leaves insufficient working space. processing efficiency theory may associate anxiety with performance interference by burdening the cognitive resources (eysenck & calvo, 1992). the working memory system is comprised of three components, i.e., phonological loop, visuospatial sketchpad, and central executive. the phonological loop rehearses the verbal medium, while the visou-spatial sketchpad processes and stores visual and spatial information. the two components are considered “slave” systems of the central executive, the apex of the hierarchy. when a person becomes anxious, the entire system is compromised. worrying is considered an irrelevant task that could affect its functions. in cases of dual tasks, an anxious individual would not utilize the high demands of their slave systems. consequently, the central executive becomes dysfunctional in terms of preparation, choice of strategy or techniques and attentional control (derakshan & eysenck, 2009). on the other hand, attentional control theory accounts for the interaction of two known attentional control systems when an individual feels anxious (eysenck et al., 2007). there is a sudden decrease of the goal-directed attentional system, which refers to control and execution 15 southeast asian mathematics education journal, volume 11, no 1 (2021) of a task and is influenced by goals, expectations and knowledge, and an abrupt increase of the stimulus-driven system, which is driven by threat internally and externally (corbetta & shulman, 2002). when anxiety impairs attention control, the adverse effects are on the functions involving attentional control. the central executive performs three functions. first, it inhibits using attentional control to tolerate irrelevant tasks. moreover, it shifts attention flexibly just to finish the task on hand. finally, it updates and monitors the working memory representations (derakshan & eysenck, 2009). there is substantive evidence that learners who possess higher levels of anxiety tend to have lower academic performance in mathematics. however, it is not yet clear that anxiety could also affect their working memory and attentional control. through the investigation of ibrahim (2018), less effective study strategies were constantly used by the students with higher anxiety. even helpful strategies were discarded as part of their study habits. hence, there is a need to understand the relationships between study strategies and learning outcomes in mathematics. furthermore, these strategies should be monitored for a specific time frame to accurately represent their strategies and how they could affect their learning outcomes. motivated learning strategies learners utilized various strategies in learning mathematics. resource management strategies mainly were utilized in the investigation of liu and lin (2010) using the mathematics motivated strategies for learning questionnaire (mmslq). findings revealed that helpseeking (m = 2.99) is the most utilized strategy while communication behaviour on the internet (m=1.51) is the least utilized strategy. in terms of gender, a significant difference exists in all dimensions of strategy in favour of males except for the help-seeking strategy in favour of females. students enrolled in cram schools after classes showed a significant difference in all dimensions. a lot of resource management strategies was also conducted. vaezi et al. (2018) found out how their strategies could predict students’ time and study environment achievement. regression analysis showed how these resource management strategies are related to their achievement. ahmed and khanam (2014) indicated that the academic achievement, time and study environment, effort regulation and help-seeking among grades 9 and 10 learners were significantly correlated. high-achieving students in the class utilize more time and study environment, peer, help and effort regulation than low achievers. science-inclined students use time and study environment and peer learning compared to humanities students in terms of specialization. hamid and singaram (2016) revealed that time and study environment are correlated with academic performance among medical students for two modules: becoming a professional and basic science. however, pearson’s product-moment correlation coefficient showed a weak relationship between the two variables. problem-solving strategies were also investigated among student and teachers. planning was revealed as the most utilized strategy (m = 3.58), while monitoring was the least utilized (m = 3.16). planning involves summarizing and reflecting on the answer, reflecting on tasks solved, drawing conclusions, relating future problems and relating problems with other problems. similar studies were conducted using the same instrument and found out how cognitive strategies prevailed as a learning strategy. in the study of baumgartner, spangenberg, and jacobs (2018), results showed that elaboration was the most preferred learning strategy for mathematics and mathematical literacy students, while peer learning was the least utilized, 16 southeast asian mathematics education journal, volume 11, no 1 (2021) respectively. significant difference exists in terms of effort regulation (t = 2.38, p <0.05) in favor of mathematics students. hamid and singaram (2016) revealed that critical thinking, time, and study environment were correlated with academic performance among medical students for two modules: becoming a professional and basic science. however, pearson’s product-moment correlation coefficient showed a weak relationship between the two variables. anxiety and learning strategies a few studies focused on anxiety and strategies were conducted to show that these variables were associated with each other. anxiety could also predict how learners utilize strategies in learning mathematics. studies show how the learners mostly utilize cognitive strategies. the study of ibrahim (2018) involved 293 students through a survey on an online platform. sets of linear regressions were used. the third model highlighted the strategies for learning mathematics as predictors of anxiety. results showed that higher anxiety was associated with higher utilization of rehearsals and organization and lesser utilization of elaboration and helpseeking. individuals with higher anxiety used less effective study strategies. helpful learning strategies were sometimes used. surprisingly, these strategies were beneficial in their mathematics achievement. similarly, kesici and erdogan (2009) revealed how rehearsal and elaboration among cognitive learning strategies significantly predict mathematical anxiety. studies have highlighted the relationship of mathematical anxiety with their academic achievement. although few studies could refute the relationship, the majority revealed the negative relationship between anxiety and achievement. in mathematics learning, the primary concern is the development of efficacy and conceptual knowledge. the relationship between achievement, conceptual knowledge and efficacy is undeniably related to each other. in contrast, a study involving older respondents did not show any relationship between these variables but considering young learners, efficacy will always determine one’s achievement. therefore, to address these problems, students need to develop learning strategies. learning strategies in mathematics learning are essential as they unconsciously allow learners to solve the problems at their pace. however, among the mentioned learning strategies for mathematics, resource management and cognitive strategies were mostly used in learning concepts and skills. furthermore, cognitive strategies predicted an individual’s anxiousness (kesici & erdogan, 2009, ibrahim 2018). the literature review on anxiety as predictors of an individual’s strategy is relatively sparse. thus, local studies are still needed to fully understand the association between anxiety and learning strategy selection and how these strategies can increase the efficiency and self-concept of filipino students in mathematics learning. the researcher investigated the association between anxiety and learning motivation strategies. the investigation of learning strategies and the level of anxiety may help explain the declining performance of the philippines for the past years and promote research-based solutions to this problem. methods 17 southeast asian mathematics education journal, volume 11, no 1 (2021) this is a prediction study that revolves around the relationship between anxiety and learning motivation strategies. the study design focuses on analysing how anxiety can establish the relationship and make predictions between the anxiety and learning motivations strategies. (kabir, 2016). the study involved 335 students enrolled in the science, technology, engineering and mathematics strand for senior high school of zamboanga del norte national high school. it is considered as one of the major public schools in the division of dipolog city, zamboanga del norte. the respondents were randomly selected through master lists retrieved from the advisers. the study was conducted from february to april 2020. survey questionnaires were used in collecting the data and were composed of 3 parts: the demographic profile, mathematical anxiety and learning motivation strategies. the level of anxiety was measured using the revised mathematics anxiety rating scale (rmars) (alexander & martray, 1989). the 25-item instrument comprises three subscales in measuring anxiety: mathematics test anxiety, mathematics course anxiety and numerical task anxiety. the mathematics test anxiety evaluates the reaction of students to any assessments conducted in mathematics class. it also measures the responses of students when they are present in any mathematics class. the numerical anxiety measures the anxiety brought by basic activities in mathematics. initial internal consistency reliability coefficients of the rmars subscales were 0.96 for the mathematics test anxiety, 0.86 for the numerical task anxiety, and 0.84 for the math course anxiety (alexander & martray, 1989). respondents will report their anxiety levels through a 5-point likert scale: 1 – not at all, 2 – a little, 3 – a fair amount, 4 – much and 5 – very much. the higher the score obtained would mean a higher level of anxiety reported by the respondent. the strategies used by the respondents in learning concepts and problem solving would be determined through the mathematics motivated strategies for learning questionnaire (mmslq) (liu & lin, 2010). for this study, the researcher would utilize the second section of the questionnaire – motivated strategies. it is a 64-item questionnaire covering four factors: cognitive strategies, metacognitive strategies, non-informational resources management and informational resource management. the components for cognitive strategies were rehearsals (6 items), organization (6 items) and elaboration (6 items). the components of metacognitive strategies included two elements: critical thinking (6 items) and self-regulation (6 items). the component of non-informational resources management could be divided into four elements: effort regulation (5items), time and study environment (8 items), peer-learning (6 items), and help-seeking (6 items). the component of informational resources management could be divided into two elements: exploratory behaviour on the internet (6 items) and communication behaviour on the internet (7 items). table 3 shows the strategies, corresponding components and item numbers. the higher mean obtained from the component means the most utilized learning strategy. respondents would respond through a 5-point likert scale: 1strongly disagree, 2disagree, 3 – neither agree nor disagree, 4 – agree and 5 – strongly agree. table 2 shows the rating scales with their corresponding description for self-report of junior high preparation, anxiety and learning strategies. survey forms through google form were distributed to respondents. the respondents were assured of the confidentiality of the data results and discussion 18 southeast asian mathematics education journal, volume 11, no 1 (2021) results table 1 shows the level of anxiety by the respondents in terms of mathematics test, numeric test, and course anxiety. table 1. level of anxiety in terms of mathematics test, numeric test and course anxiety dimension mean s. d. interpretation course 3.38 1.0216 moderately anxious mathematics test 3.26 0.8462 moderately anxious numeric test 3.04 0.3429 moderately anxious mean 3.23 0.8078 moderately anxious range description interpretation 4.21-5.00 strongly agree very high anxious 3.41-4.20 agree high anxious 2.61-3.40 either agree or disagree moderately anxious 1.81-2.60 disagree low anxious 1.00-1.80 strongly disagree very low anxious respondents’ level of mathematical anxiety was found to be moderate (x ̅ = 3.23, s.d. = .8078). among the dimensions, the course anxiety was high (x ̅ = 3.38, s.d. = 1.0216). course anxiety items covered the anxiety they have felt from course orientation and requirements. this implied that most students in the stem program were still anxious about tasks related to mathematics. table 2 presents the level of utilization of learning motivation strategies by major categories. table 2 level of utilization of learning motivation strategies by major categories categories of strategies mean s. d. interpretation meta-cognitive 3.9977 0.6368 high utilization cognitive 3.9676 0.6105 high utilization non-informational resources management 3.7256 0.5745 high utilization informational resources management 3.7778 0.8200 high utilization range description interpretation 4.21-5.00 usually very high utilization 3.41-4.20 often high utilization 2.61-3.40 sometimes moderate utilization 1.81-2.60 seldom low utilization 1.00-1.80 never very low utilization among the major categories of learning motivation strategies, the most commonly used was the meta-cognitive strategy (x ̅ = 3.9977, s.d.= .6368), while the least utilized was informational resources management. meta-cognitive strategies included critical thinking and self-regulation. table 3 illustrates the level of utilization in terms of sub-categories. 19 southeast asian mathematics education journal, volume 11, no 1 (2021) table 3. level of utilization of learning motivation strategies by sub-domains domains of strategies mean s. d. interpretation self-regulation 4.1065 0.6352 high utilization rehearsals 4.0278 0.63639 high utilization organization 3.9769 0.6443 high utilization exploratory behaviour on internet 3.9583 0.9328 high utilization help-seeking 3.912 0.5899 high utilization elaboration 3.8981 0.6448 high utilization critical thinking 3.8889 0.6915 high utilization effort regulation 3.7611 0.5635 high utilization time and study environment 3.6319 0.7263 high utilization communication behaviour on internet 3.5972 0.8165 high utilization peer-learning 3.5972 0.6961 high utilization range description interpretation 4.21-5.00 usually very high utilization 3.41-4.20 often high utilization 2.61-3.40 sometimes moderate utilization 1.81-2.60 seldom low utilization 1.00-1.80 never very low utilization among the items for learning motivation strategies, self-regulation is the most commonly utilized learning motivation strategy when dealing with mathematics and mathematics-related subjects. self-regulation refers to the ability of students to develop skills even without supervision or expectations of receiving incentives. respondents rated the statement, “if i feel confused about the math class materials, i will go over to find out where the problem is”, as high in terms of self-regulation strategies. this implies that students cultivate the value of initiative when they want to clarify the concept. ideally, this attitude is very relevant when dealing with subjects that seek constant drills. bivariate analysis was conducted to determine the relationship between respondent’s anxiety and utilization of learning strategies. table 4. relationship of anxiety and learning strategies using pearson correlation r strategies anxiety and strategies pearson r effort regulation .486** time and study environment .466* peer learning .416* help-seeking .415* exploratory behavior on internet .397* rehearsal .375* organization .392* 20 southeast asian mathematics education journal, volume 11, no 1 (2021) communication behavior on the internet .241 self-regulation .232 elaboration .220 critical thinking .121 **correlation is significant at the 0.01 level (two-tailed) *correlation is significant at 0.05 level (two-tailed) anxiety has a significant relationship between effort regulation (r = 0.486, p < 0.01), rehearsal, organization, time and study environment, peer learning, help-seeking and exploratory behaviour on the internet. furthermore, there is a positive and moderate relationship between the variables. multiple regression analysis was used to test if anxiety predicts learning motivation strategies. table 5. multiple regression analysis for anxiety and learning motivation strategies model r r2 adjusted r2 se of estimate r2 change f change df1 df2 sig. 1 .486a .236 .215 .71532 .236 10.832 1 134 .002 a predictors: (constant), effort regulation the multiple regression analysis revealed that model 1 was significant (f (1,134) = 10.832, p<0.05). furthermore, the model accounted for 23.6% of anxiety variance. the rest of the learning strategies could not account for the anxiety of the learners. respondent’s utilization of effort regulation as a learning motivations strategy indicated that when students experienced higher anxiety levels, they tended to use effort regulation to support their learning. discussion the level of anxiety of a respondent could pose a threat to their academic performance as it determines their motivation and self-efficacy. course anxiety had a higher contribution to the levels of anxiety felt by the respondents. this dimension refers to how they perceive the subject through their own interpretation. external and internal sources could also influence this. the way respondents perceive mathematics as a difficult subject might contribute to the anxiousness they have felt. hence, it is always essential to allow learners to appreciate the value of the subject. through activities that are realistic and attainable, students will be able to feel comfortable with the subject. in fact, spaniol (2017) found that students who passed the pre-calculus examination have higher self-efficacy and lower anxiety. factors contributing to their anxiety might be classroom activities that require drills on boards, oral recitation, feeling tense, and the presence of teachers (reyes, 2019). age could be another factor in the level of anxiety they have handled. the study of smith (2010), where respondents are teachers, showed a significantly low level of anxiety. the level of preparation of students could determine anxiety. in reviewing the school’s protocol on admitting students to the stem program, it was found out that there was no assessment or entrance examination which determined their admission to 21 southeast asian mathematics education journal, volume 11, no 1 (2021) the program. as the characteristics of the respondents would show, somehow, the admitted senior high school students were less confident in terms of equipping themselves with mathematical skills. in the path model analysis of shishigu (2018), prior mathematics achievement was the strongest predictor for mathematical anxiety and present achievement in mathematics. the respondents mostly utilized self-regulation. self-regulation refers to how learners attempt to control their own learning. this implies that aside from using other learning motivation strategies, self-regulation was considered the most commonly used strategy in their learning and peer learning as the least utilized strategy. in fact, the study of lavasani, hejazi and varzaneh (2011) demonstrated that the performance-approach structure has a negative effect on an individual’s level of anxiety and directly affects self-regulation. lavasani et al. (2011) revealed a positive impact of self-regulation learning strategies towards students' academic performance, motivation, and self-efficacy. respondents mostly utilize cognitive and metacognitive strategies in their learning tasks. although self-regulation was the most utilized learning motivation strategy among respondents, it had no significant relationship with mathematical anxiety. the same result goes with other cognitive and meta-cognitive strategies which have no connection with anxiety except for organization. effort regulation, rehearsal, organization, time and study environment, peer learning, help-seeking and exploratory behaviour on the internet has a significant relationship with anxiety. furthermore, resources management strategies were moderately correlated with their levels of anxiety. these strategies can be classified into informational and non-informational resources. hence, the environment, peers, and other information sources can help learners develop confidence when learning. furthermore, a positive relationship was found between anxiety and learning motivation strategies. the strength of their relationship was also found to be moderate. this was already evident that anxiety was related to the utilization of strategies. this implies that when anxiety increases, they tend to utilize specific strategies. the study results suggest that this particular group of students with moderate anxiety performed better when they used the preferred strategy. these findings are also consistent with ahmed and khanam (2014) results where high-achieving students in the class better utilized time and study environment, peer, help, and effort regulation than the low achievers. science-inclined students better utilize time and study environments and peer learning than humanities students. it can also be noticed that meta-cognitive strategies are not related to anxiety, although it is one of the highly utilized strategies by learners. the nature of subjects that employ mathematical concepts demands a learner to develop cognitive and meta-cognitive strategies as these are very helpful. however, respondents employ less valuable techniques, as affirmed by the findings of ibrahim (2018). the results revealed that learners with a higher level of anxiety tend to utilize less helpful strategies. amidst being less beneficial, respondents claimed it had helped them perform better in mathematics class. the study also suggests that even if learners utilize common strategies, anxiety allows learners to develop specific learning strategies. psychological status can also determine an individual’s accommodation of specific strategies. however, regression analysis has provided evidence that the effort regulation strategy among the investigated learning motivation strategies was predicted by mathematical anxiety. 22 southeast asian mathematics education journal, volume 11, no 1 (2021) effort regulation refers to the level of efforts pursued on a specific task. the level of effort depends on the value of the tasks and the commitment to the target. hence, the weight of academic tasks regulates their sense of effort in accomplishing the task. when one is motivated to perform the task, they use too much effort. thus, when they face uninteresting tasks, they tend to keep these tasks. however, in this case, anxiety affects the way they see the value of a task. this suggests that difficult tasks tend to motivate them and allow them to assess how significant it is for their course undertaking. therefore, the more anxious they are, the more they regulate their efforts. this contrasts with the findings of ibrahim (2018), where higher anxiety was associated with higher utilization of rehearsals and organization and lesser utilization of elaboration and help seeking. rehearsal and elaboration were also found to be the strategies related to anxiety in the study of kesici and erdogan (2009). ibrahim (2018) showed that higher anxiety was associated with higher utilization of rehearsals and organization and lesser utilization of elaboration and help seeking. on the other hand, kesici and erdogan (2009) revealed how rehearsal and elaboration among cognitive learning strategies significantly predicted mathematical anxiety. this suggests that this group of learners who are inclined with stem education prefers the use of effort regulation when they encounter higher levels of anxiety. according to mariani (2002), the central theme of learning strategies represents a bridge between one’s competency and process. these strategies are innate within the learners. there is no such thing as a good or bad strategy; it is just a matter of activating them. thus, tasks could be designed to let the learners discover these strategies. lastly, strategies could become a part of the interaction between students and learners. conclusion the findings revealed how students utilized learning strategies when they felt anxious. this is shown by the ability of stem learners to motivate themselves in achieving their goals and realizing the learning outcomes. a specific level of student’s anxiety has already proven a negative effect on their academic performance. looking at mathematical anxiety from a different perspective, we might be able to understand how human behaviours accommodate stressors and controls cognitive sources. aside from looking at possible solutions to decrease factors contributing to higher levels of anxiety, there could also be interventions that could develop learning motivation strategies. in this way, they could mediate the effects brought by the feeling of anxiousness. future research studies could be conducted where diverse learners are involved. this could be composed of learners of different strands, disciplines, and various profiles. variables such as anxiety and learning motivation strategies could also be monitored regularly to come up with observable patterns of their behaviour. comparisons of the contributing factors of anxiety towards learning motivation strategies based on their profiles could also be conducted to understand the relationship among variables better. finally, teachers and academic institutions could implement interventions to develop learning motivation strategies among students in terms of informational and non-informational resources strategies. the provision of the internet, simulations and rich sources could help them in their learning experience 23 southeast asian mathematics education journal, volume 11, no 1 (2021) acknowledgement the author expresses his sincerest thanks to zamboanga del norte national high school. references ahmed, o. & khanam, m. (2014). learning resources management strategies and academic achievement of secondary school students. the international journal of indian psychology, 2 (1). alexander, l., & martray, c. (1989). the development of an abbreviated version of the mathematics anxiety rating scale. measurement and evaluation in counselling and development, 22, 143-150. baumgartner, w. l., spangenberg, e. d., & jacobs, g. j. (2018). contrasting motivation and learning strategies of ex-mathematics and ex-mathematical literacy students. south african journal of higher education, 32 (2), 8-26. corbetta, m., & shulman, g.l. (2002). control of goal-directed a stimulus-driven attention in the brain. nature reviews neuroscience, 3, 201–215. department of education. (2019). programme for international student assessment 2018 national report of the philippines. pasig. derakshan, n., & eysenck, m. (2009). anxiety, processing efficiency, and cognitive performance new developments from attentional control theory. european psychologist, 14 (2), 168-178. eysenck, m.w., & calvo, m.g. (1992). anxiety and performance: the processing efficiency theory. journal on cognition and emotion, 6, 409–434. eysenck, m. w., derakshan, n., santos, r., & calvo, m. g. (2007). anxiety and cognitive performance: attentional control theory. journal on cognitive and emotion, 7(2), 336– 353. https://doi.org/10.1037/1528-3542.7.2.336. gurat, m. g. (2018). mathematical problem-solving strategies among student teachers. journal on efficiency and responsibility in science and education, 11 (3), 53-64. hamid, s., & singaram, v. s. (2016). motivated strategies for learning and their association with academic performance of a diverse group of 1st-year medical students. african journal for health professions education, 8, 104-107. ibrahim, a. f. (2018). the influence of individual differences in math anxiety on learning novel mathematics content. african journal for health professions education, 8(1 suppl 1):104-107. https://doi.org/10.7196/ajhpe.2016.v8i1.757. jameson, m., & fusco, b. (2014). math anxiety, math self-concept, and math self-efficacy in adult learners compared to traditional undergraduate students. adult education quarterly 64(4), 306-322. kabir, s. m. s. (2016). basic guidelines for research: an introductory approach for all disciplines. chittagong-4203, bangladesh: book zone publication. kafadar, t. (2013). examination of multiple variables of learning strategies used by students in social studies lessons [unpublished master dissertation]. kirsehir, turkey: ahi evran university. kesici, s., & erdogan, a. (2009). predicting college students' mathematics anxiety by motivational beliefs and self-regulated learning strategies. alabama: project innovation. 24 southeast asian mathematics education journal, volume 11, no 1 (2021) lavasani, m. g., hejazi, e., & varzaneh, j. y. (2011). the predicting model of math anxiety: the role of classroom goal structure, self-regulation and math self-efficacy. procediasocial and behavioral sciences. 15, 557-562. https://doi.org/10.1016/j.sbspro.2011.03.141. lavasani, m. g., mirhosseini, f. s., hejazi, e., & davoodi, m. (2011). the effect of selfregulation learning strategies training on the academic motivation and selfefficacy. procedia-social and behavioral sciences, 29, 627-632. liu, e. z., & lin, c. h. (2010). the survey study of mathematics motivated strategies for learning questionnaire (mmslq) for grade 10–12 taiwanese students. the turkish online journal of educational technology, 9 (2). mariani, l. (2002). learning strategies, teaching strategies and new curricular demands: a critical view. a journal of tesol-italy, xxix (2), fall. oxford, j. & vordick, t. (2006). math anxiety at tarleton state university: an empirical report. tarleton state university. reyes, j. (2019). mathematics anxiety and self-efficacy: a phenomenological dimension. journal of humanities and education development, 1 (1), 22-24. shishigu, a. (2018). mathematics anxiety and prevention strategy: an attempt to support students and strengthen mathematics education. mathematics education trends and research, 1(1), 1-11. smith, l. j. (2010). the relationship among mathematics anxiety, mathematical self-efficacy, mathematical teaching self-efficacy, and the instructional practices of elementary school teachers. the university of southern mississippi. spaniol, s. r. (2017). students' mathematics self-efficacy, anxiety, and course level at a community college [thesis dissertation]. walden university. star, j. r., pollack, c., durkin, k., rittle-johnson, b., lynch, k., newton, k., & gogolen, c. (2015). learning from comparison in algebra. contemporary educational psychology, 40, 41–54. vaezi, m., hatamzadeh, n., motlagh, f. z., rahimi, h., & khalvandi, m. (2018). the relationship between resource management learning strategies and academic achievement. international journal of health and life sciences, 4(1). xu, j. (2004). the casual ordering of mathematics anxiety and mathematics achievement: a longitudinal pane/analysis. journal on adolescence, 27(2), 165-179. (1 southeast asia mathematics education journal, volume 10, no 1 (2020) 55 the joyful experience in learning mathematics ricky s. yabo department of education regional office vii, cebu city, philippines ricky.yabo@deped.gov.ph abstract this study was conducted to determine the effectiveness of joyful scaffolds in teaching grade viii mathematics in the department of education of the philippines. two intact groups were used as the subjects of the study. the experimental and control group were exposed to joyful and traditional scaffolds in teaching respectively. this study includes the pre-post assessments, significant improvements, and significant mean gain differences of students exposed to traditional and joyful scaffolds in teaching mathematics in terms of performance level. the findings of the study revealed that the joyful scaffold in teaching is more effective in improving the students’ mathematics performance compared to the traditional scaffold in teaching and the skills acquired is sufficient enough to prove that these experimental group of students was able to gain experience from joyful approach and a positive change of attitude towards their view on mathematics was realized after the exposure. the cognitive progression was deepened, and attainment of knowledge was enhanced, psychomotor skills were activated and augmented through enjoyable mathematics learning activities, and students’ interests and learning manners were maximized. furthermore, the joyful scaffold in teaching in this study was immensely efficient in improving the students’ mathematics performance compared to the traditional scaffold in teaching. moreover, the outcome of this research supports the assertion that employing, relating, or incorporating several joyful media to learning boosted up students’ academic performance especially those who are detached during the classroom discussion. keywords: joyful learning, enjoyable scaffolds, operational skills, mathematical vocabulary, comprehension. introduction mathematics is a systematic application of matter consequently makes our life in order and precludes anarchy (anderssen, b., et. al., 2016). it is the foundation of all designs and inventions for without which the world cannot move even a little (hodaňová & nocar, 2016). some attributes that include ability of reasoning, creativity, hypothetical and critical thinking, ability in solving problem and even operative communication skills are fostered by mathematics (kusmaryono, 2014). everyone needs mathematics for the day-to-day life for existence hence to gain knowledge in mathematics is deemed necessary. a distinctive mathematics teaching in the philippines typically engages the question and answer nature of discussion. the mathematics secondary level curriculum in the philippines which is currently being applied promotes using an array of teaching strategies among which are practical work, discussion, problem solving, investigations and practice and consolidation, as well as cooperative learning. the numeracy component of mathematical and problem solving skills focuses on competencies necessary for effective and efficient mathematical understanding and problem deciphering. its emphasis is on how these mathematical skills would be applied to daily life situations. to this end, real-life examples the joyful experience in learning mathematics 56 are given for many of the experiences. during curriculum implementation, further emphasis will be necessary on contextualization of the mathematical competencies to align with different learning contexts, needs, and situational realities and practicalities of different learners (department of education of the philippines, 2013). problems with learning and acquiring knowledge in mathematics are global and frequently subject to profoundly embedded views (chinn, 2016). most of the times during mathematics discussion students are sleepy, inattentive and disengaged henceforth teachers are also having challenges in making the subject more appealing and captivating. a balance is required between two most important things – knowing the type of learner and choosing a suitable teaching strategy. joyful learning is a type of learning procedure or occurrence which might create students feel gratification in a learning process (udvari-solner & kluth, 2017). learning joyfully depicts a kind of emotion, articulating and causes great pleasure and a joyful insight is found to have constructive influence on the inspiration of learning (kirikkaya, e. b., et. al., 2010). joyful learning is an approach that entails a sense of pleasure, contentment, and comforting of the parties who are in the learning process. there is a bond of love and fondness between educators and learners and among learners and the learning manner will make each party seeking to give the finest to satisfy others (wei, hung, lee, & chen, 2011). learning joyfully in mathematics could make a big impact to the education of the students and makes knowledge and skills’ assimilation meaningfully (ariawan & pratiwi, 2017). joyful learning activities are engaging and appealing (minarni & napitupulu, 2017) such as mathematical games, puzzles, stories, activities relating numbers and quantities, recreational hand – on activities, realistic and life related problem solving that might help students learn holistically and develop a positive attitude, make connection with mathematics and everyday thinking, and make the mathematics classroom a place for recreation, fun, and enjoyable to stay. methods research design the research methodology used in this study was essentially the quasi experimental method. experimental and quasi-experimental research models assess whether there is an underlying relationship between independent and dependent variables. plainly defined, the independent variable is the variable of influence and the dependent variable is the variable that is being influenced (loewen & plonsky, 2015). primarily, this method determined the efficiency of the use of joyful scaffolds in the hopes of finding effective means to teach grade viii mathematics. specifically, this research intended to determine the pre-post assessments, significant improvements, and significant mean gain differences of students exposed to traditional and joyful scaffolds in teaching mathematics in terms of performance level among grade viii junior high school students of tungkop national high school, a public junior high school of the department of education of the philippines, region vii cebu province division during the school year 2018 2019. the control and experimental group were exposed to traditional and joyful scaffolds in teaching respectively. the traditional scaffolds cover the chalk talk, pasting of real objects on the board related to the topics presented and ricky s. yabo 57 visual aids on a cartolina and manila paper while joyful scaffolds in teaching include learning activities that are engaging and amusing such as mathematical games, puzzles, manipulatives, stories, activities relating numbers and quantities, recreational hands-on activities, realistic and life related problem solving. sample this study utilized 60 students which were purposively taken from the two sections in grade viii level hence the department of education of the philippines grade viii level mathematics learning competencies (lcs) were also considered. the thirty (30) students in each group were selected through probability sampling specifically simple random sampling. in this sampling each student was chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process. data collection this study utilized research instrument based and patterned to the standardized unit tests of the department of education to determine the pre-post assessments of students exposed to traditional and joyful scaffolds in teaching mathematics that were focused on the present condition to draw amplified insight into aspects which are functioning (calmorin, 1994). there were four sets of unit test that includes lcs in illustrating and differentiating linear inequalities in two variables, graphing and solving problems involving linear inequalities in two variables, solving a system and problems involving systems of linear inequalities in two variables and illustrating a relation and a function and verifying if a given relation is a function. the groupings of the content areas are based on the mathematics curriculum guide of the department of education of the philippines. each unit test is equivalent to 25 points. there were true or false, multiple choices, and computations in every unit test. the tests contents were verified and validated by the certified and authorized mathematics master teachers and education program supervisors and presented and approved by the schools division research committee (sdrc) of the department of education cebu province division, regional office vii, the philippines. there were also words and observations being considered to express authenticity and attempts to describe students in natural situation (mcdougall, 2015). moreover, the research instrument went through reliability test. reliability was based from calmorin (1994) in defining the uniformity of the research instrument testing the permanency of the response design of the respondents to be assured that the instrument is reliable, self-consistent, and unchanging. pilot test was administered to 30 grade viii students (not part of the research respondents). the assessment results were subjected to reliability alpha test using the latest version of spss. an of 0.70 and above are considered reliable. the results were all higher than 0.70 which further means that the response pattern of the respondents was dependable, consistent, and stable. both the control and experimental group were given a pre-test on grade viii mathematics second quarter lcs before having been intervened by the independent variables which are the joyful and traditional scaffolds. they were also given a post-test on grade viii mathematics second quarter lcs after having been exposed to the joyful and traditional scaffolds in teaching. http://en.wikipedia.org/wiki/randomization http://en.wikipedia.org/wiki/probability the joyful experience in learning mathematics 58 data analysis procedure this study used a z-test tool to determine the pre-post assessments of the students exposed to traditional and joyful scaffolds in teaching grade viii mathematics in terms of performance level. this was utilized to find out whether the students had attained the 60% expectation level before and after the demeanour of the study. to find out if there is a significant improvement in the control and experimental group in terms of performance level, this study employed a t-test tool for related samples. lastly, this study utilized a t-test of independent or uncorrelated means to determine if there is a significant mean gain difference in the two groups in terms of performance level. results and discussion this section illustrates the discussion, analysis, and interpretation of data on the tests’ performance in mathematics of the two groups of grade viii junior high school students as affected by the traditional and joyful scaffolds in teaching. the hypothetical mean was attained by using 60% as performance mark for the number of items on each learning competency. the critical z and t-values are ± 1.96 and 1.699 respectively at 0.05 level of significance. table 1 summary of test results on traditional scaffolds in teaching no learning competency stats traditional scaffolds in teaching pre-test post-test 1 lc 1 means 2.97 13.60 std dev 0.76 5.83 z-test value 86.70 1.32 2 lc 2 means 4.03 11.23 std dev 1.96 4.91 z-test value 30.07 4.21 3 lc 3 means 3.67 13.70 std dev 1.21 3.28 z-test value 51.29 2.17 4 lc 4 means 6.57 15.53 std dev 2.50 2.96 z-test value 18.47 0.98 total means 17.24 54.06 std dev 6.43 16.98 z-test value 36.42 1.92 table 1 shows that the computed z-test value during the pre-test is below and outside the non – rejection region thus the control group got a below average performance. this denotes that the students were not able to achieve the standard mark of the school. ricky s. yabo 59 furthermore, table 1 revealed that students in the control group during the post-test got a calculated z-test value higher than during the pre-test and it is within the non rejection region. this means that their performance is categorized as average. the skills and techniques in patterns and algebra sink in a bit to the memory of the students. the result elucidated further that the grade viii students exposed to traditional approaches in teaching progressively grasped on the skills and techniques in solving problems related to linear inequalities in two variables and function. learning mathematics activities from informal sources happening in extra-curricular training or in out-of-study atmosphere have proven to be effective and able to motivate students to be taught (hayes, 2007). the preservation of the students’ content knowledge on patterns and algebra affects a lot in their performance during the pre-test on the topics about linear inequalities in two variables and linear function. hence, the pre-test assessment of the students before exposure to traditional approaches in teaching mathematics characterized as below average. every subject in a particular class of disabilities has a different nature from that of the other class (silver, 1997). thus, each subject like mathematics also needs different learning propositions. in carrying out the learning, the teacher absolutely needs a tool that can sustain its performance so that learning can take place with interesting and fun. the aforementioned evaluation further demonstrates that students in control group during the pre-test lack of prior knowledge and techniques in solving problems related to patterns and algebra. prior knowledge is the very rudimentary foundation in learning any topic. at their early age, students should be exposed to impending facets associated to mathematics to prepare them in the future (shrestha, 2018). the basic concepts of elementary algebra should be inculcated in the mind of the students in their primary years. in this undemanding way, it will be easier for the students to grasp and link to an advanced learning related to patterns and algebra. the lack of the aforesaid realities caused the below average students’ performance during the pre-test. the abovementioned insinuations exemplified the effect of a usual intervention. students’ interest was motivated with the traditional approaches in teaching. with the use of traditional visual aids and teacher’s intervention, students were encouraged to listen and do their best during the post-test. table 2 summary of test results on joyful scaffolds in teaching no learning competency stats joyful scaffolds in teaching pre-test post-test 1 lc 1 means 2.80 22.37 std dev 1.19 2.27 z-test value 56.15 17.78 2 lc 2 means 3.03 18.90 std dev 1.13 2.98 z-test value 58.02 7.17 the joyful experience in learning mathematics 60 no learning competency stats joyful scaffolds in teaching pre-test post-test 3 lc 3 means 3.37 20.67 std dev 1.45 2.29 z-test value 43.93 13.56 4 lc 4 means 7.17 20.27 std dev 3.11 2.56 z-test value 13.79 11.28 total means 16.37 82.21 std dev 6.88 10.10 z-test value 34.73 12.04 as presented in table 2, the experimental group of students’ assessments during the pre-test is below average since the computed z-test value is outside and below the nonrejection region. they do not have yet the basic skill and technique in solving problems related to patterns and algebra. it denotes that an approach is a reaction to skills that come across in students’ life. (kundu & ghose, 2016). basic skill narrows our logic of perceiving the world of mathematics, understands, and analyses the circumstances better as one would without proper knowledge. absorbing the essential knowledge in mathematics gradually helps to grow the student’s confidence level and may establish to be of some immense help during some significant events of students’ lives. educators should continue to converse to carry out new learning goals and instructional approaches that aim on safeguarding that our students progress with the skills essential to take part in our new global society (gunter, 2007). as displayed in table 2, during the post-test, the experimental group of students attained an above average performance since the computed z-test value is above and outside the non-rejection region. this assessment implies that the experimental group of students attained a performance which is above the standard goal of the school. the joyful approach and scaffold triggered and maximized the existing knowledge of the students on the topics about linear inequalities and function hence they were able to achieved ultimate goals of learning in the second quarter of elementary algebra. booming methods for tapping student’s prior knowledge need not be difficult. vallori (2014) revealed that by merely soliciting students what they know about a topic before instruction can increase achievement. mathematics instruction should construct on students’ existing knowledge along with joyful and lively teaching techniques and strategies (forman & steen, 2000). this is because students often enjoy applicable information that can support them in mastering new content. additionally, the abovementioned results show that the grade viii students grasped already on the necessary skills needed in solving problems and exercises related to linear inequalities and function hence the post-test assessment of the students after exposure to joyful approaches in teaching elementary algebra is above average. ricky s. yabo 61 table 3 improvements of the control and experimental group no learning competency stats control experimental 1 lc 1 pre-test mean 2.97 2.80 post-test mean 13.60 22.37 mean gain 10.63 19.57 t-test value 4.78 5.43 2 lc 2 pre-test mean 4.03 3.03 post-test mean 11.23 18.90 mean gain 7.20 15.87 t-test value 4.57 5.37 3 lc 3 pre-test mean 3.67 3.37 post-test mean 13.70 20.67 mean gain 10.03 17.30 t-test value 5.14 5.40 4 lc 4 pre-test mean 6.57 7.17 post-test mean 15.53 20.27 mean gain 8.97 13.10 t-test value 5.06 5.20 total pre-test mean 17.24 16.37 post-test mean 54.06 82.21 mean gain 36.83 65.84 t-test value 4.89 5.35 table 3 presents the students’ improvement in mathematics after exposure to joyful and traditional approaches and scaffolds in teaching in terms of performance. furthermore, table 3 displays the computed t values attained by both groups in the competency which is the basis in denoting whether the students’ improvement after exposure to joyful and traditional approaches in teaching mathematics in terms of performance is significant or not significant. as shown in table 3, the improvement between the control group’s mathematics scores is significant. the post-test scores are higher than that in the pre-test. this means that the joyful experience in learning mathematics 62 the control groups’ mathematics performance improved after the intervention and exposure to the traditional approaches in teaching. accordingly, the null hypothesis of no improvement in the control group’s performance in elementary algebra after exposure to traditional approaches is rejected. thus, a significant increase in the students’ performance in mathematics is observable. these results denoted that the students have absorbed mathematics skills after exposure to the conventional technique in teaching, the lecture-discussion method. the approach significantly helped increase the student’s performance in mathematics. this means that the mathematical learning process for grade viii junior high school students that utilizes joyful approach takes place in a pleasant atmosphere and math specifically elementary algebra becomes easy. a satisfying and enjoyable learning discernment has an affirmative effect on students' learning motivation (conklin, 2014). on the other hand, the experimental group’s post-test’ performance is higher than that in the pre-test. the students’ over-all mean gain provided way a significant increase in their performance. the computed t-test mean gain difference value is greater than the critical value. this result instigated to the rejection of the null hypothesis of no improvement in the experimental group’s performance in mathematics after exposure to joyful approaches and scaffolds. thus, the students attained significantly better in the post-test than in the pre-test. this result implies that joyful approaches in teaching have strengthened the performance of the students in experimental group. the teacher-made joyful presentations have motivated the students not to absent during mathematics classes. they were stimulated also with the effort shown by the teacher in preparing the lessons joyful and interesting. the aforementioned results signify that the traditional and joyful approaches are effective in teaching mathematics. it is obvious as displayed in table 3 that the abovementioned approaches in teaching increased the mathematics performance of the students in control and experimental group. table 4 mean gain differences of the control and experimental group no learning competency stats control experimental computed t value 1 lc 1 pre-test mean 2.97 2.80 7.49 post-test mean 13.60 22.37 mean gain 10.63 19.57 std dev 6.03 2.53 2 lc 2 pre-test mean 4.03 3.03 8.18 post-test mean 11.23 18.90 mean gain 7.20 15.87 std dev 4.85 3.19 3 lc 3 pre-test mean 3.67 3.37 8.22 post-test mean 13.70 20.67 mean gain 10.03 17.30 std dev 3.74 3.08 ricky s. yabo 63 no learning competency stats control experimental computed t value 4 lc 4 pre-test mean 6.57 7.17 3.91 post-test mean 15.53 20.27 mean gain 8.97 13.10 std dev 3.77 4.38 total pre-test mean 17.24 16.37 6.63 post-test mean 54.06 82.21 mean gain 36.83 65.84 std dev 20.42 12.57 table 4 presents the mean gain difference of the grade viii junior high school students after exposure to traditional and joyful approaches in teaching mathematics. in totality, as shown in table 4, the computed t – value is less than the tabled value and outside the non-rejection region at p = 0.05 level of significance. this elucidates that the mean gains between the two groups are significantly different. hence, the hypothesis of no significant mean gain difference between the students exposed to joyful and those exposed to traditional approaches in teaching elementary algebra in terms of performance level is rejected. the joyful and traditional approaches in teaching elementary algebra specifically linear inequalities and functions significantly improved the academic performances of the students. mathematics educators are anticipated to craft ideas to inspire students by joyful activities (heywood, 2005) such as discerning, investigating, building, scheming, setting approach, and solving problems that are enfolded in mathematics games, puzzles, and hands-on activities. instructional materials are those belongings that could make easy efficient teaching and enjoyable learning that is teaching aids through which learning progression may be promoted and aggravated under the classroom condition (oyekan, 2000). activities crafted in mathematics teaching should intend to promote and prompt students to be involved and participate during the delivery of the lessons (khan, 2012). therefore, learning mathematics should be more fun and joyful both for students and teachers. it is a challenge of the mathematics teachers to think of innovations that would make the teaching of the lessons enjoyable for the students. fun in mathematics learning is the delight of pleasure particularly in math exercises and activities. learning mathematics should be an enjoyable task to do, diverting the students’ mind and body from any serious task or contributing an extra dimension to a joyful learning mathematics education. learning mathematics should be connected with recreation and play, social functions, and even seemingly ordinary activities of daily living doable to articulate, more spontaneous, playful, or active event. hence, the existence of the teaching aids in school does entail the imagination and creativity of teachers (cunningham, 2015). the joyful experience in learning mathematics 64 conclusions the joyful scaffolds in teaching mathematics in this study has promoted better learning. the cognitive progressions were deepened and attainment of knowledge was enhanced, psychomotor skills were activated and augmented through enjoyable mathematics learning activities, and students’ interests and learning manners were maximized. furthermore, the joyful scaffold in teaching in this study was immensely efficient in improving the students’ mathematics performance as compared to the traditional scaffold in teaching. the outcome of this research as reflected on the tables above supports the assertion that employing, relating, or incorporating several joyful media to learning may possibly boosted up students’ academic performance especially those who were detached during the classroom discussion. further researches might be carried out using the same experimental design applied to a bigger group of respondents and to other subject areas. the results of this study can only apply to the teacher and students in the sample and cannot be considered as representative of a larger group of teachers or students in the philippines due to sample limitations. however, the findings of this study may be beneficial to the future researchers who would probably be conducting a study on teaching technique related to this. this may serve as reference for those who would like to make further study. recommendations based and anchored on the findings of the study, educators may be perceptive in the learning styles of their students. they may be pliant enough in modifying their teaching approaches to suit to the need of the students. teachers may clinch on the promising technologies that capture the concentration and interest of the students. they may go beyond from the traditional way of teaching and integrate technologies and other appealing and joyful activities that stir the mind of the students to learn. in this simple way, students congregate and scrutinize their own information as teachers guide them in the discovery of concepts. teachers may be thoughtful in choosing the teaching method that is fitting to the interests and personality of the students and incorporate or employ joyful scaffolds in their mathematics classes. mathematics teachers may craft and develop lesson plans that would stimulate students’ desire to learn through joyful activities and exercises such as mathematics games, puzzles, hands-on activities, outdoor math activities, or any mathematics activity that would elicit the minds of the students to do discovering, exploring, creating, designing, setting some techniques, and solving math problems. acknowledgements the researcher wishes to express his supreme appreciation and admiration to the following personalities who had helped him in the pursuit of this action research, for without them this self-effacing work would not be materialized. mr. edison g. dela peña, tungkop national high school principal, for his kindness and sharing his superb insights for the fulfilment of this study; dr. joel b. umbay, minglanilla 1 public schools district supervisor, for his resonant wisdom and suggestions for the improvement of the schoolwork; ricky s. yabo 65 mr. danilo a. manguilimotan, minglanilla 2 public schools district supervisor, for his effort in looking into this humble and giving insight for the refinement of this study; dr. pamela a. rodemio, deped cebu province division education program supervisor for mathematics, for sharing her expertise and precious insights and without her the improvement and finishing point of this work would not be possible; dr. rhea mar a. angtud, former deped cebu province schools division superintendent for granting her permission to the researcher to go on with this professional activity; dr. marilyn s. andales, deped cebu province schools division superintendent, adviser of the schools division research committee, for her leadership and passion in managing the panel for the enhancement, development, and improvement of this study; dr. leah b. apao, dr. cartesa m. perico, and dr. ester a. futalan, deped cebu province schools division superintendents, chairmen of the schools division research committee, for their headship and leading the members of the sdrc in reviewing and facilitating the approval of this undertaking; mr. cesar a. restauro jr., deped regional office vii education program supervisor for mathematics, for his adeptness in sharing valuable views for the advancement of this endeavor; ms. uki rahmawati, seameo regional centre for qitep in mathematics facilitator, for her excellent and beneficial inputs during the conduct of the course on joyful learning in mathematics education that contributed to the enhancement of this study; dr. wahyudi, seameo regional centre for qitep in mathematics director, for the opportunity and blessings bestowed on the researcher to acquire knowledge and proficiency of the concepts of joyful learning in mathematics education at seameo regional centre for qitep in mathematicsin yogyakarta, indonesia that abetted to the successful implementation of the action research; his family, for their reassurances, unreserved love, prayers, support, and inspiration in fulfilling this study and professional venture; above all, to almighty god, for his 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technology (ict) to mathematics teaching and learning processes among junior and senior high school mathematics teachers in the division of tanauan city. further, the research also identified challenges encountered in ict-based teaching and learning mathematics. this study employed mixed method of research, survey questionnaire, interview, and focus group discussion (fgd) with the 77 teachers. the respondents were determined using raosoft at five percent margin of error and used stratified random sampling. the frequency, percentage, mean, standard deviation, and ttest were the statistical tools used to analyse the data. based on the findings, majority of the respondents were female, bachelor’s graduate, from teachers i to iii, 15 years below in the service and not active in attending ict-related seminars. most of them use technology in writing lesson plans, in computing students’ results and, in teaching the lesson through power point presentations. however, due to the lack of resources and little fund from the maintenance and other operating expenses (mooe), the respondents rarely use telecommunication devices such as cable, satellite, fax-machine, etc. to interact with students. there is a significant difference between the extent of the use of ict in teaching and learning mathematics and years in teaching. lack of ict facilities and trainings attended and confidence in the use of ict were the commonly identified challenges. with the aforementioned, the researchers recommended that mathematics teachers must be given more opportunities to participate ict-based seminars and trainings. keywords: mathematics education, ict-based teaching and learning, junior and senior high school mathematics teachers, qualitative and quantitative, philippines introduction technologies as link to new knowledge, resources and high order thinking skills have entered classrooms and schools worldwide. personal computers, cd-roms, on line services, the world wide web and other innovative technologies have enriched curricula and have altered the types of teaching available in the mathematics classroom. schools’ access to technology is increasing steadily every day and most of these newer technologies are now even used in traditional classrooms. the rapid development of information and communication technology (ict) brings a new paradigm in education in various aspects especially in mathematics teaching and learning, among others are the change from traditional learning to new learning, information delivery to information exchange and teacher-centred to student-centred. integration of ict into teaching and learning is not a teaching and learning with technology: ramification of ict integration in mathematics education 28 method but a medium in which a variety of methods, approaches and pedagogical philosophies may be implemented (salehi & salehi, 2012). ngeze (2017) pointed out that the use of ict in teaching and learning has brought new teaching and learning experience to both teachers and students in many countries. ict integration brings significant change in education acceleration and innovation in various countries, generally those related to the learning process, particularly in mathematics education. ict integration in mathematics education offers mathematics teachers with integrative teaching methods that motivate students’ learning, support their independent learning and active participation in the discovery of mathematics concepts and topics that helps them have deeper understanding of the mathematical ideas (baya’a & daher, 2013). also, it is stressed by the national council of teachers of mathematics (nctm) that, in mathematics teaching and learning process, technology is necessary and must be adapted (nctm, 2000). integrating ict into teaching and learning has become a great concern for many educators in developing countries like the philippines. integration of ict in the philippines plays a vital role in the educational system. the philippine government have recognized the pressing concern in improving the quality education to be able to participate in a dynamic global economy. among the critical governmental efforts are: the priority budget for education, basic education reform agenda (besra), medium term development program for 2011-2016, curriculum reform (e.g., k-12), and the ict for education (ict4e) (manaligod & garcia, 2012). however, there are some arguments that the use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. in a balanced mathematics programme, the strategic use of technology enhances mathematics teaching and learning. teachers need to be knowledgeable decision makers in determining when and how their students can use technology most effectively. schools and mathematics programmes have to provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, internet connectivity, handheld data-collection devices, and sensing probes. curricula and courses of study have to incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress. in the study of mishra and koehler (2006) tpack framework, which focuses on technological knowledge (tk), pedagogical knowledge (pk), and content knowledge (ck), offers a productive approach to many of the dilemmas that teachers face in implementing educational technology in their classrooms. the point of this theory is to understand how to use technology to teach concepts in a way that enhances students learning experiences. thus, teachers may utilize technology properly in classroom to help students develop deeper understanding of mathematics concepts. ict on the teaching and learning mathematics cannot be denied by every mathematics teacher and educator. some studies show that the use of ict in teaching and learning mathematics has a positive impact on students’ performance and learning achievement (delen, & bulut, 2011; comi, argentin, gui, origo, & pagani, 2017; eickelmann, gerick, & koop, 2017). the role of ict in the curriculum is much more than simply a passing trend. it provides a real opportunity for teachers of all phases and subjects to rethink fundamental marvelino m. niem, rizza u. veriña, & emil c. alcantara 29 pedagogical issues alongside the approaches to learning that students need to apply in classrooms. transferring of information, collecting of data and researching are the multiple benefits that students can get from ict, but it still a dream for many. most public schools in the philippines have no complete ict facilities like insufficient number of computer units, the unstable internet connection, and the lack of teachers’ skills and knowledge on the proper use of technology which results in poor student and school performance. this was evidently seen in the 2003 trends in international mathematics and science study (timms), the philippines is among the worst performer among participating countries. the country scored way below the average international score. it ranked 23rd among 25 in the elementary level and 34th in 38 countries in the secondary level for the mathematics examination. timss 2008 advanced results showed that in general, philippines performed least among ten (10) participating countries in mathematics. these observations are collaborated by world economic forum’s global competitiveness report of 2010-2011, that finds the country ranking low in the quality of science and mathematics, i.e., 112th among 139 countries surveyed (swab, 2011 cited in manaligod & garcia, 2012). due to ict’s importance in mathematics education, the researchers felt the need to study the following: (1) to determine the profile variables of the mathematics teachers in terms of: sex, highest educational attainment, years in teaching, designation and seminars attended about ict integration; (2) to what extent do the teachers integrate ict in teaching and learning mathematics; (3) to identify if there are significant differences between the teachers’ integration of ict-based teaching and learning in mathematics when grouped according to profile variables; and (4) to recognize the challenges encountered by the teachers in ict-based teaching and learning mathematics. methodology this study utilized mixed methods of research. in the qualitative part, the researchers used interview methods and fgd. however, in the quantitative part, descriptive methods were used to report the extent of the teacher integration of ict and the challenges faced by the teachers in integrating ict into teaching and learning mathematics. there were 95 public school mathematics teachers both in junior and senior high schools in 16 secondary schools in the division of tanauan city which is owned by the government wherein the students were also from the city itself. it is a highly urbanized city located at the southern part of metro manila in the province of batangas which is an hour drive from it. using raosoft sample size determination, from 95 the researchers got 77 as the sample and the respondents were selected using a stratified random sampling technique per school. the researchers initially secured the permit to conduct a study from the schools division superintendent. before handling the questionnaires, the researchers used a researcher-made questionnaire that was validated by two mathematics professors from batangas state university-main campus and two external validators, one was a master teacher in mathematics and the other one was an education program supervisor in mathematics by checking its content mathematically. after checking the validity, it was made ready for dry-run to 20 secondary school mathematics teachers from the division of lipa city. then, the reliability of the instrument was obtained using cronbach alpha statistical tool teaching and learning with technology: ramification of ict integration in mathematics education 30 and reliability co-efficient was 0.737. the research instrument was deemed valid and suitable for use. after the dry run, it was made ready for actual administration of the questionnaires. upon approval, the researchers personally administered the questionnaires at a time agreed upon by the approving authority. explicit instructions and motivations were given and explained aside from the instructions specified in the questionnaire for clarity. the instrument consisted of three (3) parts. part i is about the demographic profile of the respondents in terms of sex, highest educational attainment, years in teaching, designation and number of seminars attended related to ict for the past five (5) years. part ii and iii of the questionnaire addresses about the extent of the use of ict integration into teaching and learning mathematics and the challenges encountered by the teacher on ict – based teaching and learning in mathematics education respectively. there were fifteen (15) items on those part and rated based on the four-point likert scale equivalence. it was provided by the following description: option four (4) with a scale range from 3.25 4.00 with a verbal interpretation of very great extent or strongly agree; option three (3) with a scale range of 2.50 – 3.24 with a verbal interpretation of great extent or agree; option two (2) with a scale range 1.75 – 2.49 with a verbal interpretation of little extent or disagree; and option one (1) with a scale of 1.00 – 1.74 with a verbal interpretation of no extent at all or strongly disagree. likewise, the researchers conducted a follow-up interview and fgd for the selected secondary school mathematics teachers. all schools were represented evenly for the follow up fgd. the weighted mean, frequency distribution, percentage, mean, standard deviation, and t-test were the statistical tools used to analyse the data gathered. results and discussion demographic profile of the respondents table 1 profile of the respondents in terms of sex sex frequency percentage male 31 40.26 female 46 59.74 total 77 100 table 1 presents the distribution of respondents in terms of sex. thirty one (31) or 40.26 percent are male and 46 or 59.74 percent are female. there was a total of seventyseven respondents in the study. the results yielded that most of the respondents were female. marvelino m. niem, rizza u. veriña, & emil c. alcantara 31 table 2 profile of the respondents in terms of their educational attainment highest educational attainment frequency percentage bachelor’s degree 63 81.82 master’s degree 14 18.18 total 77 100 distribution of the respondents in terms of their educational attainment is shown in table 2 above. there are 63 or 81.82 percent bachelor’s degree holder. fourteen (14) or 18.18 percent are master’s degree holder. the results show that most of the respondents are bachelor’s degree holder. table 3 profile of the respondents in terms of years in the service years in the service frequency percentage 15 and below 57 74.03 16 and above 20 25.97 total 77 100 table 3 shows the distribution of the respondents in terms of years in service. fiftyseven (57) or 74.03 percent are from 15 years and below. meanwhile, the remaining 25.97 percent with a frequency of 20 respondents are from 16 years and above in the service. as the results show, majority of the respondents are in the service for 15 years and less. table 4 profile of the respondents in terms of their designation designation frequency percentage teacher 72 93.51 master teacher 5 6.49 total 77 100 table 4 shows the distribution of the respondents in terms of their designation. seventy-two (72) or 93.51 percent are teachers i-iii. five or 6.49 percent are master teachers i-ii. the results manifested that majority of the respondents are teachers i-iii. teaching and learning with technology: ramification of ict integration in mathematics education 32 table 5 profile of the respondents in terms of number of ict – related seminar for the past 5 years number of ict – related seminars attended interpretation frequency percentage 5 below not active 76 98.70 6 – 10 active 1 1.30 total 77 100 table 5 shows the distribution of the respondents in terms of the number of ictrelated seminars attended for the past 5 years. seventy-six (76) or 98.70 percent had attended 5 seminars and below which is verbally interpreted as not active. one or 1.30 percent had participated six to 10 seminars referred to as active. the result yielded that most of the respondents do not have enough training or seminars attended related to ict integration in teaching mathematics. as stated in table 3, most of the respondents are in the service for 15 years and below. this can be the possible reason why in terms of the number of ict-related seminars, majority or 76 respondents are not active since they only got to attend five and below. the result is alarming since the 21st century education requires more touch of technological advancement in the classroom, with ict integration in particular. the knowledge of the teachers in applying such in every lesson delivered to the students can be acquired from trainings. during the interview conducted, one teacher responded that she just had one opportunity to be part of mathematics-related ict integration training. she can barely recall how thankful she was with an expectation that she would be having another chance to attend such kind. however, the same opportunity did not knock again. with this, she just inquires to other teachers whom she believes are more knowledgeable when it comes to various matters relative to ict integration in mathematics. she also added that learning action cell (lac) is also of a great help to her since during each session teachers who have attended the seminars and trainings have the chance to cascade what they acquired. lac session is somewhat similar to a department meeting wherein the teachers were just sharing their knowledge and ideas from the previous trainings and seminars attended. on the other hand, that one respondent who was able to attend at least 6 to 10 trainings is one of the ict coordinators who happened to be a mathematics teacher as well. with this, he has more opportunity to participate in various trainings of ict as far as mathematics teaching is concerned. extent of the use of ict integration in teaching and learning mathematics table 6 shows the extent of the use of ict integration into teaching and learning mathematics. the result of the survey yielded that the indicator “i use computers in writing lesson plans in mathematics” got the highest weighted mean of 3.87 and has a verbal interpretation of very great extent. daily lesson logs can be made easier through computers since its template is already provided by the department of education. the teachers just have to fill the necessary information for each part. marvelino m. niem, rizza u. veriña, & emil c. alcantara 33 table 6 extent of the use of ict integration in teaching and learning mathematics indicators weighted mean verbal interpretation i use …  computers in writing lesson plans in mathematics.  computers to compute student’s results  power point presentation in such a way that learners will follow the lesson systematically  television/projector to teach math in class.  multi-media presentations to vary teaching methods like discussion, cooperative learning, differentiated instruction and blended learning.  scientific calculators in dealing with basic to higher mathematical calculations  slides, flash drives, diskettes, cd-rom’s for teaching mathematics in the classroom  the internet to access relevant information.  computer to generate graphs, draw columns and bar charts using spreadsheets.  basic computer software programs such as word processor and microsoft word to teach mathematics.  educational and interactive software like kahoot, quizlet, plickers, triventy, etc. to give students drill and activities to practice.  electronic graphic board or interactive white board to illustrate basic mathematical concepts.  dynamic software like google classroom, geogebra, desmos, mentimeter, etc. to teach learners mathematical concepts for better understanding and clarity in lesson proper.  social media like facebook, youtube, instagram, etc. in teaching and learning mathematics.  telecommunication devices such as cable, satellite, fax-machine etc.to interact with the pupils. 3.87 3.71 3.62 3.61 3.48 3.44 3.44 3.38 3.16 3.14 2.69 2.61 2.57 2.53 2.21 very great extent very great extent very great extent very great extent very great extent very great extent very great extent very great extent very great extent great extent great extent great extent great extent great extent little extent composite mean 3.16 great extent legend: 1 – 1.74 no extent at all 1.75 – 2.49 little extent 2.50 – 3.24 great extent 3.25 – 4.00 very great extent teaching and learning with technology: ramification of ict integration in mathematics education 34 “even if making daily lesson logs are one of the difficult things we have to prepare before seeing our students, the department still finds ways to make our lives easier in coming up with one”, one of the respondents shared during the fgd. the other respondent added that there are websites by which we can download available lesson logs and other materials for our lesson for the specific day or week. “all we need is internet connection and modify those that we downloaded according to the needs of our students”, one more respondent delivered. the indicator, “i use computers to compute students’ result” which has a weighted mean of 3.71 has a verbal interpretation of very great extent. the so-called e-class record or electronic class record is also provided to teachers. here, the teachers just have to input the scores of the students and then the final result will be yielded once every column has a complete entry. it was a great help for the teachers to make their computation easier and with lesser mistakes. on the other hand, it was found out that when it comes to the use of other telecommunication devices such as cable, satellite, fax-machine etc. to interact with the pupils the teachers, the indicator only got a weighted mean of 2.21 which is verbally interpreted as little extent. as revealed in the fgd, lack of budget to allocate in purchasing cables or internet connections. instead of putting internet, or cables, satellites, the budget is allocated for the maintenance of school, cleanliness and teacher/student developmental fund. to sum up, the respondents utilized ict in teaching and learning mathematics with a great extent as supported by the composite mean of 3.16. with this, it can be inferred that teachers really address the needs of our 21st century learners. it is never told that the traditional way of teaching has to be totally eliminated but based on the results, the teacherrespondents adjust to what is required by the k-12 curriculum. the researchers were also able to record the response by one of the teachers. he mentioned that there was a time when he entered the room with his chalk and book only, and he observed that students did not have that interest to look at the board which may imply that they were not listening since it was a mathematics class. as he continued, he then shared that the next time he entered the room with the power point presentation (ppt) together with his chalk and board, students seem to be listening well, a sign of interest to learn. the findings of the study are similar to the study of ghavifekr and rosdy (2015) which presented the degree of ict integration of teachers into teaching and learning process is high. all the teacher-respondents were able to use computer as a tool for demonstration working with presentations, they have made themselves such as ppt. on the other hand, the results are contrary to the study of bosah (2015), which revealed that there were no ict devices available for teaching mathematics in schools thus, the teachers do not make use ict devices in teaching mathematics in demonstration primary schools in anambra state. difference between the extent of teachers’ integration of ict-based teaching and learning in mathematics and the respondents’ profile table 7 shows the significant differences between the extent of teachers’ integration of ict-based teaching and learning in mathematics and the respondents’ profile. from the variable presented, sex, education, designation and seminars attended have a p-value of 0.536, 0.645, 0.105, and 0.170 respectively. since, 𝑝 > 0.05, the null hypothesis was failed to reject, which means there is no significant differences between the extent of the use of ict marvelino m. niem, rizza u. veriña, & emil c. alcantara 35 and the profile variables in terms of sex, education, designation and seminars attended. thus, the data provide insufficient evidence to conclude that the extent of the use of ict integration in teaching and learning mathematics is remarkable in those profile variables. on the other hand, the results also show that the use of ict integration of teachers whose years in teaching is 15 and below (m = 3.27, sd = 0.50) is higher than the use of ict of teachers whose years in teaching is 16 and above (m = 2.88, sd = 0.44) and has a p-value of 0.002. since, 𝑝 < 0.05, the null hypothesis is rejected. this means that the means of the two groups are significantly different from each other. thus, the data give sufficient evidence to conclude that the teachers whose years in teaching is below 15 integrate ict-based teaching and learning mathematics higher than the teachers whose years in teaching is above 16. it can be inferred that seasoned teachers need to be updated in ict-based teaching and learning mathematics and should be given more opportunities to attend seminars and trainings about ict integration into mathematics. table 7 significant differences between the extent of teachers’ integration of ict-based teaching and learning in mathematics and the respondents’ profile variables mean standard deviation p – value decision interpretation sex male 3.21 0.47 0.536 failed to reject ho failed to reject ho reject ho failed to reject ho failed to reject ho not significant not significant significant not significant not significant female education bachelor’s degree master’s degree 3.13 3.15 3.22 0.54 0.53 0.46 0.645 years in teaching 15 and below 16 and above designation 3.27 2.88 0.50 0.44 0.002 teacher master teacher 3.18 2.87 0.52 0.39 0.105 seminars not active active 3.15 3.87 0.51 -- 0.170 legend: 𝑝 ≤ 0.05 significant 𝑝 > 0.05 not significant the findings were opposed to the study conducted by ghavifekr (2015) which presented that there is no significant difference between sex and extent of use of ict. however, in the extent of use of ict and years in service it was found to be significant. challenges encountered by teacher on ict-based teaching and learning in mathematics education table 8 shows the challenges encountered by teacher on ict-based teaching and learning in mathematics education. the result manifested that respondents all strongly agreed on the three indicators: “shortage of basic infrastructure such as classrooms and internet connectivity”; “limited number and variety of subject-specific educational software available teaching and learning with technology: ramification of ict integration in mathematics education 36 in schools”; and “teachers’ lack of confidence with the use of ict in teaching-learning process”, with a weighted mean of 3.47, 3.39 and 3.36 respectively. on the other hand, the respondents agreed to the remaining indicators and the item that has the lowest weighted mean is “lack of teacher’s knowledge on how to integrate ict into pedagogical practice”. in the interview and fgd, respondents highlighted that lack of funding from the school as a contributing factor to insufficient resources. as stated, the only budget allocated to schools is for learner support material. the fgd also revealed that the absence or the slow internet connection is an obstacle to the utilization of the supposed available resources in various reliable websites. also, as disclosed during the fgd, lack of training and seminars related to ict integration in mathematics teaching is one of the complaints of the teachers since they really believe that with these, they can gain learning which they can transfer to the teaching process. this result is also supported by the data garnered from the survey questionnaire. in general, the teacher-respondents agreed on the different challenges encountered by the teacher on ict based teaching and learning mathematics listed on the table. this was substantiated in the composite mean of 2.84. with this, it can be inferred that even if there is a call for the ict integration in teaching mathematics and even to other subjects, the teacherrespondents are not ready enough in the full materialization of the integration of ict in mathematics teaching and learning. teachers still need to be trained more and the school has to have the necessary materials. table 8 challenges encountered by teacher on ict-based teaching and learning in mathematics education indicators weighted mean verbal interpretation  shortage of basic infrastructure such as ict classrooms and internet connectivity  limited number and variety of subject-specific educational software available in schools  teachers’ lack of confidence with the use of ict in teaching-learning process 3.47 3.39 3.36 strongly agree strongly agree strongly agree  inadequate ict equipment and resources like computers, laptops, projectors, whiteboards etc., available for students 3.18 agree  difficulties in scheduling enough computer time for classes as a problem in the use of ict in teaching-learning 3.13 agree marvelino m. niem, rizza u. veriña, & emil c. alcantara 37 indicators weighted mean verbal interpretation  scarcity of time needed to locate internet information, prepare lessons, explore and practice using the ict  lack of sufficient trainings for teachers to develop appropriate skills, knowledge, and attitudes regarding the effective use of ict to support learning 3.13 2.99 agree agree  absence of specific curricular standards and guidelines for integrating computers into the subject areas  lack of technical support to encourage the use of ict from administration side  lack of local school policy that will mandate teachers to use ict  teachers’ attitudes toward ict integration in instructional settings 2.97 2.91 2.65 2.60 agree agree agree agree  lack of both technical and pedagogical knowledge and skills of the teacher to use available icts in the classroom  lack of teachers’ acceptance and adoption of ict in the classroom  society, school and colleagues’ negative views about ict integration in class  lack of teacher’s knowledge on how to integrate ict into pedagogical practice 2.53 2.49 2.48 2.38 agree disagree disagree disagree composite mean 2.84 agree legend: 1 – 1.74 strongly disagree 1.75 – 2.49 disagree 2.50 – 3.24 agree 3.25 – 4.00 strongly agree as regards with the results, it seems that the teachers are challenged to teach the subject because of lack of resources like internet connection and some educational software. the teachers also lack confidence for some reason that they feel that in this 21st century education they are left behind. they also lack knowledge on integrating ict into pedagogical practice and it is shown in the number of seminars attended for the last five years. the findings of the study is similar to the study conducted by ghavifekr and rosdy (2015) which found that most teachers think ict integration is effective, but ict tools provided in school are not enough nor in good condition; training and professional development are not adequately provided for teachers; technical supports are somehow provided but can be improved from time to time; and not very good condition of computer lab in school with well-functioning tools and facilities. teaching and learning with technology: ramification of ict integration in mathematics education 38 according to jones (2004) much of the research proposes that confidence is a major barrier to the uptake of ict by teachers in the classroom. moreover, balanskat, et. al (2006) also found that teacher’s limited knowledge on ict makes them feel anxious about using ict in the classroom. lack of knowledge makes a teacher less competent that would lead to lack of confidence (bordios, 2016). conclusion based on the findings of the study, majority of them were female, bachelors graduate, from teacher i to iii, 15 years below in the service and not active in the ict-related seminars attended. most of the respondents use technology in writing lesson plans, compute students result and power point presentation in teaching the lesson. however, due to the lack of resources and little fund from the mooe, the respondents least use the telecommunication devices such as cable, satellite, fax-machine etc. to interact with the pupils or sometimes no use at all. there is a significant difference between the extent of the use of ict in teaching and learning mathematics and years in teaching. meanwhile, the remaining profile variables such as sex, education, designation and seminars have no significant difference to the extent of the use of ict in teaching and learning mathematics. it seems that the teachers are challenged to teach the subject by lacking of resources like internet connection and some educational software. the teachers also lack confidence for some reason that they feel that in this 21st century education they feel left behind. they also lack of knowledge on integrating ict into pedagogical practice and it is shown in the number of seminars attended for the last five years. acknowledgements the researchers would like to extend their deepest gratitude to the following individuals for making this research possible. dr. emil a. alcantara, ph.d., for his words of wisdom for the improvement of the study, patience in giving important advices, highly competent remarks and invaluable suggestions that made this study successful. dr. ernesto mandigma, for his detailed and constructive comments and suggestions for the enrichment of the study. edna faura – agustin, schools division superintendent, for granting the permission to gather data from the public secondary schools in the division of tanauan city. dr. bryan manalo, dr. charity aldover, mr. don s. avelino, for validating the questionnaires. principals/officer-in-charge of all the public secondary schools in the division of tanauan city for assisting us to conduct and administer the questionnaires. mrs. rodelyn o. castañas, for helping us to interpret and analyse the qualitative part of the study. co-teachers and colleagues in the profession for the sweet advices and support. researcher’s family and loved ones, for the love, moral support, encouragement and financial support, and above all, to the almighty god, for his never-ending guidance in conducting and accomplishing this study, despite all difficulties. marvelino m. niem, rizza u. veriña, & emil c. alcantara 39 references balanskat, a., blamire, r., & kefala, s. 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(2012). challenges for using ict in education: teachers’ insights. international journal of e-education, e-business, e-management and e-learning, 2(1). retrieved from: http://ijeeee.org/papers/078-z00061f10037.pdf http://www.ijiet.org/vol7/905-jr225.pdf http://ijeeee.org/papers/078-z00061f10037.pdf open access proceedings journal of physics: conference series southeast asian mathematics education journal volume 12, no. 1 (2022) 65 didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 1 rudi rudi, 2 didi suryadi, & 3 rizky rosjanuardi 1 lembaga penjaminan mutu pendidikan sulawesi selatan, makassar, indonesia 2, 3 pendidikan matematika, universitas pendidikan indonesia, bandung, indonesia email: 1 rudi.math@upi.edu abstract didactical transposition plays a pivotal role in implementing reflective practice of a mathematics teacher community. didactical transposition is a systematic approach to examine and enhance learning materials. didactical transposition within reflective practice is conducted collaboratively between teacher trainers and teacher participants to produce an effective didactical design in encountering student learning obstacles. thus, the objective of this study is to portray didactical transposition process accomplished by teacher trainers and teacher participants in the implementation of teacher reflective practice in the community. this study employed a methodological framework known didactical design research. didactical design research in accordance with a reflective framework consists of four phases which are preparation, design/reflective planning for action, design/reflective implementation in action, and design/reflective evaluation and reflection after action. research participants encompassed a teacher trainer, thirteen teachers, and thirty students. data collection procedures were completed by employing observation and documentation. this study resulted in a didactical design which was obtained through a didactical transposition process performed by teacher and teacher trainer participants. the study discovered that didactical design developed collaboratively through didactical transposition within reflective practice in community was proven effective in overcoming the obstacles of student learning due to learning materials. research findings are expected to be a model of professional learning for mathematics teachers in the community. keywords: didactical transposition, reflective practice, mathematics teacher community, proving pythagorean theorem. introduction one of existing problems in mathematics learning is students’ difficulty. several empirical studies uncovered that one of causes in students’ learning difficulty while learning mathematics is didactical obstacles (carvalho, silva, lima, coquet & clément, 2004; elia, özel, gagatsis, panaoura & özel, 2016). other research findings revealed that teachers encounter difficulties in dealing with students’ didactical obstacles (rudi, suryadi, & rosjanuardi, 2020 a, 2020b; kuzniak & rauscher, 2011). didactical obstacles occur due to learning materials, curriculum, and design utilised by teachers (brousseau, 2006; artigue, haspekian, & corblin-lenfant, 2014). therefore, it is considered significantly required to ensure that mathematical knowledge encompassed in a learning design and delivered by teachers in a classroom is aligned with curriculum and scholarly knowledge. the didactical transposition concept is in accordance with a belief that knowledge requires transformation to be effectively applied for classroom teaching objectives. didactical transposition emerged from the assumption that knowledge was initially invented not to be taught to someone else but to be utilized for purposeful ends (kang & kilpatrick, 1992) mailto:rudi.math@upi.edu didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 66 causing knowledge to transform after undergoing a process of selection, design, production, and teaching in an educational institution (chevallard & bosch, 2020). the term ‘didactical transposition’ was initiated by chevallard (1989). chevallard and bosch (2020) defined that didactical transposition encompasses four phases which are scholarly knowledge, knowledge to be taught, taught knowledge, and learned/available knowledge. the didactical transposition process possesses a vital role in the administration of a mathematics education program (winslow, 2011; arzarello, robutti, sabena, cusi, garuti, malara, martignone, 2014), in which, the mathematics education program prepares teacher candidates and builds the professionalism of in-service mathematics teachers. reflective practice in community is a form of professional learning for in-service mathematics teachers. referring to the framework of reflective practice by grushka (2005), three stages of reflective practice are elaborated; reflection on action, reflection in action, and reflection for action. regarding to the three stages, lesson study is a form of teacher professional learning implementing these reflective practice stages. didactical transposition occupies a considerable position in implementing reflective practice by teacher communities. didactical transposition is a systematic approach to scrutinize and develop learning materials (artigue, 1994). didactical transposition allows the teachers to examine whether learning materials conducted by teachers in the classroom are valid from the perspectives of learning design, students’ condition, curriculum and mathematical scholarly knowledge (bosch & gascón, 2006). didactical transposition also enables the teaching practitioners to analyse whether teachers’ knowledge on learning materials is associated with scholarly knowledge (arzarello et al., 2014). if mathematical knowledge acquired by students and teachers has been justified to be in accordance with scholarly knowledge and curriculum, it is expected that the obstacles and difficulties of students’ learning affected by learning materials can be reduced (jamilah, suryadi & priatna, 2020). one of problems associated with community reflective practice in indonesia is that it has not yet been placed its concern on students’ learning obstacles and difficulties (saito, harun, kuboki, tachibana, 2006), hence, a method dealing with students’ learning obstacles and difficulties has not yet been discovered (suratno, 2012). learning material analysis and development through a teacher reflective practice community provides as an attempt not only in dealing with students’ learning obstacles and difficulties but also to enhance teachers’ knowledge in overcoming the obstacles and difficulties of students’ learning (rudi, suryadi & rosjanuardi, 2020a, 2020b). didactical transposition of reflective practice in mathematics teacher is expected to provide solutions to the current problems of reflective practice in mathematics teacher communities in indonesia. several empirical studies have implemented didactical transposition in examining mathematics education program. after chevvallard initiated didactical transposition in mathematics education, arzarello et al. (2014), developed chevallard’s ideas into metadidactical transposition theory. the framework is applied in analysing didactical transposition process perceived from institutional dimension that is the university as a knowledgeproduction institution and the school as knowledge-user institution. then, gilles (2020) implemented this theoretical framework in establishing collaboration between researcher and teacher community. meanwhile, shinno and yanagimoto (2020) elaborated on the rudi rudi, didi suryadi, & rizky rosjanuardi 67 affirmative values of collaboration between pre-service and in-service teachers. hausberger (2017) investigated the homomorphism concept in didactical transposition performed in university whereas jamilah et al. (2020), scrutinised the didactical transposition within the education of mathematics pre-service teacher. however, the existing empirical studies on didactical transposition have not yet determined the method of didactical transposition process administered in a mathematics teacher community for reflective practice. in previous research, investigation of didactical transposition was more oriented towards researchers’ perspectives as the university representatives and teachers’ perspectives as school representatives. meanwhile, in this study, the researchers were replaced by teacher trainers from a teacher competency development institution in indonesia of which the major responsibility is facilitating the enhancement of education quality in indonesia. the teacher trainer in this particular case was performed by widyaiswara or civil servants who are educating and training in-service teachers in indonesia. the objective of this research is to identify the didactical transposition process of reflective practice in a mathematics teacher community. didactical transposition was conducted by a teacher trainer and teacher participants in the community. in this study, an investigation of didactical transposition concerned on proving the pythagorean theorem. the selection of this topic was in accordance with the fact which substantiating the pythagorean theorem was considered the most difficult topic for students (zaskis & zaskis, 2016). as elaborated in the indonesian mathematics curriculum for junior high school level, proving the pythagorean theorem is the only basic competence which requires students to prove something. methods this study employed methodological framework of didactical design research (ddr) developed by suryadi (2015) and suryadi, prabawanto & takashi (2017). ddr functions as a theoretical and methodological framework in implementing reflective practice. suryadi, et al. (2017) broke down reflective practice into three stages based on the ddr framework. the three stages encompass prospective analysis (reflection for action) in the form of hypothetical didactical design, metapedadidactic analysis (reflection in action), and retrospective analysis (reflection after action). the three stages were developed by rudi et al. (2020a, 2020b) by adding a stage which is reflective practice preparation. the application of the ddr methodological framework was due to a perspective that ddr functions to overcome students’ difficulty and update teachers’ knowledge through reflective practice (rudi et al., 2020a, 2020b). research participants encompass a teacher trainer, thirteen mathematics teachers in junior high school also as members of a mathematics teacher community in a regency in south sulawesi province, indonesia, and thirty students of grade eight in a state junior high school in indonesia. those participants were purposively selected. data collection procedures were accomplished through a teacher trainer’s journal, observation, and document study. data on scholarly knowledge and knowledge taught were collected from the teacher trainer’s journal. meanwhile, data on taught knowledge and available/learned knowledge of the teacher trainer were obtained from observation towards didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 68 the learning process of mathematical content knowledge reinforcement performed by teacher participants in the community. information in all stages of didactical transposition acquired by participants was gathered by employing observation techniques toward the learning of teacher participant in the community, while data on taught knowledge and available/learned knowledge of teacher participants were collected through observation during the learning process implementing the design. results and discussion didactical transposition by the teacher trainer the didactical transposition process by the teacher trainer employed the four stages developed by chevallard (1989, 1992) encompassing scholarly knowledge, knowledge to be taught, taught knowledge, and learned/available knowledge. scholarly knowledge is scientific knowledge enhanced and designed by university or other scientific institutions. scholarly knowledge is available in reputable research results sections of journals, and books. the teacher trainer was accessing scholarly knowledge on proving pythagorean theorem topic in literature such as books, articles, and scientific reviews from reputable scientific journals in digital form. dissemination of the research findings as the university products in digitalisation has been progressing and becoming trends in various countries (saarti & tuominen, 2020; moeini, rahrovani & chan, 2019). figure 1. didactical transposition in the implementation of reflective practice in a teacher community preparation for reflective practice rudi rudi, didi suryadi, & rizky rosjanuardi 69 didactical transposition completed by the teacher trainer at the scholarly knowledge stage functioned to ensure that materials provided to teachers were in accordance with scholarly knowledge (figure 1). at this stage, the teacher trainer implements didactical transposition to examine the used knowledge (atalar & ergun, 2018). elisha scott loomis (1852-1940) gathered 371 methods of evidence and compiled them in a book published in 1927 (ratner, 2009). apart from loomis’s book, there have been other books elaborating methods for proving the pythagorean theorem. however, most of the books cite loomis’s ideas. not only in the form of books, but other proving methods are also published in the form of articles in reputable journals. according to a reprinted book on loomis (1968), four types of methods can be applied in proving the pythagorean theorem comprising of algebra proving which implements linear equation method, geometry proving which performs comparison of the area of geometric shapes, quaternionic proving applying vector operation, and dynamic proving concerning the correlation between mass and speed. the result of didactical transposition on the teacher trainer’s knowledge to be taught is displayed in figures 2 and 3 below. figure 2. result of didactical transposition on the knowledge to be taught of chou-pei suan-ching proving (veljan, 2000). figure 3. result of didactical transposition on knowledge to be taught of james a. garfield proving (veljan, 2000). knowledge to be taught is a scholarly knowledge process resulting in curriculum-based knowledge (bosch, & gascón, 2006). knowledge to be taught can be acquired through a literature review from textbooks (jamilah, et al., 2020). in this research, the teacher trainer employed textbooks published by the teacher competency development institution in indonesia. this fact unveils the institutional role within the reflective practice of teacher community. this idea is in accordance with the perspectives proposed by chevallard (1992) didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 70 and arzarello, et al. (2014), which asserted the significance of the institutional dimension in the development process of the mathematics education program. regarding the teacher trainer’s analysis, it was presented that the proving methods by employing geometry and algebra were correlated with the curriculum and textbooks developed during the agenda of material reinforcement. the four proving methods were obtained from bhaskara, chou-pei suan-ching, tsabit ibnu qurra, and james a. garfield. chou-pei suan-ching and james a. garfield proving methods are displayed in the following figures 4 and 5. the result of the didactical transposition accomplished by the teacher trainer on the taught knowledge as presented in figure 4, the algebra proving of chou-pei suan-ching was encountering transposition to be a geometry proving. the teacher trainer provided different colors to the four right triangles. this idea tended to make the teacher participants easier in comprehending the shift/movement of the right triangles. based on the result of didactical transposition in figure 5, the teacher trainer produced a right triangle beside the trapezium. the result of the didactical transposition on taught knowledge becomes the basis in designing scaffolding for the teacher trainer and follow-up for teacher participants who possessed issues. figure a figure b figure 4. result of didactical transposition on taught knowledge, chou-pei suan-ching proving area of trapezium method i = area of triangle i + area of triangle ii + area of triangle iii area of triangle i = ½ x a x b area of triangle ii = ½ x a x b (triangle ii =triangle i) area of triangle iii = ½c x c = ½c 2 meaning that area of trapezium i = ½ x a x b + ½ x a x b + ½ x c 2 (axb = ab) = ½ (2 ab + c 2 ) area of trapezium method ii = ½ (a+b) (a+b) = ½ (a+b) 2 = ½ (a 2 +2ab+b 2 ) area of trapezium i = area of trapezium ii => ½ (2 ab + c 2 ) = ½ (a 2 + 2ab + b 2 ) (2 ab + c 2 ) = (a 2 + 2ab + b 2 ) c 2 = a 2 + b 2 (proven) figure 5. result of didactical transposition at the taught knowledge stage method 1 the area of the rectangle in figure a is the same as the area of the rectangle in figure b. the area of the white rectangle in figure a is the same as the area of the two white rectangles in figure b. the area of the white rectangle in figure a with c side is c 2 . the area of white rectangle in figure b with a and b side is a 2 +b 2 . hence, the conclusion is a 2 +b 2 = c 2 method 2 area of square in fig.a = (a+b)2…..(1) the area of square in fig.a consists of 4 triangles = c 2 +4. ½ (a.b) …. (2) from (1) and (2), we can deduce: a 2 +2ab+b 2 = c 2 +2ab, or a 2 +b 2 = c 2 rudi rudi, didi suryadi, & rizky rosjanuardi 71 knowledge acquired as taught knowledge stage was implemented in the form of teaching materials which were employed in the material reinforcement activity. not only did the teacher trainer assure the concordance of the materials with the scholarly knowledge, but he/she is also required to certify that those materials had been conformed to the teachers’ social environment and condition. during the reflective practice, the role of teacher trainer is associating scholarly knowledge and teacher characteristics (szűcs, 2018). this proposition is also embedded in the idea that teachers must possess three types of basic knowledge when teaching consisting of mathematical knowledge, mathematical knowledge for teaching, and pedagogical knowledge (sullivan, 2008). learned/available knowledge was the didactical transposition stage based on the teacher participants’ knowledge produced during the learning process. teacher participants responded to the situational design of chou-pei suan-ching proving by employing the geometry proving (figure 6) and the algebra proving method (figure 7). meanwhile, teacher participants’ responses towards proving by james a. garfield were comparable to the knowledge to be taught stage (figure 8). the result of the didactical transposition of learned/available knowledge by the teacher trainer and the teacher participants is illustrated on the following figures 6, 7, and 8. see figure 1 and 2. the white area in figure 1 is equal to white area in figure 2. *the white area in figure 1 is c 2 … (1) *the white area in figure 2 is a 2 + b 2 … (2) as the white area in figure 1 and 2 are the same, equation from (1) and (2) is formulated as followed: c 2 = a 2 + b 2 (proven) figure 6. result of didactical transposition at the learned/available knowledge stage area of white rectangle = area of white rectangle area of 4 triangles in figure 2 s figure 7. result of didactical transposition at the learned available/knowledge stage didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 72 area of trapezium abcd= area of triangle abc + area of triangle cbe + area of triangle bed (proven) figure 8. result of the teacher participants’ work in didactical transposition at the learned/available knowledge stage didactical transposition accomplished by the teacher trainer in this research concentrated more on the development of university-constructed materials rather than on knowledge construction. institutionally, the teacher trainer played a more significant role in completing duties and functions in the teacher competency development institution, compared to researchers accomplishing duties and functions in the university institution. this finding is unlike meta-didactical praxeology ideas which organise the researcher community as the representatives of university institutions (taranto, robutti & arzarello, 2020). metadidactical praxeology is a stage in investigating a mathematics education program by implementing theoretical framework of meta-didactical transposition (arzarello et al., 2014). didactical transposition by teacher participants similar to the didactical transposition administered by the teacher trainer, the elaboration on the didactical transposition by teacher participants also adhered the four steps enhanced by chevallard (1989) which consist of scholarly knowledge, knowledge to be taught, taught knowledge, and learned/available knowledge. teacher participants performed didactical transposition at the design planning as reflection for action, the design implementation as reflection in action, and the evaluation of design reflection as reflection after action. reflection for action (design planning) development of hypothetical learning trajectory (knowledge to be taught) was conducted through hypothetical didactical design development in overcoming students’ learning difficulties and obstacles (taught knowledge). reflection on action (design implementation) was implemented through content knowledge analysis understood by students (learned/available knowledge). reflection after action (design evaluation) was applied through investigation on taught knowledge and learned/available knowledge to produce empirical didactical design. teacher participants elevated their scholarly knowledge during the material reinforcement agenda. this agenda was as a part of the reflective practice preparation in the teacher community. previous discussion asserted that the material reinforcement is included in the learned/available knowledge stage in the didactical transposition conducted by teacher trainers. therefore, the learned/available knowledge in the teacher trainer’s didactical transposition also functioned as the scholarly knowledge. rudi rudi, didi suryadi, & rizky rosjanuardi 73 didactical transposition process at the knowledge to be taught step was accomplished by teacher participants while generating the hypothetical learning trajectory. in this meeting, teacher participants were discussing the learning objective, instructional theory, and its supporting activities. the formulation of activities was expected to acquire the learning objectives by concerning the local instructional theory plot. this idea is based on a perspective that research and curriculum development in mathematics education are intercorrelated and integrated, employing varied methodologies, considering correlation between children’s tasks and thoughts, and performing various learning trajectories (clements & sarama, 2004). an investigation of hypothetical learning trajectory correlates the knowledge development with pedagogical aspects such as the learning methodology (simon & tzur, 2004). this stage belongs to one of the lesson studies stages designed by suratno (2012). figure 9. formulation of local instructional theory in proving the pythagorean theorem as presented at the formulation of the local instructional theory in figure 9, teacher participants experienced didactical transposition while generating assumptions on student learning trajectory. the didactical transposition conducted by the teacher participants towards scholarly knowledge when proving the pythagorean theorem contained transformation on shapes from still to moving images through learning multimedia or puzzles. in proving the method by james a. garfield, the teacher participants also provided different colours to the three right triangles constructing the trapezium. the teacher participants’ didactical transposition at the taught knowledge stage was conducted when codifying the hypothetical didactical design. at this stage, the teacher participants developed the situational design, the prediction of students’ response, and the teacher anticipative prediction or assistance. the didactical transposition at the taught knowledge stage concentrated on the integration of the teacher participants’ content knowledge in accordance with the student learning difficulties and obstacles. the designing process is required to accommodate mathematical and didactical knowledge incorporating cognitive, interactional, and mediational aspects (pino-fan, assis & castro, 2015). at this stage, the material design is devised by adhering to the triangle interaction among teacher, material, and student (rudi et al., 2020b). the formulation result of hypothetical didactical design accommodates situational design, student response, and assistance prediction, teachers’ follow-up. students’ worksheets illustrated in figures 10 and figure 11 presented that the outcome of students’ knowledge construction was similar to what teachers possessed. nevertheless, there proving pythagorean theorem and triples puzzle powerpoint media algebra proving geometry proving algebra proving geometry proving didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 74 was a difference in conducting the mathematical representation. variety also appeared when students performed the process of proving the theorem by employing their own words. students applied different language unlike the teachers’ prediction. it implied that the developed didactical design was adequately effective to overcome students’ learning difficulties and obstacles. it is corroborated by other research findings which state that the hypothetical learning trajectory increases teachers’ professionalism (ekawati, wintarti & kurniasari, 2020; ivars, fernández, llinares & choy, 2018), and didactical design developed by concerning the hypothetical learning trajectory is effective enough to overcome students’ learning difficulties and obstacles (diana & suryadi, 2020). figure 10. students’ worksheets at the learned/available knowledge didactical transposition stage 5. determine the area of the white rectangle in figure 1 and white rectangle in figure 2. area of white rectangle in figure 1: c x c = c 2 area of white rectangle in figure 1: a x a = a 2 b x b = b 2 6. determine the correlation between the area of white rectangle in figure 1 and the two white rectangles in figure 2. (answer) the area of rectangle in figure 2 are the same, but in figure 2, the rectangle is divided into 2. 7. explain in your own words how the above process proves that the squared hypothenuse side owns the same value as the total of the other squared sides. c 2 = a 2 + b 2 1. take a look carefully at the powerpoint display presenting the shifting movement of the four congruent right triangles. 2. based on the display, the shifting process can be illustrated in the following figures. 3. determine the area of rectangle in figure 1 and figure 2. area of rectangle in figure 1: c x c area of rectangle in figure 2: b 2 + a 2 4. find the area of the 4 triangles in figure 1 and 4 triangles in figure 2. area of 4 triangles in figure 1: ½ x a x t ½ x b x a x 4 area of 4 triangles in figure 2: 2 x b x a = 2ab rudi rudi, didi suryadi, & rizky rosjanuardi 75 mathematical content knowledge is an essential component that teachers should acquire. this knowledge plays a pivotal role in developing instructional activities. without reliable mathematical content knowledge, it is difficult to enhance other knowledge components (rudi et al., 2020a). hence, the design of reflective practice in this research accommodates material reinforcement as one of reflective practice preparations. it is different from the lesson study model developed by suratno (2012) and sari, suryadi and syaodih (2018) which does not include content knowledge reinforcement of teacher participants. the didactical transposition in the reflective practice established in this research refines the pmri learning environment-based lesson study which emphasises the pedagogical aspect (fauziah & putri, 2020). figure 11. students’ worksheets at the learned/available knowledge didactical transposition step conclusion didactical transposition in the mathematics teacher community for reflective practice was accomplished by the teacher trainer and teacher participants. the teacher trainer involved in the didactical transposition practice formulated materials during the material reinforcement activities for mathematics teachers. meanwhile, teacher participants conducted didactical transposition at the design planning as reflection for action, the design implementation as reflection in action, and the evaluation of design reflection as reflection after action. the didactical transposition in the mathematics teacher community for reflective practice implementing didactical design research was effective in overcoming the student learning obstacles. during the implementation of reflective practice, the teacher trainer and the teacher participants collaboratively developed the learning material design by referring to students’ learning obstacles. collaboration between the teacher trainer and the teacher participants in the reflective practice is inseparable from the institutional dimension role. in this case, institution refers to the institution of teacher competence development as the agency housing teacher trainers and the schools as the agency accommodating teacher participants. area of triangle i, ii, and iii is equal to = l1 + l2 + l3 = area of trapezium didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 76 the didactical transposition in the mathematics teacher community for reflective practice established collaboration between teacher trainers and teacher participants to enhance mathematical and didactical knowledge of the mathematics teachers in effectively overcoming students’ learning obstacles and difficulties. acknowledgements deep gratitude is addressed to the indonesia endowment fund for education (lembaga pengelola dana pendidikan – lpdp) ministry of finance, republic of indonesia for providing funds for this research. we express sincere appreciation to the mathematics teacher community in takalar regency and students of takalar junior high school, who were willing to be participants for this research. references arzarello, f., robutti, o., sabena, c., cusi, a., garuti, r., malara, n., & martignone, f. 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(2016). prospective teachers’ conceptions of proof comprehension: eevisiting a proof of the pythagorean theorem. international journal of science and mathematics education, 14(4), 777–803.https://doi.org/10.1007/s10763-014-9595-0 didactical transposition within reflective practice of an indonesian mathematics teacher community: a case in proving the pythagorean theorem topic 80 25 southeast asian mathematics education journal, volume 11, no 1 (2021) the effects of non-digital game-based learning on brunei darussalam students’ mathematical perspectives and achievements 1 nurul aqilah mohd. yusof & 1 masitah shahrill 1sultan hassanal bolkiah institute of education, universiti brunei darussalam, brunei darussalam 1 masitah.shahrill@ubd.edu.bn abstract the purpose of this study is to investigate the effectiveness of a non-digital game-based learning approach by assessing the differences in students’ achievement score between the pre-test and the post-test on the topic of multiplication and division with indices. a paired sample t-test was used to investigate a significant difference in the students’ achievements after implementing the non-digital game-based learning intervention. a total of 35 students from two classes of nine grade students in one of the secondary schools in brunei darussalam was involved in the study. another aim of this study was to investigate the students’ perspectives on using a nondigital game-based learning approach in their learning process. this was analysed through the questionnaire and interviews. the results showed that the integration of a non-digital gamebased learning approach in the mathematics lesson did have a positive effect on the students’ achievement scores. more than half of the students believed that the game has helped them to improve their mathematical skills. keywords: non-digital game-based learning, multiplication and division with indices, achievement scores, students’ perspectives introduction by using games, the “knowledge delivered through games has a profound effect on the younger generation in every aspect, such as their mind, physical appearance, and behavior” (amdan & salleh, 2016, p. 3963). non-digital games do not require direct interaction with computers or digital devices (naik, 2017). examples of non-digital games include board games, card games, mathescape (glavaš & staščik, 2017) and sliding a picture (rondina & roble, 2019). the integration of games in teaching, including the teaching of mathematics, has a long history (naik, 2017) and has been used in several studies, such as prahmana, zulkardi, and hartono (2012) and jaelani, putri, and hartono (2013); and recent studies, for example by maharani, putri, and hartono (2019), risdayanti, prahmana, and shahrill, (2019), and brezovsky et al. (2019). some students find mathematics hard and boring, which killed their interest in learning the subject. increasing learning interest in mathematics can be accomplished in several ways, one of which is to integrate the teaching of mathematics with educational games (glavaš & staščik, 2017). the purpose of this study is to explore the effectiveness of non-digital game-based learning approach on the achievements of two classes of nine-grade brunei students by assessing the score differences between the pre-test and the post-test. this study also investigated the students’ perspectives on using a non-digital game-based learning approach in their learning process. this study aims to promote a student-centred learning environment as it is one of the 26 southeast asian mathematics education journal, volume 11, no 1 (2021) pedagogical dimensions in the national education system of brunei darussalam called sistem pendidikan negara abad ke-21, spn21. the spn21 serves as a platform to sustain the ministry’s continuous commitment in providing quality education for the nation (ministry of education, 2013). this research is guided by the following research questions: what significant statistical differences can be found in the students’ achievements before and after using the non-digital game-based learning method? and what are the students’ perspectives on the nondigital game-based learning method used in their classroom? game-based learning game-based learning is defined as a primary pedagogical tool that helps to foster soft and technical skills in a structured learning process concept since games can provide some kind of contexts and framework for the learners (sousa & rocha, 2019). it also acts as an active learning environment and encourages learning activities that engage and challenge students to achieve the learning objectives (romero, usart, & ott, 2015). in addition, in a study using games on mobile devices, the researchers stated that combining learning and game playing may improve the player’s ability within the school subject and in applying it to the real world (liu & chu, 2010). in recent years, there has been an increase of focus on games as games are exciting and create a fun learning atmosphere since the students’ experiences and learning are of interest for researchers and educators (gillern & alaswad, 2016). as mentioned by yang, chu, and chiang (2018), from educational games, learners can gain enjoyment, self-confidence and satisfaction if their skills and knowledge in game-based learning are equal to the given challenging tasks. game-based learning and achievements through game-based learning, students can generate bright interaction and facilitate their learning motivation as well as learning achievement (qian & clark, 2016). previous studies using digital game-based learning have shown that students gained improvement in terms of their achievement in a number of areas. setyaningrum, pratama, and ali (2018) studied 113 eight-grade students from three junior high schools in yogyakarta. the students who used the game-based learning with smart phones in terms of a problem-solving approach significantly outperformed those students who used a problem-solving approach using a textbook based pedagogy. meanwhile park, kim, kim, and yi (2019), studied 64 university students using an arrow-shooting game in the context of english vocabulary learning and reported a significant increase in the level of learning, motivation and engagement. naik (2017) studied the use of non-digital games to instruct first-year bsc computer science students who lacked experience of formal mathematical instruction beyond elementary levels. naik reported improved examination results and positive student feedback concerning the learning experience. turgut and temur (2017) studied the effects of game-assisted mathematics education on academic achievements in turkey using the meta-analysis approach where the results inferred that using games in teaching mathematics generally had a positive effect on the students’ achievements. as a matter of fact, previous studies have shown positive effects in using game-based learning approach, in a way that game-based learning has been implemented as one of the teaching strategies; however, the effectiveness of the approach remains unclear (law & chen, 27 southeast asian mathematics education journal, volume 11, no 1 (2021) 2016). furthermore, the fact that students have different interest may lead them to consider the games as either interactive or not, engaging or not. some students may find it more comfortable to work using drilling and conventional practice since this is the kind of learning method they have ever implemented since primary schools. another concern is the fact that having games in class would take a lot of time in the students’ learning process depending on their achievement and developmental differences, and thus at the same time, some students would be reluctant to participate in the study (fitriah, 2018). although the use of game-based learning is popular, most of the games are digital. therefore, many areas on non-digital game-based are unexplored. previous studies using digital game-based learning suggests positive results on students’ learning in terms of their performance, but there are still much to be explored on how the non-digital game-based has any effect on the students’ performance. although there was no previous study to report students’ improvement in mathematical achievement after the introduction and utilization of game-based learning method, be it the digital or non-digital games, this present study may give some indication that the assumption may be correct. methods research design the research design is based on an action research approach. action research has been used for empowering teachers since it is a significant medium (cohen, manion, & morisson, 2011). action research operates in cycles of planning, executing and fact-finding, as it is a significant feature and it is used as a way of representing action research (mcniff, lomax, & whiteheard, 1996). denscombe (2010) believed that it is a commitment to a research process where the findings application and the practice evaluation’s impact become part of the research cycle. this action research was executed in two cycles. the first cycle was conducted in class a and the second cycle in class b. sample the study was conducted in one of the government secondary schools in the brunei-muara district, with a sample of nine-grade mathematics students. there were nine students who were eliminated due to absenteeism, consequently, only data of 35 students were analysed in the study (refer to table 1). table 1 the number of participants in class a and class b male female total class a 9 6 15 class b 9 11 20 instruments the pre-test and post-test papers were created to measure the students’ achievement scores. reliability test was not conducted since the questions were from past year examination paper questions. validity of the test papers was also checked by experienced teachers. questions on 28 southeast asian mathematics education journal, volume 11, no 1 (2021) the test papers depend on the specific topic. the topic chosen was on multiplying and dividing with indices. after the post-test, a questionnaire was given to all participants to acquire their perspectives on non-digital game-based learning. interviews were also conducted to seek participants’ feedback and their preference for non-digital game-based learning. the interviewees were selected based on the students’ achievement scores. lesson interventions the first author conducted the lesson interventions in the two classes that she taught. each lesson was around 50 to 55 minutes. an instruction card was needed as well to ease students to refer back during the game lesson. the students were divided into pairs and some into a group of three. the game introduced was known as ‘break the code’ game (refer to figure 1) adapted from the ‘rules for indices (treasure hunt)’ (maths4everyone, 2018). each pair was given 16 cards. the first card was the ‘start’ card and students had to find out the answer to its question. the students found the card with the answer of the first card and answer its question, putting this second card, next to the first card. the students then had to repeat the process of finding the answer until they had answered all the questions on all of the cards. afterwards, the students would use the deciphering table to change the letters from their answers into the letters of the actual message, which is a clue to where the next set of cards were hidden. the difficulty of the game increased by providing them with more complicated mathematical expressions. figure 1. break the code set of cards. data collection mathematical achievement was collected through students’ assessment. a pre-test was given to the students each with one lesson prior to the intervention. the paired sample t-test was used to test the comparison of the pre-test and post-test scores. meanwhile, the post-test was administered a few days after the lesson intervention. the level of difficulty of both pretest and post-test paper was maintained to compare the students’ scores. the questions used in the post-test were the same as the pre-test; however, the coefficients and variables in the questions were changed. after the post-test was conducted, a questionnaire was disseminated to all students seeking their perspectives on non-digital game-based learning. a 10-minute interview, consisted of 8 29 southeast asian mathematics education journal, volume 11, no 1 (2021) questions, was conducted and only a few students were selected to be interviewed depending on their achievement scores. the interviews were recorded and transcribed. data analysis the paired sample t-test was used to test for statistical significance of mean score difference of pre-test and post-test. to give a valid result, four assumptions are required before using paired sample t-test. the first assumption is where the dependent variable, the achievement scores must be on a continuous scale. secondly, the independent variable must consist of two related groups, or, by having same subjects in both groups. in other words, same students attempted both the pre-test and the post-test. thirdly, there should be no significant outliers in the difference between the two related groups. as a result, there was no outliers in the data of this study. lastly, the distribution of the differences in the achievement scores between the two related groups should be approximately normally distributed. shapiro wilk test will be conducted first to assess the data’s normality. the effect size, which is the eta squared, was calculated for class a and class b. results and discussion comparison between pre-test and post-test of class a table 2 shows the mean and standard deviation of pre-test and post-test of class a. the results showed that mean score difference of the pre-test and post-test were 4.667 and 8.067 respectively. the increase in the mean shows that students improved in their post-test. the standard deviation of the pre-test was 2.024 and the post-test was 2.219. table 2 the mean and standard deviation of pre-test and post-test of class a n mean standard deviation std. error mean class a pre-test 15 4.667 2.024 0.523 post-test 15 8.067 2.219 0.573 the results in table 3 demonstrates a significant difference in students’ achievement scores between the pre-test and post-test with a p-value of 0.000. the mean increase in the scores was 3.4 with a 95% confidence interval ranging from 2.423 to 4.377. with an effect size of 0.799, this indicates that it has a moderate effect and it shows that the non-digital game-based learning method works for class a in particular. table 3 paired sample t-test between pre-test and post-test for class a 30 southeast asian mathematics education journal, volume 11, no 1 (2021) 95% confidence interval of the difference mean standard deviation std. error mean lower upper t df sig. (2tailed) class a 3.400 1.765 0.456 2.423 4.377 7.462 14 0.000 figure 2 shows the bar graph of total scores obtained by class a students for their pre-test. majority of the students achieved a score of 5. however, a total of six out of 16 students failed the pre-test. no students managed to get full scores since the highest score was 8, in which only one student achieved the score. figure 2. a bar graph showing the total scores of class a students for pre-test. for the post-test (refer to figure 3), all of the students improved their scores. five students managed to get a score of 10. only two students failed the test (achieving below the average score of 5). although there were still failures, every student improved their scores, since all of the score difference of each student was positive. figure 3. a bar graph showing the total scores of class a students for post-test. figure 4 shows a bar graph of the total scores obtained by each student in their pre-test and post-test. every student improved their scores as there was no decrease in the scores from their pre-test. student 15 obtained a large difference in the score, which increased from 1 to 8. student 1 and student 2 only managed to improve their score by 1. 31 southeast asian mathematics education journal, volume 11, no 1 (2021) figure 4. the total scores obtained by each student in class a for pre-test and post-test comparison between pre-test and post-test of class b table 4 shows the mean and standard deviation of pre-test and post-test of class b. the results showed that the means of the pre-test and post-test were 4.750 and 7.950 respectively. the standard deviation of the pre-test was 2.807 and the post-test was 1.820. this result also shows that students improved in their post-test. table 4 the mean and standard deviation of pre-test and post-test of class b n mean standard deviation std. error mean class b pre-test 20 4.750 2.807 0.628 post-test 20 7.950 1.820 0.407 the results displayed in table 5 indicate a significant difference in the students’ achievements from the pre-test to post-test, with a p-value of 0.000. there is a mean increase of 3.200 with a 95% confidence interval ranging from 1.989 to 4.411. an effect size of 0.617 implies a moderate effect, indicating that the non-digital game-based learning method did work in class b as well. table 5 paired sample t-test between pre-test and post-test for class b 32 southeast asian mathematics education journal, volume 11, no 1 (2021) 95% confidence interval of the difference mean standard deviation std. error mean lower upper t df sig. (2tailed) class b 3.200 2.587 0.579 1.989 4.411 5.531 19 0.000 figure 5 shows the bar graph of total scores obtained by class b students for their pre-test. in the pre-test, eight students failed where two students get zero score. similar to class a, no students managed to obtain perfect scores. three students obtained the highest score which was 8. majority of the students achieved a score of 7. meanwhile, for the post-test (refer to figure 6), only one student failed the test (achieving below the average score of 5), four students achieved the perfect score and five students obtained score of 9. figure 5. a bar graph showing the total scores of class b students for pre-test. figure 6. a bar graph showing the total scores of class b students for post-test. figure 7 shows the total scores obtained by each student in their pre-test and post-test. only one student, which was student 1, did not manage to improve in the post-test as the score decreased from 7 to 5. student 2 and student 3 maintained their scores since they scored 8 for both pre-test and post-test.. the other students’ scores showed an increase from their pre-test 33 southeast asian mathematics education journal, volume 11, no 1 (2021) to post-test. the largest score difference was 8, which was obtained by student 20, from a score of 2 to 10. figure 7. the total scores obtained by each student in class b for pre-test and post-test. from the comparison of pre-test and post-test scores, there was an increase in the mean scores for class a and class b students. through the paired sample t-test, it is conclusive that there are statistically significant differences of students’ achievements before and after using the non-digital game-based learning method. the results showed that the integration of nondigital game-based learning in the mathematics lesson did have a positive effect on the students’ achievement scores. hence, implementing non-digital game-based learning as one of the lesson interventions to use in a classroom may bring encouraging impact on students’ achievements. questionnaire and interview results the students responded to a questionnaire on their gameplay experience in the class. these questionnaire responses were supported by some of the interviewees’ responses. the results were divided into categories, which were experience on the gameplay, autotelic experience, and interaction with peers. there were five interviewees; two from class a and another three from class b. all the interviewees in class a improved their score. in other words, they gained a positive score difference. meanwhile, in class b, one interviewee improved his score in the post-test, one student’s score remained the same, and one student performed worse in post-test than in the pre-test. the interviewees were coded as a15 and a04 for class a and b18, b06 and b10 for class b. moon and ke (2020) defined peer interactions as social interactions among students in a similar age group either verbally or nonverbally. autotelic experience refers to when the player play the game for great enjoyment, instead of playing with the hope of rewards or external reasons (zheng, 2012). overall, during the interviews, when students were asked about their thoughts on the game, they found that the game was fun and enjoyable. this finding is in line with what was mentioned by gillern and alaswad (2016) that the game created a fun learning atmosphere for 34 southeast asian mathematics education journal, volume 11, no 1 (2021) students’ experiences and their learning. most of the students gave the game a rating of 8 out of 10. with this game, students may find the subject fun and interesting as the game helped them to improve their understanding of the given topic, as revealed by some of the students’ responses below: b18: it was really fun for me and my partner…because i am a very competitive person, and yatah macam (thus) … i like the game. b06: [laugh] because i don’t really like maths that much… yesterday i think it was one of the moments when i like maths. b10: because… at first i really don’t understand the game but at the end, i started to understand it gradually, and it was… so good. the students also responded to their feelings when they played the game. they believed that the game improved their skills on the topic and helped them to bond with their partner during the game. one of the students also said that the game has made him more confident. this showed that the game helped students to build more confidence in answering mathematical questions. the responses were: a15: i felt…. when i played the game i felt so excited and… that helped improve my skills. b18: it really felt macam (like)… i was challenging myself so i can improve myself and my partner and it helped me and my partner macam (like) bonding jua (also)…really fun for me. b06: hmmm what should i say? i think i was a little more confident than the usual generally, the students found this game as fun and they believed that they could improve their skills while playing the game. these concurred with yang et al.’s (2018) statement which mentioned that from educational games, the learners could gain enjoyment, self-confidence and satisfaction if their skills and knowledge in game-based learning are equal to the given challenging tasks. referring to table 6 below, these were the responses from the students on their autotelic experience. more than half of the number of students, or 86%, really enjoyed the game. this statement corresponds with the next statement, in which 83% of the students like the feeling of playing the game again. when students were asked during the interview, most of them said that they wanted to play the game again in the future. in total, 69% of the students believed that the game improved their skills and they were able to complete more difficult tasks. nevertheless, 3% of the students did not agree with the statement. probably, the questions given in the game were already challenging in the first place, hence, the given tasks are not equal to their skills. instead, this could demotivate them to play the game until the end of the session. moreover, 80% of the students found it easy to understand how to play the game. the positive responses may indicate that the game is suitable for the students. table 6 students’ autotelic experience during gameplay strongly disagree disagree neutral agree strongly agree 35 southeast asian mathematics education journal, volume 11, no 1 (2021) i really enjoyed playing the game. 0% 0% 14% 37% 49% i liked the feeling of playing and want to play it again. 0% 0% 17% 40% 43% as i played the game, my skills got improved and so i was able to complete more difficult tasks. 3% 0% 26% 43% 26% it was easy for me to understand how to play the game. 0% 0% 17% 54% 26% as this game required the students to play in pairs or a group of three, they were also asked about the interactions with their partners when playing the game (refer to table 7). there were 86% of the students who enjoyed playing the game with their partner, whereas 3% of the students disagreed with the statement. this was probably due to the fact that some students found it uncomfortable playing with some partners they were unfamiliar with, whom they referred as a mere acquaintance, instead of partnering with those whom they were comfortable with. this statement is supported by the following statement in which 68% of them agreed that their partner helped them to do better in the game. in the interview, the students said that having a partner did help them in answering the questions in the game and it also helped the student to bond with their partner. some of the responses were: b18: yes, because…for… even though it was challenging, i really, really like it. it helped me bond with my partner. again as i said,…and it was a kind of easy but challenging …yeah. b06: ummm… i did enjoy playing with him because i am used to partnering up with him. b10: yes, because he helped me a bit (with) the equation. with only 6% of them disagreed with the statement, it is believed that they found that they were not comfortable playing the game with the partner not of their choice. only 68% of the students would like to play the game with their partner again in the future. one of the responses, during the interview was “yes …because my partner helped me solved the question” (by a15). a total of 6% of the students did not agree with the statement about playing the game with the same partner. it is believed that during the interview, one of the students said they should interact with some of the other classmates. the responses were: b18: probably…but i would rather try to play it with a different person. b06: umm maybe sometime ada (yes) sometime inda (no), because i need to know some other students. some believed that it would be better to play with a different partner if they were to play the game again. this showed that students may not prefer to play with their previous partner. table 7 students’ interactions with their peers 36 southeast asian mathematics education journal, volume 11, no 1 (2021) strongly disagree disagree neutral agree strongly agree i enjoyed playing the game with my partner. 3% 0% 11% 49% 37% my partner helped me to do better in the game. 6% 0% 26% 31% 37% i would like to play the game with my partner again. 3% 3% 26% 31% 37% more than half of the students believed that the game has helped them to improve their skills. it was also found that students were able to bond with their partner during the game, hence, improving their social skills. as mentioned earlier by gillern and asward (2016), nondigital games help to improve social skills since it is a face-to-face interaction. with a face-toface interaction, this also helped to improve their interactions with the teacher (salam & shahrill, 2014; othman et al., 2015; shahrill & clarke, 2014; 2019). although there were positive responses both from the questionnaire and the interviews, some of the students still gave a few disagreements regarding the game. when it came to playing the game, some students did not feel the fast pace of time and some believed the game did not help them to improve the skills on the topic. they felt that the questions in the game were difficult to solve, thus, demotivating them to continue playing the game and just waiting for the game session to be over. lastly, although more than half of the students preferred to play with their partner again, there were those who preferred to play the game with a different partner in the future. the students also viewed interactions with their peers and the teacher to be important in their learning. obviously, students may want to play the game in a comfortable environment, in which they do not feel intimidated by their peer. thus, they would rather choose their peer whom they are comfortable with. conclusion according to the analysis of the students’ achievement scores in the tests, this present study has been able to suggest that the non-digital game-based learning method brought a positive effect on their achievement scores. as for the students’ perceptions on the non-digital gamebased learning, despite having a few students who did not agree to some of the statements, more than half actually responded positively. this showed that students do have various learning styles, in which they may or may not prefer games as one of their learning styles approaches. one of the limitations of this study is the small sample size. since this is a small-scale study with only 35 students, the generalizability is not significant. thus, further study could be done or forge links with findings in the area. furthermore, the achievement scores were only determined by one particular topic on the multiplication and division with indices, which leads to the impossibility to generalize the results. another limitation was limited to the single sampled school as the case study. therefore, more schools could be involved to expand the scope of the study. in light of the limitations of this study, a few recommendations have been suggested for future research. it is advised to conduct a wide-scale research to obtain better data accuracy of the nine-grade students. further research on different mathematics topics will need to be 37 southeast asian mathematics education journal, volume 11, no 1 (2021) conducted to see whether this particular game works or not. sometimes, the approach may not be suitable for that particular topic, and thus a different approach needs to be considered as well. the integration of the game is crucial in the mathematics lesson to have a further study on the effectiveness of the game, which provides an opportunity for the students to adapt to a different kind of learning environment in the classroom. references amdan, e. f., & salleh, s. m. 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(2017). the effect of game-assisted mathematics education on academic achievement in turkey: a meta-analysis study. international electronic journal of elementary education, 10(2), 195–206. https://doi.org/10.26822/iejee.2017236115 yang, k., chu, h., & chiang, l. (2018). effects of a progressive prompting-based educational game on second graders’ mathematics learning performance and behavioral patterns. journal of educational technology & society, 21(2), 322–334. zheng, m. (2012). fifth graders’ flow experience in a digital game-based science learning environment. the british journal of psychiatry, 111(479), 1009–1010. https://doi.org/10.1192/bjp.111.479.1009-a 40 southeast asian mathematics education journal, volume 11, no 1 (2021) southeast asia mathematics education journal volume 13, no 1 (2023) 43 usage of classroom technology and the technological pedagogical content knowledge (tpack) of mathematics teachers junymer c. plantado bicol state college of applied sciences and technology, philippines jcplantado@astean.biscast.edu.ph; junymerplantado@gmail.com abstract technological pedagogical content knowledge (tpack) is a theoretical framework in technology integration that is highly regarded in today's digital era. using a descriptive research design, this study investigated 81 secondary mathematics teachers' use of available classroom technology and level of competence in the eight tpack domains. the findings revealed that classroom technology was rarely employed. this was due to a combination of external and internal factors. while along the tpack domains, the teachers obtained the highest level of competence in ck (m = 4.29, sd = .59) and lowest in tk (m = 3.69, sd = .70). one-way anova uncovered that there was a highly significant difference in the level of competence along with tpack domains. furthermore, correlation and multiple regression analysis unveiled that the level of usage of classroom technology was discovered to be highly significant and served as the best predictor of tk (f(6, 74) = 6.17, p<.001), tpk (f(6, 74) = 6.39, p<.001), tck (f(6, 74) = 4.30, p<.001), and tpack (f(6, 74) = 2.65, p =.022). implications of the findings and recommendations are discussed. keywords: tpack, technology integration (ti), mathematics teacher introduction technology continues to progress at a breakneck pace. technology advancements and innovations have had significant impacts on almost every aspect of human life, including educational-instructional practices (muhaimin et al., 2019). the proliferation of technology has challenged the status quo of educational settings, resulting in a paradigm shift in the teaching and learning process (sasota et al., 2021). technology's opportunities and facilities for teachers have created a new landscape for classroom instruction, altering teachers' roles and expectations (ozudogru & ozudogru, 2019). teachers' knowledge and competence in teaching have been understood along the lines of shulman's pedagogical content knowledge (pck) for over 30 years. the concept of effective teaching, according to shulman's pck, is dependent on the teacher's ability to combine the domains of content and pedagogy rather than looking at each domain separately (ozudogru & ozudogru, 2019; schmid et al., 2020). however, many educators and researchers argue that this framework is insufficient, primarily because it lacks an explicit articulation of technology integration (ti), which many believe is essential in this digital era (jacinto & samonte, 2021; ozudogru & ozudogru, 2019; schmid et al., 2020). one of the recent and widely accepted theoretical frameworks that get positive acceptance along ti is the technological pedagogical content knowledge (tpack) developed by mishra and koehler (alrwaished et al.,, 2017; bakar et al., 2020; muhaimin et al., 2019; lucenario et al., 2016; oskay, 2017; wang et al., 2018). this is an extension and broadening of shulman's pck framework, which emphasized the importance of developing an integrated and mailto:jcplantado@astean.biscast.edu.ph usage of classroom technology and the technological pedagogical content knowledge (tpack) of mathematics teachers 44 interdependent understanding of the three primary forms of knowledge content, pedagogy, and technology for effective ti. although tpack research is widespread, it is pertinent to note that, despite its popularity in educational technology research, the tpack framework has received some significant criticism. according to schmid et al. (2020), tpack was primarily criticized for its lack of conceptual clarity and specificity, as well as the "fuzziness" of its boundaries, which resulted in a body of literature focusing on the development of and relationships between the tpack components from two opposing perspectives—integrative and transformative. figure 1 demonstrates the tpack framework and a brief discussion of its domains as utilized in several studies (bakar et al., 2020; koehler & mishra, 2009; mishra, 2019; oskay, 2017; ozudogru & ozudogru, 2019): figure 1. the revised version of the tpack framework and its knowledge components (mishra, 2019). • content knowledge (ck). this is "knowledge about the actual subject matter to be learned or taught." it includes an understanding of concepts, theories, ideas, organizational frameworks, evidence, and proof, as well as established practices and approaches to developing that knowledge. • pedagogical knowledge (pk). this refers to teaching methods and processes, such as classroom management, assessment, lesson plan development, and student learning. they encompass, among other things, overall educational goals, values, and objectives. • technology knowledge (tk). this refers to knowledge of various technologies, ranging from low-tech tools like pencil and paper to digital tools like the internet, digital video, interactive whiteboards, and software programs. junymer c. plantado 45 • pedagogical content knowledge (pck). this refers to pedagogical techniques, knowledge of what makes concepts difficult or easy to learn, knowledge of students' prior knowledge, and theories for specific contexts. • technological content knowledge (tck). this refers to the knowledge of how technology can generate new representations of specific content. in other words, tck implies that teachers recognize that by utilizing a specific technology, they can alter how students practice and comprehend concepts in a specific content area. • technological pedagogical knowledge (tpk). this refers to the knowledge of how various technologies can be used in teaching and how using technology may change the way teachers teach. it entails understanding the pedagogical affordances and constraints of a variety of technical tools in relation to disciplinary and developmentally appropriate pedagogical designs and strategies. • technological pedagogical content knowledge (tpack). this refers to the knowledge that teachers must have to integrate technology into their teaching in any content area. by teaching content using appropriate pedagogical methods and technologies, teachers have an intuitive understanding of the complex interplay between the three basic components of knowledge (ck, pk, tk). • context knowledge (xk). this refers to their knowledge of who they teach, where they teach, and what they teach. in other words, xk encompasses everything from a teacher's awareness of available technologies to the teacher's knowledge of the school, district, state, or national policies that they must adhere to in order to implement technology effectively. ti is a challenging, complex, diverse, and multifarious process. several studies have been conducted on ti (abas & david, 2019; bakar et al., 2020; gonzales & gonzales, 2021; hero, 2019; ibañez et al., 2021; jacinto & samonte, 2021; mercado et al., 2019; malubay & daguplo, 2018; morales et al., 2021; roble et al., 2020; sasota et al., 2021) and reported the positive impact of a technology-infused mathematics teaching and learning (gonzales & gonzales, 2021; roble et al., 2020; ibañez et al., 2021; sasota et al., 2021). however, some studies have revealed that ti implementation in the country remains low, and teachers continue to lack the necessary technical skills and knowledge for effective ti to classroom instruction (jacinto & samonte, 2021; roble et al., 2020). the issue of ti has been present in the country for decades, but it has recently gained prominence due to the growing demand among teachers on issues related to k-12 education implementation. furthermore, the call to re-examine teachers' ability and knowledge of ti has been amplified in providing quality instruction during the pandemic. teachers, as the primary carriers of the teaching process, must possess the necessary skills and continuously upgrade themselves with the necessary specialized knowledge for effective mathematics instruction despite adversities, which can be accomplished through continuous engagement in various professional development and advancement programs and activities, such as attending seminars, workshops, conferences, participating in advanced training programs, or pursuing advanced programs. several studies on tpack have recently been conducted in the country. it has been studied with one’s self-efficacy (cahapay & anoba, 2021; gonzales, 2018; jacinto & samonte, 2021; usage of classroom technology and the technological pedagogical content knowledge (tpack) of mathematics teachers 46 mercado & ibarra, 2019; ramos et al., 2020), teachers’ training and their teaching competencies (mercado et al., 2019; santos & castro, 2021; ramos, et al., 2020; gonzales & gonzales, 2021; mercado & ibarra, 2019), and level of ti implementation in the classroom (morales et al., 2021; mercado et al., 2019; malubay & daguplo, 2018; santos & castro, 2021). the findings are varied and inconclusive, particularly when it comes to determining what truly influences the level of competence of mathematics teachers across the tpack domains affecting ti. along these lines, the researcher decided to investigate the relationship between the teacher's level of competence in the tpack domains and some demographic variables such as the level of classroom technology usage. furthermore, no study on profiling the characteristics of secondary mathematics teachers and their potential implications for teaching and learning mathematics has been conducted in the target division, division of naga city, during the pandemic (the year 2020). the primary objectives of this study were to (1) determine the level of usage of available classroom technologies by teachers; (2) identify the level of competence of teachers along the tpack domains; and (3) determine the level of association between demographic profile variables and level of competence of teachers in tpack domains. methods a quantitative descriptive research design was used in this study. descriptive research design is a scientific method that entails observing and describing a subject's behavior without influencing it in any way (shuttleworth, 2019). the primary objective of this design is to "describe" individuals, situations, issues, behaviors, or phenomena in nature (siedlecki, 2020). a survey, on the other hand, was used as the primary data collection methodology. however, while the survey method can be thought of as purely descriptive in nature, this research methodology can also be used to explain (explain) and assess the influence of various factors, which can be manipulated by public action, on some phenomenon (moser & kalton, 2017). all of the respondents were secondary mathematics teachers from junior and senior high schools in naga city, camarines sur, philippines. as a sampling technique, total enumeration was employed. however, due to the pandemic's restrictions and teachers' personal reasons, only 81 out of 91 (89.01%) total mathematics teachers responded to the online survey. these 81 teachers came from the division's 11 secondary schools, including the two newly created and inaugurated high schools for the current school year. the primary instrument was a two-part questionnaire created by the researcher. part i is intended to obtain the demographic profile of the teacher-respondents, including gender, age, academic rank, educational attainment, teaching experience, and commonly used technology in mathematics teaching and learning. part ii is a tpack survey with 60 statements distributed across the eight domains of the tpack framework. the distribution of items is as follows: tk: 1 to 6; pk: 7 to 17; ck: 18 to 26; tpk: 27 to 34 (tpk offline: 27 to 30 and tpk online: 31 to 34); tck: 35 to 40; pck: 41 to 46; tpack: 47 to 55; and xk: 56 to 60. the primary instrument was a two-part questionnaire created by the researcher. part i is intended to obtain the demographic profile of the teacher-respondents, including gender, age, academic rank, educational attainment, teaching experience, and commonly used technology in mathematics junymer c. plantado 47 teaching and learning. part ii is a tpack survey with 60 statements distributed across the eight domains of the tpack framework (schmid et al., 2020). five experts in education and mathematics education research contentand face-validated the survey questionnaire. these experts rated the survey questionnaire's acceptability and validity by checking yes or no against the criteria in the validation form, which covers the meaningfulness, appropriateness, and relevance of each statement in the survey. after that, the validated questionnaire was pilot tested with 27 mathematics teachers from neighboring districts and outside the target division. the results of the pilot testing were analyzed and subjected to reliability testing. based on the computed cronbach alpha, α = .98, excellent internal consistency reliability was obtained. to determine the level of usage of common classroom technologies, a five-point scale was used with 5 being the highest (always) and 1 as the lowest (never). the mean scores were then calculated and interpreted using the guide: 4.21 – 5.00: always; 3.41 – 4.20: often; 2.61 – 3.40: sometimes; 1.81 – 2.60: seldom; and 1.00 – 1.80: never. on the other hand, to assess teachers' level of competence across the tpack domains, a five-point scale was used, with 5 being the highest (completely competent) and 1 being the lowest (incompetent). the following guide was additionally utilized to compute and interpret the means: 4.21 – 5.00: completely competent; 3.41 – 4.20: fairly competent; 2.61 – 3.40: somewhat competent; 1.81 – 2.60: slightly competent; and 1.00 – 1.80: incompetent. before beginning the study, a letter of request was provided to the schools division superintendent (sds) to seek approval and endorsement. the rationale, objectives, purpose, and timeline were all stated in the letter. it was stressed that all information derived from the study would be treated with the utmost confidentiality in accordance with the existing data privacy law. following receipt of the letter of approval and recommendation, a similar letter of request was sent to each school head and principal. however, because face-to-face survey administration is prohibited, we devised an online survey schedule and distributed the link to the online questionnaire in google forms to all interested mathematics teachers. for smooth administration of the survey and ease of monitoring, contact persons were identified upon the recommendation of the school head or principal. google forms responses were tabulated and organized in ms excel. initially, data examination and cleansing were performed, with a focus on ensuring the homogeneity and normality of the data sets. the homogeneity assumption was evaluated using levene's test for homogeneity. levene’s test of homogeneity, f(7, 640) = 6.218, p = 0.6678, indicates that the null hypothesis was not rejected. thus, the homogeneity requirement is met because there are no significant differences between the group variances. the kolmogorov-smirnov test, on the other hand, was used to ensure that the data sets were normal. this test revealed that data sets for pck and xk were not normally distributed with computed p-values of .045 and .038, respectively. furthermore, outliers were identified using the box and whisker plot, and necessary adjustments were made before subjecting the data sets to further statistical tests. the tabulated data is then statistically analyzed. the mean, standard deviation, and ranks, as well as the various demographic variables and tpack domains, were used to describe the characteristics of the teacher-respondents. pearson's product-moment correlation and spearman rank rho correlation were used to test for significant correlations between demographic variables and teacher competence levels, as well as the tpack domains. a usage of classroom technology and the technological pedagogical content knowledge (tpack) of mathematics teachers 48 multiple regression analysis was also performed to determine the impact of demographic variables on tpack domains. finally, a one-way analysis of variance (anova) was used to identify significant differences in the teachers' level of competence as well as the eight domains of the tpack framework. tukey-kramer test was utilized as a post hoc analysis test in determining which pairwise groups have means that differ significantly. results and discussion teachers’ demographic profile the majority of teachers (74.1%) are female (74.1%), between the ages of 35 and 50 (49.4%), organize teacher i iii academic rank positions (93%), have at most 10 years of teaching experience (50.6%), and do not yet have a master's degree (78%). this finding indicates that mathematics teachers are generally young, both in terms of age and teaching experience and are classified as beginning or approaching proficiency in the philippine professional standards for teachers (ppst) career stage. table 1 summarizes the technology that teachers frequently employ in the mathematics classroom. an instant messaging app, such as facebook messenger (or simply messenger), emerged first and was described as "always." the widespread use of various classroom technologies can be attributed to their utility (anwar et al., 2020), effectiveness (anwar et al., 2020; baker et al., 2018; rosmiati & siregar, 2021), and availability of the technologies (jones, 2003). considering the socio-economic status of teachers and students, as well as the perennial issue of internet connectivity in the area, one of the most viable ways for teachers to stay connected and easily communicate academic-related concerns with their students is through an instant messaging app like messenger. according to garzon et al. (2019), facebook and its products such as messenger are the most popular and widely used social networking apps in the philippines. as one of the top ten facebook users countries in the world (statista, 2017), it is safe to assume that almost all filipino students have a facebook account and use messenger. teachers can easily send instant messages and other multimedia files, such as pictures, voice recordings, video recordings, file documents, and others, to their students using this type of app at a low cost. table 1 frequency distribution of the commonly used technologies in mathematics classroom classroom technologies m sd interpretation 1. computer/laptop/desktop 4.30 .95 always 2. radio or other similar audio devices 2.20 .91 seldom 3. projector or other similar devices 3.28 1.16 sometimes 4. calculator or other similar tools 3.83 1.14 often 5. interactive whiteboard/other interactive tools 3.30 1.37 often junymer c. plantado 49 classroom technologies m sd interpretation 6. graphing calculator/similar tools/app 2.79 1.18 sometimes 7. ppt/other similar app 4.32 .88 always 8. spreadsheet/other similar web app 3.60 1.25 often 9. statistical tool/packages 2.62 1.32 sometimes 10. lms (google classroom, edmodo, etc.) 3.33 1.34 sometimes 11. online assessment apps/programs 2.32 1.02 seldom 12. video conferencing app 3.80 1.32 often 13. facebook messenger/messaging app 4.72 .68 always overall 3.42 1.12 often radio and other similar audio device use, on the other hand, ranked at the bottom and was interpreted as "seldom." the low use of radio and other audio devices was attributed to their inherent limitations, even during face-to-face teaching. teachers would rather employ other devices that offer the same features as well as extras like video playback. overall, the composite mean score of teachers in the level of use of classroom technologies indicates that teachers "often" use technology when teaching mathematics. this finding was consistent with the findings of roble et al., (2020) and abas and david (2019), in which the extent of ti implementation was considerably low. while there are some areas where teachers received high mean scores, the results show that there are some areas where teachers need to improve in terms of ti, particularly in this era of distance learning education. a literature review revealed some common factors that could explain the low level of ti implementation in the classroom. hamutoglu and basarmak (2020) revealed that beliefs towards teaching-learning activities, beliefs towards expert support, technological self-efficacy beliefs, family resistance, assessment, and pedagogical self-efficacy beliefs are some internal barriers while lack of vision, lack of money, lack of training, infrastructure, content, and time are all part of external factors that serve as ti barriers. level of competence along tpack domains table 2 summarizes the characteristics of mathematics teachers across the tpack framework domains. the teachers with the highest mean score in ck, as illustrated in the table, were interpreted as "completely competent." the high mean ck score demonstrates that mathematics teachers have a very high level of understanding and knowledge of mathematical concepts, theories, ideas, organizational frameworks, knowledge of evidence and proof, and established practices and approaches to developing mathematical knowledge. furthermore, pck and pk were ranked second and third, respectively, and were both interpreted as "fairly competent." these results represent that teachers are well-versed in various teaching methods usage of classroom technology and the technological pedagogical content knowledge (tpack) of mathematics teachers 50 and processes, as well as pedagogical techniques for teaching mathematical concepts. these findings are consistent with other studies that discovered high levels of competence or confidence in teachers along ck, pc, and pck (muhaimin et al., 2019; ozudogru & ozudogru, 2019; alrwaished et al., 2017). table 2 teachers' characteristics along the tpack domains domains m sd interpretation tk 3.69 .70 fairly competent pk 4.06 .59 fairly competent ck 4.29 .59 completely competent tpk 3.83 .67 fairly competent tck 4.01 .70 fairly competent pck 4.13 .63 fairly competent tpack 3.87 .69 fairly competent xk 3.96 .69 fairly competent overall 3.97 .66 fairly competent on the other hand, the teachers got the lowest mean score at tk and whose score was interpreted as "fairly competent." teachers at this level are characterized as having a basic understanding of the various educational technologies that can be used in mathematics instruction. it is also worth noting that other domains with tk as a sub-component, such as tck, tpk, and tpack, ranked relatively low in the survey. the teacher's lack of competence in tck, tpk, and tpack can be attributed to their lack of competence in tk. tk is the weakest component among the domains that require more attention and consideration, according to the integrative perspective (schmid et al., 2020). this current observation was similar to previous studies in which tk was discovered to lag among the domains of teachers in the tpack framework (gonzales, 2018; malubay & daguplo, 2018). except for ck, the teacher's overall level of competence is explained as "fairly competent" across all domains. the obtained overall mean score indicates that teachers have a fair level of knowledge or skills in integrating various technologies in teaching and representing mathematical content in their community. while there are some areas with promising results, it cannot be denied that some domains, particularly with ti, require improvement. this finding is consistent with the previous finding regarding the extent to which various classroom technologies are implemented (jacinto & samonte, 2021; malubay & daguplo, 2018; roble et al., 2020). in addition, an analysis of variance (anova) was performed to determine whether there is a difference in the levels of competence of mathematics teachers across the eight domains. the junymer c. plantado 51 test was highly significant; thus, it can be concluded that there was a significant difference between the levels of teacher’s competence along the eight domains, (f(7, 638) = 6.48, p<.01). post hoc analysis employing the tukey hsd test revealed that there was a highly significant difference between tk and pk; tk and ck; tk and pck; ck and tpk, but not between tk and tck; ck and tpack; and ck and xk. taken together, the findings demonstrate that teachers' levels of competence across the eight domains differ statistically, with teachers' levels of competence between tk and ck appearing to differ the most. correlation between demographic variables and tpack domains table 3 depicts the correlation matrix between demographic variables and teachers' level of competence in the eight domains. the majority of the coefficient values are negative, indicating a negative association. a negative association indicates an indirect relationship between the two variables under consideration. moreover, it can also be seen that these correlation coefficient values, .01 < |r| < .49, indicate a very weak to low degree of association. in general, these values imply that the strength of association between the demographic variable and the level of competence of teachers across the tpack domains being compared is "unsubstantial" or "near negligible." these findings are consistent with the results of malubay and daguplo (2018), who discovered that the majority of variables in the respondent's profile, such as gender, age, number of years in service, and number of training, have a weak linear relationship with the different tpack domains. table 3 teachers' characteristics along the tpack domains domains tk pk ck tpk tck pck tpack xk 1. sex -.21 -.11 -.13 -.12 -.02 -.09 -0.10 -0.19 2. age -.32** -.19 -.23* -.38** -.42*** -.22* -.37*** -.38** 3. rank -.38*** -.08 -.11 -.34** -.28* -.10 -.27* -.16 4. teaching experience -.42*** -.26* -.25* -.44*** -.44*** -.26* -.39*** -.39** 5. educational attainment .01 .05 .02 -.02 -.02 -.01 -.01 -.01 6. use of classroom technology .43*** .17 .15 .49*** .38*** .19 .45*** .23* note: *p <.05, **p <.01, ***p <.001, n = 81 it is interesting to note, however, that when significant association was examined among those pairs with positive correlation, the test was discovered to be highly significant between utilization of classroom technology and tk, (rs(79) = .43, p<.001); tpk, (rs (79) = .49, p<.001); tck, (rs (79) = .38, p<.001); and tpack, (rs (79) = .45, p<.001). furthermore, the test emerged usage of classroom technology and the technological pedagogical content knowledge (tpack) of mathematics teachers 52 to be significant between the use of classroom technology and xk (rs (79) = -.23, p<.001). this implies that the extent of utilization of classroom technology is directly associated with the tk, tpk, tck, tpack, and xk of the teachers. furthermore, multiple regression analysis was performed to determine whether demographic variables (explanatory variables) and tpack domains (outcome variables) can influence or predict teacher competence. the demographic variables (sex, age, academic rank, teaching experience, highest educational attainment, and frequent use of classroom technology) were hypothesized to positively predict tk, pk, ck, tpk, tck, pck, xk, and tpack. the test was highly significant for tk (f(6, 74) = 6.17, p<.001), tpk (f(6, 74) = 6.39, p<.001), tck (f(6, 74) = 4.30, p<.001), tpack (f(6, 74) = 4.89, p<.001); significant for xk (f(6, 74) = 2.65, p =.022). only the frequent use of common classroom technologies was determined to have a strong and positive influence on tk (β =.34, t = 3.39, p< .001); tpk (β =.41, t = 4.08, p< .001), tck (β =.28, t = 2.66, p< .001), and tpack (β =.38, t = 3.68, p<.001) when each predictor's individual contribution was closely examined. while none of the predictors appeared to correctly predict xk. thus, regression analysis revealed that frequent use of common classroom technologies served as the best predictor, and an increase in classroom technology use would also increase the teacher's level of competence in addition to tk, tpk, tck, and tpack. this differs from the study of malubay and daguplo (2018), in which the tpack domains were considered and investigated as explanatory predictors rather than demographic variables. conclusion this study serves as a baseline for profiling teacher characteristics about the tpack framework and technology integration. the findings revealed that the majority of mathematics teachers are female, relatively young in terms of age and teaching experience, mostly in entrylevel positions, and do not yet organize master's degrees. in terms of the use of various classroom technologies, however, the findings show that teachers "often" used these technologies in mathematics instruction. as a result, the use of classroom technologies was relatively low, which can be attributed to a variety of internal and external factors. teachers have been discovered to have a very high level of competence in ck for the tpack domains, indicating that teachers have a solid foundation on the various mathematical concepts, theories, ideas, organizational frameworks, evidence, and proof knowledge, as well as established practices and approaches. the findings, on the other hand, revealed that teachers have the lowest tk competence. this is consistent with the earlier finding about the low level of common classroom technologies. furthermore, other domains with tk as a component rank relatively low, indicating that tk is the teachers' weakest domain. overall, the teachers' level of competence in ti using the tpack framework is just adequate. the test was highly significant for tk, tpk, tck, and tpack when the overall effect of the teacher's demographic characteristics on their level of competence was investigated along with the eight tpack domains. when all six demographic variables were employed as predictors, the test was also significant for xk. only the frequent use of classroom technologies was found to have a positive influence among the six predictors when the individual effect was junymer c. plantado 53 examined. as a result, the extent and frequency with which classroom technology was utilized served as the best and strongest predictor of tk, tpk, tck, and tpack. as a result, specialized training focusing on improving teachers' competence in technologyintegrated mathematics instruction is recommended, which can be accomplished effectively through responsive, localized, and bottom-up professional development programs and activities. in addition, a similar study involving mathematics teachers from various private schools could be conducted to compare and validate the current findings. this will give you a broader perspective on the topic. acknowledgements the researcher would like to extend its gratitude to the people who made this study possible. first, to all the secondary mathematics teachers of the division for their effort and participation in this study. second, to the deped officials (sds, school heads, focal persons, etc.) who approved and endorsed the study. third, to the college administrators and officials who evaluated, approved, and funded this humble piece of work. also, special thanks to the research assistant, research enumerator, and editor for their contributions to the success of this study. references abas, m., & david, a. 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(2018). preservice teachers' tpack development: a review of literature. journal of digital learning in teacher education, 34(4), 234-258. https://doi.org/10.1080/21532974.2018.1498039 southeast asian mathematics education journal volume 12, no. 1 (2022) 37 vanuatu's typical approach of mathematics vs the tuam approach of mathematics kency obed sawah naruto university of education, tokushima, japan kobed2011@gmail.com abstract there are multiple factors contributing to the low level of mathematics in basic education in the republic of vanuatu. results through the vanuatu standard test of achievement (vansta) in 2017 and 2019 unveiled that there were gaps in the performance of mathematics which cause the overall achievement to stagnant which were below the expected minimum standard (curriculum development unit, 2020). this study investigated the current situation of the teaching mathematics approach in the country recognised as the ‘i do-we doyou do’ teaching model. in comparison, the study also examined the influence of the ‘tryunderstand-apply-master’ (tuam) discovery learning process on students in vanuatu. the study compared these two teaching approaches through pre and post-test interventions among the control and experimental group of two grade five classes. the findings of the study discovered a possibility that the tuam discovery learning process could be effective in improving the mathematics level in the basic education in the republic of vanuatu. keywords: i do–we do–you do, try-understand-apply-master (tuam). introduction in this 21 st century, students’ learning should be the core of learning and teaching processes in mathematics education. teachers in the 21 st century have to concern that students possess potential, and they require the suitable opportunity to develop. thus, it is necessary to concentrate on the approaches of implementing a mathematics curriculum. one possibility is implemented for teachers to move away from procedural memorisation but move toward mathematical thinking processes, mathematical reasoning, and mathematical problem-solving skills which provides the best opportunities for students to explore and investigate their mathematical potentials (cai & howson, 2013). concerning the nature of the current approach of teaching as an instructional strategy, through the ‘i do-we do-you do’ strategy, learners are able to memorise mathematical ideas to acquire new knowledge instead of investigating the ideas to construct the new knowledge themselves. hence, the tuam strategy was employed for this comparison study as it is evident to be among teaching approaches which enhance stimulating children’s mathematical thinking processes through a discovery learning process. takahashi (2006) elaborated that allowing students to learn through the discovery process is a method of encouraging them to utilise their minds in formulating new knowledge rather than depending on the teacher’s ideas in solving problems. cai and howson (2013) asserted that the abilities to think independently, and critically, to learn, to be creative, and to learn how to learn, are the best qualities that teachers should achieve in encouraging their students to develop. jaleniauskienė and jucevičienė (2018) initially explained that in this 21st century, learners ought to be prepared with an extensive set of abilities rather than procuring a limit occupation-specific information. considering vanuatu's typical approach of mathematics vs the tuam approach of mathematics 38 such characteristics, the nature of the tuam strategy corroborating the awareness of scrutinizing its’ effectiveness in the vanuatu context in comparison to the typical ‘i do-we do-you do’ strategy which elevating memorisation through a typical lecturing lesson. literature review the ‘i do-we do-you do’ approach mathematics is taught and learned through different approaches. in vanuatu, mathematics is frequently taught by employing the ‘i do-we do-you do’ model of teaching. according to vanuatu's latest intended curriculum, this approach of teaching is defined as: “i dothe teacher explains, models, and demonstrates the topic which will be learned and the way to think during this step. the student’s duty is to pay attention and listen carefully to the teacher. we do – students work with the teacher as more examples are illustrated. it provides the students further opportunity to be encouraged until they demonstrate the skills and knowledge necessary to move to the final, independent step. you do – it is the final step of the model in which students possess the opportunity to demonstrate mastery in the skill or knowledge by performing independently” (vanuatu ministry of education and training, 2018). the approach reflects vanuatu’s traditional ideology of education. it is depicted from the country’s cultural and traditional method of learning: the conviction that learning occurs by observing and copying from peers, parents, and adults or elderly people (sanga, niroa, matai, & crowel, 2004). it is vigorously believed that children learn best when they observe and copy what others conduct. the approach is considered to some extent as a scaffolding process of learning. in the first step, the ‘i do’ stage, teachers in vanuatu employed examples and demonstrations to navigate students’ minds in learning new mathematical material. based on vygotsky’s theory of constructivism, they considered providing examples during a lesson as a method of manipulating students to overcome problems on their own. concerning that when instructions on the way to solve a problem is not displayed before allowing students to solve the problem themselves provides them with opportunities to entertain their exploratory hypothesis at the expense of encountering initial failures (sinha et al., 2020). the try-understand-apply-mastered (tuam) approach japan on the other hand is teaching mathematics through the tuam discovery learning process in reverse of the ‘i do-we do-you do’ teaching model. this problem-solving strategy is elaborated by takahashi (2006) with the initials t-try, u-understand, a-apply, and m-master. takahashi (2006) explained that the first step in this teaching approach which he called as try with the initial ‘t’ is when a problem is portrayed and the first attempt is provided for students to explore and discover the solutions to the problem. the next step in this approach is understand with the initial ‘u’. in this stage, the students are guided through discussion to compare different solutions to the problem. after comparing the solutions, it is implemented the apply stage with the initial ‘a’. students are provided other similar problems with the opportunity to implement the method they have confirmed during discussion. when students apply the skills that they attain from the first and second step, in kency obed sawah 39 this stage, they also master the skills and it is where the initial ‘m’ for master occurs as the final step in the process. mayer (2004) asserted that this concept of the discovery learning process is well recognized as the constructivist approach of learning whereby learners are demanded to be active participants who identify to construct reasonable and structured knowledge themselves under the teacher’s guidance. furthermore, bakker (2018) did not intend to limit the term discovery to the action of investigating something unknown to mankind but rather was willing to incorporated all forms of acquiring knowledge for oneself by the utilisation of one’s mind. this approach provides the best opportunity for students to construct new knowledge by employing their minds. it is a process of training one’s mind to think critically and enhance speed, accuracy, and confidence in mathematical concepts. trninic (2018) further demonstrated this approach as the notion of student-as-explorer. he emphasised that through this strategy, knowledge is obtained by the student rather than the teacher. he also highlighted that as an explorer, one is an active organiser of experiences who produces firm understandings by frequently constructing them anew. he elaborated more that students begin constructing knowledge of arithmetic principles through the discovery process. ojose (2008) corroborated the previous idea that through discovery learning, children are able to enhance mathematical reasoning skills when there are investigating ideas. this approach of teaching allows students to work freely in a learning environment (mayer, 2004). the students are not restricted to construct learning processes. developing mathematical reasoning skills through investigation is moreover encouraged for them (ojose, 2008). the opportunities they possess here enhance their mathematical understanding when they extract relevant information from a problem statement (ojose, 2008). unlike the ‘i dowe do-you do’ approach, the tuam strategy corroborates the scaffolding theory by vygotsky in its rightful manner. it exposed scaffolding as temporary instructional support which elevates cognitive reasoning (byun, lee, & cerreto, 2013). table 1 summarises the characteristics of the two teaching approaches based on the above discussion. table 1 summary of teaching approaches characteristics tuam i do-we do-you do student engagement  students are challenged and encouraged by the teacher to think deeply and explore variety of approaches to a solution.  formal and informal discussions are structured and facilitated based on the learners’ ideas.  teacher integrates the application of inquiry skills into learning experiences.  inquiry skills are applied and refined collaboratively and individually by learners with accountability.  students are challenged and encouraged by the teacher through modelling of problem solutions.  formal and informal discussions are structured through examples and demonstrations.  examples are employed by students to implement and refine the inquiry skills. instructional relevance  the student communicates knowledge and understanding through interactions.  complex and authentic problems are addressed by students collaboratively.  the teacher provides prompt feedback when students are attempting problems.  teacher turns classes into powerpoint/lecture shows.  the teacher instructs and demonstrates on how to solve problems to students.  the teacher provides prompt feedback during the discussion vanuatu's typical approach of mathematics vs the tuam approach of mathematics 40 characteristics tuam i do-we do-you do knowledge of content  only essential support for students who are struggling with mathematical content is provided by the teacher.  the student explores content knowledge collaboratively and individually with accountability.  students who are struggling with mathematical content receives solutions from the teacher. the teacher demonstrates the understanding and in-dept knowledge of mathematics to students. learning climate  as learners, student accepts that their learning is their responsibility.  active participation and authentic engagement are comprehended by students.  a sense of accomplishment and confidence is displayed by students.  educational risk in learning is acknowledged by students.  learning opportunities encourages students to participate and accept that learning is a process and mistake is a natural aspect of it.  students’ work is valued, appreciated and employed as a learning tool in the learning environment.  students depend on the teacher for learning.  student avoids taking educational risks in learning.  learning environment lacks the opportunity to allow students participating in contested activities and accepting that learning is a process and mistake is a natural aspect of it.  teacher’s work is valued, appreciated and employed as learning tool in the learning environment. the objective of this study is to investigate the effectiveness of the tuam discovery learning process in elementary schools in vanuatu in comparison to the typical ‘i do-we doyou do’ teaching approach. specifically, the study concerns on how the approach to solve mathematical problems changes students’ thinking processes. the investigation addressed the following research questions; 1. what is the current situation of the ‘i do-we do-you do’ teaching model in elementary mathematical education in vanuatu? 2. what influence does the tuam discovery learning process possess on students? 3. can the tuam discovery learning process influence students’ learning of mathematical concepts in elementary schools in vanuatu? methods data collection the sample groups selected for the study were two grade 5 classes of which one grade was treated as an experimental group and the other grade as the control group. even though the attendance was not consistent throughout the interventions, the enrolment of each grade ranged from 35 to 52 each day. the grades and the schools involved were randomly selected. there were five lessons of intervention in each grade similarly in both groups. all lessons were in accordance with multiplication which was the topic to be delivered during this period of interventions based on the curriculum document for this particular grade. the specific topics encompassed were new topics for students at this level, nevertheless, the concept of multiplication had been explained in their previous grades. thus, the pre-test results obtained reflected students’ prior knowledge of the multiplication concept studied in their previous grades, whereas the post-test mirrored students’ performances during interventions. table 2 demonstrates the topics included in these five lessons. kency obed sawah 41 table 2 topics covered during interventions # topics 1 multiplication – factors 2 multiplication – commutativity & associativity 3 multiplication – with and without trading 4 multiplication – converting addition into multiplication and vice versa 5 multiplication – as the inverse of division the interventions encompass pre-and post-performance tests, as well as interviews. this paper provides the findings through pre-and post-performance tests. all questions displayed in the test were merely word problems which required students to perform mathematical operations in solving each problem. students’ responses to the questions were evaluated as correct or incorrect responses, nevertheless, the responses were also analysed to evaluate students’ thinking processes in each question. table 3 illustrates the sample of the pre-and post-test administered for interventions. table 3 sample questions pre and post-tests 1 mum is earning 80 vt in 8 hours. how much is she earning in one hour? 2 there are 8 boxes of pencils on the table with 7 pencils in each box. how many pencils altogether are on the table? 3 there are 5 blue boxes of reading books on the teacher’s table. in the blue box, there are 4 red small boxes with 6 reading books in each box. how many reading books are there altogether? 4 there are 421 ropes of fish. each rope has 4 fish in it. how many fish are there altogether? 5 on a farm, jim planted 465 rows of watermelon seeds. in one row, he planted 32 seeds. how many seeds altogether did jim plant? 6 these students received the same number of awards at the end of the year. adelpha 4 received awards, jaylin 4 awards, jeremy 4 awards, and hendry 4 awards. how many awards altogether were awarded to these students? 7 a rope is 72 meters long. if you cut it equally into 8-meter pieces, how many pieces of rope will you get? the interventions incorporated five lessons in each class of both groups. the same lessons applied in the experimental group were also utilised in the control group. however, the lesson structures were different. in the experimental group, the lessons were structured in accordance with the tuam approach. moreover, in the control group, the lessons were structured based on the ‘i do-we do-you do’ approach. figure 1 displays the summary of the lesson outlines. vanuatu's typical approach of mathematics vs the tuam approach of mathematics 42 figure 1. the summary outline of the lessons results the primary data for this research paper was collected through pre-and post-performance tests. observations of students’ responses throughout the treatment were also investigated to identify the status of the current approach and the influence of the experimented approach. the data were evaluated based on the t-test normal distribution model utilising excel and rsoftware. the t-test model under normality distribution was selected for this analysis as the study compared two dependent variables represented by a control group and an experimental group. it was necessary to administer this model to examine if the sample size of the study was normally distributed or not. the t-test under normality was employed to assess the null and the alternative hypothesis defined for the study. initially, the null hypothesis of the study was understood as ‘the tuam strategy will impact more positively on students’ achievement than the typical ‘i do-we do-you do’ strategy. on the other hand, the alternative hypothesis was elaborated as the control group performed better than the experimental group. the t-test under normality distribution was administered to discover if the null hypothesis can be rejected. the results of the testing are presented through tables, graphs, and images. generally, the mean results according to figure 2 discovered that students of both groups possess naturally enhanced even though the experimental group performed much higher. noticing that the mean result obtained increases from 1.073 to 1.694 in the experimental group, whereas the mean result increases a little from 0.861 to 1.029 in the control group. such is an indication that both strategies are able to positively influence students' mathematical thinking processes. particularly, it indicates that the current teaching approach can positively affect students’ performances regardless of the limitation it owns. however, the difference in achievement deeply discovered that the tuam strategy is much more effective than the current strategy regarding the trend in improvement. understanding that the difference in outcome in the pre-test was 0.21 presenting the experimental group which accomplished higher than the control group. the results after the treatment revealed that the difference in outcome during the post-test accelerated to 0.67 which indicates that the experimental group still performed much higher than the control group. when analysing these results with respect to the counterfactual trend, the difference-in-difference of the outcome resulted to be 0.45 which illustrates that the experimental group performed higher kency obed sawah 43 than the control group with a big difference that caused the accelerated results. these results revealed that there is a more positive impact on the approach of teaching in the experimental group than that of the control group. figure 2. the difference-in-difference in the outcome the results in table 4 further highlighted that there was no significant difference in students’ performances before the treatment, but changes occurred which presented the improvement in students’ performances after the interventions. understanding that in the pretest, the p-value of 0.338>0.05 displays no statistically significant difference. it indicates that the performances in both groups were the same even though students from the experimental group accomplished much higher results. in the post-test, the p-value of 0.036<0.05 presents that there was a statistically significant difference. changes in performance arose at different level as a result of the interventions. it indicates also that in the experimental group, the skewness of 1.157 was gained from the pre-test, and 0.88 was attained from the post-test, and the kurtosis of 0.752 was also acquired from the pre-test and -0.107 in the post-test. in the control group, the skewness of 0.032 was obtained from the pre-test and 1.033 was attained from the post-test, and the kurtosis of -0.069 was also acquired from the pre-test and 0.049 in the post-test. these sample characteristics signified that the test scores accomplished were approximately normally distributed. generally, the findings unveiled that there was no significant difference in students’ performances before the treatment, but after the treatment, there was a statistically significant difference. there was a more positive impact of teaching in the experimental group than that of the control group. however, the approach of intervention in the control group was also accepted to gain a positive impact on students’ achievements as uncovered through these results. table 4 basic statistics results pre-performance achievement n p-value mean standard deviation skewness kurtosis variance min. max. exp. gr. pretest 43 0.338 1.073 1.27 1.157 0.752 1.62 0 5 cont. gr. pretest 36 0.861 0.59 0.032 -0.069 0.352 0 2 vanuatu's typical approach of mathematics vs the tuam approach of mathematics 44 post-performance achievement n p-value mean standard deviation skewness kurtosis variance min. max. exp. gr. posttest 40 0.036 1.694 1.43 0.885 -0.107 2.05 0 7 cont. gr. post-test 34 1.029 1.17 1.033 0.049 1.363 0 4 analysis of students’ thinking process the questions that administered during the pre-and post-treatment for investigation in both groups were all about word problems. there were seven questions as illustrated in table 3 above. the same questions employed in the pre-test were also applied in the post-test similarly in both groups. table 5 displays the summary of individual questions. generally, the result here revealed that there was an improvement in students’ achievements for all questions in both groups. these results emphasised that the current teaching approach possessed positively impacted students’ performances in such situations. however, when observing the difference-in-difference of these questions, the results unveiled that a significant difference in q.3 and q.7 whereby the experimental group performed better than the control group with a higher difference-in-difference in the outcome. in q.3 the difference-in-difference in outcome in the experimental group was 31.4% indicating that the experimental group performed better than the control group. moreover, in q.7 the difference-in-difference, the outcome was 44.7% which signifies that the experimental group still performed better than the control group. table 5 displays the summary of these results. this finding implies that although both teaching approaches own a positive impact on students’ mathematical performances, the experimental group possesses a more positive impact on students’ thinking processes than that of the control group. figure 4 further demonstrates students’ thinking process in these questions. table 5 summary of correct responses of individual questions by percentages (%) # experimental group control group diff-in-diff pre-test post-test difference pre-test post-test difference 1 16.7 35 18.3 11.1 8.8 2.3 16 2 31 40 9 2.8 29.4 26.6 17.6 3 4.8 45 40.2 0 8.8 8.8 31.4 4 19 32.5 13.5 0 17.6 17.6 4.2 5 0 2.5 2.5 0 0 0 2.5 6 38.1 32.5 5.6 20 26.5 6.5 0.9 7 2.4 50 47.6 0 2.9 2.9 44.7 based on the results in table 5, below is a discussion of students’ responses to q.3 and q.7 for both groups. figure 3 presents the responses to q.3 of student a from the control group and student b from the experimental group. the responses were received from the same student before and after treatment. this question was depicted from the topic kency obed sawah 45 ‘multiplication – commutativity and associativity’. it was displayed as: ‘there are 5 blue boxes of reading books on the teacher’s table. in the blue box, there are 4 red small boxes with 6 reading books in each. how many reading books are there altogether?’ the question aimed to examine students’ understanding of multiplication as commutativity and associativity. according to figure 3, the pre-test result displayed a limited understanding of the mathematical content from both students before the intervention as usual. even though the responses before the intervention were incorrect, it might be indicated that both students interpreted the situation as a multiplication problem by considering the operation they employed. it is also an implication of students’ previous knowledge of multiplication in their previous grades. initially, in previous grades, the ‘i do-we do-you do’ strategy became a common teaching practice for mathematics course. hence, there is a possibility to state that the student’s previous knowledge reflected the typical common teaching practice at some point. however, during the post-test, both students performed better but at different levels of understanding, identifying that both students employed a diagram in demonstrating their thinking process by adding the number of books in the small boxes first and then adding the results to obtain the number of books in the big box which was correctly 120. they illustrated the meaning of multiplication as a repeated addition. however, student b from the experimental group moved further to express his understanding of multiplication as commutativity and associativity by switching the order of multiplier and multiplicand several times and still obtaining the same result. it indicates that the tuam strategy was able to impact the child’s mathematical thinking processes compared to the ‘i do-we do-you do’ strategy. however, the current teaching approach was accepted to possess some positive impact on students’ thinking processes even though the tuam strategy initially owns a more positive impact, identifying that student a was able to perform the operation by employing a diagram to solve the problem. it can be implied that the typical ‘i do-we do-you do’ strategy can cause changes in students’ mathematical thinking processes to some extent. figure 3. students’ way of thinking – q.3 – students a and b figure 4 demonstrates students’ responses to q.7 from both groups. the responses were acquired from the same student before and after treatment. this question was extracted from the topic ‘multiplication – the inverse of division’. it was illustrated as; ‘a rope is 72 meters long. if you cut it equally into 8-meter pieces, how many pieces of rope will you get?’ the rationale behind this question was to investigate students’ understanding of multiplication as the inverse of division. based on figure 4, the pre-test results emphasised both students’ vanuatu's typical approach of mathematics vs the tuam approach of mathematics 46 limited understanding of this concept before treatment, understanding that student c from the control group understood the problem situation as an addition considering the operations which he preferred to apply. student d from the experimental group on the other hand comprehended the problem as subtraction as he preferred to conduct the operation as subtraction. these results manifest students’ understanding level of the situations in the problem before the treatment. after the interventions, figure 4 demonstrated that students performed better but with different levels of understanding. firstly, the results in figure 4 revealed that both students interpreted the problem as a multiplication problem. they both assumed that multiplying the length of each piece of rope by the unknown number of pieces would be equal to the length of the long rope. as a result, even though the integer 9 was not displayed in the story problem, by multiplying the length of each piece of rope by the unknown number of pieces, the result would be equal to the length of the long rope that was 72 meters. after interpreting the problem situation, student c answered the question based on his memorisation of multiplication tables considering the provided response, understanding that no additional information apart from the answer was illustrated. it produces an assumption that since the student understands the situation as a multiplication problem, it is not necessary to investigate other possible interpretations of the solutions apart from retrieving the memorisation of the multiplication tables. however, student d from the experimental group moved further to confirm this assumption by switching the order of multiplier and multiplicand several times but still obtaining the same result. hence, these results demonstrated how the tuam strategy impacted students’ mathematical thinking processes deeply compared to the ‘i do-we do-you do’ strategy. thus, it can be implied that the current teaching approach is able to encourage students in solving problems based on memorisation while the tuam strategy encouraged students in investigating possible solutions to the problem. figure 4. students’ way of thinking – q.7 – students c and d discussion the results of this study revealed that students’ mathematical understanding and thinking process in both groups enhanced over the course of interventions. the qualitative analysis of students’ responses implied that the treatment provided during interventions provided a significant impact on students’ thinking process which allowed them to produce various solutions to a particular problem. the findings discovered that there was a positive impact on kency obed sawah 47 students’ achievements in both groups. initially, the improvement in both groups portrayed that the current teaching approach influenced positively on students’ mathematical thinking processes but to a certain level of understanding when compared with the experimented approach. specifically, the quantitative analysis revealed that although there is a positive impact in both groups, the results display that the experimental group performed much higher than the control group, identifying that in the pre-test the p-value 0.338>0.05 demonstrates no statistically significant difference. however, in the post-test, the p-value of 0.036<0.05 displays that there was a statistically significant difference. these results uncovered that there was no significant difference in students’ performances before the treatment, but after the treatment, a statistically significant difference was discovered which indicates the experimental group which performed much higher not by chance. hence, the findings revealed corroborated that when allowing students to investigate mathematical ideas, the opportunities they possess will enhance their mathematical understanding when they extract relevant information from a problem statement (ojose, 2008). unveiling that when observing the difference-in-difference in the outcome of students’ achievements, figure 2 presented that the experimental group performed better than the control group. it indicates that the students owned an opportunity to be an explorer of mathematical ideas during interventions (trninic, 2018). the knowledge they obtained during self-discovery allowed them to practically demonstrate it during the post-performance test which produced the accelerated results as highlighted by bakker (2018), mayer (2004), and trninic (2018). the difference in outcome in the pre-test was 0.21% illustrating the experimental group which performed higher than the control group and after the treatment, the difference in outcome during the post-test elevated to 0.67% which indicates that the experimental group still performed much higher than the control group with a difference-indifference in the outcome of 0.45. the result unveiled that the experimental group performed higher than the control group with a big difference which made me the results accelerated. based on takahashi’s (2006) illustration of the tuam strategy, students presented that they were able to master the mathematical solutions well during their discovery processes and applied them later in the test which produced an accelerating result. it indicates that the tuam strategy possesses the capacity of encouraging students to explore mathematical solutions and implement them in any situation necessary. on the other hand, the improvement of results in the control group implied that students can also learn through observation. sanga et al. (2004) asserted that children learn best when they observe what others perform. however, these findings highlighted that learning through observation is less effective than learning through self-discovery. furthermore, the qualitative results in figures 3 and 4 above uncovered that the tuam strategy provided more significant impact to students’ mathematical thinking process than the typical ‘i do-we do-you do’ strategy. the results evidently revealed that where there was a limitation in understanding of multiplication content, the treatments provided through the interventions made students able to obtain the concepts accurately at different levels, understanding that both students in the control group demonstrated a correct response and so as the experimental group. however, reflecting on the nature of the ‘i do-we do-you do’ strategy, there is a possibility that students’ response was in accordance with their memorisation of the multiplication tables when considering the response of student c. there vanuatu's typical approach of mathematics vs the tuam approach of mathematics 48 is also a possibility that by implementing learning through observation (sanga et al., 2004), students were not able to scrutinise deeply the situation of the problem as demonstrated in both responses of student a and c. in response to research question 1, these findings discovered that there is a positive impact of the typical approach ‘i do-we do-you do’ on students’ achievement in mathematics. to some extent, the ‘i do-we do-you do’ typical approach is able to influence students positively in obtaining a correct understanding of mathematical content based on enlightenment by the teacher (sanga et al., 2004). the results emphasised that students’ achievements in the control group were enhanced based on the imitation. the students were able to acquire mathematical knowledge based on the observation of the teachers’ demonstrated ideas. they learned and improved based on what the teacher demonstrated. in other words, if there was no explanation or example of a mathematical problem provided, it would be difficult for students to obtain a piece of new mathematical knowledge. on the other hand, concerning research questions 2 and 3, the findings unveiled a possibility that the tuam approach was also able to influence students’ mathematical achievements. most specifically, by providing students opportunities to attempt mathematical problems, they will be encouraged to explore possible solutions to the problem as asserted by takahashi (2006), ojose (2008), bakker (2018), mayer (2004), and tminic (2018). thus, the findings of the study corroborate that the tuam strategy is able to positively influence students’ mathematical thinking process when allowing them to acquire a conceptual understanding of mathematical content. conclusion learning and teaching mathematics should not be based on a particular teaching approach. on the other hand, it is necessary to implement other learning approaches that involve students as the centre of learning which are evident to help stimulate students’ mathematical thinking. although the typical ‘i do-we do-you do’ teaching approach continuously impacts students’ mathematical thinking, this study revealed that the tuam teaching approach is also able to positively impact students’ mathematical thinking. conclusively, the findings through this research recommended that in this 21 st century, the tuam discovery learning strategy can be promising for mathematical education in vanuatu on two bases. firstly, it is a student-centred approach whereby students are able to construct freely their learning and develop their mathematical thinking. when the opportunity to solve a mathematical problem is initially provided for students, it allows them to develop their mathematical thinking process as they attempt and solve mathematical problems. although they will encounter misconceptions, their mistakes can be a lesson for them in exploring better solutions to mathematical problems. secondly, it encourages collaborative learning whereby students interact and learn from each other as well as from the teacher. it provides students the opportunity to discuss their mathematical reasoning and mathematical arguments, and justify their mathematical perspectives. as a result, even though both teaching pedagogies play crucial roles in children’s learning in different ways, in this 21 st century, the tuam discovery learning strategy is highly recommended for the vanuatu mathematics education. kency obed sawah 49 however, there were some limitations to these findings. firstly, there were merely five lessons displayed for interventions in both groups. there may be further discovery if the number of lessons increases. furthermore, the questions provided for pre-and post-treatment were all about word problems. the results might have changed if there were mathematical operations with direct integers. finally, students’ attendance during interventions was not consistent. some students faithfully attended the lessons but did not attend the postperformance test while some students who took the tests were not attending the classes regularly. these limitations may produce some impact on the results. acknowledgment the study recognised the efforts of students, teachers, and school principals who were involved in the study. their participation and contributions during the implementation are very much appreciated. it is also a pleasure to acknowledge professor satoshi kusaka of the naruto university of education who kindly guided the process of the study as well as other professors for their constructive ideas towards the process of conducting the study. the study is also delighted to acknowledge japan international cooperation agency (jica) for providing financial assistance towards this study. finally, all 2020 to 2023 international students of the global education course of naruto university of education 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(2018). mathematics teacher's guide, year 4. port vila: ministry of education. southeast asia mathematics education journal volume 13, no 1 (2023) 57 mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills mark joseph d. pastor & lily ann c. pedro mariano marcos state university, the philippines mjpastor@mmsu.edu.ph abstract this study aimed to determine the relationship between the secondary mathematics teachers’ level of ict expertise, level of ict use and students’ problem-solving skills. it employed two data collection tools: the mathematics teachers' survey questionnaire (mtsq) and the mathematics assessment on problem solving (maps). the samples encompassed 40 grade 7 mathematics teachers and 2,439 grade 7 students from three school divisions in ilocos norte, philippines. the results demonstrated that teachers are highly competent in basic ict skills and applications, and they have positive beliefs about the use of ict in teaching. however, because preparing ict-enriched instruction takes more time, they only use ict in teaching once or twice a week on average. according to the study's findings, teachers who are younger and have attended more ict-related training are better equipped with ict skills, use ict in classroom instruction more frequently, and have a better disposition towards ict integration in teaching. the study further discovered that when teachers believe they have a high level of ict expertise, they are more likely to use ict in their classrooms. similarly, when teachers are more knowledgeable about using ict, they are more inclined to support ict integration in the classroom. notably, the study reveals that students' problem-solving skills are significantly related to teachers' level of expertise, level of ict use, and their beliefs about ict integration. keywords: ict expertise, ict use, ict beliefs, problem solving skills, mathematics education introduction information and communication technologies or ict are now ubiquitous in all aspects of everyday life. they have rapidly become one of the fundamental building blocks of modern civilization, and many countries now consider understanding ict and mastering the fundamental skills and concepts of ict to be part of the core of education, alongside reading, writing, and arithmetic (khvilon & patru, 2002). icts have had an impact on research, teaching, and learning in the subject of education (yusuf, 2005). previous research into teachers' use of icts identified staff development as one of the contributing variables to using icts effectively in the classroom. teachers and students recognize ict skills and then apply them in the teaching and learning process in response to global educational problems by investigating and implementing ict by employing effective approaches (buyong, 2002). not only do individuals and schools value the role of technology; governments increasingly recognize it as an important component of education reform and devote significant resources to it (bulman & fairlie, 2016). many countries have been encouraging the use of technology in various subjects, including mathematics, over the last three decades (alhejoj, 2020). math education requires strategies for problem solving. teaching students how to solve problems has always been a challenge for mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills 58 educators. thus, integration of ict plays a significant role in improving students' problemsolving skills. a substantial body of research literature has demonstrated that ict can assist children in developing higher-order mathematical thinking and problem-solving skills (drigas & karyotaki, 2016; leite, 2019; parno et al., 2021). however, educational systems face new challenges in the twenty-first century. as a result, it is critical to develop a workforce capable of serving both the country and the rest of the world. information is essential in the creation of knowledge, and a nation's capacity for knowledge acquisition and application will determine its future strength. the educational system should be capable of developing students' critical thinking abilities, internalization of moral principles, improved communication skills, and improved information-seeking abilities. according to the national council of teachers of mathematics (nctm), effective teachers maximize the potential of technology to develop students' understanding, stimulate their interest, and increase their mathematical proficiency (nctm, 2015). as a result, the main issue is how teachers integrate technology, the necessary knowledge, and their motivation or challenges to do so. consequently, today’s math teachers must plan and administer best practices for ict to engage students in their learning, allow them to develop critical thinking as well as enhance their math skills to construct, and expand (niess, 2013). technological, pedagogical and content knowledge (tpack) is an elastic framework for technology integration that involves three types of knowledge that instructors must connect in order for technology integration to be successful in education. tpack is regarded as a leading tool or map for understanding teachers' knowledge of how to effectively integrate ict in their classrooms and for assisting in the development of this knowledge as well as ict practice (alhejoj, 2020). given these digital world demands and ict-related studies and frameworks, it is critical for a developing country like the philippines to ensure that the next generation of filipino graduates gains a broader range of skills and that all filipinos have equal access to and benefit from the digital economy. the philippine government has been working to reform the educational system since 2011. the department of education's (deped) k-12 educational reform program, in particular, aims to align philippine education with global standards, aligning its vision with that of international education organizations and agencies. enhancing the mathematical skills of filipino students thus requires a strategic focus on teachers. compelling evidence from international experience indicates that teachers who possess strong content knowledge are the main determining factor behind high-performing students (glewwe et al., 2011). previous research, however, has demonstrated that ict integration is a complex phenomenon (mackey & mills, 2002; ng et al., 2010) and that technology or computer use among teachers is a difficult process (chen, 2010). many mathematics teachers are still struggling to implement icts as an instructional teaching-learning methodology (kaleliyilmaz, 2015). within years of implementing various technology initiatives in educational systems, ismail et al. (2007) reported that teachers’ level of technology integration was still low. evidence from research has consistently shown that school teachers have not advanced to higher levels of ict use and expertise, which is detrimental to the effective integration of ict in teaching practice (castillo, 2007). mark joseph d. pastor, lily ann c. pedro 59 ict use in schools has become a focus of educational research (eickelmann, 2011). problem-solving skills acquired through ict use are similar to those required for successful mathematical competence in secondary schools (senkbeil & wittwer, 2008). however, it is critical to comprehend how ict use can influence learning and achievement (voogt, 2008), which is not yet clear. furthermore, little research has been conducted on how mathematics teachers integrate technology into their classroom instruction (bray & tangney, 2017). while various studies have been conducted to determine the impact of ict on teaching and learning math, it remains to be determined whether the extent of ict use by teachers, as well as their levels of ict expertise and beliefs about ict integration, are predictors of students' problem-solving skills. in this study, ict use in general and at school, particularly for mathematics learning, is incorporated into an analysis to obtain a comprehensive picture of the relationship between teachers' levels of ict expertise and use, their beliefs about ict integration, and students' problem-solving skills. the objective of this study was to determine the relationship between mathematics teachers' levels of ict expertise and use and their beliefs about ict integration and students' problemsolving skills. this study specifically sought solutions to the following problems: 1. what are the socio-demographic characteristics of mathematics teachers in terms of: a. age; b. number of years teaching mathematics; c. educational attainment; and d. ict-related training attended? 2. what is the teachers’ level of ict expertise in the classroom? 3. what is the teachers’ level of ict use in the teaching and learning process? 4. what are the teachers’ beliefs about ict integration? 5. is there a significant relationship between each of the socio-demographic characteristics of the mathematics teachers and their: a. level of ict expertise; b. level of ict use; and c. beliefs about ict integration? 6. is there a significant relationship between mathematics teachers’: a. level of ict expertise and use; b. level of ict expertise and their beliefs about ict integration; and c. level of ict use and their beliefs about ict integration? 7. what is the student’s level of problem-solving skills in mathematics? 8. is there a significant relationship between the student's problem-solving skills and teachers’: a. level of ict expertise; b. level of ict use; and c. beliefs about ict integration? mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills 60 methods research design a descriptive-correlational research design was employed in this study. it is descriptive because it describes the teachers' socio-demographic profile, level of ict expertise and use, and beliefs about ict integration. it is correlational as it established possible relationships among teachers' socio-demographic profiles, their level of ict expertise and use, their beliefs about ict integration, and students' problem-solving skills. population and sampling procedures the population of the study involved forty (40) grade 7 mathematics teachers from 25 public secondary schools in the city schools division of laoag and batac and schools division of ilocos norte; and 2439 grade 7 students who were selected through cluster sampling. research instruments data from this research were obtained through two instruments, encompassing mathematics teacher’s survey questionnaire (mtsq) and mathematical assessment on problem solving (maps). the mathematics teacher's survey questionnaire (mtsq) was divided into four sections: the first section collected socio-demographic information from the teacher-respondents, involving age, number of years teaching mathematics, educational attainment, and participation in ict-related training. the second section of the questionnaire, which assesses teachers' level of ict expertise, was adapted from the questionnaire used by alharbi (2014) in his study on the use of ict in secondary school teaching in kuwait. the third section of the questionnaire inquired about teachers' use of ict in the classroom. items in this portion were culled from the combined survey instruments of umar and hassan (2015) and kamau (2014). the instruments were modified to update several technological devices that were already obsolete in the teaching and learning process. the fourth section of the questionnaire encompassed a series of statements that assessed teachers' beliefs about ict integration. these statements were drawn from several studies (alharbi, 2014; kamau, 2014; and moila, 2006) on teachers' beliefs about ict integration. a group of mathematics experts validated the instrument by checking the completeness of the items and whether it could provide answers to the problems raised in the study. the cronbach alpha was calculated to be 0.826, indicating that the instrument was reliable for the study. the mathematical assessment on problem solving (maps) is a 50-item teacher-created test with 40 multiple-choice items and 10 constructed-response items. the test items were employed to assess the three problem-solving cognitive domains used in the 2015 timss: a) knowing facts, procedures, and concepts; b) applying knowledge and understanding; and c) reasoning. this teacher-created test was validated in two stages. the first phase involved a committee of math content experts, including teachers and administrators, evaluating all questions to ensure their appropriateness for measuring the problem-solving skills of grade 7 mathematics students. the second phase involved the validation of the pre-identified cognitive domain of each item by another set of math experts in the locality. to evaluate students' mark joseph d. pastor, lily ann c. pedro 61 answers to each problem in the given test, the 2015 timss scoring guide was utilized. the questions covered grade 7 mathematics topics such as number sense (rational numbers), number sense (exponents, powers, and ratios), and algebra. data analysis the research data was interpreted and analyzed using frequency and percentage distributions, means, and pearson's r. the first method is used for evaluating the information on the respondents' profiles. the weighted means were computed and interpreted to describe the teacher's level of ict expertise using the following range of values with corresponding descriptive interpretations. range of means descriptive interpretation 4.18 – 5.00 expert 3.34 – 4.17 advanced 2,51 – 3.33 average 1.68 – 2.50 beginner 0.85 – 1.67 newcomer 0.00 – 0.84 unfamiliar similarly, the weighted means of teachers' ict use were calculated and interpreted using the following range of values and descriptive interpretation. range of means descriptive interpretation 2.26 – 3.00 high 1.51 – 2.25 moderate 0.76 – 1.50 low 0.00 – 0.75 no integration the weighted means for the teacher's beliefs about ict integration, on the other hand, were interpreted using the following range of values with corresponding descriptive interpretation: range of means descriptive interpretation 4.21 – 5.00 very highly favorable 3.41 – 4.20 highly favorable 2.61 – 3.40 moderately favorable 1.81 – 2.60 slightly favorable 1.00 – 1.80 not favorable the students' problem-solving skills were determined using their problem-solving test scores, and they were categorized by employing the following scale range with their corresponding level of competency. range of means descriptive interpretation 46 – 60 advanced 31 – 45 high 16 – 30 intermediate 0 – 15 low pearson's r was employed to examine the relationship between the teacher's sociodemographic characteristics, level of ict expertise, level of ict use, beliefs about ict integration, and problem-solving skills. mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills 62 results and discussion teachers’ socio-demographic characteristics generally, teachers are young both in age (x̅ = 35.08) and teaching experience (x̅ = 9.98). the majority of teachers (21 or 52.50% of the total) are bs graduates with master's degrees. the majority of them (21 or 52.50%) have yet to participate in any ict-related training. this is consistent with a walet (2014) survey, which revealed that few teachers are trained in countries where ict is scarce, such as the philippines, myanmar, and kyrgyzstan. in his research, bonifacio (2013) discovered that public schools typically send only a few teachers to computer literacy training. teachers’ level of ict expertise generally, teachers perceived themselves to be highly competent in the basic ict skills and applications required in the integration of ict in teaching mathematics. table 1 mathematics teachers’ level of ict expertise ict skill mean descriptive interpretation 1. the fundamentals of operating a personal computer/laptop (keyboard, mouse, turning on, shutting down, and so on) 4.33 expert 2. managing files (moving, deleting, copying files, etc.) 4.25 expert 3. utilizing a word processor (microsoft word or equivalent software) 4.13 advanced 4. utilizing a spreadsheet processor (microsoft excel or equivalent software) 3.78 advanced 5. creating presentations by combining files from various sources (such as sound or video files) 3.38 advanced 6. utilizing presentation software (microsoft powerpoint or equivalent software) 3.63 advanced 7. manipulating lcd projectors during the presentation of topics 3.63 advanced 8. manipulating television and/or dvd players during the presentation of topics 3.63 advanced 9. editing pictures or raw videos 3.15 average 10. utilizing digital cameras 3.55 advanced 11. internet browsing on the computer 4.05 advanced 12. internet browsing on the mobile phone 3.90 advanced 13. searching for information on the internet 4.05 advanced 14. downloading files from the internet 3.90 advanced 15. employing email (reading and sending emails) 3.68 advanced 16. utilizing different social media sites (facebook, twitter, etc.) 3.90 advanced 17. creating/using chatgroups and forums in teaching (facebook group, twitter, etc.) 3.18 average 18. publishing a personal blog (blogspot, wordpress, etc.) 2.35 beginner mark joseph d. pastor, lily ann c. pedro 63 19. designing a web page or personal site 1.88 beginner 20. producing a learning software 1.80 beginner overall mean 3.51 advanced legend: range of means descriptive interpretation 4.18 – 5.00 expert 3.34 – 4.17 advanced 2.51 – 3.33 average 1.69 – 2.50 beginner 0.85 – 1.67 newcomer 0.00 – 0.84 unfamiliar the obtained composite mean of 3.51 is equivalent to a descriptive rating of advanced level. teachers are most experts in basic operations of personal computers (x̅ = 4.33) and managing files (x̅ = 4.25). this is similar to the findings of del rosario (2015), who discovered that teachers use their personal computers in the classroom regularly and are implementing creative uses of technology such as data management and presentation. however, teachers scored the lowest in designing web pages (x̅ = 1.88) or personal sites and producing learning software (x̅ =1.80) which suggests that they have to upgrade their knowledge and skills in more advanced ict applications. several studies have revealed a need for teachers to improve their ict knowledge and skills (mckenna, 2015; park, 2016), and additional improvements are required to enrich the effectiveness of their ict utilization by identifying supplementary information they considered when planning instructional experiences (browne, 2019). teachers’ level of ict use the perceived levels of ict use were determined by investigating teaching and learning material, generating a lesson plan, creating activity sheets, teaching mathematical concepts, and assessing students' performance. the component preparing activity sheets recorded the highest level of use among teachers (x̅ = 2.19) followed by searching for teaching and learning material (x̅ = 2.02), preparing a lesson plan (x̅ = 1.98), and teaching mathematical concepts (x̅ = 1.81). teachers use ict least frequently in evaluating students’ performance (x̅ = 1.60). overall, teachers perceived themselves to be moderate users (x̅ = 1.87) of ict that is, they only use ict in the classroom once or twice a week. according to kiru (2018), teachers in various countries used ict in mathematics instruction with varying degrees of frequency. given the likelihood of differences between countries in aspects such as the role of ict in teaching and learning or government initiatives in various countries, such differences in teachers' ict use are plausible (kiru, 2018). table 2 mathematics teachers’ level of ict use ict in classroom activities mean descriptive interpretation i. searching for teaching and learning material 2.02 moderate ii. preparing a lesson plan 1.98 moderate iii. preparing activity sheets/worksheets 2.19 moderate iv. teaching mathematical concepts 1.81 moderate mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills 64 v. evaluating students’ performance 1.60 moderate overall mean 1.87 moderate legend: range of means descriptive interpretation 2.26 – 3.00 high three to five times a week 1.51 – 2.25 moderate once or twice a week 0.76 – 1.50 low once or twice a month 0.00 – 0.75 no integration no integration teachers’ beliefs about ict integration generally, teachers organized positive beliefs about the use of icts in education (x̅ = 3.92). they highly favor the benefits of utilizing ict in enhancing classroom instruction and developing students’ knowledge and skills. table 3 mathematics teachers’ beliefs about ict integration. statement on ict integration mean descriptive interpretation 1. the use of ict makes my teaching more interesting 4.25 very highly favorable 2. the use of ict makes my preparation of lessons faster 4.03 highly favorable 3. the use of ict decreases students’ motivation* 1.95 slightly favorable 4. the use of ict improves my classroom management as a teacher 4.03 highly favorable 5. the use of ict motivates my students to get more involved in learning activities 4.18 highly favorable 6. the use of ict promotes the development of interpersonal skills of students (e.g., ability to relate or work with others) 4.08 highly favorable 7. the use of ict promotes the development of communication skills of students (e.g., writing and explaining mathematical solutions) 3.85 highly favorable 8. the use of ict helps accommodate my students’ personal learning styles 3.93 highly favorable 9. the use of ict encourages my students to develop their problem-solving skills 3.85 highly favorable 10. ict often prevents teaching because of interruption in work or software* 2.58 slightly favorable 11. the use of ict has brought positive impact on my students’ learning 4.08 highly favorable 12. the use of ict improves my students’ test and exams results 3.90 highly favorable 13. the use of ict makes it more difficult to control the class* 2.10 slightly favorable 14. the use of ict promotes collaborative learning among my students 4.00 highly favorable 15. the use of ict gives the teachers the opportunity to be learning facilitators instead of information providers 4.00 highly favorable 16. the use of ict makes teachers feel more competent as educators 4.05 highly favorable 17. the use of ict is difficult to use while teaching mathematics* 2.45 slightly favorable mark joseph d. pastor, lily ann c. pedro 65 18. the use of ict creates a platform for me to communicate with other teachers sharing common problems 3.85 highly favorable 19. ict-integrated instruction is more effective than the traditional method of instruction 3.73 highly favorable 20. the use of ict gives many problems in managing classrooms that use ict* 2.33 slightly favorable overall mean 3.56 highly favorable *negative statements legend: range of means descriptive interpretation 4.21 – 5.00 very highly favorable 3.41 – 4.20 highly favorable 2.61 – 3.40 moderately favorable 1.81 – 2.60 slightly favorable 1.00 – 1.80 not favorable specifically, they favored the highest that ict makes teaching more interesting (x̅= 4.25) while they disagreed the strongest that ict decrease students’ motivation (x̅ = 1.95). this finding is consistent with the findings of umar and hassan (2015), who discovered that teachers generally agree that the use of ict has improved their classroom teaching practices. furthermore, andre (2020) revealed that teachers have a positive attitude toward ict integration and are willing to learn new ict skills. level of students’ problem-solving skills except for 357 students, all were at least intermediate in problem solving. there were 250 students who attained advanced level, 764 students attained high level, and 1068 students attained intermediate level. however, with a mean of 29.50 and a standard deviation of 9.09, the overall level of problem solving skills was intermediate. table 4 distribution of students according to their level of problem-solving skills range of percentage score level of achievement f percentage 46 – 60 advanced 250 10.25 31 – 45 high 764 31.32 16 – 30 intermediate 1068 43.79 0 – 15 low 357 14.64 total 2439 100.00 mean 29.50 overall level of problem-solving skills intermediate in general, the findings indicate that students can apply basic mathematical knowledge in simple situations, but they require significant effort to formulate, grapple with, and solve complex problems. as a result, teachers must continue to teach students how to apply their understanding and knowledge to a wide range of relatively complicated but contextualized problems. mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills 66 correlation analysis of mathematics teachers' socio-demographic characteristics and their level of ict expertise, level of use, and beliefs about ict integration. the teachers’ level of ict expertise is significantly related at the 0.01 level of significance with their age (r = -0.409) and number of years teaching mathematics (r = -0.449). this suggests that younger math teachers have a higher level of ict expertise than older ones. similarly, teachers who are relatively new to the profession have a higher level of ict expertise than those with more years of teaching experience. it should also be noted that teachers' participation in ict-related training is significantly related to their level of ict expertise (r = 0.389), indicating that more training tends to increase their level of ict expertise. there was, however, no established relationship between teachers' level of ict expertise and their educational attainment (r = 0.259). data analysis revealed a statistically significant positive relationship between teachers' level of ict use and their number of ict-related training (r = 0.469) at the 0.01 level. at the 0.05 level, teachers' ict use is significantly related to their age (r = -0.319) and number of years teaching mathematics (r = -0.393). this implies that older teachers or teachers at the top of their profession have a lower level of ict use in the classroom. table 5 relationship between the socio-demographic characteristics of mathematics teachers and their level of ict expertise, level of use and beliefs about ict integration socio-demographic characteristics level of ict expertise level of ict use beliefs about ict integration age -0.409** -0.319* -0.376* number of years teaching mathematics -0.449** -0.393* -0.492** educational attainment 0.259 0.295 0.046 number of ict-related training 0.389* 0.469** 0.387* at the 0.01 level of significance, teachers' beliefs about ict integration are significantly associated with their number of years teaching mathematics (r = -0.492), age (r = -0.376), and the number of ict-related training (r = 0.387). this means that younger teachers, both in terms of age and teaching experience, have more positive beliefs about ict integration than older teachers. it also indicates that teachers who have participated in various ict-related trainings have a more positive attitude toward ict integration and how ict may successfully enhance classroom management and the teaching process. correlation analysis between the teachers’ level of ict expertise, level of ict use and their beliefs about ict integration and students’ problem-solving skills at the 0.01 level, there is a significant relationship between teachers’ level of ict expertise and level of ict use (r = 0.796); between teachers’ level of ict expertise and beliefs about mark joseph d. pastor, lily ann c. pedro 67 ict integration (r = 0.514); and between teachers’ level of ict use and their beliefs about ict integration (r = 0.509). table 6 relationship between the students’ problem-solving skills and teachers’ level of ict expertise, level of ict use and their beliefs about ict integration. student’s problem solving skills probability i. level of ict expertise 0.552** 0.000 ii. level of ict use 0.632** 0.000 iii. beliefs about ict integration 0.345* 0.029 ** significant at 0.01 probability level (2-tailed) *significant at 0.05 probability level (2-tailed) students’ problem solving skills is significantly correlated with each of the following teacher characteristics: level of ict expertise (r =0.552), level of ict use (r = 0.632) and beliefs about ict integration (r = 0.345). the results are consistent with the findings of leite (2019), who discovered that teachers believe that the use of technology enhances students' problemsolving abilities. furthermore, the study's findings indicated that teachers can improve students' problem-solving skills by utilizing technology as a tool. conclusion the objective of this study was to investigate the mathematics teachers' levels of ict expertise and use, as well as their beliefs about ict integration, and to determine possible relationships between these factors and students' problem-solving abilities. several major conclusions can be drawn from the analysis of data and findings by employing the descriptive correlational design. according to the study's findings, teachers' socio-demographic characteristics such as age, number of years teaching mathematics, and number of ict-related training are indicators of teachers' levels of ict expertise and use, as well as their beliefs about ict integration. the younger the teachers are in age and profession, the more open they are to embracing technological innovations. furthermore, the more training teachers attend, the better equipped they are to implement ict in the classroom. this suggests that ongoing ict professional development for teachers is critical in enhancing their technological skills and ensuring effective ict integration in the mathematics classroom. this, in turn, can positively impact students' problem-solving abilities. it is also possible to conclude that there is a significant relationship between and among teachers' levels of ict expertise, use, and beliefs about ict integration. teachers with a higher level of ict expertise integrate educational technology into their teaching more frequently. furthermore, teachers who favor ict as an effective tool for efficiently acquainting students with math tend to incorporate ict in introducing mathematical concepts to students, thereby improving their problem-solving abilities. as a result, encouraging positive attitudes toward ict integration among teachers can lead to increased adoption and effective application of mathematics teachers’ levels of ict expertise and use and their beliefs about ict integration and students’ problem-solving skills 68 technology in the classroom, ultimately enhancing students' problem-solving abilities and overall learning experiences. moreover, teachers’ level of ict expertise, level of ict use and their beliefs about ict integration is significantly related to the students’ problem-solving skills. when teachers are equipped with ict skills and frequently use ict in their teaching, students' problem-solving abilities improve. furthermore, when teachers consider the positive impact of ict on improving the quality of mathematics instruction, students' problem-solving skills enhance. this suggests that effective integration of ict tools and resources can create an engaging and interactive learning environment that encourages students' critical thinking and problemsolving abilities. furthermore, when teachers recognize the positive impact of ict on the quality of mathematics instruction, students' problem-solving skills escalate. based on the findings and conclusion, the following recommendations are suggested. teachers must keep up with technological advancements in teaching, particularly in terms of aligning technologies with content and pedagogy and developing the ability to use ict creatively to meet the specific learning needs of their students. they will be able to align instruction with standards, particularly those that embody 21st century knowledge and skills such as critical thinking and problem solving. teachers should be encouraged to use ict not only in the classroom to teach mathematical concepts, but also in the development of technology-enhanced assessments that assess student mastery of higher order thinking skills. as a result, educational interventions such as ictrelated teacher training should be implemented to persuade teachers of the benefits of ict in teaching and learning. to encourage teachers to use ict regularly, the department of education and local government education authorities must provide computers, internet access, and other ict infrastructure in all government schools. school administrators must provide strong support for their teachers' personal and professional development in terms of ict literacy and educational technology-related teaching pedagogy. they must provide technical assistance, such as conducting ict-related training during the induction seminar and workshops that may assist their teachers in continuing to integrate ict into their classroom practice. teacher training institutions must provide knowledge, experiences, and supervision to preservice mathematics teachers to prepare them to design and implement ict in the classroom. these institutions' teacher preparation goals should center on student mastery of academic content and knowledge, as well as mastery of 21st-century skills such as critical thinking, problem-solving, communication, technology literacy, collaboration, and creativity. policymakers and developers have to develop curricula and programs that highlight various ways teachers can seize opportunities for integrating ict tools and innovative teaching strategies into their classroom instruction. the curriculum should emphasize opportunities for students to apply technology skills across content areas while adopting a problem-solving approach to learning. by implementing these recommendations, educators, policymakers, and institutions can promote the effective integration of ict in mathematics education, enhancing students' problem-solving skills and preparing them for 21st-century challenges. future researchers can build on these findings by investigating additional factors influencing teachers' ict integration mark joseph d. pastor, lily ann c. pedro 69 and the long-term impact of ict use on students' academic performance and overall development. furthermore, this study primarily examined the relationships between mathematics teachers' ict factors and students' problem-solving skills. other variables and factors that may influence students' problem-solving abilities, such as teaching methods, curriculum design, or student characteristics, were not thoroughly investigated, despite their importance. future researchers could build on this research by conducting a more in-depth examination of the numerous factors that influence students' problem-solving abilities in mathematics, including the role of ict integration. acknowledgements the researchers wish to express their utmost gratitude to the department of science and technology – science education institute (dost-sei) of the philippines, through the national consortium in graduate science and mathematics education (ncgsme) for funding the conduct of this study. references alharbi, e. 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(2005). information and communication technology and education: analysing the nigerian national policy for information technology. international education journal, 6(3), 316-321 microsoft word seamej.journal.vol1.draft 5 edited by wahyudi jan 2012 br.docx allan leslie white school mathematics teachers are super heroes allan leslie white <al.white@uws.edu.au> university of western sydney, australia. abstract hollywood has produced many super heroes such as superman, batman and wonder woman. recently it released a film titled 'waiting for superman' which shows a young boy imprisoned within a system and classroom that does not stimulate his learning while actively destroying his motivation and engagement with the educational process. the film implied the task of fixing the problem was so great that only superman could fix it. so what are the criteria for a super hero? firstly it is someone with extraordinary powers beyond those of most mortals. in this paper i will propose that most mathematics teachers meet the criteria and are super heroes who combat the spread of darkness and ignorance of mathematics. i will present evidence to prove that most mild mannered mathematics teachers are really super heroes in disguise. mathematics teachers have super powers. they have the power to understand and value mathematics, something that is beyond the vast majority of the population. what is the basis of their power? it is their mathematics pedagogical and content knowledge. not only can they do mathematics, but they can construct a learning environment where their students develop conceptual knowledge and deep learning. they use the latest developments in technology to assist their battle with the forces of darkness and innumeracy. while more mathematics has been invented in the last 50 years than in the preceding years of human development, teachers are expected to keep abreast of this new knowledge. hollywood may be waiting for superman, but the real super heroes are every day engaged in the battle to reveal to their students the power and the beauty of mathematics that can transform their lives. introduction the history of movies produced in hollywood reveals a number of super heroes such as superman, spiderman, batman and wonder woman. hollywood recently released a film titled "waiting for superman". the film documented the story of a young boy imprisoned within a dysfunctional school system and classroom that failed to stimulate his learning and actively destroyed his motivation to learn and his engagement with the educational process. the film implied that the task of fixing the problem was so great that only superman, a super hero could fix it. sadly, the boy despaired when incorrectly told by his mother that super heroes did not exist. what are the criteria for super heroes? super heroes share a number of common traits contained in the following questions and answers: (1) what do super heroes do? usually super heroes battle the forces of ignorance and darkness which threaten human civilisation. (2) what is special about super heroes? all super heroes have super powers that other humans do not have or abilities that are beyond those of ordinary humans and these help the heroes overcome the forces of darkness. (3) how do you identify super heroes? most super heroes on the surface are difficult to identify and are usually mild mannered, plain looking, conservatively dressed and do not stand out in a crowd. yet all super heroes have either key words: super heroes, behaviourism, external examinations, pedagogical content knowledge, integration of information communication technologies 3 school mathematics teachers are super heroes special costumes (superman's cape), places (such as bat cave) or devices (spiderman's web) that assist them in their battles. this paper provides hope to the boy in the film by arguing that super heroes do exist and that mathematics teachers are super heroes. this is in conflict with one premise of the documentary, "waiting for superman," which blamed the weaknesses of public education on uncaring and incompetent teachers and in the united states of america there have been frequent media attacks on teachers and their unions by politicians and others. teacher bashing is almost a national past-time. the teachers are portrayed as forces of darkness rather than super heroes. this paper attacks this view and will provide evidence to demonstrate that most mathematics teachers are super heroes. it will organise the evidence using the structure of the three questions listed above to justify this claim. super heroes battle the forces of darkness if all super heroes battle the forces of evil and darkness that threaten human development, then what are these forces that are faced by mathematics teachers? while there is a huge list, i will dwell on only three. the treatment of the three is not meant to be an exhaustive or comprehensive but rather a brief snapshot as deeper treatments are available elsewhere. the three selected are: firstly there is the crippling after effects of behaviourism on the teaching and learning process in school mathematics classrooms; secondly there are many issues involving large scale examinations and their use by authorities as measures of quality assurance; and thirdly, there is a collective impact of the lack of mathematics knowledge in the general population and the explosion of new mathematics that underpins modern technological innovations. the darkness of behaviourism behaviourism as a philosophical tradition has had a large influence upon mathematics education, particularly on the western tradition. skinner's (1953) theory that by using cause and effect behaviour could be manipulated by conditioning; bloom's (1956) taxonomy of educational objectives; and gagne's (1967) work on learning hierarchies were all highly influential upon mathematics teaching and teacher training programmes. while there were some good outcomes, some of the negative aspects of this movement were the use of behavioural objectives, outcomes based education, mastery learning, programmed learning, an over emphasis on skills drill and practice, and a focus on large scale skills based testing (often multiple choice questions) as opposed to testing understanding and the application of knowledge. it was common to hear that the essence of behaviourist teaching was contained in the saying "a long journey consists of many small steps", and this assumed that a child could master any skill as it just depended on the teacher making the steps small enough and giving 4 allan leslie white the child enough time. however, some students proved this assumption incorrect and became known as slow learners. yet the truth was that no matter how small the steps or how much time was devoted to practice by these students, they would not learn until the approach was changed. the behaviourist pedagogical approaches became problematic and there was a need to change (clements, 2003). for example. the first aspect of reducing a task to small steps requires the teacher to reduce a student’s role by 'emptying' the task of much of its cognitive challenge (brousseau, 1984). a task is broken into a number of smaller steps and if the student answered each step, then the teacher tended to believe that the student had learnt what had just been taught, and assumed that the student would construct the whole from the parts. in that sense, the students were presumed to have learnt what they were expected to learn from the original question. cognitively challenging questions were removed from the classroom and replaced by bite-size portions. when teachers adopted this style, in an attempt to help students tackle higher-level mathematics tasks, they denied their students the opportunity to formulate and apply strategies of their own (clements, 2004). the result was that the students failed if given unseen or novel problems because there was no one to 'cut them up' or tell them which mathematical tool to use. skemp (1976) in his work on instrumental and relational understanding highlighted why this approach failed. from his research studies he showed that if a, b and c are steps in a learning hierarchy and the teacher instructs students on how to go from a to b and then from b to c then the students usually failed to acquire a holistic understanding. thus they mostly did not see how a, b and c were related, nor could they return from c to a. a clear local example comes from the chang mai district thailand, where vaiyavutjamai (2004) investigated why so many students failed to learn the material covered in mathematics lesson. she focussed on the questions asked by the experienced teachers during 16 lessons of six form 3 algebra classes. the sample involved 4 teachers from two government schools and a total of 231 students (across the three streams of high, medium and low). she reported the prevalence of the 'cognitive empting' process across the 16 lessons. a high level question was followed by a sequence of low level questions by the teacher to give structure to students' thinking. low level questions required very brief answers and were usually chorused by the class. detailed examples of this emptying process can be found elsewhere (see clements, 2004). the research showed that after having participated in the form 3 lessons on linear equations and in-equations, many of the 231 students were still struggling to cope with the elementary questions. the retention data showed that in regards to a long term view, the approach failed except for the high stream students. as one teacher taught all three streams, it appears that this result has more to do with the students than the method. 5 school mathematics teachers are super heroes what is surprising is that both students and teachers were happy with this approach because they had become accustomed to this way of teaching, even though the results were poor. brousseau's (1984) research into didactical contracts helps explain why behaviourist pedagogies are resistant to change. the didactical contract encompasses the conscious and subconscious beliefs, behaviours and relationships that guide and control what teachers and students do within mathematics lessons. his research shows that once teachers and students become accustomed to an approach then they resist any change. teachers and students develop sets of ingrained actions that arise from, yet simultaneously determine, didactical contracts. the teachers and their students have reciprocal expectancies, and their actions tend to become economical in the sense that they are guided by expectations of what can and cannot be done in 'normal' lessons (jaworski & gellert, 2003). although these expectations generate common classroom practices, it is usually the case that neither the teacher nor the students subject the expectations to reflection or scrutiny. a teacher attempting a different pedagogy is usually subjected to cries of ‘is this in the test?’ similar findings have been reported in brunei (lim, 2000), and new zealand (nz) schools. barton (2003) maintained that in nz classroom settings, didactical contracts influenced teachers’ aims, methods, behaviours, content covered, and choice of procedures for assessing learning. so the darkness of lingering behaviourism influences many mathematics classrooms, through the process of cognitive emptying and unchallenged didactical classroom contracts. who is there to fight this dark influence upon the teaching and learning of our children and release students to enjoy and understand the power of mathematics in their lives? the darkness of external examinations the usual goal of assessment is to provide useful, timely and appropriate information that is fair and equitable and helpful to teachers in making plans for improving the classroom or system’s learning and teaching cycle. assessment data could consist of the mathematical understanding of an individual student, the achievement of a group or a class, or the overall achievement of a system. assessment data was collected at different points in the learning and teaching cycle. however, more recently in some countries there has been a developing confusion where teachers have been asked to meet the principles of assessment listed above, yet are told that externally imposed testing will be used to rate the effectiveness or quality of schools or teachers, and used in the distribution of resources. usually this external testing is merely a measure of how many facts can be stuffed into the students' short term memory, to be regurgitated on a multiple choice examination and then promptly forgotten. the information gained from these examinations is not usually helpful in improving the teaching and learning process. the use of standardized test results are such a misleading indicator of teaching or learning and successful efforts to raise scores can actually lower the quality of students' 6 allan leslie white education. for example, in atlanta the large scale testing regime is under attack. the pressure upon teachers and educational authorities has lead to cheating and even fraud (torres, 2011). in 2009, in 56 schools, accounting for 78% of teachers and principals were found to have cheated. it is the position of this paper that these teachers are also super heroes. the reason is because they were fighting to help their students by trying to soften the impact of an unfair and inequitable method of distributing resources and they were not seeking personal gain. there are many other things wrong with large scale testing and some are listed: "the us tests have been criticised for narrowing the curriculum to reading and maths and multiplechoice formats... 'we have learnt about the potential negative effects of very narrow tests, particularly when they are put in a high-stakes context,' said professor darling-hammond" (patty, 2011, p.1). however, this section will confine itself to briefly discussing one. it concerns the faulty assumption (linked to behaviourism) that in the process of the mastery of skills, the students come to an understanding. it assumes that if students show a high degree of mastery on a test then they have a good understanding of the underlying mathematical concept. while this may happen with some students, there were many students where this assumption proved false. the work of erlwanger (1975) showed that elementary american students who passed mastery tests were unable to apply the mathematics and developed a mechanistic view of mathematics. the eminent dutch mathematician freudenthal (1979) attacked the concept of mastery learning. researchers ellerton and olson (2005) conducted a study of 83 grades 7 and 8 north american students completing a test comprising items from illinois standards achievement tests. their findings indicated a 35% mismatch with students who gave correct answers with little or no understanding and others who gave incorrect answers but possessed some understanding. what a wonderful system for allocating resources when 35% of the results are not reliable. who but a super hero would stand up against such injustice? surely large scale testing cannot be all bad, what of the apparent success of some countries on predominantly skills based international comparison tests? there are some countries with outstanding performances from their confucian-heritage students on international comparative studies (e.g., on timss or pisa). while these countries have achieved high results, the authorities are concerned with the poor attitudes and engagement of their students towards mathematics and the small number who choose to continue studying mathematics at university. zhao gave a keynote address at an east asian education forum and claimed: the east asian students suffer, actually. there is psychological stress, there is a lot of direction, a lack of social experiences and therefore emotional development, he said. the concern about consequences of the approach typified by self-styled 'tiger mum' amy chua spreads beyond the suffering of individual students. east asian educators are not at all happy with what they have achieved; they look at what they have not achieved. they look at the children's lack of confidence, for example, creativity, entrepreneurial spirit and imagination,' (stevenson, 2011, p. 1). 7 school mathematics teachers are super heroes australia is also feeling the negative effects of external testing through the national assessment program literacy and numeracy (naplan) which examines students in years 3, 5, 7 and 9. we're seeing a great deal of stress, anxiety, and concern among kids who are being kept in at lunch, sitting practice tests on the weekends, and are under increasing pressure to perform because the teachers and schools have so much riding on the children's performance (o'keefe, 2011, p. 8). parents and society expect teachers to resist these negative influences upon their children’s education and to remain knowledgeable of recent philosophies, their pedagogies, and to integrate them into their classroom practice. is this not a task for a super hero? surely it would take someone with super human strength and patience to weather the demands of external testing and concentrate on assessing their students in ways that go beyond basic recall and memory to include diagnostic procedures, investigations, problem solving, creativity and the ability to generalize principles and apply them to novel problems. the darkness of ignorance a third darkness results from the combination of two influences that produce confusion, dissatisfaction and disempowerment. the first influence results from the speed of technological change which is underpinned by developments in mathematics. most members of the general population are either unaware or do not understand the mathematics that is used. how many know that there has been more mathematics invented in the last 50 years than in all the preceding years of human knowledge, or that a mathematical monster fractal curve forms the aerial in their mobile phone? before fractals, the geometries available were useful in the human built environment, where straight lines, right angles and circles were useful. it took fractals to be able to describe the seemingly chaotic world of nature. an iterative expression, using the power of the computer can produce a representation of a tree. hollywood hires mathematicians to produce their special effects in movies. the second influence results from the poor numeracy skills of the general population. skills learned in schools many years ago are forgotten and technological devices are relied upon to fill the void. in figure 1 above, the pill seller is trading upon the general lack of mathematical knowledge in the population to peddle his miracle drugs. careful readers will detect the pills do not improve spelling. 8 allan leslie white manifes of the algorith investig docume teachers super h mathem mathem applicat particip models what po their qu have to superpo integrat the resulti sts itself in a amount of hms when gations ena ents are ex s are requir ero? having bri matics clas matically lit tions of ma pate in polit . super her owers super all super h uest to overc o help them owers i wil tion of info ing confusi a number o f time stud it is clai abled by tec xpected to r red to prepa efly establi srooms wh terate adul athematics, tical discus roes need to r heroes use s eroes have come ignor m win their ll briefly de rmation co figure ion from t f ways. for dents spend imed that chnology ( reflect mod are their stu ished some here teach ts who sh be able to ssions that d o daily con in their bat super hero super powe ance and st battle are m escribe are: ommunicatio e1. mathema the impact r example, o d becoming more time (see wolfra dern develo udents for a of the dar hers toil t hould know decode pop draw upon nfront this c ttles. oes have su ers that othe tupidity. th many and i : mathemat on technol atics in a bo of these t one develop g highly p e should b am, 2010). opments (su a future wo rk forces at to meet t w examples pular texts statistics a challenge a uper power er humans d e super pow i will limit ical pedago logies (icts ottle two influen pment has b proficient w be devoted the mathe uch as frac orld. surely ttacking the the challen s of techno that contai and results and the nex rs do not have wers that ma t this discus ogical conte s) into the c nces upon een the que with compu d to mathe ematics cur ctal geomet this is a ta e quality of nge of pr ologically n mathema from mathe xt section co which help athematics ssion to thr ent knowled classroom; schools estioning utational ematical rriculum try) and ask for a f school roducing relevant atics and ematical onsiders them in teachers ree. the dge; the and, the 9 school mathematics teachers are super heroes use of diagnostic assessment to uncover rich data to direct the teaching and learning process in mathematics classrooms. pedagogical content knowledge teaching is a process of continual striving for excellence, a quest for the perfect lesson and an understanding that it can never be achieved. there is always something, upon reflection, that could be improved to meet the individual needs of the students. it is the combination of reflection, professional learning and experience that contributes to the gradual accumulation of pedagogical knowledge and super power. it is the teachers' mathematical pedagogical content knowledge, the special knowledge that teachers have, which gives them the power to construct a learning environment whereby their students develop conceptual knowledge and deep learning. teachers' mathematical pedagogical content knowledge is an area of considerable research. while it is beyond the scope of this paper to give this area the treatment it deserves, it is necessary to make some brief points. the initial work of shulman (1986) and colleagues proposed that a basis of mathematics teacher professional knowledge would contain: (i) mathematics content knowledge both substantive and syntactic; (ii) general pedagogical knowledge that included generic principles of classroom management; (iii) mathematics curriculum knowledge including materials and programmes; (iv) mathematical pedagogical content knowledge that included forms of representation, concepts, useful analogies, examples and demonstrations;(v) knowledge of learners; (vi) knowledge of educational contexts, communities and cultures; and (vii) knowledge of educational purposes. shulman's work stimulated the growth of further research and other frameworks, such as hill, ball and shilling’s (2008) model which focused on conceptualising the domain of effective teachers' unique knowledge of students' mathematical ideas and thinking (see figure 2). figure2. domain map for mathematical knowledge for teaching (from hill, ball, & schilling, 2008, p. 377). 10 allan leslie white as teachers gain experience and develop their pedagogical content knowledge they move beyond the traditional approaches that are greatly criticised by researchers. for example, clements (2004) is scathing in his argument: that traditional 'teacher-telling' approaches to teaching mathematics are so ingrained in the cultures of school mathematics programs that students and teachers alike believe, mistakenly, that such methods are maximally useful in assisting students to learn mathematics (p. 1). his research showed a prevalence and popularity for a 'teacher telling approach' that follows the lesson structure of: teacher review; teacher models examples; student seatwork where they practice similar examples to those modelled by the teacher. when he analysed the classroom discourse patterns, the data revealed that teachers asked questions that were of a low cognitive level and very few high level questions. in response many teachers develop their pedagogical content knowledge by adopting a mathematics curriculum that focuses on real-life problems that still exposes students to the abstract tools of mathematics, especially the manipulation of unknown quantities. they incorporate mathematical investigations as they are fundamental both to the study of mathematics itself and to an understanding of the ways in which mathematics can be used to extend knowledge and solve problems in many fields. teachers recognise there is a world of difference between teaching "pure" mathematics, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical or scientific formula models and clarifies a real-world situation. in this way super heroes reveal the power and majesty of mathematics as ways of making sense of the world. it is the teachers' deep passion for their discipline and appreciation and concern for their students that drives them to seek to improve their teaching strategies with the goal of improved student learning. integration of information communication technologies the second super power i wish to briefly mention which is allied to the first is the integration of icts into the classroom. an examination of the research literature, it could be argued, produces five broad categories or metaphors of teacher response (white, 2004). these metaphors describe how teachers tend to view icts as either: a demon; a servant; an idol; a partner; or, a liberator. in terms of this paper i would argue that teachers who may belong to the first category are casualties who need gentle care and encouragement, those in category two and three are developing their powers, and those in the last two are the powerful super heroes. ict as demon. the evidence for this approach is observable in those teachers who actively oppose and subvert any attempt to integrate ict into the curriculum. they are either afraid or unwilling to learn and so conduct a campaign of active or passive resistance. if 11 school mathematics teachers are super heroes compelled by the authorities, they will do the minimum and often the result leads to surface integration and sometimes to an inappropriate use of ict. some teachers’ resistance has resulted from the frustration of being sent to a professional development program on the use of a software package only to return to school where the software is not available. ict as servant. these teachers are accepting of icts but adopt a conservative position towards the technologies being used. the technology is new yet the pedagogy remains much the same as in the past. icts thus are a tool for enhancing students’ learning outcomes within the existing curriculum and using existing learning processes. ict as idol. this approach promotes ict as a tool for use across the curriculum where the emphasis is upon the development of ict-related skills, knowledge, processes and attitudes. it is more focused upon teaching about computers rather than with computers. there are many examples of professional development programs that give teachers intensive experience of software packages but fail to assist teachers with the teaching and learning implications. the difficult task of using the icts in the classroom is given very surface treatment. thus teachers then struggle to integrate what they have learnt. ict as partner. there are teachers who have seriously attempted integrating ict into their classrooms. these classrooms are where students are actively engaged in gathering data, aggregating their data with those gathered by other students, and making meaning of their results. here, icts are integral to the pedagogy that will change not only how students learn but what they learn. it means the use of icts in the teaching of mathematics moves beyond pointing to how icts can support, improve, and provide new ways of teaching to how icts change the way mathematics is expected to be performed. ict as liberator. this is a radical approach where integration is a component of the reforms that will alter the organisation and structure of schooling itself. there are over one hundred virtual schools already existing in the u.s.a. as evidence of this trend. the link between the first and second super power is obvious, with the second being a powerful addition to the first. the second power is important in allowing students to develop an understanding of the concepts before developing their procedural proficiency. diagnostic assessment procedures the earlier remarks concerning the usefulness of assessment to the teaching and learning process within the classroom are particularly true for diagnostic assessment. even external examinations may have some small value: while we generally accept the usefulness of diagnostic assessments, both internal and external to an individual class or school at all levels of schooling... when external assessments are conducted we seek to emphasise their diagnostic applications, even though many tests are of limited value, particularly at student level" (alegounarias, 2011, p. 10). 12 allan leslie white diagnostic tests give teachers the data to find targets to aim at with their super powers. while there is a wealth of mathematical diagnostic procedures available to the classroom teacher i will only briefly mention two. the first is an innovative online resource called smart (specific mathematics assessments that reveal thinking stacey et.al., 2009). based on the victorian mathematics developmental continuum (stacey et al., 2006), the site offers a set of online tests covering most topics commonly taught in victoria; years 7 to 9. teachers choose one of the available smart tests appropriate to their class. students are given a password so they can attempt the test in class or at home. responses are marked online, and teachers receive the patterns of results electronically analysed with diagnosis when requested online. this feedback includes a summary of the findings, along with information on the common misconceptions in the topic and relevant links to the syllabus (smart, 2010). the second concerns word problems (usually written textbook problems) and newman’s error analysis (nea) which focuses upon the mathematical and literacy aspects of problem solving. nea was originally designed to assist teachers diagnose the nature of the difficulties experienced by students working with mathematical word problems, but it has developed further super powers involving pedagogical and classroom problem solving strategies. nea provides teachers with a framework to determine where misunderstandings occur and where to target effective teaching strategies to overcome them. moreover, nea provides a nice link between literacy and numeracy. newman (1977, 1983) maintained that when a person attempted to answer a standard, written, mathematics word problem then that person had to be able to overcome a number of successive levels: level 1 reading (or decoding), 2 comprehension, 3 transformation, 4 process skills, and 5 encoding (see figure 3 for the interview prompts). along the way, it was always possible to make a careless error and there were students who gave incorrect answers because they were not motivated to answer to their level of ability. studies typically reported approximately 70 per cent of errors made by year 7 students (first year of secondary school) were at the comprehension and transformation levels. these researchers also found that reading errors accounted for less than 5 per cent of initial errors, and the same was true for process skills errors being mostly associated with standard numerical operations (ellerton & clements, 1996). 13 school mathematics teachers are super heroes figure 3. problem solving classroom poster (english & indonesian). unfortunately there is not space to include all the pedagogical strategies in this paper but they are available elsewhere (see white, 2009; 2011). having identified the need for super heroes and described some of their super powers with which they fight the forces of darkness and ignorance, it is now time to consider how to identify them. identifying super heroes in the movie world, most super heroes are mild mannered, ordinary looking, who do not stand out in a crowd. this is true in real life, as most mathematics teachers appear to be just ordinary humans. yet all super heroes have special costumes, places, attributes or devices which make them special. where will i find and how will i identify a super hero? one identifying trait is that mathematics teacher super heroes gather at professional conferences, meetings or associations. these mild mannered, ordinary looking super heroes will be found devoting their free time to professional teaching organisations, or collaborating in professional teacher learning activities. the southeast asian ministers of education organisation (seameo) regional centre for quality improvement of teachers and educational personnel (qitep) and the pppptk (p4tk) matematika yogyakarta indonesia are two organisations dedicated to producing super heroes. conclusion schools are collaborative enterprises and the quality of mathematics teaching and school performance depends upon whether the institutional systems support mathematics 14 allan leslie white teachers' efforts. mathematics teachers are key contributors to improving education and every effort should be made to bring teachers together to help each other become more effective professionals. thus the formation of regional centre for quality improvement of teachers and educational personnel in mathematics (qitep) is a wonderful initiative for the encouragement and development of super heroes. institutions like qitep are not to be found in other parts of the world (except perhaps recsam in malaysia, or nismed in the philippines). i conclude this paper with a story of a business man complaining about education to a mathematics teacher. he asked, "you're a teacher, so be honest. what do you make?" the teacher had a reputation for honesty replied, you want to know what i make? well, i make kids work harder than they ever thought they could. i make a small achievement feel like a medal of honour. i make kids sit through 40 minutes of class time when their parents can't make them sit for 5 minutes without television. you want to know what i make? i make kids wonder. i make them question. i make them apologize and mean it. i make them have respect and take responsibility for their actions. i teach them how to write and then i make them write. i make them read, and use their brains to reason. i make them see the wonder, the beauty, and the power of mathematics and use it to make sense of their world. i make my classroom a place where all my students feel safe. finally, i make them understand that if they use the gifts they were given, work hard, and follow their hearts, they can succeed in life. only a super hero could do all this and more. i wish to thank all the hard working mathematics teachers for the super human efforts they make for their students. in my eyes they are all super heroes. references alegounarias, t. 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(2010). conrad wolfram: teaching kids real math with computers. ted talks, nov., 15. retrieved 1 september 2011 from http://www.youtube.com/watch?v=60ovlfaupjg 17 southeast asian mathematics education journal volume 11, no 2 (2021) 67 predictors of employability of the bachelor of secondary education major in mathematics graduates 1alcher j. arpilleda, 2anthony mark joseph m. manile & 3pacita i. soringa 1,2st. paul university surigao, philippines 3st. joseph learning center, philippines 1alcher.arpilleda@spus.edu.ph 2amj062792@gmail.com 3pacitasoringa@yahoo.com abstract the goal of every program provided by a school or university is to make it more responsive to the needs of fast-changing employment demands. this goal also entails enhancing the students' skills to be prepared for their future jobs. this study identified the predictors of employability of the bachelor of secondary education major in mathematics graduates of st. paul university surigao academic year 2005-2015. descriptive survey research was employed using a researcher-made questionnaire encompassing the personal information, employment status, and exit skills with seven identified subparts. the questionnaire elicited employability through the skills of 81.82% of the participants. utilizing the employment status variables and the exit skills, linear regression analysis extracted the predictors of employability. the findings revealed that the exit skills of the graduates are the predictors of employability. thus, the school should create programs and activities which enhance the students' skills, particularly the predictors of employability identified in this study. keywords: employability, employment, exit skills, predictors introduction in the status-quo of education, graduates' employment is the basis of many universities or higher education institutions to evaluate on how their program performs in creating their students fruitful and productive in their field (smith et al., 2000 as cited by boholano, 2012). knight and yorke (2003) defined employability as an assemblage of achievements, skills, understanding, and personal attributes that will support graduates to attain employment and be successful in their selected profession. meanwhile, skills help one acquire employment, but achievements prove that one is able to implement his or her skills in practice (carr, 2016). cardona and andres (2014) argued that skills are able to organize part of the context of being employable. in most cases, students have enrolled in any higher institution to acquire new knowledge and skills applicable in the workplace. the responsibility of university or college in training students is not limited to imparting academic skills but also building their character as members of society's workforce. after finishing the degree in education, graduates should prepare themselves for the licensure examination for teachers (let) in most cases. however, other graduates frequently look for other jobs, domestic or international, to possess experience before the teaching career or before obtaining the let. the institution has no more control or contact with the graduates regarding their employment status except when it is established graduates’ social gatherings. mailto:kristiyajati@p4tkmatematika.org predictors of employability of the bachelor of secondary education major in mathematics graduates 68 tracer studies were conducted before by the college of teacher education faculty to determine the let performance and the employment rate of the graduates of education faculty. in may 2006, a study on let performance of spu surigao education graduates from 2000 to 2005 was performed. in august 2009, a study on the let performance and employment rate of education graduates from 2000 to 2008 of st. paul university surigao was also administered. however, the current tracer study was conducted to trace the predictors of employability of bachelor of secondary education major in mathematics graduates of st. paul university surigao a.y. 2005-2015. particularly, the objective is to determine the graduates' profiles, employment status, and exit skills. the findings of the present study provide as the basis of the researcher to enhance or update the curricula of the bachelor of secondary educationmathematics program to be responsive to the needs of fast-changing employment demands and also to improve the skills of the students for them to be prepared in their future jobs. this study is also deemed necessary to unleash the skills of mathematics teachers further to uphold enhancing the mathematical performance of the filipinos as the students lagged behind other countries in the trends in international mathematics and science study 2019 (timss), an international assessment for mathematics and science for grade 4. conceptual framework of the study the study was anchored on the concept of cardona and andres (2014) that despite the graduates' status, the prior challenge is not only that they are employed or employable but also that their employment best utilizes their education and that their skills generate part the context of being employable. they also explained that the skills of the graduates are able to build part of the context of being employable. then, the researchers perceived the program outcomes of bachelor of secondary education as the basis for the skills of the education graduates. cardona and andres (2014) stated that leadership, communication, information technology, problem-solving, critical thinking, human relations, creativity, decision-making, technical, and research are able to establish part of the context of being employable. furthermore, the program outcomes (po) of bachelor of secondary education, as stated in the university handbook, also discusses about skills like communication, numeracy, critical thinking, leadership through facilitating learning, problem-solving, planning and decision making, creativity, innovativeness, human relations through instilling filipino cultural values, and research skills. based on the two sources, the researchers considered the following skills of graduates: communication. it refers to the skills of the respondents in dealing with and communicating with the people around them. it associates writing, speaking, and listening. as explained in the college of teacher education program outcome 1 (ctepo1), graduates are equipped with a basic and higher level of literacy, communication, and learning skills required for higher learning. leadership. it refers to the respondents’ skills in managing and supervising a team and cooperating in every task that a group should perform. as stated in ctepo2, leadership skill is employed in demonstrating and facilitating the learning processes in their students. alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 69 decision making. it refers to the respondents’ skills in planning a certain activity and making a decision towards the implementation of the plan. as to the context of teachers, as specified in ctepo4, graduates demonstrate the acquired teaching process skills, encompassing curriculum development, lesson planning, materials development, educational assessment, and teaching approaches. these teaching process skills involve decision-making. problem solving. it refers to the respondents’ skills in handling the problems which occurs along his/her working journey. as presented in ctepo1, graduates are equipped with a basic and higher level of critical thinking required for higher learning. problem-solving skill is also essential in lesson planning, educational assessment, as stipulated in ctepo4. creativity. it refers to the respondents’ skills in creating something new beneficial for the betterment of himself/herself, his/her workplace, and the entire community. it is declared in ctepo9 that graduates are able to create, innovate, simulate, and evaluate in enhancing student learning. human relations. it refers to the respondents’ skills in associating other people and interacting with them in times of work. it is explained in ctepo11 that graduates act upon filipino cultural values in the exemplification of the values of discipline, honesty, social responsibility, thrift, hard work, compassion, availability, accountability, teamwork, and respect for elders, wise conservation, and utilization of natural resources, with christ as a model. research. it refers to the respondents’ skills in conducting an investigation or research study which helps improve the workplace or the employment status of the respondent. it is also stated in ctepo12 that graduates should conduct research and utilize results in response to the demands of 21st-century teaching and learning. methods research design the study employed a quantitative descriptive design using a survey technique to allow the researcher to obtain quantifiable information on the predictors of employability of the bsedmathematics graduates (2005-2015) of st. paul university surigao. the design was considered appropriate because it is probably the best method available to collect data from tracing the employment status and exit skills that could predict the employability of the spu surigao bsed mathematics graduates. participants the participants of this study were the graduates of bachelor of secondary education major in mathematics of st. paul university surigao from 2005 to 2015. the researchers considered the year 2005 as the start of the scope of participants because, in that year, the first batch graduated from st. paul university surigao as it transitioned from san nicolas college. instrument the main instrument administered to solicit information was a researcher-designed questionnaire, a self-assessment survey wherein the participants provided personal and predictors of employability of the bachelor of secondary education major in mathematics graduates 70 professional information and rate themselves in terms of their skills. the tool is appropriate as the participants understood very well of their performance regarding the skills encompassing in the study. the questionnaire consisted of three (3) parts. part 1 is the profile of the respondents comprising of the name, age, sex, year graduated, let performance, current and home address, contact number, and e-mail address. part 2 of the questionnaire is on the respondents' employment status, encompassing present work, employment type, employer type, company/institution/school name, and workplace. part 3 of the questionnaire is on the skills of the graduates, incorporating communication, leadership, decision making, problem-solving, creativity, human relations, and research. the researchers were able to formulate the indicators for each skill through the concepts provided by cardona & andres (2014), program outcomes of the bachelor of secondary education, and the work ethics presented in the university student handbook, 2015 edition. three experts validated the questionnaire for content validity. data gathering procedure the researchers provide a letter of request to the dean for approval to conduct the study. upon approval, the researchers administered the questionnaire to the respondents. considering that the respondents were not living in the same place, the researchers sent the questionnaires through social media (facebook, yahoo, or gmail). after the participants answered the questionnaire, they sent it back to the researchers. the data were collected, tallied, analyzed, and interpreted the result. data analysis in tracing the predictors of employability of the bsed mathematics graduates of st. paul university surigao, the researchers employed several statistical tools to examine the data of the study. the first one is frequency count and percentage distribution, which were utilized to describe the participants' profile and employment status. we also administered descriptive statistics in the form of mean and standard deviation, to determine the skills of the graduates. the following were the basis for interpretation of data: table 1 basis for data interpretation scale range verbal interpretation qualitative description 4 3.25 – 4.00 strongly agree very skillful 3 2.50 – 3.24 agree skillful 2 1.75 – 2.49 disagree less skillful 1 1.00 – 1.74 strongly disagree least skillful then, we employed pearson product moment correlation of coefficient, to calculate the significance of the relationship between the exit skills and each variable of the participants’ profile; age and let performance scores, as well as the significance of the relationship between the graduates’ skills and their employment status as to the work, employment type, and employer type. the variables were coded into numerical data. for determining the alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 71 significance of the relationship between exit skills and the variable sex, the researcher employed point-biserial coefficient of correlation. last but not least, multiple linear regression analysis was administered to determine the graduates' exit skills predictors in which the relationship between one dependent variable and two or more independent variables was examined. it generates important predictors between one or more independent variables and the dependent variable (frost, 2013). the predictors were the exit skills of the graduates, considered as independent variables, while the employment characteristics are the dependent variables. further, the forward stepwise method was used because it selects a subset of predictor variables for the final model. it is best implemented to obtain the best predictor and apply all possible algorithms to the variables in a provided subset. it is tractable, and it provides a good sequence of a model (wuencsch, 2014). results and discussion profile of the bsed math graduates table 2 displays the profile of the bachelor of secondary education major in mathematics graduates in terms of age, sex, and let performance. table 2 profile of the bsed math graduates categories f (27) % age 20 -24 years old 6 22.22 25-28 years old 11 40.74 29-32 years old 8 29.63 33-36 years old 2 7.41 sex male 9 33.33 female 18 66.67 let performance passed 21 77.78 failed 1 3.70 not yet enrolled 5 18.52 as to age, of the 27 participants, 6 or 22.22% are 20-24 years old; 11 or 40.74% are 25-28 years old; 8 or 29.63% are 29-32 years old, and 2 or 7.41% are 33-36 years old. it means that most of the graduates answering the survey questionnaire were 25 to 28 years old. based on the data collected, the age with the highest number of graduates is 32 years old and was all from batch 2005, considering that 2005 owns the highest number of graduates. as presented in the table, the age bracket with the highest frequency is 25-28 years old, and this bracket is composed of the graduates of batches 2009-2015. among these batches, batch 2012 possesses the highest frequency with three graduates. predictors of employability of the bachelor of secondary education major in mathematics graduates 72 regarding sex, of the 27 participants, 9 or 33.33% are males, and 18 or 66.67% are females. in this study, more female participants graduated and answered the survey questionnaire than males. based on the data obtained, batch 2005 owns the highest number of female graduates with five graduates. regarding the let performance, of the 27 participants, 21 or 77.78% passed the licensure examination for teachers; 1 or 3.70% failed, and 5 or 18.52% of the participants have not yet enrolled the let. based on the data collected, one failed in the licensure examination for teachers and was from batch 2009, while those who have not taken the let yet were one from batch 2015, 2011, 2013, and two from batch 2005. other graduates who were part of the study already passed the let. as the data display, batch 2005 has the highest frequency in every profile variable, and it is true because it has the highest number of graduates with six participants. employment status of the graduates table 3 presents the employment status of the graduates as to the present work, employment type, and employer type. table 3 employment status of the graduates f (27) % work teacher 22 81.48 tele sales adviser 1 3.70 administrative assistant 1 3.70 office secretary 1 3.70 domestic helper 1 3.70 army captain 1 3.70 employment type permanent 17 62.96 casual 2 7.41 temporary 6 22.22 probationary 2 7.41 employer type public 16 59.26 private 11 40.74 in terms of present work, of the 27 participants, 22 or 81.48% are teachers; 1 or 3.70% is a tele sales adviser, administrative assistant, office secretary, domestic helper, and army captain. most of the participants are teachers. most of the participants possess their work which matches the undergraduate course they have enrolled. from the data collected, most of the participants are recently teaching with batch 2005 as the batch with the highest frequency with four graduates, while the rest of the participants possess different works and were from batches 2005, 2011, 2013, and 2015. as to the employment type, of the 27 participants, 17 or 62.96% are permanent; 2 or 7.41% are casual and probationary; 6 or 22.22% are temporary. it implies that most of the participants alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 73 are permanent in their employment. based on the data collected, batch 2005 owns the highest number of permanently employed graduates with five graduates. regarding the employer type, of the 27 participants, 16 or 59.26% are working with a public employer, while 11 or 40.74% are working with a private employer. it indicates that most of the participants are employed in a public institution or company. based on the data, batch 2005 owns the highest number of graduates employed in a public institution with five graduates. as the data present, batch 2005 has the highest frequency in every variable under the employment status, and it is true because 2005 possesses the highest number of graduates with six participants. exit skills of graduates table 4 displays the exit skills of the graduates as to communication skills, leadership skills, decision making, problem-solving, creativity, human relations, and research. table 4 exit skills of the graduates skills m sd qd 1. communication skills  i know how to write communication letters (memorandum, business letter, etc.) 3.31 0.55 vs  when i talk with someone in the workplace, i use english or filipino. 3.08 0.69 s  when someone is talking, i listen attentively to what he/she is saying to hear the point of the message. 3.73 0.45 vs  when i speak or listen, i look into the eyes of the person i am conversing with to produce the interaction more successful. 3.65 0.49 vs  i attempt to visualize what other people say so that i know how to respond. 3.65 0.49 vs mean: 3.48 0.53 vs 2. leadership skills  i empower people by sharing leadership responsibilities. 3.50 0.51 vs  i inspire people to achieve set goals. 3.50 0.51 vs  i build and maintain collaborative partnerships for the improvement of my workplace and the community. 3.69 0.47 vs  i challenge people to take risks with tranquil daring. 3.23 0.65 s  i manage my works, operations, and resources for the effectiveness and efficiency. 3.69 0.47 vs mean: 3.52 0.52 vs 3. decision making  i implement a well-defined process to structure my decisions. 3.42 0.58 vs  i take the time required to select the best decision-making tool for each specific decision. 3.35 0.63 vs  i consider a variety of potential solutions before i make my decision. 3.54 0.51 vs  i observe various planning practices to make sound decisions. 3.58 0.50 vs  i determine the most important factors to the decision and use such to evaluate my choices. 3.62 0.50 vs mean: 3.50 0.54 vs 4. problem solving  i take time to investigate how things are working. 3.81 0.40 vs predictors of employability of the bachelor of secondary education major in mathematics graduates 74  i am flexible in which i can easily adapt and adjust when a problem arises and stays focus to think well of the solution. 3.65 0.56 vs  i maintain an attitude of openness to ideas that could help solve the problem. 3.85 0.37 vs  i attempt to look at problems from different perspectives and generate multiple solutions. 3.73 0.45 vs  i handle problems with utmost prudence. 3.46 0.71 vs mean: 3.70 0.50 vs 5. creativity  i provide new ideas or concepts and go beyond conditioned setups. 3.42 0.50 vs  i enable self-discovery. 3.50 0.51 vs  i optimize and share resources. 3.54 0.51 vs  i create and innovate alternative strategies for the improvement of the institution. 3.42 0.58 vs  i consult others to enhance knowledge and wisdom to be able to create new things. 3.65 0.49 vs mean: 3.51 0.52 vs 6. human relations  i share a healthy rapport with the people around me. 3.85 0.37 vs  i respect the social decorum of the group. 3.85 0.37 vs  i treat and value individuals with dignity as children of god. 3.88 0.33 vs  i feel comfortable with different kinds of people in my workplace. 3.54 0.65 vs  i promote positive relationships with compassion and tenderness. 3.77 0.43 vs mean: 3.78 0.43 vs 7. research  i continue my learning by enduring with the new research and studies that could help enhance my workplace. 3.46 0.51 vs  i continue to pursue information to obtain a broad perspective. 3.58 0.50 vs  i conduct research and utilize the results in response to the demands of society. 3.12 0.77 s  i learn how to process events and experiences as means to perfection. 3.31 0.55 vs  i critically evaluate data to produce the information needed. 3.35 0.63 vs mean: 3.36 0.59 vs grand mean 3.55 0.52 vs legend scale range verbal interpretation (vi) qualitative description (qd) 4 3.25 – 4.00 strongly agree very skillful (vs) 3 2.50 – 3.24 agree skillful (s) 2 1.75 – 2.49 disagree less skillful (ls) 1 1.00 – 1.74 strongly disagree least skillful (ns) as to communication skills, among the five indicators, the item when someone is talking, i listen attentively to what he/she is saying to clearly hear the point of the message (m=3.73, sd=0.45) owns the highest mean and is illustrated as very skillful, while the item with the lowest mean is when i talk with someone in the workplace, i use english or filipino (m=3.08, sd=0.69) which is characterized as skillful. on average, communication skills (m=3.48, sd=0.53) are described as very skillful. it indicates that the graduates are well-trained in communication skills except employing english or filipino in the workplace. as to leadership skills, among the five indicators, the items with the highest means are i build and maintain collaborative partnerships for the improvement of my workplace and the community; i manage my works, operations, and resources for the effectiveness and efficiency (m=3.69, sd=0.47) and are characterized as very skillful, while the item i challenge people to take risks with tranquil daring obtained the lowest mean (m=3.23, sd=0.65), which is alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 75 construed as skillful. on average, leadership skill (m=3.52, sd=0.52) is portrayed as very skillful. it indicates that the leadership skills of the graduates are honed well by the school and are improved in their workplace aside from challenging people to take risks. as to decision making, among the five indicators, the item with the highest mean is i determine the factors most important to the decision, and use such to evaluate my choices (m=3.62, sd=0.50) and is characterized as very skillful, while the item that obtained the lowest mean is i take the time required to select the best decision-making tool for each specific decision (m=3.35, sd=0.63) which is construed as very skillful. on average, decision-making (m=3.50, sd=0.54) is described as very skillful. it manifests that the activities or programs provided to the graduates enhanced their decision-making skills. as to problem-solving, among the five indicators, the item with the highest mean is i maintain an attitude of openness to ideas which could help solve the problem (m=3.85, sd=0.37), and is characterized as very skillful. in contrast, the item i handle problems by utmost prudence received the lowest mean (m=3.46, sd=0.71), construed as very skillful. on average, problem-solving (m=3.70, sd=0.50) is described as very skillful. it implies that their problem-solving skills improved with the different activities in school and their workplace. as to creativity, among the five indicators, the item with the highest mean is i consult others to enhance knowledge and wisdom to be able to create new things (m=3.65, sd=0.49), and is illustrated as very skillful, while the items with the lowest mean are i provide new ideas or concepts and go beyond conditioned setups; i create and innovate alternative strategies for the improvement of the institution (m=3.42, sd=0.50), which are portrayed as very skillful. on average, creativity (m=3.51, sd=0.52) is described as very skillful. it indicates that their work and activities conducted provided them the opportunity to express their creativity. as to human relations, among the five indicators, the item with the highest mean is i treat and value individuals with dignity as children of god (m=3.88, sd=0.33), which is portrayed as very skillful. in contrast, the item i feel comfortable with different kinds of people in my workplace obtained the lowest mean (m=3.54, sd=0.65), which is likewise characterized as very skillful. on average, human relations (m=3.78, sd=0.43) is illustrated as very skillful. it indicates that the school has trained them on the proper way of relating to others, which is true as it is reflected on the school's core values and where they practiced those values. as to research, among the five indicators, the item with the highest mean is i continue to pursue information to gain broad perspective (m=3.58, sd=0.50) and is characterized as very skillful, while the item with the lowest mean is i conduct research and utilize the results in response to the demands of society (m=3.12, sd=0.77), and is portrayed as skillful. on average, research (m=3.36, sd=0.59) is described as very skillful. it implies that their skills in the research were performed when they were in their undergraduate studies. still, in their workplace, research is not that required. it is reflected on the result of the study, particularly on the item i conduct research and utilize the results in response to the demands of society. on average, the exit skills of the graduates (m=3.55, sd=0.52) are portrayed as very skillful. although there are three indicators described as skillful, it did not affect the average of each skill and even the average of all the skills. thus, it implies that the school has provided predictors of employability of the bachelor of secondary education major in mathematics graduates 76 activities that enhanced their skills, and they were trained to be prepared in their careers when they graduate. relationship between the profile of the participants and the exit skills table 5 presents the significance of the relationship between the profile and the exit skills of the participants. as to the hypothesis explaining that there is no significant relationship between the profile of the participants and their exit skills, findings revealed that the age and let performance of the participants have no bearing with their exit skills. however, it is also presented that the sex of the participants is significantly associated with the decision making (r=-0.45 and p-value =0.0196). the finding displays that there is a moderately small correlation between sex and decision-making. other skills own a very small correlation with the profile of the participants, as displayed in the statistical computations. table 5 relationship measure between the profile of the participants and their exit skills profile exit skills r p-value age communication skills -0.25 0.2141 leadership skills -0.15 0.4706 decision making -0.38 0.0526 problem solving -0.25 0.2139 creativity -0.30 0.1309 human relations -0.17 0.4111 research -0.29 0.1442 sex communication skills -0.10 0.6241 leadership skills -0.02 0.9199 decision making -0.45 0.0196 problem solving -0.38 0.0578 creativity -0.12 0.5577 human relations -0.20 0.3390 research -0.30 0.1423 let performance communication skills 0.09 0.6537 leadership skills 0.30 0.1327 decision making -0.03 0.8739 problem solving 0.25 0.2131 creativity 0.02 0.9047 human relations 0.12 0.5465 research 0.14 0.4808 based on the statistical computations, the communication skills, leadership skills, decision making, problem solving, creativity, human relations, and research skills are not according to their age (r = -0.25, -0.15, -0.38, -0.25, -0.30, -0.17, and -0.29; and p-values= 0.2141, 0.4706, 0.0526, 0.2139, 0.1309, 0.4111, and 0.1442, respectively), thus, the null hypothesis is not rejected. it indicates that the skills of a graduate do not matter as to his or her age. from the same table, the sex of the participants possesses a significant relationship to decision making (r= -0.45 and p-value= 0.0196), which leads to the rejection of the null hypothesis. it implies that the sex of the participants influences his/her decision-making skills. alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 77 this finding is approved by the study of desvaux et al. (2010) in which they discovered that one is better in decision-making than the other, and men make decisions better with executions than women. however, the sex of the participants does not bear any relationship with the other exit skills as to communication skills, leadership skills, decision making, problem-solving, creativity, human relations, and research skills (r= -0.10, -0.02, -0.38, -0.12, -0.20, and -0.30; p-values= 0.6241, 0.9199, 0.0578, 0.5577, 0.3390, 0.1423, respectively); thus, the null hypothesis is not rejected among these skills. it implies that the sex of the participants has no influence on the exit skills except on decision making. table 5 also presents that the communication skills, leadership skills, decision making, problem-solving, creativity, human relations, and research skills of the participants have no significant relationship to the let performance of the graduates (r= 0.09, 0.30, -0.03, 0.25, 0.02, 0.12, and 0.14; and p-values= 0.6537, 0.1327, 0.8739, 0.2131, 0.9047, 0.5465, 0.4808, respectively); thus, there is no significant relationship between the profile of the graduates and their exit skills. it reveals that the let performance of the graduates has no bearing on their exit skills. it is true because let measures the graduates' knowledge of the courses they have enrolled, which excludes the skills mentioned in this study. in other words, decision-making is the only indicator that bears a relationship with the profile of the graduates, particularly to the sex of the participants. relationship between the employment status and exit skills of the graduates table 6 displays the significance of the relationship between the employment status and the exit skills of the graduates. table 6 relationship measure between the employment status and exit skills of the graduates employment characteristics exit skills r p decision work communication skills -0.04 0.8296 do not reject ho2 leadership skills 0.19 0.0344 reject ho2 decision making 0.02 0.9096 do not reject ho2 problem solving 0.21 0.0298 reject ho2 creativity -0.15 0.0444 reject ho2 human relations 0.05 0.7929 do not reject ho2 research -0.04 0.8478 do not reject ho2 employment type communication skills -0.08 0.6911 do not reject ho2 leadership skills -0.07 0.7416 do not reject ho2 decision making 0.28 0.0152 reject ho2 problem solving 0.19 0.3389 do not reject ho2 creativity 0.07 0.7412 do not reject ho2 human relations 0.03 0.8745 do not reject ho2 research 0.19 0.3342 do not reject ho2 employer type communication skills 0.40 0.0401 reject ho2 leadership skills 0.20 0.3244 do not reject ho2 decision making 0.30 0.0278 reject ho2 problem solving 0.39 0.1438 do not reject ho2 predictors of employability of the bachelor of secondary education major in mathematics graduates 78 creativity 0.32 0.1011 do not reject ho2 human relations 0.27 0.1816 do not reject ho2 research 0.26 0.1926 do not reject ho2 as to the hypothesis which asserts that there is no relationship between the employment status and the exit skills of the graduates, findings unveil that the communication skills, decision making, human relations, and research skills of the participants are not significantly associated with work (r= -0.04, 0.02, 0.05, and -0.04; and p-values= 0.8296, 0.9096, 0.7929, and 0.8478, respectively). these statistical computations produce the acceptance of the null hypothesis of no significant relationship between the employment status and the exits skills of the graduates. however, leadership skills, problem-solving, and creativity possess a significant relationship to the employment status of the graduates (r= 0.19, 0.21, and -0.15; and p-values= 0.0344, 0.0298, and 0.0444, respectively); thus, the null hypothesis is rejected between the exit skills and employment status of graduates. the result is true because leadership skills are essential if somebody is willing to be productive and effective in the work (mcgurgan, 2021); the ability to solve problems is critical in any work; creative thinkers can also create new views for the company. from the same table, it is presented that the type of employment has no significant relationship to the communication skills, leadership skills, problem-solving, creativity, human relations, and research skills of the participants (r= -0.08, -0.07, 0.19, 0.07, 0.03, and 0.199; and p-values= 0.6911, 0.7416, 0.3389, 0.7412, 0.8745, and 0.3342, respectively); thus, the null hypothesis is not rejected between the employment status and the exits skills of the graduates. however, decision-making is significantly associated with the type of employment (the computed r is 0.28 and the p-value is 0.0152); thus, the null hypothesis is rejected between the employment status and the exits skills of the graduates. it implies that decision-making skill depends on the type of employment of the graduate. it is true because decision making skill is beneficial to employers as they save time and money. employees with this skill are mostly preferred by the employer and are provided high positions. table 6 also displays that communication and decision-making skills possess a significant relationship to the employer type (r= 0.40 and 0.30; and p-values= 0.0401 and 0.0278, respectively), leading to the rejection of the null hypothesis between the employment status and the exits skills of the graduates. it indicates that the communication and decision-making skills of the graduates have a bearing on their employer type. the result is true because decisionmaking skills are beneficial to employers as they save time and money; employers look into how people work and communicate with others. however, leadership skills, problem solving, creativity, human relations, and research skills of the participants have no significant relationship to employer type (r= 0.20, 0.39, 0.32, 0.27, and 0.26; and p-values= 0.3244, 0.1438, 0.1011, 0.1816, and 0.1926, respectively). the statistical computations lead to the acceptance of the null hypothesis between the employment status and the exits skills of the graduates. it signifies that exit skills, except communication and decision making, have no bearing on the employer type. based on the statistical computations, it unveiled that leadership skills, problem-solving, and creativity possess a significant relationship to work; decision making is significantly alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 79 associated with the type of employment; communication skills and decision making have a significant relationship to the employer type. exit skills as predictors for employability of graduates table 7 presents the regression model for the predictors of employability. the exit skills were implemented as predictors of the employability of the graduates. the researchers employed multiple linear regression analysis to identify the predictors for the graduates' employment. table 7 regression model for the predictors of employability dependent variable predictors beta st. err. of beta β st. err. of β p-level adjust ed r² work constant -0.32 2.50 0.8983 0.1519 problem solving 0.61 0.25 2.25 0.93 0.0226 creativity -0.58 0.25 -1.84 0.80 0.0298 leadership skills 0.24 0.23 0.88 0.86 0.3156 employment type constant -0.62 1.61 0.7042 0.0402 decision making 0.28 0.19 0.67 0.46 0.1520 employer type constant -1.31 1.07 0.2323 0.1484 communication skills 0.36 0.18 0.51 0.26 0.0639 decision making 0.24 0.18 0.27 0.21 0.2033 the statistical computations display that problem solving, creativity, and leadership skills are predictors for work (adjusted r2=0.1519 and p-values=0.0226, 0.0298, and 0.3156, respectively). the three predictors were able to account for 15.19% of the variance in the employment status of the graduates in terms of work. these three skills are considered predictors because the adjusted r2 produced by the statistical tool is an indicator of whether the variable/s (exit skills) is/are to be considered as predictors or not. it also indicates that among the seven exit skills, problem-solving, creativity, and leadership skills possess the highest adjusted r2, which is the basis for a variable to be a predictor. the computation of the predictor for work supports table 5 on the significant relationship of leadership skills, problem-solving, and creativity to work. there is a significant relationship between these three skills and work, and these skills are also as predictors for work. as displayed in table 7, the problem-solving and leadership skills owned considerable positive regression weights, which indicates that participants with higher scores on these skills were expected to have higher employability after controlling for the other variables in the model. the creativity skill possesses a significant negative weight (opposite in sign from its correlation with the criterion), which implies that those participants with higher creativity scores were expected to have lower employability (a suppressor effect). leadership skill is considered as one of the predictors for work because it is essential to be productive and effective. being knowledgeable about the work is not enough, but one must also share knowledge with the people in the workplace (leviticus, 2016). predictors of employability of the bachelor of secondary education major in mathematics graduates 80 rihal (2017) argued that effective leadership is established upon a solid foundation with a clear mission, a vision for the future, a specific strategy, and a culture conducive to success. hence, it is a crucial skill required in the workplace. moreover, leadership was considered a basis element for a well-coordinated and integrated provision of care, both from the patients and healthcare professionals (sfantou et al., 2017). one of the first research to empirically rank-order skill demand is the study of rios et al. (2020). they discovered that in high demand skills by the employers include oral and written communication, collaboration, and problem-solving skills, with particular emphasis on the pairing of oral and written communication. their study emphasized that problem-solving skill is one of in-high demand skills required by employers in which it is considered a predictor in the present study. soft skills like creativity, empathy, judgment, and the ability to motivate others, as chui, manyika and miremadi (2015) discovered, remain unique to humans. creative thinkers tend to create new ways of conducting things as they are innovative. these are added value to the work environment, making systems and procedures more efficient and effective. one will land a work that requires creativity if he/she is creative. it is the reason why creativity is a predictor for work. it is also presented in table 6 that decision-making is a predictor for the employment type (adjusted r2=0.0402 and p-value=0.1520). it implies that decision-making is one of the skills that employers consider as the worker's employment type. the decision-making skill possessed significant positive regression weight, which indicates that participants with higher scores on this skill were expected to own higher employability after controlling for the other variables in the model. the result for the predictors of the employment type supports table 6 on the significant relationship of decision making to the type of employment. there is a significant relationship between decision-making and employment type, and this skill is also a predictor for the employment type. table 7 also displays that communication skills and decision-making are the predictors for the variable employer type (adjusted r2=0.1484, p-values=0.0639, and 0.2033, respectively.) the communication and decision-making skills owned significant positive regression weights, that indicates that participants with higher scores on these skills were expected to have higher employability after controlling for the other variables in the model. findings on the predictor for the type of employer support the result in table 6 on the significant relationship of communication skills and decision making to the employer type. there is a significant relationship between communication skills and decision making and the type of employer, and these skills are also predictors for the employer type. the result corroborates with the findings of suarta et al. (2017), in which they discovered that communication, problem-solving and decision-making, and teamwork skills are employability skills with the highest importance level. decision-making is considered a predictor for employment and employer type because it is crucial in the workplace and for employers to save time and money. it also requires the recording of information that can be referred to when planning future projects. good communication skills help ensure that staff members understand the instructions and expectations. hold regular meetings with the staff to discuss progress on projects, alcher j. arpilleda, anthony mark joseph m. manile & pacita i. soringa 81 achievements, and departmental, and company news (leviticus, 2016). it is the reason why communication skill is a predictor for the employer type. conclusion based on the findings revealed in this study, the researcher concluded that most of the bachelor of secondary education major in mathematics graduates from 2005-2015 are females and let passers and teachers in a public institution. the exit skills of the graduates are characterized as very skillful, which indicates that they were able to effectively employ the skills acquired from the school to their workplace. decision-making is the only skill that bears a relationship with the profile of the graduates, particularly sex. leadership skills, problemsolving, and creativity possess a significant relationship to work. decision-making is significantly associated with the type of employment. communication skills and decisionmaking possess a significant relationship to the employer type. the skills such as problemsolving, creativity, leadership skills, communication, and decision-making are predictors of the employability of the graduates. on the other hand, human relations and research are nonpredictors of employability. therefore, the exit skills of the graduates are the predictors of employability. based on the conclusions, the researchers recommend that the school creates programs and activities to enhance students' skills, particularly problem-solving, creativity, leadership skills, communication, and decision-making, which are considered predictors of employability. they may focus on the non-predictors of employability, specifically human relations, and research, because these skills are still required in the workplace, particularly on the part of the teachers. since the study focuses merely on the exit skills as predictors of employability of the bachelor of secondary education major in mathematics graduates, it is recommended to the future researchers to expand this study by encompassing (1) exit skills as predictors of employability of the teacher education graduates; and employment status and let performance of the education graduates. acknowledgments we wish to convey our heartfelt gratitude to the graduates of bachelor of secondary education major in mathematics of st. paul university surigao, batches 2005-2015 for their participation in the conduct of the study. we would also like to thank st. paul university surigao for allowing us to conduct this study, along with the administration, faculty and staff. references boholano, h. b. 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(2021) discovered that the majority of their subjects' students had moderate or low critical thinking skills. this result is consistent with that revealed by wayudi et al. (2020). apart from these two studies, several studies also have demonstrated the need to enhance student’s critical thinking skills (agnafia, 2019; hidayat et al., 2019; hidayati et al., 2021; li et al., 2021; ridho et al., 2020). similar to the problem of critical thinking skills, many students still have low creative thinking skills. a study conducted by rasnawati et al. (2019) unveiled that the vocational high school students who were their subjects had low creative thinking skills. rachman and amelia (2020) also found similar results, specifically, the creative thinking skills of high school students who were their subjects were lacking. several other studies corroborate the findings of these studies. (kadir et al., 2022; siregar, 2019; suparman & zanthy, 2019). mailto:yosepdwikristanto@usd.ac.id integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 2 the existence of problems related to students’ critical and creative thinking skills indicates the need for learning innovation. vincent-lancrin et al. (2019) provide learning design principles to develop students’ critical thinking skills and creativity. to begin, such learning must pique students' interest and be challenging. the learning should also help students develop technical skills and enable them to create actual products or artifacts. furthermore, the learning environment must allow students to co-design components of a product or solution. it implies that learning must be open to a wide range of student's interests, ideas, and abilities, as well as provide space for student agency. the principle of respect for diverse perspectives in dealing with problems is as follows. furthermore, learning also needs to provide space for the unexpected. finally, the learning also needs to provide space and time for students to reflect as well as to give and receive feedback. giving and receiving feedback not only encourage students to improve their work but also facilitate them to learn (kristanto, 2018). one learning approach that follows these principles is steam education. steam education is a learning approach that integrates science, technology, engineering, arts, and mathematics. steam education makes students more appreciative of various fields of knowledge simultaneously. it sparks the development of their critical and creative thinking skills in re-imagining new and old real-world problems (b. wilson & hawkins, 2019). the steam approach is innovative because it is considered up-to-date in the industry 4.0 era, which can support critical and creative thinking skills through project-based learning (lu et al., 2022; shatunova et al., 2019). this project-based steam learning is based on real-world problems and can teach students how to research, propose, and select solutions, as well as design and create products (chistyakov et al., 2023; diego-mantecon et al., 2021). generally, implementing the steam approach administers an engineering design process (edp) (ozkan & umdu topsakal, 2021). although a variety of edp cycles is found in the literature (haik et al., 2017, p. 9; hubka, 2015, p. 31), these cycles typically include problem clarification, program assembly for needs, design planning, prototype construction, testing, and optimization, product analysis, and product presentations to clients or target groups (vossen et al., 2020). these stages can be simplified into five: asking, imagining, planning, creating, and improving (hester & cunningham, 2007). the edp can bridge science and mathematics concepts in making or using technology while also considering aesthetics in the steam approach. according to the literature, the steam approach has the potential to develop or improve students' critical and creative thinking skills. this approach can provide students with the opportunity to create products that will help them develop their creativity and problem-solving skills (katz-buonincontro, 2018). the implementation of steam teaching and learning by wilson et al. (2021) for elementary and middle school students illustrated that this approach effectively increased critical and creative thinking skills. furthermore, numerous other studies have discovered similar results, which indicate the steam approach can help students develop critical and creative thinking skills (alkhabra et al., 2023; anggraeni & suratno, 2021; engelman et al., 2017; priantari et al., 2020; rahmawati et al., 2019). problem-solving is a central activity in steam education. the problem-solving activities can be supported by learning designs that support the development of computational thinking (ct) dimensions (barr & stephenson, 2011; wu & su, 2021). decomposition, pattern recognition, abstraction, and algorithm are the ct dimensions. (google, 2023). decomposition epiphany princess mariana, yosep dwi kristanto 3 is the process of breaking down a complex problem into smaller problems in order to make the problem easier to understand, handle, or manage. the search for similarities between different problems is referred to as pattern recognition. focusing on important information while ignoring irrelevant details is what abstraction entails. the final dimension, algorithm, refers to the process of creating steps or rules to solve problems. the four ct dimensions can be embedded in steam learning activities (barr & stephenson, 2011). ct support in teaching and learning is often carried out using computers, especially programming. it is because programming includes making computer-readable instructions so that the computer can complete specific tasks or problems (wang et al., 2022). it is in line with one of the dimensions of ct, namely the algorithm. programming is also essential to support critical tasks related to ct (grover & pea, 2013). the programming activities are also often integrated into steam education, such as using scratch (oh et al., 2013; tan et al., 2020) and lego mindstorm (ding et al., 2019; ruiz et al., 2019). ct support in teaching and learning can also be implemented without the use of a computer. this strategy is appropriate for implementation in schools that lack technological infrastructure (brackmann et al., 2017). thus, integrating ct and steam education has a greater potential to be widely implemented. furthermore, this integration in learning that does not use computers or other expensive technology makes it easier for teachers or other practitioners to adopt or adapt it (padmi et al., 2022). in summary, on the one hand, critical and creative thinking skills are two essential skills for students to live and contribute productively in the 21st century. on the other hand, many students still lack these two skills. steam education that supports the development of ct, which hereinafter we refer to as steam-ct, can potentially develop students’ critical and creative thinking skills. such teaching and learning can be implemented without a computer so that learning activities can be widely adopted or adapted. therefore, the present study aimed to analyze students’ critical and creative thinking skills in the steam-ct approach, which did not use computers or other digital technologies. methods the present study employed a descriptive qualitative method. this method is employed to achieve the research objectives because it is appropriate for describing events or experiences and seeking in-depth knowledge of the phenomena being studied (kim et al., 2017; neergaard et al., 2009). learning design the steam-ct approach in the present study provided experiences for students to design and develop seesaw miniatures that are fun, efficient, and safe. the training was conducted over four meetings. at each meeting, respectively, the students (1) imagine and design a seesaw; (2) create the designed seesaw; (3) test and present the seesaw; and (4) improve and reflect on the seesaw. during the first meeting, students imagined and designed a seesaw that meets three criteria: fun, efficiency, and safety. students were guided to learn art, simple machines, the types and strengths of the constituent materials, and linear functions while decomposing the integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 4 characteristics of the seesaw. with this knowledge, the students devised a list of the tools and materials required, sketched the design, and planned the sequential steps that would be used to construct the seesaw. students made seesaws at the second meeting, using the tools and materials planned and the design sketches drawn at the first meeting. students did this by listing and explaining what needs to be considered when building a seesaw. in addition, the students were asked to analyze and explain what influences the balance of the seesaw. students tested and presented their seesaws at the third meeting. they tested the seesaw and evaluated it to discover if it was enjoyable, efficient, and safe. they also analyzed areas for improvement and observed the seesaw patterns of other groups to inspire them to improve their own. following that, the students presented their seesaws in classical. in the fourth meeting, students improved their seesaw and reflected on their learning experiences. the learning activities at this meeting began with decomposing the steps to improve the seesaw. after that, students identified the variables so that the seesaw fits the fun, efficient, and safety criteria. finally, students reflected on their learning experiences to abstract the factors that support successful seesaw development. they also modelled the seesaw using linear functions. table 1 illustrates the learning experiences mapping in each meeting with steam content and ct dimensions. the learning design was discussed with the mathematics, natural sciences, and arts teachers of the students who were the subjects of the present study. table 1 mapping learning activities, steam content, and ct dimensions meeting learning experience steam content ct dimensions 1 imagining and designing seesaws simple machine (natural science); simple product engineering (craft); model image (arts and culture); straight line equations (mathematics) decomposition, algorithm 2 creating seesaws simple machine (natural science); creating simple products (crafts) decomposition 3 testing and presenting seesaws simple machine (natural science); testing and communicating of phenomena (informatics); testing and presenting of engineering works (craft) pattern recognition 4 improving seesaws and reflecting on learning experiences simple machine (natural science); engineering procedures (craft); application of linear functions (mathematics) decomposition, abstraction research subject the subjects of the present study were 26 eighth-grade students, consisting of 14 boys and 12 girls. all of the subjects came from one class at a private junior high school in yogyakarta, indonesia. the subject selection was conducted by first discussing with the teachers so that the selected students were usually active and had good verbal skills. thus, the data obtained from epiphany princess mariana, yosep dwi kristanto 5 these subjects can provide rich and valuable information about their critical and creative thinking skills (campbell et al., 2020; kelly, 2010). data collection the data in the present study were the students’ answers in their worksheets and the seesaw construction they created. the sequence of the questions and instructions in the worksheet is adjusted to the edp cycle (see appendix a). the questions and instructions in the worksheet are also structured following indicators of critical and creative thinking skills, as shown in table 2. critical thinking skills indicators are obtained by synthesizing critical thinking skills indicators from ennis (2015), sihotang et al. (2012), and wade (1995). formulating the problem, gathering facts, planning, devising a strategy, and providing additional explanation were the obtained critical thinking skills indicators. the indicators of creative thinking skills were synthesized from treffinger et al. (2002), mahmudi (2010), and guilford (1976). the synthesis obtained four indicators: fluency, flexibility, authenticity, and detailedness. these indicators were used to create tasks in student worksheets as well as guidelines for scoring students' products. table 2 depicts the mapping of indicators of critical and creative thinking skills, student worksheet tasks, and student products. table 2 mapping of critical and creative thinking indicators, student worksheet tasks, and student product skill indicator student worksheet’s tasks student’s product critical thinking formulating the problem i.5 gathering facts i.1, i.2, i.3, iv.3 planning i.4, i.5 devising strategy ii.1, iv.1 purpose providing further explanation ii.2, iii.1, iii.2, iv.2, iv.3, iv.4 purpose creative thinking fluency ii.1, iv.1 flexibility i.4, i.5 authenticity i.4, i.5, iii.2, iv.3 design and construction detailedness ii.2, iii.1, iii.2, iv.2, iv.3, iv.4 relevance data analysis the data analysis process began with the development of a rubric for scoring student answers on worksheets and the products they created. the rubric is divided into two parts: a rubric for students' worksheet answers and the resulting seesaw product. each item on the worksheet has a maximum score of 10 or 15. the difference in the maximum score indicates a difference in cognitive demand in each question. as an illustration, we consider question number i.4, which asks students to name the tools and materials they will use, had a smaller cognitive demand than question number iv.1, which asks them in groups to discuss and write down the strategies they need to improve their works. it aligns with our findings in the results and discussion that students experience difficulties developing integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 6 improvement strategies. thus, questions i.4 and iv.1 have a maximum score of 10 and 15, respectively. we administered the same considerations to determine the maximum score for the other questions. to demonstrate how we score students’ answers on the worksheet, identify the example of one group’s answers shown in figure 1. figure 1. a sample of students’ answers on the worksheet based on figure 1, the group mentioned its improvement strategy and its purpose clearly and fluently. therefore, we suggested that the group has demonstrated devising strategy and fluency skills. there are three scoring aspects that we use for students’ seesaw products, i.e. relevance, design and construction, and purpose of the product. product relevance was related to detailedness, product design and construction were related to authenticity, and product purpose was related to devising strategy and providing further explanation. we scored each aspect of the scoring with a range of 1 to 4. as an illustration of the scoring process that we carried out on student products, consider figure 2 below. figure 2. a sample of students’ seesaw product figure 2 depicts a product that is distinct from those produced by other groups. the originality comes from the use of various colors and decorations while maintaining a balance of seesaws. thus, we proposed that the group's work was authentic. we provided this product with a four for product form or authenticity. we scored the other aspects using similar criteria. after completing the scoring process, we utilized descriptive statistics, specifically percentages, to summarize the scores for each indicator of critical and creative thinking skills. in addition, we use thematic analysis to identify key themes in students' worksheet answers. braun and clarke (2006) proposed a procedure for conducting thematic analysis. epiphany princess mariana, yosep dwi kristanto 7 results and discussion the results of the analysis of students’ critical and creative thinking skills are presented and discussed in the following three sections. students’ critical thinking skills table 3 displays the average score of critical thinking skills based on the students’ answers on the worksheet and the seesaw product presented for each indicator. based on these five indicators, the student's critical thinking skills averaged 73.97. table 3 students’ critical thinking skills indicator average formulating the problem 72 gathering facts 75.5 planning 86 devising strategy 66 providing further explanation 70.38 average 73.97 the skill of planning is the indicator of critical thinking skills with the highest average score. two themes of planning skills can be discovered in the student's worksheet answers. first, students can meticulously plan the tools and materials to be used. second, students can design each design's function or usability. translation: write down the tools and materials that will be used (tools and materials provided: glue gun and popsicle sticks) answer: tools: a hot glue gun (to hold the sticks together) markers (to draw) scissors/cutter (to cut cardboard and sticks) materials: popsicle sticks (as the arm of the seesaw) a cardboard/paperboard (as base and pedestal) loads (coins) (to check the seesaw’s balance) figure 3. planning for tools and materials in one group’s answer. in the worksheet, all students described the tools and materials in detail. figure 3 depicts the response of one group of students. the diagram shows that the group was able to not only plan tools and materials in detail but also provide a comprehensive classification of tools and materials. in other words, they can create categories and then determine who the members of those categories are. furthermore, the group provided functional descriptions of the tools and materials they intend to employ to construct a seesaw. integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 8 translation: draw a seesaw design as detailed and attractive as possible! write down the reasons too! answer: board length = 30 cm board width = 2 cm fulcrum height = 6 cm, fulcrum width = 3 cm load weight = 3 grams (1 coin) the length and height of the handle = 2 cm reason a. the length is 30 cm, thus, it is not too short b. the width is 2 cm, thus, it is not too thick c. the height of the fulcrum is 6 cm, thus, it can seesaw d. the weight of the load is 3 grams, thus, it balances e. the length of the handle is 2 cm, thus, it is not too long f. the handle length of the horizontal part is 2 cm to match the length of the handle figure 4. mentioning the function of the design from one group’s answer. the second theme is that students design the designs' function or usability. almost every group created a seesaw based on the size and design of the seesaw drawn. the group includes reasons for each size and explains its function, especially for those shown in figure 4. however, developing strategy skills is the indicator of critical thinking skills that receives the lowest score. there are three themes associated with strategy development: (1) product development strategy, (2) evaluation awareness, and (3) improvement strategy. figure 5. the initial and final product of one group. figure 5 demonstrates the seesaw product before (left) and after (right) revision. during the trial and presentation, the students who constructed the seesaw saw their mistakes and received feedback from the teacher. the feedback relates to the balance, comfort, and aesthetics of the seesaw. nonetheless, these students have not used the feedback to develop improvement strategies and have not used these strategies to improve the seesaw. epiphany princess mariana, yosep dwi kristanto 9 students’ creative thinking skills table 4 displays the average score of creative thinking skills based on the worksheet answers and seesaw products of each indicator. based on these four indicators, the average creative thinking skills of students are 73.05. table 4 students’ creative thinking skills indicator average fluency 57 flexibility 86 authenticity 78.83 detailedness 70.38 average 73.05 four indicators of creative thinking skills are evaluated, with each indicator receiving a different average score. flexibility is the creative thinking skills indicator with the highest average score. there are two themes that emerged from students' work in terms of their flexibility: (1) variations in answers and problem-solving; and (2) flexibility in creating appealing designs. translation: draw a seesaw design as detailed and attractive as possible! write down the reasons too! answer: tools and materials cardboard (as a base) scissors (for cutting) toothpick (as support reinforcement) sticks (as seesaw) glue (as adhesive) weights (plasticine) the seesaw has a balanced length and a moderate fulcrum, so the board does not rise too high (safe seesaw). figure 6. students work showing the theme of variation figure 6 displays students who provide a variety of answers and problem-solving ideas, as well as flexibility in creating appealing designs. the students documented the tools and materials. surprisingly, these students documented the function of each tool and material. this answer is also unique in that it mentions "play dough" as the weight. it means that the students come up with different problem-solving ideas for seesaw weights. integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 10 furthermore, figure 6 demonstrates that these students are adaptable in creating appealing designs. these students present three variations of the image, each with a unique perspective and function. the students each contribute a unique nuance to the product designs. figure 7 depicts the finished seesaw, whose design is depicted in figure 6. even though the seesaw is imperfect, in that it is not balanced and the weight is not the same as planned, the students have created a seesaw that is nearly identical to the design they worked on. figure 7. seesaw product the fluency indicator, on the other hand, receives the lowest score for creative thinking skills. several factors contribute to the current study's average student fluency score, which is still relatively low. these factors include (1) insufficient problem solutions and (2) a lack of understanding of the errors. translation: write the group’s strategy for fixing the seesaw! create a new seesaw figure 8. example of low fluency figure 8 portrays one of the students' incomplete answers in writing solutions, as well as a lack of comprehension of the errors. the student devised a plan to improve the seesaw by replacing it with a new one. these students may develop the notion that the seesaw they construct must be replaced due to numerous errors. these students, however, did not provide a detailed solution to improve it. the students are aware of errors in their previous designs but are unable to write down ideas for how to correct them fluently. discussion the present study has described the students’ critical and creative thinking skills in integrative steam-ct teaching and learning. based on the analysis of critical thinking skills, the students are able to make a reasonable plan. it is because they are given a space to make a plan through one of the edp stages, namely planning (national research council, 2012). planning is an essential activity in learning. it is because planning necessitates students to consider their objectives and devise strategies to achieve them (eilam & aharon, 2003). such epiphany princess mariana, yosep dwi kristanto 11 planning can trigger the desired learning behaviors and ultimately lead to higher learning outcomes (raković et al., 2022). based on an analysis of creative thinking skills, the students are adaptable in providing alternative solutions and can create an appealing design. the students have provided reasons and functions for the design aspects on which they are working. their design drawing is also visually appealing. it is inextricably associated with the critical role of the arts in steam integrative learning, which encourages student creativity (liao, 2016). the present study also found that several aspects of critical and creative thinking skills need attention. this research shows that some students still lack detail in providing solutions to problems. in general, the students are less aware of the errors made. the students need to evaluate the errors so that the errors can be corrected and not repeated. in addition, they also need to use the feedback they receive for improvement. therefore, evaluation practices supported by students’ feedback literacy are essential for solving problems (carless & boud, 2018; ifenthaler, 2012). it can be corroborated in an integrative steam-ct approach by providing peer feedback activities (chang et al., 2021; kristanto, 2018). this feedback practice supports the growth of students’ critical thinking skills and creativity (vincent-lancrin et al., 2019). there are several limitations to the current study. following the characteristics of the research method used, namely descriptive qualitative, this study only directly describes the critical and creative thinking skills of students who are the subject of this study. thus, the findings of this study cannot be generalized to different contexts and settings. second, this study uses students’ answers on worksheets and their final product. thus, the description of students’ critical and creative thinking skills presented here is their skills during the learning process. conclusion the current study explained students' critical and creative thinking skills in innovative steam and ct teaching and learning practices. this practice has sparked students to be able to make plans to solve problems, be flexible in providing solutions, and create aesthetic product designs. nonetheless, this study also found that it was necessary to support students in carrying out in-depth evaluations so that they could provide accurate solutions. in addition, students also need to be supported in acquiring feedback literacy. therefore, we recommend that the steam-ct approach needs to provide space for students to develop their feedback literacy. acknowledgements we want to express our sincere gratitude to beni utomo, m.sc. and adhi surya nugraha, s.pd., m.mat. for their invaluable comments and feedback on our initial manuscript. we also thank scholastica lista febriantari, s.pd., dina ari puspita, s.pd., and antonius w., s.pd. for their generosity in validating our lesson plans. finally, we are grateful to all the research participants who generously gave their time and effort to this project. integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 12 references agnafia, d. n. 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(2021). visual programming environments and computational thinking performance of fifthand sixth-grade students. journal of educational computing research, 59(6), 1075–1092. https://doi.org/10.1177/0735633120988807 epiphany princess mariana, yosep dwi kristanto 17 appendix a. student worksheet worksheet i: let’s think about and design a seesaw! let's think about it! a playground in one of the cities has a variety of children's toys. the seesaw is one of the park's children's toys. how do we make seesaws that are enjoyable, efficient, and safe? task i.1: identify the characteristics of a fun seesaw! task i.2: identify the characteristics of an efficient seesaw! task i.3: identify the characteristics of a safe seesaw! let’s design! let's design and decide on the tools and materials to use now that we've identified fun, efficient, and safe seesaws! task i.4: write down the tools and materials utilized (tools and materials provided: hot glue gun and popsicle sticks) task i.5: draw a seesaw design in detail and as attractive as possible! write down the reasons too! worksheet ii: let’s make seesaws! let's build a seesaw! let's build a fun, efficient, and safe seesaw using the pre-made designs and the tools and materials provided! task ii.1: what factors do you consider when creating seesaws? for instance, the pedestal's location, the length and width of the board, or the weight of the load) task ii.2: according to the group, what influences the seesaw to be balanced? worksheet iii: let’s test and present the seesaw! let us test and then present! check the results of the seesaw product to see if they are in accordance with the success indicators, consult with the teacher, and present it to the class! task iii.1: write down the evaluation results and analyze your group’s mistakes! task iii.2: write down the improvement/improvement efforts that the group will do in the seesaw project! worksheet iv: let’s fix and reflect on the seesaw! let’s fix the seesaw! it’s time to fix the seesaw based on the results of trials, evaluations, and improvement efforts. task iv.1: write down the group’s strategy for fixing the seesaw! task iv.2: which combination of variables influences the correctly constructed seesaw? let’s reflect! reflect on the results of doing a seesaw project with your group mates! task iv.3: draw the final design of the finished product, and determine the following: (a) the seesaw gradient, if one side is loaded; and (b) the straight-line equation of the seesaw if one side is loaded. task iv.4: write down your conclusions after doing a seesaw project! integrating steam education and computational thinking: analysis of students’ critical and creative thinking skills in an innovative teaching and learning 18 symposium 1 southeast asian mathematics education journal, volume 11, no 1 (2021) the classification of mathematical literacy ability in cognitive growth learning viewed from multiple intelligences aprilia nurul chasanah department of mathematics education, universitas tidar, indonesia aprilianurul@untidar.ac.id abstract this research aims to determine the quality of the cognitive growth mathematical learning model towards mathematical literacy ability and to describe the classification of mathematical literacy ability viewed from the theory of multiple intelligences. this descriptive qualitative study involved 30 eight-grade students with the inclination on each type of multiple intelligence as the research subjects. the data were collected through test, observation, and interviews. the research revealed the following points: (1) the quality of mathematical learning using the cognitive growth model was in the good category; (2) mathematical literacy ability in cognitive growth learning viewed from multiple intelligences theory can be classified into: verbal-linguistic, logical-mathematical, and fourth-level musical intelligences; third-level visual-spatial intelligences, second-level intrapersonal intelligences, and first-level bodilykinesthetic, interpersonal, and naturalistic intelligences. based on these results, this study concludes that the mathematical literacy on different topics compiled based on the indicators of mathematical literacy in cognitive growth learning model has been well improved. keywords: cognitive growth model, mathematical literacy ability, multiple intelligences theory introduction in terms of national education, law no. 20 of 2003 on national education system, article 3 states: "the national education functions to develop the capability, character, and civilization of the nation for enhancing its intellectual capacity, and is aimed at developing learners’ potentials so that they become persons imbued with human values who are faithful and pious to one and only god; who possess morals and noble character; who are healthy, knowledgeable, competent, creative, independent; and as citizens, are democratic and responsible.” (depdiknas, 2003). in addition, it is expected that the students can use mathematics as a way of reasoning (logical, critical, systematic, and objective ways of thinking). according to gagne (1985), the indirect object of learning mathematics is that the students should have the ability to solve various problems. gagne's opinion and the purpose of the curriculum on mathematics highlight that in order to solve a problem, the students need to have adequate reasoning abilities that can be obtained through learning mathematics. the low quality of indonesian human resources currently is due to the poor quality of education, especially related to mathematics as seen from various indicators. at the national level, mathematics learning in schools is evaluated through the standard computer-based national examination, while, at the international level, students’ mathematical abilities are assessed by two methods of assessment: timss (trend in international mathematics and science study) and pisa (program for international student assessment). in terms of mathematics abilities, indonesian students ranked 36th out of 40 countries in 2011 in the trends in international mathematics and science study (timss), and in 2015 indonesian students ranked 45th out of 50 countries with the score of 397 that is far below the international average score of 500. 2 southeast asian mathematics education journal, volume 11, no 1 (2021) the low quality of education can also be seen in the 2015 pisa report which ranked indonesia 62nd for science, 63rd for mathematics, and 64th for reading out of 70 countries (oecd, 2016). similar performance can be seen in 2012 where indonesia ranked 64th for science and mathematics and 61st for reading out of 65 countries. the average scores for science, mathematics, and reading was 403, 386, and 397 respectively in 2015, and 382, 375, and 397 in 2012. mathematics literacy is very important because it emphasizes the students' ability to analyze, reason and communicate ideas effectively about the mathematical problems they encounter (oecd, 2009). this is what connects mathematics studied in the classroom to various real-world situations. according to the oecd (2012), mathematics literacy is the ability to formulate, implement, and interpret mathematics in various contexts. in this case, it includes mathematics reasoning and uses mathematics concepts, procedures, facts and tools to describe, explain, and predict phenomena/events. based on observations of eighth grade students at the ihsanul fikri islamic junior high school in magelang city, and smp n 8 magelang, it was evident that the questions given to students were still at the basic level. the teachers had not yet provided more varied questions, especially related to mathematical literacy. the students solved many standard problems without deep understanding. as a result, their mathematical literacy abilities and strategic competencies did not improve. this is supported by rusmining, waluya and sugianto's research (2014) which advised mathematics teachers that they should begin to introduce students with problems related to mathematics literacy. many efforts have been made to improve the students' ability in terms of achieving better performance on the mathematics problem questions in pisa. solving these problems not only emphasizes the scope of learning achievement, but also considers the students’ psychology and characteristics as inseparable elements. the students' mathematics literacy skills can be viewed from various dimensions. the dimensions of individual differences include the ability to think logically, creativity, cognitively, and intelligently. the theory multiple intelligences is a theoretical framework for defining, understanding, developing, and assessing different intelligences. the teachers apply this as a framework for teaching and learning in class. learning mathematics is not a simple task. the teachers must try to be creative in the learning process (gouws, 2007). the concept of multiple intelligences focuses on the aspects of uniqueness for each child. this fact is supported by rafianti's research (2013) stating that improving the students' understanding of mathematics concepts and reasoning ability using multiple intelligence-based mathematics learning was better than those receiving conventional learning methods. the learning quality must also be considered and one of the influential factors is the accuracy of the learning model. based on the observations made on the mathematics teachers of grades vii and viii at one of the junior high schools in magelang, central java, indonesia, most of teachers still use the basic learning models. they rarely used new models to help the students in learning, especially for learning geometry. some teachers until now still teach using traditional methods, which emphasizes training or practice and procedural questions. thus, the teachers function as the center or source of all the materials, which only gives the teacher a room to be active in the learning process, while treating students as the passive recipients of the material. this situation is one of the main causes of the low quality of students' understanding of mathematics (ali & jameel, 2016). the cognitive growth model is one of the learning methods that can improve the students’ mathematic literacy abilities. according to piaget in joyce (1992), the cognitive growth model aims at improving the students’ thinking abilities (cognitive). it attempts to match the stage of learning development and improve the students' mathematic literacy abilities. the role of students in this model is to generate responses and ask for justification in 3 southeast asian mathematics education journal, volume 11, no 1 (2021) conveying the results in the learning process. through this process, the teachers are also required to prepare the materials well and condition of the class so that learning activities are appropriate for the learning objectives. this encouraged the researcher to examine the quality of cognitive growth learning upon the students’ mathematic literacy abilities and to classify the mathematic literacy abilities of eighth grade junior high school students. the syntax of cognitive growth learning refers to joyce's opinion listed in the following table 1. table 1 syntax of cognitive growth model phase description phase 1 confrontation with stage-relevant tasks the integration of tasks/problems according to the stage, and the students’ orientation on the problem to study; it is intended that students are ready to think more critically in the next learning phase. phase 2 inquiry organizing the students to raise their sensitivity and improving their critical thinking ability, performing in group formation activities in a class. analyzing and evaluating the process; the learning process that has been implemented is evaluated/reflected to improve the learning activities, while the results are criticized and discussed together in the class phase 3 transfer phase the integration of tasks/problems according to the stage, and the students’ orientation on the problem to study; it is intended that students are ready to think more critically in the next learning phase. based on the views of this syntax, the cognitive growth model fits in with the stages of learning development and improves mathematics literacy. this research aims to provide additional knowledge about learning mathematics, especially to improve mathematical literacy. besides, it is also expected to provide inputs to the educators for more innovative learning using cognitive growth model. the research mainly focuses to answer the following questions: (i) how does the quality of cognitive growth learning influence the students’ mathematic literacy abilities?; and (ii) is the classification of mathematic literacy abilities of the eighth-grade students related to the multiple intelligences? thus, the purpose of this study is to determine the quality of the cognitive growth model in the mathematics learning process compared with the students’ mathematic literacy abilities, and to describe the students’ mathematic literacy abilities in terms of the theory of multiple intelligences. research methods this is a descriptive qualitative research on the quality of the cognitive growth learning model and the classification of the students' mathematics literacy ability of eight-grade students based on the multiple intelligences theory. the research subjects were the eighth-grade students of junior high school in magelang city and the research subjects were selected based on the results of multiple intelligences tests. two students were selected for each level because the data were analyzed using constant comparative method, and the selection had a snowball effect in which the next subject was selected based on on the analysis of the previous subject. if there was no subject to occupy a particular level, the process was done repeatedly until one subject was selected. this study used an interview as the main research instrument. the researcher carried out the interview based on the interview guidelines. to conduct the interview, the researcher 4 southeast asian mathematics education journal, volume 11, no 1 (2021) acted as a planner, data implementer and collector, analysers, data interpreter, and the reporter of the research results. other instruments were in the form of mathematics literacy questions, interview guidelines, observation sheets, lesson plans and syllabus, and multiple intelligences tests. pisa-based mathematics literacy questions this study used the mathematics question sheet taken from the mathematics questions from the pisa criteria (oecd, 2013). the question sheet was in the form of word problems taken from realistic daily life problems. this instrument was validated by experts, consisting of four mathematical education experts/mathematicians; two lecturers of mathematics education of semarang state university and two mathematics teachers in magelang. validation was done to find out that the use of language and construction of the questions was in accordance with the indicators. the assessment of the validators revealed that the question was in accordance with the formulation of the research problem in terms of the construction of the questions, the language of the questions, and the subject matter. interview guideline the interview guidelines in this study contained a list of questions to be asked orally by the researcher to the students to uncover the students’ literacy level based on the mathematics literacy indicators. these interview guidelines were validated by three experts, consisting of mathematics education experts. some improvements were made to the interview guidelines during the validation process. mathematics literacy-based learning materials learning materials were validated by three mathematics education experts by considering various aspects, namely: syllabus indicators; lesson plans that must be in accordance with the mathematics literacy competencies; and the learning objectives. the instruments were revised according to the validators’ advice. multiple intelligences test the questionnaire used a likert scale and was adopted and modified from a multiple intelligences measurement tool known as roger’s indicators of multiple intelligences (rimi) test. the questionnaire was modified to adjust to local conditions using easy-to-understand language for the respondents who were still at junior high school level. the multiple intelligences test was assessed based onthe number of the students’ correct answers on each item. results and discussion results of multiple intelligences test the multiple intelligences test was aimed to determine the type of the students’ intelligence and was used as a consideration in choosing the subjects to have an in-depth interview about the mathematics literacy ability. the multiple intelligences test was assisted by psychologists and accompanied by eight observers in each category of multiple intelligences. based on multiple intelligences test results, the distribution of multiple intelligences of eight-grade students of class b can be seen in table 2 below. 5 southeast asian mathematics education journal, volume 11, no 1 (2021) table 2. multiple intelligences classification of eight-grade students of class b students’ category number of students percentage verbal/linguistics 3 10.00 % logical mathematics 4 13.33 % visual/spatial 5 16.67 % kinesthetics 2 6.67 % musical 5 16.67 % interpersonal 4 13.33 % intrapersonal 4 13.33 % naturalists 3 10.00 % total 30 100.00 % the quality of mathematics learning using cognitive growth model on the achievement of mathematics literacy abilities the quality of mathematics learning using the cognitive growth model for mathematical literacy abilities was rated in the good category. the learning quality is classified as good if 3 minimum domains are met in the good category, namely planning and preparation, classroom management and organization, and assessment (mac gregor, 2007). the three domains can be specified as follows. planning and preparation the measurement of the learning quality on the preparation stage is carried out using a validity test on the minimum device in the good category. table 3. data summary of validation test results no. learning device score category 1. syllabus 3.070 good 2. lesson plans 3.670 very good 3. students’ worksheets 3.780 very good 4. students’ jobsheets 3.580 very good 5. material supplement 3.625 very good 6. math literacy abilities test 3.580 very good 7. multiple intelligence test 3.070 good classroom management and organization the learning management in the classroom had an average score of 3.93, which belongs to the good category. the results of the assessment of the learning outcomes obtained from the observation process are shown in table 4. table 4. results of learning organization assessment no. learning quality average score category 1. observation 1 3.81 good 2. observation 2 3.96 good 6 southeast asian mathematics education journal, volume 11, no 1 (2021) 3. observation 3 4.21 very good average 3.99 good assessment the assessment attempts to measure the achievement of the learning objectives and is obtained from the results of the mathematics literacy ability test (in indonesian, tes kemampuan literasi matematika, abbreviated as tklm) and the students’ response to the questionnaire. the average score for the tklm is 72.31 which belongs to the good category. the results of students’ response analysis show that the percentage of students' positive responses to all aspects was higher than 50%, so more than 50% of the students gave a positive response toward the learning process. classification of mathematics literacy abilities on cognitive growth learning viewed from multiple intelligence of the eight-grade students data of mathematics literacy test from try-out 1, 2 and 3 is presented in table 5. table 5. results of mathematics literacy test no. notes tklm 1 tklm 2 tklm 3 1 average score 60.39 76.54 80.01 2 lowest score 52.00 53.00 56.25 3 highest score 70.00 72.70 81.39 4 number of level 1 students 21.00 10.00 8.00 5 number of level 2 students 8.00 9.00 5.00 6 number of level 3 students 1.00 8.00 5.00 7 number of level 4 students 0.00 3.00 12.00 8 number of level 5 students 0.00 0.00 0.00 9 number of level 6 students 0.00 0.00 0.00 table 4 portrays an increase in the average score of mathematics literacy ability, from 60.39 in tklm 1 to 76.54 in the second trial, and to 80.01 in the third trial. there was an upsurge in the number of students in the three levels from tklm 1 to tklm 2, which decreased slightly between tklm 2 and tklm 3, and rose again since the students could successfully achieve level 4. mathematics literacy abilities of verbal/linguistics type based on the results of tklm, the verbal type students demonstrated diverse abilities on the math literacy questions of the written tests. from the results of the in-depth interviews, the average verbal/linguistic ability of students was scored at level 4 and was classified as good at level 4. 7 southeast asian mathematics education journal, volume 11, no 1 (2021) figure 1. level of mathematics literacy abilities of verbal/linguistic students. the results indicate that the verbal/linguistic students' mathematical literacy abilities from the indicators of communication, mathematising, representation, reasoning and argument, solving problems for devising strategies, using symbolic, formal and technical language and operations, using mathematics tools are at the good category (level 4). however, the verbal/linguistic students could not evaluate the solutions for the mathematics literacy problems, although their advantages were in solving mathematical literacy problems from indicators of communication and reasoning in the argument process as verbal students give more responses. during the interview they provided a complete description at level 4, even though the draft answers were not as complete as what the verbal type students convey. this result resonates with the finding from mannamaa, et al. (2012), which stated that the students who have high verbal abilities are able to convey problems of mathematics stories. mathematics literacy abilities of logical-mathematical students based on the results of the tklm, the logical-mathematical students showed homogeneous abilities on the written test. from the in-depth interviews, the average ability of the mathematics literacy was at level 4 and was classified as good at the level 4. figure 2. level of mathematics literacy abilities of logical-mathematical students. the mathematics literacy abilities of the logical-mathematical students on the indicators were classified as level 4. however, at level 5 they began to experience difficulties in the process of solving the literacy problems that had not been fully implemented and the representation had not been fulfilled. thus, the logical-mathematical students' understanding at level 5 in the representation process were still low. this indicator corresponds with pisa result, which indicate that students at level 4 were able to work effectively using implied models in concrete situations, but had difficulties in facing obstacles or making assumptions. 8 southeast asian mathematics education journal, volume 11, no 1 (2021) mathematics literacy abilities of visual/spatial students based on the results of the tklm, the visual-type students presented diverse abilities on the written test. from the results of the in-depth interviews, the average ability achievement of visual/spatial students was at level 3 and were classified as good at level 3. figure 3. level of mathematics literacy abilities of visual students. the prominent aspect with the visual students lies in the images created. spatial visual intelligence is the ability to visualize two or three-dimensional objects (images) to solve mathematical problems in daily life. from the interviews, the visual students revealed their preference of the pictorial literacy questions because these type of questions made them easier to find out information about the problem. this result is in line with the study by ningsih (2014), which stated that students with visual spatial intelligence learn more effectively by looking at pictures/images. in a study conducted by boakes (2009), spatial/visual was stated to be an important part of geometrical thinking. mathematics literacy abilities of kinesthetic students based on the results of the tklm, the kinesthetic type students idicated a homogeneous ability on the written test at level 1. the in-depth interviews portrayed that the average ability of students’ achievement was also classified at level 1. figure 4. level of mathematics literacy abilities of kinesthetic students. mathematics literacy abilities of musical students based on the results of the tklm, the musical students had heterogeneous abilities on written tests. the in-depth interviews indicated that the average ability of students’ achievement was categorized at level 2. 9 southeast asian mathematics education journal, volume 11, no 1 (2021) figure 5. level of mathematics literacy abilities of musical students. the notable aspect of the musical students lies in their analysis and representation. they could convey ideas in solving mathematical literacy problems, which corresponds with the study by damar (2012), which stated that there was a positive and significant relationship between musical and mathematical abilities. mathematics literacy abilities of intrapersonal students based on the results of the tklm, the intrapersonal students had various abilities on written tests. the in-depth interviews revealed that the average ability of students’ achievement was at level 2. the mathematics literacy skills of the intrapersonal students were at level 2. figure 6. level of mathematics literacy abilities of intrapersonal students. the intrapersonal students convey ideas well in solving the problems which resembles the finding of febriyanti (2018), which stated that students with intrapersonal intelligence communicate well in writing mathematics. during interviews, the students expressed their preferences on literacy problems in the form of simple questions because simple questions made them easier to find out information from the questions. mathematics literacy abilities of interpersonal students based on the results of the tklm, the interpersonal students had a homogeneous ability on written tests. the in-depth interviews demonstrated that the average ability of students’ achievement was at level 1, and thus the mathematics literacy abilities of the interpersonal students were low. 10 southeast asian mathematics education journal, volume 11, no 1 (2021) figure 7. level of mathematics literacy abilities of interpersonal students. this research, however, contradicts the results of hidayati (2014), which stated that students' mathematics learning achievement with high intrapersonal intelligence is better than those having moderate and low intrapersonal intelligence. this study is supported by the results of the in-depth interviews, in whichone student could reach level 4. mathematics literacy abilities of naturalist students based on the results of the mathematical literacy ability test (tklm), the naturalist students had a homogeneous ability on written tests of mathematics literacy questions. from the results of in-depth interview, the average achievement ofnaturalist students in the mathematics literacy was at level 1. figure 8. level of mathematics literacy abilities of naturalist students. the students could not provide complete arguments or explanations, which is in accordance with gardner (2011), who explained that the naturalist students have less ability in delivering their ideas for solving the mathematics problems but they are outstanding when asked to look for the data from the surrounding environment. conclusion and suggestion this research acknowledges the small sample size but can generate the following conclusion points. (1) the quality of mathematics learning in the cognitive growth model of mathematical literacy abilities was in the good category, and this result is evident in the three domains of quality learning criteria that include (a) planning and preparation, (b) classroom management and organization (process), and (c) assessment (evaluation). (2) the classification of mathematics literacy abilities of the cognitive growth learning model in terms of the multiple intelligences, are verbal/linguistic, logical/ mathematical, and musical-typed students classified at level 4, visual/spatial students categorized at level 3, intrapersonal-typed students classified at level 2, while kinesthetics, interpersonal, and naturalist students are at level 1. 11 southeast asian mathematics education journal, volume 11, no 1 (2021) the cognitive growth model may be useful in monitoring the students’ mathematics literacy abilities. however, further research is needed to expand the observed dimensions, for example in terms of the students’ ability to think logically, creatively, and cognitively. these dimensions are estimated to influence the students’ mathematics literacy abilities. references ali, h. h., & jameel, h. t. (2016). causes of poor performance in mathematics from teachers, parents and student’s perspective. american scientific research journal for engineering, technology, and sciences, 15(1), 122-136. boakes, n. j. (2009). origami instruction in the middle school mathematics classroom: its impact on spatial visualization and geometry knowledge of students. research in middle level education, 32(7), 1-12. damar, k. (2012). hubungan antara kemampuan musikal dengan kemampuan matematika siswa smpn 1 minggir sleman [unpublished master thesis]. universitas negeri yogyakarta. depdiknas. (2003). sistem pendidikan nasional. jakarta: depdiknas. febriyanti, r. (2018). students’mathematical communication abilities in mathematical problem solving viewed from intrapersonal and interpersonal intelligences. mathedunesa, 7(1), 93-100. gardner, h. (2011). the theory of multiple intelligences: as psychology, as education, as social science [address]. honorary degree from josé cela university in madrid, spain, and the prince of asturias prize for social science. gagne, r.m. (1985). the condition of learning and theory of instruction (4th ed.). new york: holt, rinehart and winston. gouws, f.e. (2007). teaching and learning through multiple intelligences in the outcomesbased education classroom. african review journal, 4(2), 60–74. hidayati. (2014). eksperimentasi pembelajaran matematika dengan model pembelajaran kooperatif jigsaw dan teams games tournament (tgt) ditinjau dari kecerdasan intrapersonal siswa. jurnal elektronik pembelajaran matematika, 2(2), 152-162. joyce, b., & weil, m. (1992). models of teaching (4th ed.). london: allyn and bacon. mac gregor, r. r. (2007). the essential practices of high-quality teaching and learning. bellevue, wa, us: the center for educational effectiveness, inc. männamaa, m., kikas, e., peets, k., & palu, a. (2012). cognitive correlates of math skills in third-grade students. educational psychology, 32(1), 21-44. ningsih. (2014). kecerdasan visual spasial siswa smp dalam mengkonstruksi rumus pythagoras dengan pembelajaran berbasis origami di kelas viii. jurnal ilmiah pendidikan matematika, 3(1), 203-211. oecd. (2009). learning mathematics for life: a perspective from pisa. paris: oecd publishing. https://static1.squarespace.com/static/5c65be2cfb22a5044c764343/t/5c6c783f15fcc04fbd6e61c4/1550612543679/473+-+madrid+oct+22+2011.pdf https://static1.squarespace.com/static/5c65be2cfb22a5044c764343/t/5c6c783f15fcc04fbd6e61c4/1550612543679/473+-+madrid+oct+22+2011.pdf 12 southeast asian mathematics education journal, volume 11, no 1 (2021) oecd. (2013). pisa 2012 results in focus what 15-year-olds know and what they can do with what they know. paris: oecd publishing. retrieved from http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf. oecd. (2016). pisa 2015: pisa results in focus. paris. pisa-oecd publishing. oecd. (2012). pisa results: what students know and can do – student performance in mathematics, reading and science. paris: oecd publishing. rafianti. (2013). penerapan model pembelajaran berbasis multiple intelligences untuk meningkatkan kemampuan pemahaman konsep, penalaran matematika, dan selfconfidence siswa mts. bandung: universitas pendidikan indonesia. rusmining, waluya, s. b., & sugianto. (2014). analysis of mathematics literacy, learning constructivism and character education. international journal of education and research, 2(8), 331-340. southeast asian mathematics education journal volume 11, no 2 (2021) 119 the innovative learning of square and rectangle employing macanan traditional indonesian game 1padhila angraini, 1rully charitas indra prahmana* & 2masitah shahrill 1mathematics education, universitas ahmad dahlan, yogyakarta, indonesia 2sultan hassanal bolkiah institute of education, universiti brunei darussalam, brunei darussalam 2rully.indra@mpmat.uad.ac.id abstract geometry is one of the essential mathematics materials, such as square and rectangle. however, most elementary school students experience difficulty in understanding it due to the abstractness of the geometric material. furthermore, teachers teaching in the remote areas of indonesia still encounter problems exploring this abstract material to make students understand caused by the limited learning resources. on the other hand, students in remote areas are familiar with traditional games, such as macanan. hence, the objective of this study is to design learning activities utilizing macanan game in assisting students understand the concept of the perimeter and area of squares and rectangles for fourth-grade students. a design research approach was implemented and performed at one of the elementary schools in jambi, indonesia. the research results presented that macanan could be a context for a starting point in this learning design of both square and rectangular learning. employing this context, it would be fun, enjoyable, and easy to understand the perimeter and area of squares and rectangles for students. this game could be a context for teachers in remote areas in teaching geometry and be a reference for identifying other contexts which can make mathematics learning easy in remote areas. keywords: design research, indonesian traditional games, square and rectangle, ethnomathematics. introduction mathematics is a basic science which is highly essential in life (cozzens & roberts, 2020; phoenix, 2018). it is a subject that must be provided to students, from elementary to secondary education (rowland, 2012; graham & fennell, 2001). however, teachers in schools frequently teach mathematics directly in an abstract form unassociated with daily life or real-world contexts (chong, shahrill, & li, 2019; risdiyanti, prahmana, & shahrill, 2019). hence, students generally experience difficulties, anxiety, meaninglessness, misunderstanding, and misconception with the learned mathematics (nurhayati, chang, naaranoja, 2019; maschietto & trouche, 2010). in essence, mathematics is a human activity (gravemeijer, 2020; ernest, 2013), and learning mathematics should be taught in accordance with activities in students' daily lives. through the activities, students will be able to comprehend the essence of mathematics, expeditiously understand the concept, make meaning, and take advantage of mathematics in solving problems in the students' daily lives (chong & shahrill, 2016; freudenthal, 1991; phonapichat, wongwanich, & sujiva, 2014). therefore, it is essential to teach mathematics with relevant contexts associated with student activities. besides the concern on how mathematics is taught in schools, school conditions also influence students’ mathematical processes (acharya, 2017). in indonesia, not all regions are well developed, wherein some areas are still underdeveloped or frequently referred to as the remote the innovative learning of square and rectangle employing macanan traditional indonesian game 120 areas (hendayana, supriatna, & imansyah, 2010). these areas are commonly far from urban dwellings, road access to these places is customarily difficult to reach, and internet access is not available (hendayana et al., 2010; kurniati, arafat, & mulyadi, 2020). it makes school facilities and infrastructure difficult to provide and hinders mathematics learning process of students (kurniati et al., 2020; febriana et al., 2018). on the other hand, one of the solutions is employing materials around the classrooms or schools to uphold the mathematics learning process for the students so that the mathematics learning process can be conducted optimally in limited conditions (kusumah & nurhasanah, 2017; hadi, 2002; zulkardi & putri, 2019). thus, the materials discovered locally can be utilized as context by the students and implemented as starting points in learning mathematics, hence, students will easily understand. geometry is one of the most crucial basic mathematics topics, including the concepts of points, lines, shapes, spatial numbers, and others (laborde, 2015). these concepts are the basis for developing object visualization skills and learning other mathematical concepts in accordance with visual and spatial matters (laborde, 2015). in daily life, geometry provides means to describe, analyze, and observe mathematical structures (jones & tzekaki, 2016). besides, geometry studies also contribute to developing mathematical reasoning skills, critical and logical thinking, and analytical skills (gunhan, 2014). however, students still find it difficult in learning geometry. several researchers revealed that in studying geometry, students frequently make mistakes in drawing or visualizing the shapes illustrated, determining the formulas and results, identifying objects based on facts, utilizing symbols, using rules, and interpreting story problems related to geometry in mathematical form (riastuti, mardiyana & pramudya, 2017; noto, priatna, & dahlan, 2019; özerem, 2012, angraini & prahmana, 2019). there are many factors causing students’ difficulties, such as teaching mathematics without relating it to students’ daily lives, the method which is merely memorizing formulas, and also the low quality or not provided facilities and infrastructure encompassing the lack of teaching material resources and teaching aids (riastuti et al., 2017; noto et al., 2019; özerem, 2012; angraini & prahmana, 2019). one of the remote areas in indonesia is in the village of pemunyian, jambi province. this village is located in an oil palm plantation area which is far from the urban area. road access is hard to reach by public transportation, and communication access such as telephone or internet signal is not available. since access is strenuous to achieve, facilities and infrastructure in schools are hardly to provide. hence, it has also made learning mathematics difficult at one of the schools at pemunyian village, sunshine elementary school (a pseudonym). based on the researchers’ observations, students possess predicament in learning mathematics and they do not yet understand some basic mathematics materials, such as number operations and geometry. in mathematics, there is an approach known as pendidikan matematika realistik indonesia (pmri) which is an adoption of the realistic mathematics education (rme) initiated by hans freudhental from utrecht, the netherlands (sembiring, hadi, & dolk, 2008; sembiring, 2010; fauziah, putri, zulkardi, & somakim, 2018). pmri is characterized by utilizing context as a starting point for learning mathematics to guide students to understand mathematical concepts from informal to formal forms (sembiring et al., 2008; hadi, 2017; nasution, putri, & zulkardi, 2018; sembiring, 2010). the use of familiar and close contexts to students’ daily lives makes them easier to imagine mathematical concepts and identify meaningful relations between abstract ideas and the practical applications in the real world (gravemeijer & doorman, 1999; van den padhila angraini, rully charitas indra prahmana & masitah shahrill 121 heuvel-panhuizen, 2005; risdiyanti et al., 2019). students also experience easy understanding on the mathematical phenomena from their own perspective and experiences allowing them to obtain meaning from the mathematics they are learning (unesco, 2008). in remote areas, mathematics learning can be maximized by the pmri approach and utilizing the local context in students’ daily lives and culture as starting points for learning (kusumah & nurhasanah, 2017). therefore, researchers are concerned in developing understanding of square and rectangular shapes in geometry learning designs employing local contexts which are familiar and close to students, which is the macanan traditional game. the game was applied at sunshine elementary school in optimizing mathematics learning in remote areas. the macanan game is a traditional javanese game which is also frequently played in jambi. in this game, some players play as humans and other players play as the tiger, and they attempt to catch each other (dharmamulya, 2008). risdiyanti and prahmana (2018) explained that there is ethnomathematics in the macanan game regarding the concepts of odd numbers, geometry, flat shapes, and congruence. furthermore, entertaining macanan game also encompass social, cultural, and moral values that students can learn in order to form good student character (prahmana & d’ambrosio, 2020). several previous researchers have developed mathematics learning by performing other games such as learning designs using numbers in congklak traditional game (muslimin, putri, & somakim, 2012); social arithmetic learning using the kubuk manuk game (risdiyanti et al., 2019); learning number operations employing tepuk bergambar game (prahmana, zulkardi, & hartono, 2012); learning number operations utilizing bermain satu rumah game (nasrullah & zulkardi, 2011), learning measuring applying patok lele game (wijaya, 2008), and learning time using gasing game (jaelani, putri, & hartono, 2013). however, there is a few studies which has implemented the learning design in schools particularly in the remote areas. therefore, the researchers developed a learning design, employing traditional game contexts, to be implemented in remote areas in optimizing mathematics learning in those areas. moreover, the learning design is expected to be a reference for mathematics teachers, schools, and researchers and also, may contribute to the cultural perspectives of mathematics education in indonesia particularly and other countries generally. the previous studies about squares and rectangles were examined by haris and putri (2011) which utilized the context of woven bamboo to teach the area of a square. it was implemented for third-grade students in state elementary school of 119 palembang. furthermore, wahyuni (2014) was designing the learning trajectory of square and rectangle properties applying the context of lapis legit cake implemented in 7th-grade students at islamic junior high school of hasanah pekanbaru. on the other hand, the context of plaid pattern cloth (haryani, putri, & santoso, 2015) and the reallotment activities (fitri & prahmana, 2018) have been implemented to teach the concept of square and rectangular area for junior high school students. however, all research studies were conducted in urban areas. methods this study employed a research design method in enhancing the quality of learning practices in the classroom specifically in remote area schools. research design is defined as a systematic and flexible method for improving the quality of education (simonson, 2006). besides, the research design is also defined as a method to develop or validate a learning theory the innovative learning of square and rectangle employing macanan traditional indonesian game 122 (gravemeijer & van eerde, 2009). this research was conducted through a hypothetical analysis of interactive schemes encompassing students' thinking strategies in the classroom. there are three stages implemented in this research, which are preliminary design, experimental design, and retrospective analysis (bakker, 2004; gravemeijer & cobb, 2006). the objective of preliminary design is to formulate a learning trajectory elaborated and refined in the experimental design stage (plomp & nieveen 2013). at this stage, the researcher performs observations on the curriculum applied based on school conditions and designs a hypothetical learning trajectory (hlt), accommodating learning objectives, learning activities, and conjectures or alleged student thinking strategies (prahmana, 2017). this conjecture provides as a new one which develop in every lesson, is flexible and can be revised during the experimental design stage to adapt to the conditions in the classroom (van den akker et al., 2006). after the hlt was designed in the preliminary design stage, the hlt was implemented at the experimental design stage to explore students' thinking strategies. furthermore, in the retrospective analysis stage, the conjectures in the hlt are compared with the results of implementation in the classroom (plomp & nieveen, 2013). the researchers analyzed the data with the teacher and supervisor to increase the validity of the study. the results are interpreted as a learning trajectory utilizing macanan traditional game. in the research design, it is not a design which works but how and why the design can work. a retrospective analysis of the hlt design was compared with the learning that has been performed, and the results explain how the square and rectangular concepts were generated from the traditional game, such as a macanan traditional game. remote school context square and rectangular shape learning designs employing macanan traditional game were implemented in the elementary school, pemunyian village, jambi province, sumatra island, indonesia. the distance between the nearest city of muara bungo city to the school is about 52.18 km. it can merely be reached by a land route along 97.1 km which is unpaved road with a travel time of about 2 hours to 3 hours (figure 1). figure 1. location of sunshine elementary school from muara bungo city. muara bungo city sunshine elementary school padhila angraini, rully charitas indra prahmana & masitah shahrill 123 the roads in the pemunyian village area are in the form of rocky soil, making it difficult for people to get in and out of the village. public transportation cannot reach the village due to the location far from the city, difficult access to public transportation, and non-existent communication access like telephone and internet. regarding the description, sunshine elementary school is known a remote and underdeveloped area. these conditions tremendously affect the sunshine elementary school particularly the quality of learning facilities and infrastructure, the educators, and the learning. figure 2 contains photos of the sunshine elementary school from the front view and in the classroom. in this school, learning resources such as books are seriously limited because access to purchase books in the town is undoubtedly far, and the availability of books in the muara bungo city is also limited. learning resources from the internet cannot be attained by students because there is no internet access at all. figure 2. front view of sunshine elementary school (left) and classroom (right). parents of students possess a telephone, but the signal is eminently weak and they must walk long distances and up the hill to obtain a signal. therefore, students primarily depend heavily on teachers' learning resources. however, the teachers’ academic qualifications at this school are also low. some teachers’ education is senior high school, and the others graduated from varied majors of bachelor degree but are teaching all courses. in other words, teachers in these schools did not graduate from primary school teacher education department. these conditions have limited the students’ learning process, and it is frequently problematic to understand the lesson due to the limited learning facilities and infrastructure as well as inadequate teacher qualifications. results and discussion when conducting research at the school, the researcher performed observations to identify curriculum and conditions of students in grade four. then, the researcher discovered that many students did not understand some basic mathematics material such as number operations regarding addition, subtraction, multiplication, and division. researchers also unveiled that there were still students who were not good at reading. besides, the prerequisite materials for learning square and rectangle are like elements in the plane figure that should have been learned by students, but there were still many students who did not understand. it makes it difficult for researchers to implement square and rectangular learning designs utilizing the macanan game. in solving the problems, before the researcher implemented the learning design, the researcher the innovative learning of square and rectangle employing macanan traditional indonesian game 124 provided treatment by providing private lessons to students as an expectation that students could understand the prerequisite materials first before studying square and rectangular material. after students understood the prerequisite material, the researcher applied the square and rectangular learning design employing macanan traditional game as a starting point for learning. the learning design consists of four activities, which are activity 1, playing macanan traditional game, activity 2, identifying the elements and properties of squares and rectangles, activity 3, determining the perimeter of the square and rectangle, and activity 4, determining the area of the square and rectangle. the summary of these activities is presented in table 1. table 1 summary of square and rectangular learning activities employing macanan traditional game activity and learning trajectory concepts of square and rectangle the learning descriptions activity 1 playing macanan traditional game the shape of square and rectangle students play the macanan game students find plane shape in the macanan game field students present and discuss the result of finding plane shape in front of the classroom activity 2 identifying the elements and properties of square and rectangle the elements and properties of square and rectangle students cut, past and color the plane shapes in the macanan game field. students identify elements and properties of square and rectangles students present the results of identifying elements and properties of square and rectangle, then continued to class discussion activity 3 determining the perimeter of the square and rectangle the perimeter of square and rectangle students solve problem about perimeter of the square students solve problem about perimeter of the rectangle activity 4 determining the area of the square and rectangle the area of square and rectangle students solve problem about area of the square students solve problem about area of the rectangle activity 1. playing macanan traditional game the first activity is to play macanan traditional game. the game of macanan is played by two people employing field games (figure 3) and rock games. the stones are considered people and a tiger. people have to catch the tiger to win, or the tiger eats all people to win. they start from either end of the field, and take turns of moving from one vertex to another. people and the tiger should walk according to the line in the tiger game field. the way people catch the padhila angraini, rully charitas indra prahmana & masitah shahrill 125 tiger is to surround the tiger and have to jump it successfully. when the tiger wants to catch people, it must jump over the people in the game field. figure 3. field of macanan game. the objective of this activity is to introduce and equate students' perceptions of the context of macanan traditional game to help them understand squares and rectangles. activity 1 comprises of three activities, which are activity 1.1, students play the macanan game; activity 1.2, students find plane shapes in the macanan game field, cut and past them on the template provided; and activity 1.3, students present and discuss the result of finding plane shape in front of the class. activity 1.1. students play the macanan game the teacher began activity 1 by exploring students' knowledge about the macanan game utilized as a learning context, then the teacher explained the activities to be performed. then, the teacher guided students to create groups of two people. the teacher distributed the student activity sheets (sas) and invited students to simulate the macanan game in front of the class. the simulation can be illustrated in figure 4. figure 4. students and teachers were applying macanan game. once students understood the way to play the macanan game, each group played the game at their tables (figure 5). before starting the game, the students implemented suit to decide to be the tiger (the winner) or the people (the loser). then, the students were playing in accordance with what was understood during simulations with the teacher. the innovative learning of square and rectangle employing macanan traditional indonesian game 126 figure 5. one group of students was playing the macanan game. activity 1.2. students find plane shape in the macanan game field after the students finished playing the macanan game, the teacher guided them to find planes shapes in the tiger field. students succeeded in finding plane shapes: squares, rectangles, and triangles (figure 6). figure 6. students were finding plane shapes on the macanan game field. activity 1.3. students present and discuss the result of finding plane shape in front of the class the teacher guided students to conduct a class discussion. the teacher instilled the students' responses about the context of the macanan game by inviting students to look back again at the macanan game field. then, the teacher demanded students to present different kinds of plane shapes on the playing field (figure 7). figure 7. the teacher invited students to look back again at the macanan game (left) and student with teacher were having a class discussion (right). padhila angraini, rully charitas indra prahmana & masitah shahrill 127 students then answered with the plane shapes they had discovered: triangles, squares, and rectangles. this activity can be portrayed in dialog 1. dialogue 1 teacher : pada bidang permainan macanan terdapat bangun datar dengan bentuk apa saja? [in the macanan game field, what kinds of plane shapes are there?] student : segitiga, segiempat. [triangle, square] teacher : berapa banyak bangun yang ada pada bidang permainan ini? [how many planes shapes are there in the macanan game filed?] student : segitiga, segiempat, segitiga lagi, persegi panjang. [triangle, square, triangle again, rectangle] activity 2. identifying the elements and properties of square and rectangle activity 2 aims to help students understand and identify the elements and properties of squares and rectangles. the elements comprise of lines, diagonals, points, and corners. the objective of this activity is also to help students understand the relation between elements of square and rectangular. activity 2 consists of several activities which are activity 2.1 containing students’ cutting, pasting, and coloring the plane shapes in the macanan game field; activity 2.2, identifying elements and properties of square and rectangles; activity 2.3, entailing students presenting the results of identifying elements and properties of square and rectangle, then a class discussion. before these activities began, the teacher allowed students to play the macanan game first in order to maintain and increase students’ motivation so that learning mathematics is fun for students. activity 2.1. student cut, past, and color the plane shapes in the macanan game field in this activity 2.1, the teacher began the activity by distributing paper in which there was a picture of the macanan game field. the teacher instructed students to cut the plane shapes that they have discovered in the game field. then, the pieces of the plane shapes were being re-attached to the student worksheet. after being attached to the student worksheet, students provide different colors to the plane shapes. figure 8. students were pasting the piece of plane shape (left) and students were coloring the plane shape (right). the innovative learning of square and rectangle employing macanan traditional indonesian game 128 figure 8 displays students who were pasting and coloring a piece of plane shape on the worksheet. the results of activity 2.1 can be identified in figure 9. figure 9. the student result in activity 2.1. activity 2.2. students identify elements and properties of square and rectangles in activity 2.2, the teacher demanded the students to identify the square and rectangular elements such as sides, diagonals, angles, points, and lines. students made a list of these elements and properties in the column provided in the worksheet. students could identify the elements of squares and rectangles. they were able to display the elements in form of images, defined in the student’s own language, and employed these elements to identify the properties of squares and rectangles. the results were that the students were successful in identifying a square's properties, such as having the same size of each side, having four angles of 90 degrees, and having two diagonals of the same length. the students were also able to identify a rectangle's properties, such as having two long and two wide sides, two diagonals of the same length, and four angles of 90 degrees. it is presented in figure 10. figure 10. the result of identifying the properties of square and rectangle. padhila angraini, rully charitas indra prahmana & masitah shahrill 129 activity 2.3. students present and discuss the results of identifying elements and properties of square and rectangle after activity 2.2, students presented the results in front of the class. it aims to explore the various thinking strategies of students. students produced a picture of the pieces of a plane shape from activity 2.1, then displayed the squares and rectangles' elements and properties in the shapes they had drawn. student presentations are presented in figure 11. figure 11. students were presenting the results of identifying elements and properties of square and rectangle in front of class. the teacher guided them to conduct class discussions. it aims to equalize perceptions about squares and rectangles' elements and properties. snippets from class discussions are demonstrated in dialogue 2. dialogue 2 teacher : take a look at the pictures you have made, what can be connected from one line to another line? do you know which side is called and how many are they? student : four. (students presented the line which connects the intersection points on the square) teacher : when it is known that there are four sides, which side are these? (the teacher provided the corner notation to the square image) student : side is the line connecting a to b, b to c, c to d and d to a. (see figure 11) activity 3. determining the perimeter of the square and rectangle activity 3 aims to help students understand the perimeter of squares and rectangles. this activity began with the teacher presenting a picture of the macanan game field and a student worksheet. then students solved some problems on the student worksheet associated with the macanan game field on the student worksheet. these problems are as follows. problem 3.1. determining perimeter of the square yogi and varif were playing the game with the position of the rock as displayed in figure 12. yogi was as the tiger and varif was as the people. the stones in the figure were illustrated with dots. the tiger was illustrated with red dot, while black dots represented people. it was 87 diskusi kelas setelah siswa menyelesaikan las, kemudian siswa dan guru melakukan diskusi kelas yang dipimpin oleh guru. hal ini bertujuan untuk mengeksplorasi berbagai strategi berfikir siswa sehingga diperoleh persamaan persepsi siswa yang sesuai dengan tujuan pembelajaran. pada diskusi kelas ini guru memerintahkan salah satu kelompok untuk mempresentasikan hasil pengerjaan di depan kelas. hal ini dapat dilihat pada gambar 4.37. gambar 4.37 siswa mempresentasikan hasil pengerjaannya 3. aktivitas 3: keliling persegi dan persegi panjang pada aktivitas ketiga ini, siswa akan mempelajari keliling persegi dan persegi panjang. permainan ini telah dimodifikasi dengan aturan yang telah disesuai dengan tujuan pembelajaran. modifikasi permainan ini bertujuan agar siswa dapat memahami tentang keliling persegi dan persegi panjang. selain itu pada aktivitas ketiga ini siswa mengetahui hubungan-hubungan antara setiap sisi yang ada pada persegi dan persegi panjang. sehingga, pada akhir pembelajaran siswa dapat menyimpulkan keliling persegi dan persegi panjang adalah penjumlahan sisi-sisi yang ada pada bangun tersebut. kegiatan awal pada kegiatan awal ini guru akan membuka pembelajaran terlebih dahulu dengan mengucapkan salam. kemudian, ketika salam selesai guru akan mengingatkan kembali pembelajaran sebelumnya kepada siswa. selanjutnya, guru akan menyampaikan tujuan pembelajaran pada aktivitas ketiga ini, yang mana tujuan pembelajaran pada aktivitas ini yakni siswa dapat menentukan keliling persegi dan persegi panjang. dilain hal, pada pembagian kelompok sama dengan pertemuan sebelumnya, akan tetapi ada beberapa orang yang berbeda dikarenakan the innovative learning of square and rectangle employing macanan traditional indonesian game 130 then connecting the dots on the drawing of the macanan game area to obtain a plane shape and then provided the shape color. from the plane shape identified, the students were asked whether they could determine the perimeter of the plane shape they discovered on the macanan game field if they understood that the distance between the points was 4 cm. then, the students were also demanded to answer the question: what is the perimeter of the plane shape and what is the formula? figure 12. the macanan game field on problem 3.1. after the students obtained an image of the macanan game field, the students connected the dots on the plane. the result of connecting these points was a square shape. then, students colored the square shape. after that, students solved the problem and wrote the answers to the student’s worksheet which can be identified in figure 13. figure 13. students’ answers of problem 3.1 for one square of sides 4 cm. figure 13 displays that students comprehended the perimeter of a square and could determine it correctly even though they still employed the basic method by adding up all the sides of the square. hence, it can be concluded that the student has built the understanding on the perimeter of a square and strategies to determine it. answers and the formula of the square perimeter are presented in figure 14. padhila angraini, rully charitas indra prahmana & masitah shahrill 131 figure 14. students’ answers for the perimeter and the formula of the square sides 5 cm. figure 14 illustrates that the students answered in their own language, the perimeter of a square is the sum of the sides of the square. the students determined the perimeter formula by adding the sides of the square 4 times. the students were also able to explain the formula in formal form. based on this answer, it can be concluded that the students had understood the definition and formula of the perimeter of a square. problem 3.2. determining perimeter of the rectangle jeni and selfi were playing the game with the position of the rock presented in figure 15. jeni was as the tiger and selfi was as the people. the stones in the figure were illustrated with dots. the tiger and the people were illustrated with black dot and red dots respectively. they connected the dots on the drawing of the macanan game area to obtain a plane shape and then provided the shape color. from the plane shape discovered, they students had to answer whether they could explain what was meant by the long side and the wide side, what the perimeter of the plane shape was, and what the formula was. figure 15. the macanan game field on problem 3.2. the result of connecting and coloring the rectangle was subsequently employed by students to solve the problem and write the answers to the students’ worksheet as displayed in figure 16. it indicates that students have answered using their own language that the long side is the longest side, and the wide side is the shortest side. the innovative learning of square and rectangle employing macanan traditional indonesian game 132 figure 16. students’ answers of problem 3.1. students also understood how to determine the perimeter of a rectangle and write the formula in formal form. however, students still found difficulties in defining the perimeter of a rectangle. the teacher then clarified that the definition of the perimeter of a rectangle was determined by adding the sides of the rectangle or by adding twice the length plus twice the width of the rectangle. activity 4. determining the area of the square and rectangle activity 4 aims to help students understand the area of squares and rectangles. this activity began with the teacher providing a picture of the macanan game field and student worksheet. then students solved several problems on the student worksheet associated with the macanan game field on the student worksheet. these problems are as follows. problem 4.1. determining area of the square jeni and selfi were playing the game with the position of the rock as follows (figure 17). jeni was as the tiger and selfi was as the people. the stones in the figure were illustrated with dots. the tiger and the people were illustrated with black dot and red dots respectively. they connected the dots on the drawing of the macanan game area to obtain a plane shape and color the shape. from this plane shape, they were demanded to answer whether they could determine how many tiles there were in the macanan game field, given the side length was 6 cm, what the area of the plane shape was, and if it was identified that the side length was 7 cm, what the area of the plane shape was. figure 17. the macanan game field on problem 4.1. after connecting the points, a square shape is obtained, and then students colored the rectangle. then, students solved the problem and wrote the answers to the students’ worksheet which can be observed in figure 18. padhila angraini, rully charitas indra prahmana & masitah shahrill 133 figure 18. students’ answers of problem 4.1 question 1 and 2 for square sides 6 cm. in figure 18, the student answered that there were 4 squares following the boundaries of the stone arrangement. the four tiles on the square represented the area of the square. students were able to identify the area of a square if they understood the length of the side was 6 cm. as seen in figure 18, students determined this by making 6 columns and 6 rows on a square so that several tiles were obtained in the square. then, they assigned numbers to each of these 1 cm squares and counted the number, which was 36 tiles. students were also able to determine the area of a 7 cm square, that was 49 square cm (figure 19). in this answer, the students determined the area of a square and wrote it in a formal form guided by the teacher. it can be identified in dialogue 3. dialogue 3 teacher : now, now that you know that the length of the side of a square is 4 cm, the area of a square is 16 cm, the question is where can you obtain 16 cm from? student : (students answered by repeating what was identified) teacher : the length of the side of a square is 4 cm so the area of a square is sixteen, to obtain sixteen, what operation do we have to use? student : multiplication. teacher : so that the area of a square is four times four, then, what length is 4 cm? student : side length. teacher : what is the side represented by? student : denoted by s. teacher : thus, it can be written that the area of a square is equal to s times s. figure 19. students’ answers of problem 4.1 question 3. the innovative learning of square and rectangle employing macanan traditional indonesian game 134 problem 4.2. determining area of the rectangle jeni and selfi were playing the game with the position of the rocks shown (figure 20). they connected the dots on the drawing of the macanan game area to obtain a plane shape and color the shape. from the plane shape discovered, they were demanded to answer whether they could determine how many tiles there were in the macanan game field, given the side length is 6 cm, what the area of the plane shape was, if they identified that the long side was 8 cm and wide side was 4 cm, what the area of the plane shape was. figure 20. the macanan game field on problem 4.2. after the students obtained an image of the macanan game field, the students connected the dots on the plane. the result of connecting these points was a square, then students colored the rectangle. after that, students solved the problem and wrote the answers to the students’ worksheet which can be observed in figure 21. figure 21. students’ answers of problem 4.1 questions 1 and 2 for rectangle 8×4 cm. figure 21 illustrates that students could determine the number of tiles in a rectangle, that was 8 tiles. students could also calculate the area of a rectangle by drawing a rectangle then making 8 columns and 4 rows. after that, they counted the number of tiles, 32, so that the rectangle area was 32 square cm. padhila angraini, rully charitas indra prahmana & masitah shahrill 135 figure 22. students’ answers of problem 4.1 question 3 rectangle 14×7 cm. in figure 22, students were able to shift their strategy to determine the area of the rectangle by utilizing multiplication between the long side and the wide side. therefore, based on the students' answers to figures 21 and 22, it can be explained that students understood the perimeter and area of the rectangle and its formula. the implementation results of the learning trajectory revealed that the macanan game possesses a role as a starting point in learning squares and rectangles. in activities 1 and 2, the macanan game area can be employed for starting points in identifying elements and properties of squares and rectangles, by identifying flat shapes in the macanan game area, then cutting them out and sticking to the sas which has been provided then identifying the elements and properties. in activities 3 and 4, the macanan game area is utilized as a starting point to discover the formula for the perimeter and area of squares and rectangles. the formulas can be generated from informal to formal forms by applying the squares in the macanan game area. as the principle in realistic mathematics education (rme) is to use a context that can formulate concepts from informal to formal forms and thinking strategies for students (sembiring et al., 2008; hadi, 2017; prahmana, 2017; nasution et al., 2018, sembiring, 2010). it is also supported by passarella (2021), who argued that the other rme’s principles, which are the heuristics of didactical phenomenology, guided reinvention, and emergent modelling, might keep student understanding to reinvent the mathematics concept in solving an experientially significant problem. furthermore, the results of the learning evaluation and also the answers in the sas show that students can understand geometric concepts regarding elements, properties, perimeter, and areas of squares and rectangles. students are also able to visualize an object with or without the help of the macanan game context. success in learning geometry depends on students who can understand formulas, visualize objects, observe mathematical structures, and implement problems in everyday life problems in learning geometry (laborde, 2015; jones & tzekaki, 2016; gunhan 2014). in this study, students can visualize objects, identify and use formulas, and perform mathematical reasoning. the result of a design research displays that with the local context and the rme approach, mathematics learning in remote area schools with limited facilities and infrastructure can still be conducted optimally. in fact, in remote areas, there are still many unique cultures and things in daily life which can be utilized as a context for learning mathematics. the innovative learning of square and rectangle employing macanan traditional indonesian game 136 conclusion this study has implications for rural areas that are clearly different from students' and schools' conditions in urban areas. recently, not many researchers have designed learning with the pmri approach implemented in remote areas. this research provides a contribution to a new knowledge of how pmri is applied in indonesian remote areas, although the sample size prevents generalization of the findings. furthermore, it contributes a small direction for mathematics knowledge and research in indonesia. for further 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(2019). new school mathematics curricula, pisa and pmri in indonesia (pp. 39–49). https://doi.org/10.1007/978-981-13-6312-2_3. https://doi.org/10.1007/s11858-008-0125-9 https://doi.org/10.1007/978-981-13-6312-2_3 southeast asia mathematics education journal, volume 10, no 1 (2020) 13 improving viii grade students’ mathematical problem solving ability through realistic mathematics education i made ari purwadi smp laboratorium undiksha singaraja, indonesia abstract this paper aims to describe the improvement of students' mathematical problem solving (mps) ability on the topic of simultaneous linear equations system through realistic mathematics education (rme). nineteen students of viii grade in smp laboratorium undiksha in the first semester of the academic year 2019/2020 participated as the research subject. this classroom action research consists of three cycles and every cycle contains three meetings. the data was collected by observation and essay-test before being analysed using descriptive statistics. the finding is that the average score of the students’ mps ability increased from 71,92 in the first cycle, to 73,2 in the second cycle, and to 75 in the third cycle. as demonstrated, the implementation of rme could increase students’ mps ability. hence, the implementation of rme has made students more motivated, more active during discussion and learning process, solve problems more easily, and communicate their understanding better. keywords: rme, mathematics, problem solving. introduction mathematics is an important part of civilization and possesses vital role in the development of science, engineering, and technology. aside from that, mathematics is also crucial in nurturing competent workforce. therefore, education system should put mathematics as one of the top priorities and ensure that every child will learn it. problem solving is one of the skills that can be nurtured through mathematics. national council of teachers of mathematics (nctm) lists five process standards in mathematics; one of them is problem solving. in addition, to enhance the quality of education and human resources, indonesian government through the curriculum 2013 put emphasize on the development of 21 st century’s skill, which includes problem-solving (kemendikbud, 2018; phonsa et al., 2019; tican & deniz, 2019). problem-solving is a higher-order thinking skill because it applies to multiple skills and should be learned by the students during the process of learning (demitra & sarjoko, 2018). if the students want to be a successful problem solver, they need to deeply understand the strategies and techniques of problem solving (marchis, 2013). furthermore, they also have to be able to think logically, analytically, systematically, critically, and creatively (surya & putri, 2016). problem solving requires students to engage in a problem which solution method is not obvious, thus demanding them to choose and use appropriate knowledge (nctm, 2000). unfortunately, reality shows that many students are currently struggling to solve mathematical problems (hendriana et al., 2018). for the students participating in this study, their mathematics achievement during the past year has not met the curriculum requirement, improving viii grade students’ mathematical problem solving ability through realistic mathematics education 14 specifically to achieve 70 for the minimum score of mastery criteria. this situation demonstrates the lack of the students’ problem solving ability (hendriana et al., 2018; surya et al., 2016). based on the preceding problem, using an innovative learning approach will help the students to minimize any problem they may have about mathematics problem solving on the topic of simultaneous linear equations system. therefore, selecting a learning approach which allows the students to construct their knowledge through their explorations on social and mathematics phenomena is the right way to do. one of the innovative strategies which can be applied in the mathematics classroom is the realistic mathematics education (rme) approach. rme was initiated in 1971 as part of wiskobas (primary mathematics) project in the netherlands (treffers, 1993). hans freudenthal, one of the key people of the project, was a firm believer that mathematics is a part of human activity (treffers, 1993). it must be connected to reality, stay close to the students, and should be relevant to the society (hadi, 2020; julie et al., 2014; yet et al., 2017). the main characteristic of rme is the central role of rich and realistic problem situation (van den heuvel-panhuizen & drijvers, 2014). these situations elicit the emergence of mathematical concepts in the beginning of the lesson, and in the end of the lesson, become the context in which the students can apply their freshly acquired knowledge. rme emphasizes the students' exploration and starting from the real world in the process to develop mathematical understanding (hadi, 2020). van den heuvel-panhuizen (in hadi, 2020) further stated that the students should learn mathematics by applying the mathematical concept in daily-life situations which make sense to them. it means the context must be meaningful and problems have to be imaginable, hence the word “realistic” in rme. students have to be allowed to reinvent mathematics concepts by doing it in the process of learning (ardiyani et al., 2018; fauzi & waluya, 2018; yet et al., 2017). thus, the role of the teacher in rme learning is as a facilitator who can establish the link between students’ world to the world of mathematics. following the explanation above, some researches showed that rme is effective in learning mathematics. first, research by widyastuti and pujiastuti (2014) found that rme is effective to enhance the students’ mathematical conceptual understanding and disposition. second, research by yet et al. (2017) concluded that rme has a positive influence on the students’ mathematics self-report. third, research by arsaythamby and zubainur (2014) revealed that rme is effective to help the students increase their learning activities. fourth, research by fauzi and waluya (2018) reported that rme has good effects in developing the students’ mathematical communication. fifth, research by idris and silalahi (2016) concluded that rme can improve the elementary school students’ ability in solving contextual problems. based on the aforementioned fact, the author considers it important to investigate learning approach that might be able to increase the students’ mathematical problem solving. this article aims to describe rme in the context of junior high school mathematics classroom in indonesia and how it affects the students’ mathematical problem solving. the research question to be answered in this paper is how does rme affect junior high school students’ mathematical problem solving (mps) ability? i made ari purwadi 15 methods this classroom action research conducted at smp laboratorium undiksha in the first semester of the academic year 2019/2020. nineteen seventh grade students participated in this study. the research method employed here is action research. in action research, teacher pairs practical teaching implementation with a research procedure, as a personal attempt to understand classroom phenomenon while being involved in the process (hopkins, 2008). it is conducted to improve the mathematics learning quality especially to the system, work methods, processes, content, and classroom learning situations/atmosphere (ni’mah, 2017). this research consists of three cycles with four phases in each cycle namely planning, implementation, observation, and evaluation or reflection (djajadi, 2019). the detailed activities of each phase are described as follows: planning after identifying the problem and initial reflection was carried out, the researcher compiled the action plan which consists of; (1) designing learning scenario and tool (lesson plans and worksheets), (2) developing instruments (validation sheets, observation sheets, problem-solving test); (3) preparing media and teaching aids; and (4) validating instruments. implementation the activities in the implementation phase were as follows; (1) implementing the action plan; (2) examining the learning trajectory, and (3) improving or enriching the learning process. there are three meetings in this phase. observation and data collection this phase was conducted at the same time as implementation phase. the observer used observation sheets as a guide. furthermore, the data on the students’ mathematics problem solving ability was also collected through an essay test. evaluation-reflection the evaluation and reflection were conducted on each cycle. the reflection was used as a basis for planning the next phase. the data collected in this study consist of the learning process and students' mathematics problem solving ability. these data were collected through observation sheets and essay tests. the questions in the mps test use real-life contexts and there is no obvious clue on mathematics concepts needed to solve it. the learning process data were selected for relevant details and described in relation to the research question. the data for the students' mathematics problem solving ability were analysed using descriptive statistics. results and discussion firstly, the researcher prepared the lesson plans and worksheets for nine meetings (three meetings for each cycle). the lesson plans, worksheets, and tests are designed according to rme approach. all of them designed for guiding students to find the mathematics concept independently and the teacher just being a facilitator. improving viii grade students’ mathematical problem solving ability through realistic mathematics education 16 there are three rme principles that was used as a reference by the researcher in designing the lesson plans and worksheets, namely: 1) guided reinvention through mathematization, 2) didactical phenomenology, and 3) emergent modelling. these principles manifest themselves in the worksheet through 1) use of phenomenological exploration or contexts, 2) the use of model, and 3) intertwining of various learning strands (hadi, 2020). the learning material employs local-cultural contexts which are experientially real to students and does not contain information on how to use the formula directly. the activity designed by the teacher also have to engage students in the discussion process (putri et al., 2015). the teacher also applied the classroom socio-norms. the teacher emphases that the communication occurred not only between teacher and students but also among the students. teacher ‘s role to explain the context or problem or concept are minimal and the students are given more chance to explain and answer the question. in other hand, the students had to be active in the learning process. secondly, the researcher conducted a preliminary test to determine the students’ initial ability on problem solving. the researcher analysed the test result by statistics descriptive. the researcher used minimum completeness criteria of 70 in reference to the 2013 curriculum. based on the test, the highest score was 79, while the lowest was 40. according to the completeness criteria, ten students did not meet the target score and nine students did. the average score was 65,5. the research result in every cycle is as follows. first cycle the first cycle of this research was done in three meetings and it began with learning about how to make mathematical model of a context. at the beginning of the learning, the teacher instructed students to make five groups and stimulated them to recall the concept of algebra as the pre-requisite knowledge. teacher started the discussion by showing a context, followed by the students’ discussion about how to transform the real word problem into a model with mathematical symbol. the following is a context that was displayed by the teacher: “the price of 4 pencils and 5 notebooks is idr 23,000, while the price of 2 pencils and 3 notebooks with the same type is idr 13,000. make the mathematics model and what is the price of one notebook” the following is a brief segment illustrating group 1 discussion: student 1 : (s1 makes the model by drawing 4 pencils and 5 notebook) figure 1. strategy of student 1 by using pictorial. i made ari purwadi 17 students 2 : why don’t you consider the pencil is “p” or “a” or “x”? it makes them simpler, like mine. do you remember what we learn in algebra class? you can assume the unknown thing by variable. figure 2. suggestion from student 2. student 1 : wait, i’m confused, i think it will be easier if i draw the question. i want to guess the price of this pencil by the real price in the market. yesterday, i bought my pencil idr 2.500 and i think i must try and guess until get idr 23.000. (thinking) student 2 : well, this is mine (s2 shows his work) figure 3. mathematics model of students 2. student 1 : (after 10 minutes) i give up, i can’t find the appropriate price. i will try to do it like you. student 2 : yeeahhh, i got it, (s2 gets the model “4p+5b = 23.000 and 2p+3b = 13.000”). student 3 : why don’t you use pencil = x and book = y? student 2 : ya, it is variable, you can use other alphabets. student 1 : i changed my model, okay, done!!! we get 4p+5b = 23.000 and 2p+3b = 13.000 with p = price of one pencil and b = the price of one notebook. student 1 : mr. ari, are we correct? (show the model to the teacher) teacher : hmmm, great job, we will discuss with other groups soon. improving viii grade students’ mathematical problem solving ability through realistic mathematics education 18 student 2 : now, we must get the price of one pencil. wait, i will look at the example. i learned this material last night. we can use elimination method? figure 4. work of students 2 teacher : are you sure? check it again! student 2 : wait sir, (write the solution) student 3 : i found the price of one notebook (show her work) figure 5. strategy (pictorial) belong to student 3 student 2 : woo, great idea. teacher : excellent, how if you change into mathematics symbol? student 3 : ya sir, i will try. student 3 : (15 minutes later, s3 shows the work) i made ari purwadi 19 figure 6. student 3’s work (symbolic). students refined and adjusted the models. the teacher observed that the students can make various and different models. in the next meeting, they learned how to formulate the mathematical model until generalizing. lastly, students determine the value of variables without inducing the techniques. the example of the question in the mps test of the first cycle is as follows. “hadi buys 7 cups of coffee and 4 pieces of toast with the cost is idr 51.000 and budi buys 5 cups of coffee and 4 pieces of toast with the cost is idr 41.000. find the cost for each item.” figure 7 showed the sample of student’s answer. figure 7. student’s work at the first cycle. the highest score of the test was 90, while the lowest was 56. seven students (37%) did not meet the target score, while 12 students (63%) did. the average score was 71.92. this score increased from the preliminary test. the first cycle's reflection reveals several problems in carrying out the lesson as follows; (1) only a few students participated in discussion, (2) the students needed a lot of teacher's guidance, (3) the high-ability students dominated the discussion, (4) some students were still shy and reluctant to offer their comments and suggestions, as well as (5) the scaffolding from peers did not optimally run. based on the reflections, to enhance the learning quality in the second cycle, the researcher made some improvement by (1) evaluating the learning context and problem by redesigning problems to be more familiar with the students, (2) requiring all students to engage in discussions and giving rewards to the most active groups and students, (3) asking the students who assist their friend not to give the whole information at the beginning but improving viii grade students’ mathematical problem solving ability through realistic mathematics education 20 only tried to help by giving the necessary clue, and (4) at the end of the meeting, the students were encouraged to reflect and make conclusion. second cycle the second cycle began from introducing elimination technique for solving system of linear equation with two variables. it started from elimination because in the first cycle, the students can find the ways to eliminate and subtracted one variable on the model. teacher redeveloped the worksheet in order to make students can understand the concept easily. students discussed their model and the plan of solution with other students. after that they refined and adjusted until found the solution. the teacher observed that some students can make unique ways to find the steps of elimination technique. the teacher shows the picture on the slide and students are instructed to get the price of one umbrella and one cap. figure 8. first context at the 5 th meeting (modified from holt, 2006). the following is a brief segment that illustrates a discussion in group 2: student 5 : eliminating in bahasa is “menghilangkan”, so we have to eliminate one variable on the equation. student 6 : it is similar with how to get the price of the notebook that we learned last week, yes? student 5 : okay, i will try to change the pattern. at the first line there are two umbrellas and a cap but in the second line there are one umbrella and two caps. when the umbrella in the first line change by one cap the price will decrease 4.000 to be 76.000. so that, if the pattern is continued, we will get 3 caps with price 76.000-4.000 = 72.000. then a cap is 24.000. teacher : excellent, other students have other opinion? student 6 : if the first line is changed to 3 umbrellas it will equal to 84.000, thus the price of an umbrella is 28.000. here is the example of the question in the mps test of second cycle. “the sum of two numbers is 36 and their difference is 9. find the two number.” idr 80.000 idr 76.000 i made ari purwadi 21 figure 9 showed the sample of student’s answer. figure 9. student’s work at the second cycle. the highest score of the test in the second cycle was 78, while the lowest was 42. four students (21%) did not meet the target score, while 15 students (79%) did. the average score of students' mps was 73,2. the average score increased 1,77% from the previous cycle. the highest score achieved was significantly smaller than the first cycle, more students met the target score. several problems were identified during the reflection namely (1) the high-ability students still dominate the discussion; and (2) the scaffolding from the peers did not run optimally. these problems were taken into consideration during the refinement of the learning activity, which includes (1) allowing the passive students to present their work in front of the class; and (2) emphasizing to assist students only by providing clues to solve the problem. third cycle the third cycle also conducted in 3 meetings and it began by an introduction to the substitution technique. the topic of discussion was mixed (elimination and substitution) technique for solving system of linear equation with two variables. teacher instructed the students to recall the value of function and how to enter the element of domain to the function to obtain the range. the lesson started with the following problem “two t-shirts cost idr 200.000, one t-shirt and one trouser cost idr 400.000. find the cost of each item”. the following is a brief segment that illustrates a common response: student 9 : ah i see, i buy 2 t-shirts and equals idr 200.000 so one of them is equal to idr 100.000 (thinking). teacher : all right, then? student 9 : let’s change the price of one t-shirt into the second model. aha, i got it. one of it 100.000 so we can write 100.000 + trouser = 400.000. yes, one trouser equal to 300.000. teacher : excellent, other students have another opinion? or similar ways? teacher : anybody know the synonym of “change”? student 10 : yes, change is equal to substitute. teacher : right, we can use it to determine the price of the item. in the third meeting the students applied the mix method to solve problems. here is the example of the question in the third cycle mps test. “the sum of two numbers is 36 and improving viii grade students’ mathematical problem solving ability through realistic mathematics education 22 their difference is 9. find the two number.” figure 10 showed the sample of student’s answer. figure 10. student’s work at the third cycle. the highest score of the test in the second cycle was 100, while the lowest was 45. two students (21%) did not meet the target score, while 17 students (79%) did. the average score of students' mps was 75. the average score increased 2,45% from the previous cycle. the highest score achieved was significantly higher than the second cycle and the number of students who met the target score increased. the finding suggests that the students' mathematical problem solving ability increased throughout the cycles. this finding is relevant to widyastuti and pujiastuti (2014) which revealed learning with rme can improve fifth-grade students' ability to understand mathematics concepts and logical thinking. saleh et al. (2017) also stated that rme can improve students' mathematical reasoning ability. furthermore, the students' engagement during classroom activity also increased. this finding can be attributed to the use of contextual problems, which is in line with the reality principle of rme (van den heuvel-panhuizen & drijvers, 2014). the learning activities designed in this study was consistently started by presenting contextual problems to the students, which are familiar to the students and relevant to the society they live in. contextual problems help fostering an idea that mathematics is close to real life and useful for solving daily life problems (de lange, 1987). this understanding can motivate students to learn mathematics. purwadi et al. (2019) also revealed that learning mathematics by presenting problems and real objects will make students feel that mathematics is useful. besides, learning also emphasizes understanding the processes rather than learning algorithms. at the beginning of learning, the teacher provided students with realistic problems and then allowing them to understand the problems and the context clearly. the student attempted to describe to others the problem in their own words based on their understanding. students can use various models such as tables, charts, pictorial or visual images, and cartesian coordinates in solving the problem. this is in line with the activity principle of rme, which states that students should be active participants of the learning process and in-charge of their own understanding (van den heuvel-panhuizen & drijvers, 2014). afterward, the teacher led students to discuss their different strategies that culminates in a single solution. furthermore, the teacher also provided the students with hands-on activity with real objects. it made students more engaged in learning activities. widyastuti & pujiastuti (2014) also revealed that through rme students will better see mathematics as an effort to solve daily problems thus it can attract the students' interest and motivation to learn math. i made ari purwadi 23 in addition, the researcher also observed the importance of selecting problems or contexts that evoke the students’ interest and engage them in the learning process. in this initial stage, researchers try to provide non-routine problems of shopping and selecting items to buy in a supermarket or store. it is important to remember that the problems must be in the student's zone of proximal development (vygotsky, 1978, p. 86). meaning, the problems are difficult enough that they cannot solve them easily with their prerequisite knowledge and thus, push them to devise new strategy and develop new knowledge, but not too difficult such a way that it causes frustration and despair. conclusion based on the results and discussion above, it can be concluded that the students' mps ability and engagement increase with rme activities. however, students' mps abilities were not yet optimal since the average score of students' mps was still lower than 80. this research is expected to provide a good practice in the application of rme to improve students' mps ability. the author also hopes to contribute to the achievement of educational goals and enrich the results of existing research. in order to get optimal results, the future research needs collaboration with other learning tools or media because creative and innovative learning is very important to support the success of learning. the limitation of this study is the small number of participants and the specific context. the findings here may not be generalizable nationwide, however cities and provinces with similar characteristics may obtain similar result. references ardiyani, s. m., gunarhadi, g., & riyadi, r. 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(2017). the effects of realistic mathematics education on students’ math self reports in fifth grades mathematics course. international journal of curriculum and instruction, 9(1), 23. improving viii grade students’ mathematical problem solving ability through realistic mathematics education 26 southeast asia mathematics education journal, volume 10, no 1 (2020) 41 the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division nanang setiadi sd bandut sedayu, bantul, yogyakarta n4n4n6std@yahoo.com abstract this paper discusses the use of realistic mathematics education (rme) as an alternative approach to enhance indonesian 5th-grade students’ ability in the topics of multiplication and division. the study presents the analysis of indonesian 5th-grade students’ difficulties in applying vertical multiplication and division, that is in reapplying the steps of the method. furthermore, it describes a mathematics teaching learning practice to stimulate students in constructing their strategies, mathematical models, and number sense in solving mathematical problems that involve multiplication and division. in implementing rme, the steps taken to improve the learning process were: (1) analysing in detail the difficulties of students in vertical multiplication and division, (2) providing contexts of mathematical problems that can stimulate students to think mathematically, (3) holding a class mathematics congress, and (4) conducting a test to measure students’ achievement. the implementation of rme has helped the 5th-grade students to improve their ability in multiplication and division. there were more students whose grades passed the minimum passing score. moreover, there was an increase in the class average test score. keywords: rme, 5th-grade students, multiplication, division, vertical method. introduction multiplication and division are two numeracy skills learned in primary school. these skills are developed once students have mastered the skills of addition and subtraction, which become initials to learn about multiplication and division. slavin (2005) argues that multiplication is the sum in a quick way. meanwhile, karim et al. (1996) states that multiplication is a repeated sum. conversely, division can be interpreted as a repeated reduction. one of algorithms to do multiplication and division is by using vertical method. it is popular worldwide. however, there are some difficulties when children, including indonesian primary students, applying the procedure. several researches (ahmad & sivasubramaniam, 2010; hasan, 2012; rosyadi, 2016) indicate that primary students are dealing with hardship in using vertical multiplication and division because they are confused with the place value in the algorithm. in line with the relevance studies, the difficulty in doing vertical multiplication and division is also confronted by students in sd bandut, bantul, yogyakarta. the score of mathematics final assessment of grade 5 students in sd bandut on the first semester were low. only 2 out of 26 students were able to achieve the mathematics minimum passing score, that is 70. it reflects that there are problems in learning mathematics. one of which was the difficulty in applying vertical multiplication and division mailto:n4n4n6std@yahoo.com the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division 42 to solve problems for calculating speed, water debit, and scale. on the other hand, the 5thgrade students are expected to be proficient in operating the vertical method to solve more complex mathematical problems. another problem is about students’ perception of mathematics. the 5th-grade students in sd bandut consider mathematics as a difficult and scary subject. the negative perception influences the development of students’ learning (yusmanita et al., 2018). to bring mathematics closer to students’ life, we plan to use suitable mathematical activities that is based on realistic mathematics education (rme) approach. not only real or related to students’ daily life, mathematical activities in rme can also be imagined by students (van den heuvel-panhuizen & drijvers, 2014). mathematical activities in this study focus on contexts. according to fosnot and dolk (2001), contexts play an important role to stimulate students to solve mathematical problems. the context of the mathematical activities is designed to enable students to construct their knowledge in mathematics. in addition, the learning context must be able to encourage students to come up with different strategies in solving mathematical problems. by using contexts, we expect students are encouraged to build their mathematical concepts. freudenthal (1991) mentioned that rme can encourage students to build their own knowledge by connecting students’ prior knowledge related to their experience in the daily life to the mathematical problems. moreover, mathematics is a human activity to reason about problems so that the solution makes sense. according to the aforementioned background, we want to determine students' obstacles in applying vertical multiplication and division. furthermore, we would like to explore the impact of implementing rme to students' ability in multiplication and division. methods the participant of the study was 26 grade 5 students in sd bandut which consist of 15 boys and 11 girls. there were four inclusive students who had slow learning barriers that require special attention. the study adapted action research as its methodology, which its purpose is to improve the quality of teaching learning practices (gall et al., 2003). in addition, creswell (2012) elaborates an action research as follows: action research designs are systematic procedures used by teachers (or other individuals in an educational setting) to gather quantitative and qualitative data to address improvements in their educational setting, their teaching, and the learning of their students. in some action research designs, you seek to address and solve local, practical problems, such as a classroom-discipline issue for a teacher. in other studies, your objective might be to empower, transform, and emancipate individuals in educational settings. (p. 22) the following are the steps applied in this study: 1. collecting data in the form of students’ scores in the pre-test of multiplication and division; 2. analysing students’ problems in applying vertical method for multiplication and division; 3. applying rme; and 4. conducting a test to measure the students’ achievement. nanang setiadi 43 results and discussion problems encountered the following are the difficulties of the 5th-grade students of sd bandut in operating vertical multiplication and division. 1. inaccuracy in multiplication figure 1. example of an inaccuracy. figure 1 shows a student’s solution for a question given: "mr. rian made a catfish pond with the size of 12 m length, 5 m width, and 30 m height. what is the pool volume?" solving the problem above involves multiplication of three numbers: 30, 12, and 5. the student did 30 × 12. the result was accurate, 360. however, the multiplication did not require a carrying step. the next multiplication was 360 × 5. at this stage, the student seemed to have difficulty with the multiplication of 5 × 3 plus the carried number, 3. the result should be 18 but it became 33. thus, final result was inaccurate. 2. incorrect in arranging the results of multiplication figure 2. answer lines arrangement. the question of the solution illustrated in figure 2 is "a cuboid water reservoir is 13 meters length, 11 meters width, and 18 meters height. what is the volume?" the problem was solved by multiplying 13, 11, and 18. for 13 × 11, the student did not experience difficulties in finding the accurate result, 143. the next step was to multiply 143 by 18. here, the student found difficulty in arranging the results of multiplication of 143 × 8 and 143 × 1. the results of the multiplications are 1,144 and 143. in writing the result, the student should have only shifted 143 to larger number place. regarding the multiplication of 143 × 1, the number value of 1 is tens place. therefore, there must be only one 0 put behind 143. however, the student put 00 which made the addition of the two answer lines inaccurate. the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division 44 3. students do not understand under what condition they need to carry the number figure 3. the carrying step. in the multiplication shown in figure 3, the student solved 13 × 11 × 18 with inaccurate results. the reason was the mistake in doing the second multiplication, 143 × 18. the multiplication began by multiplying 8 to 3 with a result of 24. the student wrote digit 4 and carried 2. next, the student multiplied 8 by 4 with the result of 32. the student added 32 with 2 as the carried number and the result was 34. student should have written 4 and carried 3 because there was another multiplication, 8 × 1. however, the student wrote 34, which became error in result because it was too big. 4. students forget to do one step multiplication figure 4. the steps of the student. at the completion of the problem shown in figure 4, the student had no difficulty in the concept of carrying in multiplication. however, the results of the multiplication became inaccurate because the student made a mistake in the second multiplication of 361 × 19. it was notable because the result was correct. a problem arose when multiplying 361 by 1. the result should have been 361, but it was written 61. 5. students have difficulty in operating vertical division with 0 remainder figure 5. remainder of 0. nanang setiadi 45 figure 5 shows the student's solution to the question: "cuboid toy packages with the length of 5 cm, the width of 3 cm, and the height of 2 cm will be put into a big cuboid with 30 cm length, 11 cm width, and 10 cm height. count how many packages of toys can get into the big cuboid?" in solving the problem, the student calculated the big and small cuboid volumes through multiplication. the results of these multiplications were accurate: 3,300 cm3 and 330 cm3. the next step was to divide 3,300 by 330 to find out how many toy packages could fit into the big cuboid. multiplication with these numbers is easy if the student knew that 3,300: 330 is equal to 330 × 10. on the other hand, the student used the vertical division by initially dividing 330 by 330 with 1 as the result. the student subtracted 330 with 330, and the result is 0. the problem arose when there was 0 left over to divide. after bringing a copy down, the remainder became 00. the student copied the remainder 00 behind 1, so that the final result was 100. 6. students’ steps were inaccurate in division figure 6. the student’s accuracy in division. as seen in figure 6, the student divided 1,000 by 8 with vertical division. the division began with dividing 10 by 8 and resulted with 1 and a remainder of 2. the result 1 was written above, while 2 as the remainder was written below. then, digit 0 from 1,000 was copied and brought down to change 2 to 20. next, the student divided 20 by 8. at this stage, an error occurred. he wrote 12 as the result without any remainder. therefore, according to the student, 1,000 by 8 was 112. this result was inaccurate considering 112 × 8 is not 1,000. by knowing the facts above, it can be concluded that the vertical multiplication and division is a difficult arithmetic operation for the 5th-grade students in sd bandut. the method had been studied for approximately three years. however, many students had not been able to apply the methods properly. the students’ difficulties in multiplication were carrying numbers, arranging the answer lines of the multiplication, predicting the result of the division, and working with 0 as a remainder. thus, the teacher the 5th-grade students in sd bandut should look for more efficient ways to help students carry out multiplication and division operations. the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division 46 the implementation of rme to help students in multiplication and division considering the findings about students’ difficulties in applying vertical operations, rme offers solutions. the following are the steps for implementing rme to overcome the multiplication and division difficulties. a. using a context for mathematical problems here are contexts given to students: 1. the context of counting fruits figure 7. using the fruit context (fosnot & dolk, 2001). for the 5th-grade students, the context shown in figure 7 enables them to use multiplication for the rows and the columns to find out the result. in reality, many 5th-grade students in sd bandut did not memorize the result of 6 × 9. many students tried to count one by one. the context was chosen because it is impossible for students to solve the multiplication by using vertical method. to work with the vertical multiplication, two digits are needed for at least one of the given numbers. the fruit context is rich to emerge different strategies. this will lead students to count efficiently. using the context, students were involved in the discussion of counting the fruits in the picture below: figure 8. context of fruit grouping (fosnot & dolk, 2001). the context of the calculation in figure 8 is intended to stimulate students to do different kinds of grouping. the grouping strategy helps them to calculate a larger number. therefore, in order to solve 6 × 9, students are expected to group numbers that make them easy to calculate. 2. the context of counting cakes figure 9. the cake context (fosnot & dolk, 2001). nanang setiadi 47 students were given an additional context about counting cakes as presented in figure 9. the cakes were placed in two parts: the top and bottom. each section had the same capacity of 20 cakes. the students were provided with three cakes placed in three separate containers. the first, the second, and the third container contained 36 cakes, 16 cakes, and 20 cakes respectively. students were expected to be able to find out that 36 is a grouping of 20 plus 16 through the context. another possibility was that students would be able to find out that 36 is the result of 40 − 4. this context shows that the use of vertical multiplication is unnecessary. in fact, multiplication can be solved by addition and subtraction. there is no carrying process in the multiplication. using the context, we can reduce the students’ error in doing multiplication. 3. the multiplication with bigger numbers the activity continued with an investigation of multiplication with a bigger number, that was 16 × 16. students were asked to determine the results of the multiplication by using other strategies than the vertical multiplication. they were expected to be able to generalize the method that has been found previously on multiplication 6 × 9 to solve multiplication 16 × 16. in this context, students' flexibility to multiplied numbers was developed. students started to look for efficient ways to solve the problem accurately. b. the context of division the first context given to students was the following story: "fitra suseno wants to put 28 pieces of cake in plates to prepare for the party. he wants to put 7 pieces of cake on each plate. he wants to know how many plates he needs". in the above context, students were shown with a simple context about dividing the same amount of food on each plate, which is 7 pieces of cake each. based on this context, students were expected to be able to understand the meaning of the division by grouping strategy. the context of the story actually involves the division of 28:7. however, the completion of the context will lead students to solve the division problem by reducing it repeatedly until it runs out and resulted in 4. the second context provided was also in the form of a story as follows: "bandut primary school will hold a tour. the number of teachers and students is 372 people. the capacity of 1 bus is 50 people. how many buses does the school have to order for accommodating everyone?" the second context presents more challenges to students because in the second context involves 372 divided by 50 with a remainder. this context helps students develop their logic in the completion. students were expected to be able to discover the concept of division by breaking 372 into 50s. as the expected results, students got 7 groupings of 50 and the remainder was 22. the next challenge was to reason the number of buses that must be ordered, whether 7 or 8 buses. thus, we expected the grouping strategy would appear. the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division 48 c. math congress in the classroom mathematical class congresses were conducted after all students had found their own ways to solve multiplication and division problems. this congress was a presentation from students about the methods they use. presentation is an important part of rme because students are not only learning to share but also to develop concepts of thoughts and to organize them to be conveyed to others. students who are presenting must be prepared with all possible questions and answers. it makes students to be able to build deeper knowledge from others. the mathematics congress provides a room for students to discuss the methods they use. and to inspire each other about strategies in multiplication and division. d. the multiplication and division test the development of student knowledge was measured by the re-examination of multiplication and division of cube and cuboid volumes. the test results were analysed and compared with the pre-test. the indications of the development of students' abilities can be seen from the increase of the average score and the increase in the number of students who passed the minimum passing score. based on the activities that have been carried out with the rme approach, the results of the learning improvement were as follows: a. the students’ discovery through the context of counting fruits figure 10. the discovery of the first multiplication model. the completion found in figure 10 shows that the student used the repetitive addition strategy to count the fruits in the picture. figure 11. the discovery of the second multiplication model. figure 11 presents the strategies used by the student in grouping 10 and 20. the idea was very helpful to facilitate the calculation. nanang setiadi 49 figure 12. the discovery of the third multiplication model. in figure 12, the student used a strategy similar to the previous one. he broke down the 6 × 9 into 3 × 9 plus 3 × 9. the interesting part was at the step of 27 + 27, the student changed it into 20 + 20 plus 7 + 7 to make it easier to add. figure 13. the discovery of the fourth multiplication model. the strategy used in figure 13 was interesting because the student changed 6 × 9 to 6 × 10. the result of multiplication was 6 × 10 minus 6 × 1. it was an elegant strategy because the multiplication 6 × 9 is close to 6 × 10. by changing it into 6 × 10, the multiplication became easier. after the activity in the context of fruit was finished, students were invited to conduct an investigation with a multiplication of a bigger number, it was 16 × 16. at this stage, the students were expected to be able to generalize the models that had been developed to solve 9 × 6. the following are some of the students' findings and their explanations. figure 14. the first multiplication model of 16 × 16. as shown in figure 14, the student changed the multiplication of 16 × 16 into 16 × 20 then subtracted it by 16 × 4. the interesting point was when subtracting 320 by 64, the student used the number line model. students who used models apparently were able to do multiplication such as 16 × 20 and 16 × 4. the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division 50 figure 15. the second multiplication model of 16 × 16. the solution presented in figure 15 was interesting and efficient. the students changed the multiplication of 16 × 16 to 10 × 16 plus 5 × 16 and 1 × 16. this model helps them to do multiplications. 10 × 16 is usually easy for students. meanwhile, the result of 5 × 16 is half of 10 × 16. in general, this model is indeed an efficient strategy in solving multiplication. b. students’ discoveries through the context of calculating plates to hold snacks and ordering buses the following are students' discoveries and their explanations: figure 16. the first strategy in distributing passengers. as figure 16 shows, the student drew food groupings with the number in which per group is 7. the result obtained is 4. figure 17 illustrates the student's discovery in the bus context which involved 372:50. figure 17. the second strategy in distributing passengers. in finding the solution, the student drew 8 buses as a result of the division. the model was situational because it used the images mentioned in the context of the given problem. although the result was correct, the solving model was less efficient because drawing takes more time. figure 18. the third strategy in distributing passengers. nanang setiadi 51 the solution shown in figure 18 used a repeated subtraction with 22 as the remainder. the use of the model was efficient because it only subtracted 50 for 7 times. however, the model was easy for the students who already had mastered the subtraction. figure 19. the fourth strategy in distributing passengers. the solution presented in figure 19 was done by drawing buses containing 50 passengers each. the passengers were added up to 350. the remainder of 22 passengers were accommodated by an additional bus. thus, the number of bus that needs to be ordered was 8. the division solution model used repeated addition. figure 20. the fifth strategy in distributing passengers. the solution illustrated in figure 20 was attractive and efficient. the settlement was done by breaking the division of 372:50 to three groups of 100: 50 and a group of 50:1 with a remainder of 22:50. in the context of booking a bus, the result was 7 plus 1 bus for 22 passengers only. c. the comparison of the test results before and after rme and its analysis figure 21. the list of the exam scores. 0 20 40 60 80 100 120 s t u d e n t 1 s t u d e n t 2 s t u d e n t 3 s t u d e n t 4 s t u d e n t 5 s t u d e n t 6 s t u d e n t 7 s t u d e n t 8 s t u d e n t 9 s t u d e n t 1 0 s t u d e n t 1 1 s t u d e n t 1 2 s t u d e n t 1 3 s t u d e n t 1 4 s t u d e n t 1 5 s t u d e n t 1 6 s t u d e n t 1 7 s t u d e n t 1 8 s t u d e n t 1 9 s t u d e n t 2 0 s t u d e n t 2 1 s t u d e n t 2 2 s t u d e n t 2 3 s t u d e n t 2 4 s t u d e n t 2 5 s t u d e n t 2 6 a v e r a g e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 students' math scores before rme after rme the implementation of realistic mathematics education (rme) to support indonesian 5th-grade students to learn multiplication and division 52 figure 21 shows the data for exams before and after the implementation of rme. the data presented are the mathematics test scores of 5th-grade students in sd bandut in the topic of cube and cuboid volumes. test 1 was the pre-test that carried out before the implementation of rme. meanwhile, test 2 was the result of post-test. based on the data above, there are some differences that can be seen from the results of exams before and after the implementation rme. in the pre-test, there were three students passing the minimum passing score, that was 70. in the post-test, there were seven students passing the minimum passing score. the highest score in the pretest was 90, while the highest score in the post-test was 100. there was an increase in the average score, that was 26.54 in the pre-test to 38.85 in the post-test. moreover, there were less students who scored 0 (from seven students in pre-test to three students in the post-test). the increase of students’ score is related to the chosen mathematical problems that enable students to use different strategies in solving problems. the strategies that students used are the reflection of their prior knowledge. while working on the problems, students relate their strategies to the mathematical concepts, in which influences students’ understanding about the topic (hiebert, 1984). to sum up, the implementation of rme has helped the 5th-grade students of sd bandut in improving their multiplication and division ability. conclusion based on the discussion, we conclude that the implementation of rme has been able to improve the ability of the 5th-grade students of sd bandut in solving multiplication and division problems. students were given an opportunity to build their own knowledge and to gain a deeper understanding on the concepts of multiplication and division. these affect students to think logically in solving multiplication and division problems and to emerge different strategies other than vertical multiplication and division. all in all, the implementation of rme were able to increase the creativity of the 5th-grade students of sd bandut in solving multiplication and division problems. references ahmad, n., & sivasubramaniam, p. (2010). multiplication and the reference sum method. international conference on mathematics education research. 8, pp. 72-78. procedia social and behavioral sciences. doi:https://doi.org/10.1016/j.sbspro.2010.12.010 creswell, j. w. (2012). educational research: planning, conducting, and evaluating quantitative and qualitative research (4th ed.). boston: pearson education inc. fosnot, c. t., & dolk, m. (2001). constructing multiplication and division. portsmouth: heinemann. freudenthal, h. (1991). revisiting mathematics education: china lecturers. dordrecht: kluwer academic publishers. gall, m. d., borg, w. r., & gall, j. p. (2003). educational research: an introduction (7th ed). boston: pearson education. nanang setiadi 53 hasan, q. a. (2012). rekonstruksi pemahaman konsep pembagian pada siswa berkemampuan tinggi. seminar nasional matematika dan pendidikan matematika. yogyakarta: jurusan pendidikan fmipa uny. retrieved from https://core.ac.uk/download/pdf/11066981.pdf hiebert, j. (1984). children's mathematics learning: the struggle to link form and understanding. the elementary school journal, 84(5), 496-513. doi:https://doi.org/10.1086/461380 karim, m. a. (1996). pendidikan matematika 1. malang: depdikbud. rosyadi, w. (2016). analisis kesulitan belajar operasi hitung pembagian pada siswa kelas iv sdn di kecamatan winong kabupaten pati. semarang, jawa tengah, indonesia. retrieved from https://lib.unnes.ac.id/24887/1/1401412370.pdf slavin, s. (2005). matematika praktis untuk sekolah dasar kelas i dan kelas ii. bandung: rekarya jaya. van den heuvel-panhuizen, m., & drijvers, p. (2014). realistic mathematics education. encyclopedia of mathematics education, 521-534. doi:10.1007/978-94-007-4978-8 yusmanita, s., ikhsan, m., & zubainur, c. m. (2018). penerapan pendekatan matematika realistik untuk meningkatkan kemampuan operasi hitung matematika. jurnal elemen, 4(1), 93–104. retrieved from https://core.ac.uk/download/pdf/229259259.pdf https://core.ac.uk/download/pdf/229259259.pdf southeast asian mathematics education journal volume 12, no. 2 (2022) 125 the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 1rosni othman, 2masitah shahrill, 2roslinawati roslan, 3farida nurhasanah, 2nordiana zakir & 2daniel asamoah 1sekolah rendah kapok, muara, cluster 4, bandar seri begawan, brunei darussalam 2sultan hassanal bolkiah institute of education, universiti brunei darussalam, bandar seri begawan, brunei darussalam 3teacher training and education faculty, universitas sebelas maret, surakarta, indonesia 1masitah.shahrill@ubd.edu.bn abstract questioning is one of the critical repertoires in dialogic teaching. teachers who set dialogic classrooms need to be able to use questioning effectively. effective questioning techniques by teachers improve teacher-student instructional dialogues in primary school mathematics classrooms. in this study, the questioning practices of three primary school mathematics teachers were analysed in their journey to incorporate dialogic teaching. data were gathered through lesson observations, video recordings and teacher interviews. the three teachers’ classroom discourses were transcribed verbatim, and teachers’ questions were analysed to find out the types of questions, how the teachers asked the questions and the feedback given to the student’s responses. findings from this study indicated that the three teachers used effective questioning techniques in ensuring dialogic teaching, with focusing, genuine enquiry, and closed testing questions being the most predominant. the teachers portrayed positive attitudes towards dialogic teaching and shared their comprehensive understanding of the approach. keywords: dialogic teaching, questioning techniques, teachers’ perceptions, primary school mathematics. introduction in dialogic teaching, teachers create opportunities for learners to participate actively in classroom interactions, share their ideas and construct a common understanding of the concepts (mercer & dawes, 2014). the dialogic teaching approach has been implemented in the educational systems of brunei darussalam (henceforth referred to as brunei) since 2017 via the literacy and numeracy coaching programme (lncp). the dialogic approach was implemented in brunei on the score that classroom interactions in various subjects, including mathematics, were dominated by teachers’ closed-ended questioning. this limited students’ opportunities to construct their mathematical thinking and understanding (salam & shahrill, 2014; shahrill & clarke, 2014; shahrill, 2018). mathematics has been emphasised in this present study because students have been observed to be passive in the mathematics classroom and mostly take information from teachers. again, teachers are more interested in completing the mathematics syllabus for examination purposes (shahrill, 2018; shahrill & clarke, 2019), and not much classroom communication that promotes higher-order thinking. zakir (2018) similarly reported that preschool classrooms’ culture was very “rigid and mostly on teacherdirected teaching” (p. 235). this arguably may impact students’ learning styles as they go through their educational activities. there was, therefore, the need to re-organise teaching mailto:kristiyajati@p4tkmatematika.org the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 126 approaches and lessons in schools based on the needs of students so that they could share their learning experiences and be active participants in classroom mathematics interactions. the ministry of education (referred to as the ministry) initiated the dialogic teaching initiative to improve the literacy and numeracy of students in brunei. in 2014, the newly revised primary mathematics curriculum was introduced through the numeracy initiative project organised by the ministry’s curriculum development department (cdd). this was meant to meet the requirements of the national education system for the 21st century (sistem pendidikan negara abad ke-21, spn21). the curriculum stressed the development of robust mathematical concepts and skills such as problem-solving, critical thinking, collaboration, and communication. all primary mathematics teachers were involved in the project to familiarise themselves with the new “yes! maths” curriculum package. this was based on a collaboration between the cdd and marshall cavendish education private limited, singapore. in addition, the brunei numeracy national standards framework outlined the criteria for the seven expectation levels of students. the expectation levels were aligned with the programme for international student assessment (pisa) proficiency levels, in which brunei has participated since 2018. the framework provides teachers, parents, the ministry, and other stakeholders a form of measurement to match students’ progress and achievement to the standards to know at which level students are in their learning. others include the centre for british teachers (cfbt), which collaborated with the ministry to create the ‘teaching for mathematics mastery’ framework. it consisted of the requirements for effective mathematics teaching in brunei. one of the emphases was on the teaching of mathematics content dialogically. this was conducted based on the work of alexander (2017) through the lncp. from the programme, international coaches have been deployed to primary and secondary schools to support the teachers in their development to be effective teachers. the coaches provide professional development for mathematics teachers in the schools with the anticipation of training them on the effective use of the mathematics framework through effective questioning and classroom collaboration. the ministry has been determined to ensure classroom interaction through the dialogic teaching approach, especially in mathematics classrooms at the primary school level, since 2017. however, there have not been systematic attempts to assess teachers’ questioning, which is critical in ensuring a dialogic teaching approach in primary school mathematics classrooms. the only study on dialogic teaching in brunei focused on primary school science teachers and the neglect of other mathematics teachers (roslan, 2014). other studies (shahrill & clarke, 2014) considered “teachers’ talk” in senior high school mathematics classrooms. they did not comprehensively assess primary school teachers’ use of dialogic teaching. meanwhile, the lncp has been organising the teacher professional development (tpd) programmes to equip primary school mathematics teachers to develop effective questioning techniques to ensure dialogic instructions. international coaches have also been assigned to conduct the tpd in schools to help mathematics teachers to develop their questioning techniques. since 2017, tpd has focused on the types of questions used, who the teacher addresses the questions to, and how the teacher responds to students’ initial responses (based on alexander, 2017). arguably, since the implementation of the revised mathematics curriculum emphasises dialogic teaching, there is a need to assess teachers’ questioning techniques, which are vital in ensuring a dialogic teaching approach in primary school rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 127 mathematics classrooms in brunei. in this study, we provide evidence for the shifts in the types of questioning and how these shifts impacted the nature of classroom interactions. we, therefore, aimed to assess the effective questioning techniques employed by primary school mathematics teachers as they move towards dialogic teaching. the study answered the following research questions: what questioning techniques are used by primary school mathematics teachers to incorporate dialogic teaching? and what are the perceptions of teachers in incorporating dialogic teaching in primary school mathematics instructions? literature review the nature of dialogic classrooms dialogic teaching means teaching strategies should support continuous interaction between teachers and students and among students and their tasks (yıldırım & uzun, 2021). alexander (2017) premised his conception of dialogic teaching on questioning. he argues that initial and extended questions are one of the features of classroom interaction in a dialogic classroom. alexander stated, “questions are structured to provoke thoughtful answers, and no less important answers provoke further questions, and are seen as the building blocks of dialogues rather than its terminal point” (p. 42). alexander further presented four repertoires in implementing dialogic teaching, which have been adopted in brunei classrooms: talk for everyday life, learning talk, teaching talks, and classroom organisation. the four categorisations were further regrouped into the role of teachers’ questioning and the role of students’ talks in the dialogic classroom. teachers’ role in a dialogic classroom although dialogic classrooms can be challenging as it comes with inclusion problems (rapanta et al., 2021), teachers, as facilitators of their students’ learning, should develop the needed competencies to implement it in mathematics classrooms. this is because using the approach in the mathematics classroom improves students’ performance and reduces their mathematics anxiety (ozbek & uyumaz, 2020). the role of the teachers is crucial because they create an environment conducive for the students to feel comfortable and safe to participate in the discussion actively. the teachers’ role in classrooms is often associated with questioning. in the teaching and learning of mathematics, there should be an effective use of questioning to encourage students’ participation to elicit their mathematical thinking and understanding (cdd, 2017). alexander (2017) believes teachers can still use traditional teacher talk in their classrooms, incorporating rote, recitation, instruction, and exposition. these types of teacher talks play their roles in the classroom discourse. however, teachers need to ensure that their talks are not dominated only by these actions throughout the lesson. alexander (2017) emphasised teachers’ need to include discussion and scaffold dialogues in their talks. opening the opportunities for discussion will allow the students to express their ideas verbally. when the students translate what they think into verbal language, they are more likely to understand the discussed concept (vygotsky, 1978). scaffold language, in this sense, involves interactions that inspire students to think, questions that require students to demonstrate their mathematical thinking, and involves students’ answers which are built upon the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 128 by following up with another question to expand their thinking and understanding. it also improves feedback, which pushes the students to their fullest potential, extended contributions, and exchanges that link together, forming collective understanding (roslan et al., 2018). the emphasis of dialogic teaching in brunei’s classroom is on the use of good questioning techniques to be employed by the teachers to elicit students’ mathematical thinking and reasoning. as outlined in the teaching for mathematics mastery framework, there are different types of questions that the teachers can use to elicit different outcomes. these include focusing, genuine inquiry, closed and open testing, leading, statements, and rhetorical questions (cfbt, 2017). the teachers use questioning to challenge and build up the students’ understanding by providing follow-ups to their answers. one way to follow up on the students’ answers is by acknowledging whether they are correct. giving positive acknowledgement even when students have answered incorrectly is an element of effective teaching (alexander, 2017). aside from acknowledging, the teachers shall build on the students’ understanding or try to correct misunderstanding by probing the students’ thinking (alexander, 2017; shahrill, 2013; salam & shahrill, 2014; marmin et al., 2021). as part of dialogic teaching, alexander theorises that classroom organisation is one of the repertoires of dialogic teaching. organising the class effectively will allow the teacher to easily select the targeted students, groups, or pairs to contribute their input regarding the discussed concepts. the teachers will also enable volunteers to take part by raising their hands to indicate their choice to participate. the role of students in a dialogic classroom in a dialogic classroom, the students are expected to be interactive and actively participate in class discussions (asterhan et al., 2020). their talks shall reflect their mathematical understanding and reasoning. however, previous studies have reported the pedagogical style of several mathematics teachers does not allow students to talk in the classroom. lee (2016), for example, found in singapore that the teacher mostly dominated the classroom discourse. however, after the intervention, by incorporating dialogic teaching elements, the classroom discourse changed to encourage students’ participation. shahrill (2009), salam and shahrill (2014), and shahrill and clarke (2014) also encountered the dominance of the teacher talks in the lessons that they observed in bruneian classrooms. although there were shreds of evidence of teacher and students’ interactions, they were brief and were not expanded to form wholeclass discussions. zakir (2018) calls the inadequate students’ participation in instructional dialogues a “cultural issue in brunei, where children are not used to an adult asking them for their perspectives” (p. 221). based on alexander (2017), students should learn to narrate, explain, analyse, speculate, imagine, explore, evaluate, discuss, argue, justify, and ask questions. this suggests that in a dialogic classroom, the students should be given opportunities to expand their thinking and construct their knowledge (wegerif, 2019). for gillies (2020), students should play the role of “engaging in constructive discussion with their peers in inquiry group tasks, compare findings, and express their opinions” (p. 1). this implies that under the teacher’s guidance, mathematics classrooms should be such that there is enough room for instructional dialogues, collaboration, and free expression of opinion and views about mathematics concepts by students compared rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 129 to mere spoon-feeding. the classroom environment should be conducive for students to play these crucial roles in the dialogic teaching approach. questioning techniques employed by teachers oral questioning is vital in teacher-student instructional dialogues (desli & galanopolou, 2017). in mathematics classrooms, mahmud et al. (2020) indicate that oral questioning encourages systematic and focused thinking abilities in students. this is because the kind of questions the teacher poses determines the way students think and prepare responses. effective oral questioning skills increase students’ inquiry and allow them to explore mathematical concepts. according to celik and guzel (2016), teachers can understand how their students master mathematical concepts, give the needed feedback, and plan interventional strategies through effective questioning. oral questioning accounts for almost 60% of instructional time (farrell & mom, 2015); meanwhile, instructional outcomes are predicted by teachers’ questioning behaviour (maphosa & wadesango, 2017). according to the cfbt (2017), teachers’ questioning techniques focus on the type of questions to be asked, how teachers ask the questions and the feedback given to the students’ responses. in brunei primary and secondary schools, these questioning techniques have been adopted by lncp to ensure dialogic instructional pedagogies. the questioning techniques’ descriptions based on cfbt (2017) are provided in appendices a, b and c. methods study context the present study was conducted in a primary school in brunei. the education system in brunei was established following the british education system and used a bilingual education policy (muhammad & petra, 2021). the english language (the second language) is used as the medium of instruction, although malay is the first language (sharbawi & jaidin, 2020). the classroom culture in brunei is different from other western counterparts because the country’s philosophy and the malay islamic monarchy (or melayu islam beraja, often shortened as mib) play a vital role in education. two social values that are part of the philosophy of the country and the islamic culture are respect for old age and humility. for this reason, students are seen to respect their teachers or elders by not questioning their knowledge as the belief that they (elders) are more knowledgeable (zakir, 2018). for this reason, classroom interactions have followed such traditional routes because students feel they may be challenging or disrespecting their teachers when they exchange conceptual words with teachers. this has extended to mathematics classrooms at the primary school level, where teachers closely question students to neglect students’ interaction with teachers or among themselves. to solve this problem, there was a need to adopt alexander’s dialogic teaching in the mathematics classroom. research design the study uses the qualitative case study design. the design helps to comprehensively understand opinions and experiences about a particular phenomenon and generate new ideas the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 130 (creswell, 2014). a case study is a valuable way to gather information which involves more minor participants (zainal, 2007). although case studies can be multiple or single (bennet & elman, 2007), the present study uses a single case study since the study focused on a specific phenomenon in a particular context (crowe et al., 2011; teegavarapu et al., 2008). the qualitative case study design is deemed appropriate because there is a need to understand how primary school mathematics teachers use effective questioning to ensure dialogic instructions. there is also the need to examine how their perceptions of dialogic teaching affect and inform their practices. to obtain comprehensive information based on these parameters, this study observed and interviewed primary school mathematics teachers on how they ensure dialogical teaching. this signifies the use of multiple data sources, where data is converged in a triangulation fashion, to ensure the reliability of findings (yin, 2003). participants the participants were purposely sampled since they possessed the characteristic of interest of the study (creswell, 2012). the three sampled participants were year 6 primary school mathematics teachers who were directly involved in the lncp as a local coach (lc), potential local coach (plc), and learning partner (lp). the lc and plc are responsible for supervising the implementation of dialogic teaching in their school. therefore, they were well informed on dialogic teaching as a pedagogical approach in mathematics classrooms. an lp is a person that the international coach (ic), lc, and plc coach under the lncp. yin (2003) opines that the nature of case study design makes sample size irrelevant but cautions that emphasis must be on getting in-depth information on the case. however, observing and interviewing three participants was sufficient since they were directly involved in dialogic training programmes and could provide in-depth information on how they implement it in their mathematics classrooms. table 1 provides a summary of the teachers’ demographic details. table 1 participants’ demographics participant 1 participant 2 participant 3 gender female male female age 35 34 37 highest qualification ma education (uk) ba primary education (brunei) ba primary education (brunei) teaching experience 9 8 14 teaching experience in mathematics 2 8 14 role in lncp potential local coach (plc) local coach (lc) learning partner (lp) note: the teachers’ roles in lncp are used from this point of the study instruments and data collection lesson observations and interviews were conducted over four months. by employing the non-participant observation technique, observations were made in five consecutive lessons for all the participants. lessons were video-recorded to provide retrievable data of the real context of the lessons, and this was done to observe behaviours and patterns of interactions (goldman rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 131 & mcdermott, 2007). it also helped capture the participants’ facial expressions, gestures, and body language during the lessons. the video recordings were necessary for the transcribing process of the lesson discourse. field notes were also taken during the lesson observations. three video recorders, which recorded the participants from several different angles within the same classroom, were set up in each class simultaneously. the first video recorder could capture a wide-angled frame and be placed where the full view of the class could be obtained. the second video recorder was focused on the teacher, whereas the third video recorder was focused on the students. the dialogues during the video-recorded lessons were transcribed based on the categories in the lncp tools. the lncp tools were adapted and modified to suit this study. the tools are the extensions of the teacher performance appraisal (tpa) form that evaluate the performance of a teacher and their students’ achievement in a particular lesson, hence, the name tpa+. the content of the tools is as follows: (a) tpa+ lesson observation support document (see appendix d): it specifically looked at the type of questions the teachers asked, how the teachers addressed the question to the students, how the teachers responded to students’ answers and the type of students’ responses. the transcriptions for the lesson observations were recorded in this document. (b) tpa+ observation tool (see appendix e): this tool was used to tally how many times the abovementioned criteria were used. at the end of the lesson observation sequences, a face-to-face interview was conducted with each teacher using a structured interview guide. the interviews were conducted to gain insights into the teachers’ experiences and perceptions of incorporating dialogic teaching in their mathematics classrooms. the interviews also included video-stimulated recall interviews based on one to two chosen lessons. this helped recall the classroom discourses occurring during the lessons and identified when the teachers believed that dialogic teaching had happened during class interactions. the video-stimulated recall interviews supplemented the researchers’ lack of involvement in the lesson and could avoid the biased evaluation of the contexts just by observing and reviewing the video recordings (shahrill, 2017; xu & clarke, 2018). also, the interviews were audio-recorded and transcribed for further analysis. data analysis the data from the interviews were analysed using thematic analysis. this helped in “identifying, analysing, and interpreting patterned meanings or ‘themes’ in the qualitative data” (braun & clarke, 2014, p. 95). the transcriptions from the interviews were read and analysed severally to determine the themes or categories. triangulation of data sources was utilised to ensure the trustworthiness of the findings. this was to check the consistency of data obtained from the lesson observations and interviews. an ic was invited specially to become a member checker to ensure the transcription of the lesson observations, together with the second and third authors. some parts of the video recordings were given to the ic, which enabled entering the data into the lncp tools. the reason was to enter accurate data into the lncp tools. excerpts from the interviews have also been quoted verbatim to validate further the key findings that emerged from the interviews. although this was a qualitative study, the transcriptions from the lncp tools for the questioning techniques were analysed and the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 132 quantified to frequency and percentage counts. this step is to identify the questioning techniques that were more and less dominant or either, in the teachers’ classrooms. this made it easier to compare the three classrooms. frequencies and percentages were also used to depict situations in each of the teachers’ classes and show the distribution in each category. the percentages were calculated by taking the frequencies of occurrences for each criterion from a category, dividing by the total number of occurrences and multiplying by a hundred. findings and discussion questioning techniques employed by primary school mathematics teachers the three categories from analysing the questioning techniques employed by the three teachers were: the types of questions, how teachers ask the questions, and the feedback given to the students’ responses. the transcriptions from 15 lessons were analysed based on these categories. table 2 illustrates the frequency (n) of the questions that occurred during each lesson for the three teachers. table 2 frequency distribution of the number of questions asked by the teacher teachers lesson number n topics plc 1 2 3 4 5 122 119 75 72 104 understanding ratio equivalent ratio the ratio of three quantities ratio and fractions solving world problems on ratio total 492 lc 1 2 3 4 5 40 49 47 32 41 expressing one quantitative as a percentage of another quantity finding the percentage of a quantitative solving word problems on percentage understanding ratio equivalent ratio total 209 lp 1 2 3 4 5 87 56 49 2 6 calculating average calculating total solving word problems on average meaning of average relevance of average total 200 from the five lessons observed from all three teachers (in table 2), the plc asks the most questions (492), and this is followed by the lc (248 questions) and the lp (200 questions). although the teachers used the same lesson duration (60 minutes) and same year group to ensure uniformity, the number of questions asked by the three teachers varied due to the different lesson structures. the number of questions also differed due to the teachers’ time spent on the whole class interactions. the whole class interactions occurred mainly during the starter activities, main teacher input, and plenary. the plc spent an average of 10 minutes in each of her lessons on starter activities, 35 minutes on main teacher input, 20 minutes on group rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 133 work, and 5 minutes on the plenary. the lc spent an average of 5 minutes on his starter activities in each lesson, 20 minutes on main teacher input, 20 minutes on pair work or group work, 10 minutes on individual work, and 5 minutes on plenary. in comparison, the lp spent an average of 15 minutes on her starter activities, 25 minutes on main teacher input, 15 minutes on group work or pair work, and 5 minutes on the plenary. the type of questions asked the types of questions the teachers ask are analysed based on the eight questions identified by the cfbt (2017). table 3 illustrates the frequencies (n) and percentages (%) of the type of questions asked by the teacher participants in their consecutive lessons. table 3 frequencies and percentages of the type of questions types of questions plc lc lp n (%) n (%) n (%) focusing question 309 (62.8) 101 (48.3) 118 (59.0) genuine enquiry 79 (16.1) 73 (34.9) 56 (28.0) closed testing 74 (15.1) 35 (16.8) 22 (11.0) open testing 8 (1.6) 0 (0) 0 (0) statements 1 (0.2) 0 (0) 0 (0) leading 21 (4.2) 0 (0) 2 (1.0) rhetorical 0 (0) 0 (0) 2 (1.0) guess-my-mind 0 (0) 0 (0) 0 (0) total 492 (100) 209 (100) 200 (100) from table 6, the plc predominantly asks focusing (62.8%), genuine enquiry (16.1%), and closed testing questions (15.1%). the lc asks focusing (48.3%), genuine enquiry (34.9%), and closed testing questions (16.7%). similarly, the lp asks focusing (59.0%), genuine enquiry (28.0%), and closed testing questions (11.0%). table 6 implies that all three teachers (plc, lc, lp) predominantly utilise similar questions: focusing, genuine enquiry, and closed testing. how teachers ask questions this category examines the way the teachers address questions to the students. how teachers ask questions was analysed based on the five categories identified by the cfbt (2017). table 4 illustrates each category’s frequencies (n) and percentages (%) of occurrences. table 4 frequencies and percentage of how the teachers ask questions categories plc lc lp n (%) n (%) n (%) whole class (orally) 297 (60.4) 45 (21.5) 136 (68.0)) whole class (physically) 27(5.5) 63 (30.2) 15 (7.5) individual volunteers 82 (16.7) 21 (10.0) 29 (14.5) specific students/pair/group 86 (17.4) 80 (38.3) 18 (9.0) not directed to anyone 0(0) 0 (0) 2 (1.0) total 492 (100) 209 (100) 200 (100) the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 134 from table 4, the plc mostly directs questions to the whole class orally (60.4%), specific students/pairs/groups (17.4%), volunteers (16.7%), and the whole class physically (5.5%). the lc targets specific students/pairs/groups (38.3%), the whole class physically (30.1%), the whole class orally (21.5%), and volunteers (10.1%). similarly, the lp directs questions to the whole class orally (68.0%), volunteers (14.5%), specific students/pairs/groups (9.0%), and the whole class physically (7.5%). irrespective of the various roles played by the teachers in lncp towards inculcating dialogic teaching, their ways of asking questions follow the same trends. they mostly direct questions to the whole class (orally), specific students/pairs/groups, individual volunteers, specific students/pairs/groups, and the whole class (physically). this shows the varied questioning skills of the teachers as they incorporate dialogic teaching. feedback given to the students’ responses this section explores the feedback practices of the teachers based on the six types of feedback outlined by the cfbt (2017). table 5 shows the frequencies (n) and percentages (%) of the three teachers’ feedback on the students’ answers. table 5 the percentage of the feedback given to students categories plc lc lp n (%) n (%) n (%) acknowledge 139 (28.3) 87 (41.6) 40 (20.0) follow up 242(49.2) 78 (37.3) 80 (40) compare 6(1.2) 5 (2.4) 3(1.5) adding to 77 (15.7) 33 (15.8) 54 (27.0) re-voicing 27(5.5) 6 (2.9) 20 (10.0) rephrasing 1 (0.2) 0 (0) 1 (0.5) no response required 0 (0) 0 (0) 2 (0.1) total 492 (100) 209 (100) 200 (100) from table 5, the plc predominantly uses follow-up (49.2%), acknowledgement (28.3%), and adding up to students’ responses (15.7%). similarly, the lc uses acknowledgement (41.6%), follows up (37.3%), and adds up (15.8%). the lp follows up (40.0%), adds up (27.0%), and acknowledges students’ responses (20.0%). this implies that the teachers predominantly utilise similar feedback practices, namely: follow-up, acknowledgement, and adding to, with the follow-up being the most prevalent. effective questioning that may lead to teacher-student dialogues was the motivation for the lesson by the lc. the lesson is full of follow-up, acknowledgement, and adding up to most of the responses from the students. the lc also allows an individual student and the whole class to respond to questions. this atmosphere presented in the lc lesson characterises dialogic teaching. teachers’ perceptions of dialogic teaching in mathematics classroom after completing the lesson observation sequences in their respective schools, interviews were conducted with each teacher. this was to gain insights into the teachers’ perceptions of rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 135 implementing dialogic teaching in mathematics instructions. the themes that emerged from teachers’ perceptions covered the following: the importance of questioning techniques in dialogic teaching, dialogic teaching focusing on the students’ centred learning, students’ confidence by utilising dialogic teaching, and dialogic teaching not impeding the teachers and students’ examination preparations. generally, the three teachers report positive perceptions towards dialogic teaching. the teachers reported positive perceptions of questioning and saw it quintessential in ensuring dialogic teaching. they also perceive that dialogic teaching encourages child-centred learning and students’ confidence and does not impede teachers and students’ examination preparations. the excerpts from these themes presented below validate these findings: importance of questioning technique in dialogic teaching generally, the teachers perceived that questioning is key in dialogic teaching. according to the teachers, the different types of questions that they posed helped students participate more in class. to validate this finding, the plc responded on the importance of questioning in dialogic teaching as follows: for dialogic teaching, there should be three main things to note: the types of questions where the teachers shouldn’t be too focused on close-ended and should be more openended to see how students respond to them. then the second thing is how we pose questions in classes, where we’ll try to move from asking it to the whole class and then try asking individually or, better, asking volunteers from students to answer but make sure that we give enough wait time before we call any students to answer. thirdly, it will be on how to respond. how we respond to answers that students give. that is, we should acknowledge any answers that they have provided, and for good answers, maybe we can try to ask the students to explain to their friends and then teachers as well use the answers as prompters or guides to the next question that we want to ask and also, we can use the answers by praising students or rewarding and then avoid misconceptions we can re-explain what students give us and give the correct concept or from the misconceptions that they did. (plc) the excerpt from the plc depicts that effective questioning techniques in dialogic teaching help the students to talk, explain and share ideas. to the plc, more open-ended questions should be asked in the mathematics classroom, and feedback should be provided to ensure a dialogic class. she confirms the types of questions used by the teacher, how the teacher asks the questions, and the feedback the teacher gives to the students’ responses. the lc also mentioned the importance of effective questioning techniques in dialogic teaching and commented as follows: ok, one of them is questioning techniques. so, how the teachers respond to the students’ answers and also the type of answers the students give. how are we going to correct the students by rephrasing or re-voicing? the strategies on how the teachers question the students. other than that, the questions can also be differentiated, so sometimes the challenging questions are for the high-ability, less challenging ones for the middle and low-ability students. so, basically, how the teachers question the students. (lc) the lc emphasise a unique mechanism apart from the questioning techniques. according to him, the need to differentiate the questions so that students from different abilities could participate in the lesson is relevant in dialogic teaching. the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 136 dialogic teaching focusing on the students’ centred learning the teachers perceive dialogic teaching as encouraging students’ centred learning. in dialogic teaching, the focus is on the students’ talk to construct the meaning of what they are learning. the teacher participants revealed that throughout their teaching, they observed that their students were able to talk, share their ideas and explain their ways of finding solutions. the following are the excerpts of the teachers’ interview responses: dialogic teaching is one way of teaching that we involve all the students in their learning; make them share their ideas either with their peers or in groups. (lp) it changed from teacher-centred to student-centred. students tend to give more opinions and ideas during the lesson. they are exposed to public speaking in terms of presenting their findings and methods. most importantly, less teacher talk. (lc) dialogic teaching encourages students to talk such that students should be doing more talking and explaining during the lesson, and in other words, it’s more towards studentcentred during the teaching and learning process. (plc) the responses imply that the teachers have similar perceptions of dialogic teaching concerning its ability to encourage child-centred learning. the teachers perceive dialogic teaching as a platform for the students to be active in their learning, emphasising the students’ contributions during the instructional process. teachers become facilitators and effectively monitor students’ learning. this indicates that the teachers generally have positive attitudes towards dialogic teaching. students’ confidence by utilising dialogic teaching the three teachers believed that dialogic teaching increases students’ confidence and changes the way they participate in class. to validate this stance, the lp had this say: in a way, it is good for the students as they can gain confidence in presenting their ideas. (lp). the plc also commented similarly as follows: after the use of dialogic teaching, students are more open to answering questions, and they are not afraid to volunteer. that is the one that clearly can be seen from my experience, and then they talk more during their activities. (plc) the lc also shared the same view as follows: from what i can see, the students tend to participate more. i've just taught them this year. at the beginning of the year, it was difficult for them to answer my questions and contribute their ideas. so, slowly i build them up. now, i can see the difference from january until now. now, they are braver in presenting their answers. (lc) referring to the responses of the participants on students’ confidence, the teachers perceive that dialogic teaching encourages them to create a student-friendly environment in which the students feel safe and at ease to share their thoughts, ideas, and opinions. this could be seen from the three teachers’ classes during the observation. the students volunteered to share and explain their ideas. they gave reasons during their explanation and not just presenting the solutions to the mathematical problems. these observations confirm the teachers’ reportage on the ability of dialogic teaching to encourage confidence in students. rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 137 dialogic teaching is not a hindrance examination preparations the participants did not see dialogic teaching as a hindrance in preparing students for external examinations. they commented that dialogic teaching helped promote the students’ understanding of mathematical concepts as they prepared for examinations. the following is a comment made by the lc: as a year 6 teacher and an lc, it helps me to teach more concept-based teaching apart from chalk-and-talk. the students themselves find the methods on their own with a bit of guidance from the teacher, which is more to inquiry too. this helps the students to master the concept easily, and this allows them to improve their academic achievement, where they can think independently and creatively since they are already exposed to this approach. (lc) the lc perceives dialogic teaching as encouraging the students to learn and improve their achievement. according to him, dialogic teaching promotes inquiry-based learning, which makes the students independent in creating their methods or ways of solving mathematical problems. the lp also reported as follows: as a year 6 teacher, dialogic teaching is also good, but towards the end of the semester, it is good to have a small group with a balance of dialogic and drilling of past paper questions. (lp) according to the lp, there should be a balance between dialogic teaching and ‘drilling’ of past paper questions while students prepare for external examinations. this implies that in dialogic instructions, solving past practical questions should be added to instructions. discussions findings from the present study indicate that the teachers employed varied questioning techniques in their quest to ensure dialogic teaching. the results show that teachers use focusing questions, genuine enquiry, and closed testing in providing dialogic teaching. roslan (2014) reported a similar finding indicating that teachers use different prompts for the students to give quality answers in the dialogic discourse. the kinds of questioning used are a positive move towards dialogic teaching, especially genuine enquiry questions. it allows students to explain how they get their solutions even though they can be incorrect. teachers use it to assess the level of understanding of the students and can identify if there are any misconceptions or mistakes in the process. genuine enquiry questions allow the students to construct their understanding by verbalising their thinking process (vygotsky, 1978). this is corroborated by alexander (2017), who advocated that in ensuring dialogic teaching, students should actively contribute their ideas and thinking to the whole class. even though closed testing was used as one of the top three frequently used types of questions, most teachers used it to recall multiplication and division facts at the start of the lesson. this implies that rote learning or recitation can occur in dialogic teaching, as it is unavoidable to ask the students these types of questions. the use of focusing questions as part of the predominant questioning techniques used by teachers is seen as scaffolding the students’ learning and allowing them to understand their learning. this further confirms alexander’s (2017) assertion that discussions and dialogues are essential for students to succeed at their own pace in dialogic instructions. students are, therefore, probed further through instructional the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 138 conversations to address their learning gaps, as exemplified in vygotsky’s (1978) zone of proximal development (zpd). in this study, we found that the teachers also directed their questions to the whole class orally, the whole class physically, individual volunteers, and specific students or pairs or groups. thus, teachers primarily emphasise instructional dialogues and classroom conversations that characterise dialogic teaching. gillies (2020) found that science teachers’ ability to use varied questioning in dialogic instructions exposes students to constructive discussions with peers to compare their findings on a task. gillies’ position has been reaffirmed among the students observed in the mathematics classrooms as their teachers employed dialogic instructions. it should be acknowledged that in asking questions using these approaches, there is the likelihood of targeting the higher-ability students to neglect lowachieving students. this notwithstanding, emphasises oral questioning and asking questions in pairs allows students to contribute and get their ideas across. this is particularly important in the mathematics classroom because students can participate in the teaching and learning process and will not remain passive throughout the lesson (cfbt, 2017). the findings of this study also show that the teachers did not have the same pattern in giving feedback to the students. this can be explained based on their differences in concepts taught, teaching experiences, and their status in employing dialogic teaching. irrespective of this, all the teachers used feedback practices such as acknowledgement, follow-up, and adding, with fellow-ups being the predominant feedback practices. this is a positive indication of dialogic teaching. for instance, acknowledging students’ answers, whether correct or not, signals to the students that their contribution to learning matters. it motivates the students to try, even though their ideas or solutions are incorrect. most importantly, the students become active in contributing to the whole class discussion and strive to develop mathematical knowledge by themselves (vygotsky, 1978). throughout the observation, it was evident that students were active in mathematical instructions and were engaged in the classroom discourse, as teachers used varied questioning techniques and feedback to ensure dialogic mathematics instructions. this evidence supports that instructional dialogues provide teaching and learning responses in which students contribute to the classroom discourse (garcía-carrión et al., 2020). in contrast and as an improvement on zakir’s (2018) findings that children in brunei are not used to an adult asking them for their perspectives, the use of dialogue teaching improved mathematical dialogues with their teachers. further, the study findings move away from the passive learning portrayed in previous studies in mathematics classrooms (hanafiah, 2008; salam & shahrill, 2014; shahrill & clarke, 2014). the three teachers unanimously believed that the most critical element in dialogic teaching is the questioning techniques, indicating that questioning is the tool that drives mathematics classrooms towards dialogic teaching. this corroborates research findings from previous studies that effective questioning techniques are essential for students to be active learners (salam & shahrill, 2014; shahrill & clarke, 2014; shahrill, 2018, marmin et al., 2021). the teachers in this study portrayed a positive perception towards dialogic teaching, as they believed in its benefits. in addition, their students displayed more activeness and confidence to participate in the teaching and learning process. consistent with previous studies, ozbek and uyumaz (2020), for instance, reported that dialogic instructions reduce students’ anxiety in rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 139 mathematics classrooms. teachers’ quests to use effective questioning skills bring the students on board, and they are free to share mathematical experiences due to teacher-student collaborations. the teachers also believed that dialogic teaching is not a barrier in preparing students for external examinations as it supports and strengthens their conceptual understanding. we believe the teachers’ positive perceptions are due to the extensive tpd programmes and in-house mentoring on dialogic teaching. this positive perception exhibited by mathematics teachers indicates that they are more likely to implement collaborative instructions in their mathematics classrooms. this will allow the students to challenge themselves to think mathematically and provide reasoning for their views (bozkurt & polat, 2017). the study found that effective questioning techniques are fundamental in dialogic teaching in mathematics classrooms. although the teachers played different roles regarding lncp (plc, lc, lp), they all ensured instructional dialogues through effective questioning skills and reported similar questioning strategies. with their respective positions, irrespective of the topics they taught, duration, and teaching experiences, all the teachers incorporated dialogic instructions in their mathematics classrooms. after the launch of the lncp, this is the maiden study to investigate mathematics teachers’ questioning techniques in ensuring dialogic teaching at primary school levels in brunei. the following recommendations are made; firstly, although the study focused on whole-class interactions between the teachers and students, it emphasised teachers’ questioning and perception in ensuring dialogic teaching. it will be beneficial to focus on students’ talk to determine the dialogic of their talk in mathematics classrooms. future researchers may want to focus on using audio recorders to record students’ talk during their interactions with pairs or groups. it can provide insights into the mathematical thinking and reasoning that the students discuss while completing their tasks together. secondly, investigating students’ perceptions towards dialogic teaching may also shed light on its impact on their learning process. a simple questionnaire may be appropriate in this quest and improve the generalisation problems. thirdly, as shown in this study, the three mathematics teachers covered different topics in mathematics. future studies may want to conduct investigations on the same study variables but focus on various subjects to study the patterns (if any) of the different questioning techniques adopted by the teachers involved in tpd in dialogic teaching (other than the plc, lc, and lp). conclusion the types of questions used by the teachers are crucial in eliciting different student responses. the way the teachers addressed the questions and provided feedback to the students can be seen as encouraging and motivating them (students) to share their ideas, thinking, and solutions to a mathematical problem. with teachers’ positive perception toward dialogic teaching, they will have the opportunity to explore its advantages. the findings imply that the ability to use questioning effectively leads to instructional dialogues, which has been confirmed in the primary school mathematics classrooms used in this study. this can impact the students’ learning and teachers’ teaching practices as it brings different opportunities for both the teachers and students to explore. the teachers can use higher-order thinking questions to develop students’ critical and creative thinking abilities. the students would be allowed to be the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 140 active in the construction of their mathematical knowledge. the present study’s findings reiterate the importance of the tpd programmes on dialogic teaching. this study informs the ministry and policymakers on the successful implementation of dialogic teaching, especially in mathematics classrooms, although there is more room for improvement. recommendations we recommend future studies exploring the challenges of using dialogic instructions in primary school mathematics classrooms. lastly, the present study provides evidence that primary school mathematics teachers use effective questioning techniques in ensuring dialogic teaching and have positive perceptions of it. we recommend that the ministry maintain its tpd on dialogic instructions. irrespective of the subject of instruction, policies should be strengthened so that teachers adopt effective teaching practices other than questioning to ensure dialogic pedagogy. acknowledgements we gratefully acknowledge the contributions made by the 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(2018). the impact of educational change processes in brunei preschools: an interpretive study (unpublished doctoral dissertation). university of sheffield, united kingdom. the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 144 appendix a descriptions of the types of questions with examples types of questions sample questions closed testing the question has only one answer or demands a “yes” or “no” answer. this type of question is primarily used in the examination. for example: is 3×4 = 12? or what is 3×4? guess what is in my mind only the ‘guess what is in my mind’ answer is acceptable. any other way to come up with the answer is not encouraged or promoted. it does not support dialogic teaching and learning. rhetorical the teacher will ask a question where the answer is not necessary, or the teacher will answer her/his question. the focus is on the teacher rather than on the students. no response is desired or allowed. leading often leads the students to answer what the teacher is thinking in her/his mind. does not encourage creativity, promote independent thinking or find an alternative way to solve a problem. for example, equal fractions are called…. open testing the question has more than one answer or more than one way to solve the problem. the answer to the question is given; the students are encouraged to write a question for the answer. for example, i have 4 edges, what might i be? focusing the intention is to guide and or scaffold learning so that students can achieve some degree of success. for example: which column do i write the ten in? (focus is on writing the tens in the correct colum) genuine enquiry the teacher does not know what the student is thinking, or the teacher does not have the answer. it promotes lateral thinking and supports dialogic teaching and learning. this can often lead to hot types of questions. for example: how do you know? statements the statements are often written or discussed with the students to promote higher-order thinking and or making connections to prior learning or planning to a new unit. it can be used in the starter activity or the plenary as a way to consolidate or extend learning. for example, all triangles have three sides. discuss. or a square is a quadrilateral. discuss. appendix b descriptions of how targeted teachers ask questions categories descriptions whole class (orally) the teacher asks the whole class and is commonly used in a whole-class discussion. whole class (physically) students need to show their answers physically, such as using mini whiteboards, number cards, concrete materials, and body parts individual volunteers the teacher asks for volunteers or the students voluntarily answer the questions. specific students/pair/group the teacher targets specific students, pairs, or groups to answer the questions. not directed to anyone occurs when rhetorical questions are used which do not need students to answer. rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 145 appendix c descriptions and samples of feedback types of feedbacks description sample acknowledge the teacher gives encouraging words to the students even though their answers are wrong. good job, nice try, excellent, thank you follow up the teacher shows interest in the students’ answer and pursues another question to get them to rethink it through. it often involves genuine enquiry questions. how did you get the answer? can you tell us more? why did you say so? compare the teacher compares what is being said or done by two or more students. this can help the students to rethink or spot any mistakes that they could do. what is the difference between your answers? do you think your work matches our step to success? adding to the teacher adds up some more information to the students’ answers. student: the ratio is 4 to 5. teacher: the ratio of the circles to the stars is 4 to 5. re-voicing the teacher reiterates students’ answers by correcting the terms/grammar used or translating it from malay fifteen minus three or take away three from fifteen ‘pasal di sini ada dua nombor’ (because there are two numbers here). rephrasing the teacher paraphrases the student’s answers or asks other students to do it. can you tell us what she had said just now but in your own words? appendix d the tpa+ lesson observation support document (adapted and modified from literacy and numeracy coaching programme materials) question types (qt) closed testing (c); guess-what’s-in-my-mind (gm); rhetorical (r); leading (l); open testing (o); focusing (f); genuine-enquiry (ge); statements (s) qt what questions do teachers ask? how do teachers question students? circle one student response type: circle one how do teachers respond to students’ answers? circle one whole class orally whole class-physically individual volunteer(s) specific student/pair/group short utterance full sentence discussion of ideas comparison/ conjecture/ generalisation acknowledge compare re-voicing follow up adding to rephrasing the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 146 whole class orally whole class-physically individual volunteer(s) specific student/pair/group short utterance full sentence discussion of ideas comparison/ conjecture/ generalisation acknowledge compare re-voicing follow up adding to rephrasing whole class orally whole class-physically individual volunteer(s) specific student/pair/group short utterance full sentence discussion of ideas comparison/ conjecture/ generalisation acknowledge compare re-voicing follow up adding to rephrasing rosni othman, masitah shahrill, roslinawati roslan, farida nurhasanah, nordiana zakir & daniel asamoah 147 appendix e the tpa+ observation tool (adapted and modified from literacy and numeracy coaching programme materials) tpa+ observation tool school teacher observed no. date observed class observed tick to record final judgements q teaching content dialogically 5 4 3 2 1 1 what are the types of questions that the teacher asks? 2 how do teachers question students? 3 how do teachers respond to students’ answers? q1. teach content dialogically: what types of questions do the teachers ask? question types tally ineffective questions (guess-what’s-in-my mind/rhetorical/leading) testing questions (closed & open) focusing genuine-enquiry: statements: q2. teach content dialogically: how do teachers question students? mini whiteboard etc. (show me) targeted student individual volunteer chorusing q3. teach content dialogically: how do teachers respond to students’ answers? acknowledge: yes. that’s it. you got it. no, good try. not quite. thank you. follow up: how did you get that answer? why do you think that? can you tell us more about that? compare: do you feel that supports or contrasts with what zayden thought? thank you, elon. that takes us back to what zayden said. adding to: the teacher develops the students’ responses by adding more detail/further information. re-voicing: the teacher repeats the student’s answer & may correct grammatical/vocabulary errors that the student made. rephrasing: can you tell us what elon just said and tell us a bit more about it in your own words? the questioning techniques of primary school mathematics teachers in their journey to incorporate dialogic teaching 148 southeast asian mathematics education journal volume 12, no. 1 (2022) 11 enhancing mathematics learning by integrating growth mindset principles in ninthgrade supplementary materials 1 christian r. repuya & 2 jedh esterninos 1 bicol state college of applied sciences and technology, philippines 2 university of nueva caceres, philippines 1 crrepuya@astean.biscast.edu.ph abstract this paper determined the supplementary materials in ninth-grade mathematics which may be enhanced to integrate growth mindset principles to improve students’ procedural fluency and foster their growth mindset in mathematics. this study employed a descriptive-developmental method by administering a quasi-experimental design and mixed-method research approach to determine the research questions. the respondents in this study were the 60 ninth-grade students at a state secondary high school in the philippines. the study implemented the validated researcher-made procedural fluency test and growth mindset questionnaire in determining students’ performance in procedural fluency and mindset in mathematics, respectively. thematic analysis was employed to investigate students’ responses in focus group discussions (fgd), informal interviews and learning journals in scrutinising the learning experiences and mindsets of the students. findings displayed that the supplementary materials which can be developed in incorporating growth mindset principles were motivational activities, reflection activities, and instructional videos. utilising the developed supplementary materials elevates students’ procedural fluency. it influences students to shift from fixed mindsets to growth mindsets in mathematics by providing them with significant learning experiences that help students enhance a growth mindset. furthermore, the implementation group performed better than the comparison group, particularly with developing growth mindsets. the study results are limited to the participants encompassing; similar research employing the developed supplementary materials to other learning areas with a larger sample is recommended for more generalisable results. keywords: growth mindset in mathematics, motivational, procedural fluency, reflection, instructional videos introduction studying mathematics is tremendously crucial in people’s lives. mathematics education begins in the early stage of education. its objective is to assist students acquire the significance of mathematics applications in the real world by enhancing their cognitive ability to recognise forms, patterns, and relationships. however, despite its significance, mathematics is still the least preferred and hated subject among students in general, and encourages for fewer students than other subjects. it is caused by the students who perceive mathematics as a complex subject, particularly following instructions. it results in difficulty obtaining the subject and trouble recalling its equations and methods to discover a solution to a problem (gafoor & kurukkan, 2015). correspondingly, students who possess a negative attitude towards their teachers and mathematics also perceive it as boring, and they are not encouraged (mohapi, 2015). enjoyment and attitude in learning mathematics significantly determine students’ performances. furthermore, the factors influencing the students to like or dislike mathematics encompasses students’ social-psychological, environmental, instructional, and abilities enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 12 factors. moreover, the teacher’s instructional strategy, institutional sources, weak understanding and evaluation approaches, and failure in comprehending directions are the factors of failure to pass mathematical examinations (mazana, montero, & casmir, 2019). in the philippines, the international examination results in the program for international student assessment (pisa) 2018 also presented a low growth mindset among students (organization for economic co-operation and development (oecd), 2019). the pisa 2018 (oecd, 2019) results revealed that only 19% of students in the philippines attained level 2 or higher in mathematics. only 31% of students in the philippines possessed a growth mindset. as a result, the philippines performs low in the pisa examination, with an average score of only 350 points in mathematics, science, and reading during school year (sy) 2018-2019, which place them in the rank 77 th out of nearly 80 nations (oecd, 2019). unfortunately, the national achievement test (nat) results formulated this global performance at the local levels. in the 2012 nat, the philippines performed low, with a mean percentage score (mps) of only 46.37 in mathematics. the enhanced performance was not acquired compared with the past years (47.82 in 2006 and 50.7 in 2005) and an overall rating of 48.57 during sy 2011-2012. this dilemma in mathematics paused threats to the future of mathematics education. this evidence indicates that the students are not motivated and own a low interest in learning mathematics because they possess a low growth mindset. the students believe that learning mathematics is complicated as they own low procedural fluency. procedural fluency refers to a student's understanding of procedures, as well as when and how to employ them correctly, and also the ability to administer procedures flexibly, precisely, and efficiently (inayah, septian, & suwarman, 2020). in response, the department of education (deped) (2020) encourages teachers to design lessons, learning materials, and instructional materials to assist students become motivated and have elevated understanding of the topics. mendoza, caranto, and david (2015) recommended that supplementary materials as integrating instructional videos effectively acquire the student's interest. integrating instructional video into teaching/learning is a method in teaching which utilises videos that allows teachers to deliver lessons and encourage students through video recordings. gieras (2020) asserted that designing engaging instructional videos which encourage interactions and engagement to corroborate students’ learning can be a very well effective medium for assisting instruction in remote, hybrid, and flipped or blended learning conditions. moreover, harackiewicz, smith, and priniski (2018), when students improve their interest in studying mathematics, their motivation in learning is also enhanced. in her article, campbell (2017) argued that success necessitates students to engage in the individual growth process. she highlighted on how difficult it might be to be open in producing incorrect and altering course all the time. it is normal to be involved to what we believe works and what we are accustomed to. she emphasised that in establishing a growth mindset, students should educate themselves to identify “deficient” as a shortfall in their learning or experience rather than a weakness in their capability. dweck (as cited in boaler, 2013) corroborated that intelligence and smartness could be acquired, and that brain grows from training with a growth mindset. christian r. repuya & jedh esterninos 13 conley (2014) implemented a growth mindset when providing students feedback, elaborating that praising them for intelligence made them less willing to pursue educational opportunities because they are anxious about losing their “smart” identity if they experienced low. furthermore, because avoiding academic risks entails avoiding learning, applauding students undermines their academic achievement. the researcher employed the power of the word “yet” associated with a growth mindset. for instance, in praising students, “your sentence structure does not yet match the tone you are attempting to achieve”. the word “yet” accepts negative comment while at the same time conveys confidence that the students receive shortly (a teacher as this one possesses a growth mindset). in addition to giving feedback, allowing students to perform reflective learning activities help them aware of their performance. in their study, weng, puspitasari, rathinasabapathi, and kuo (2021) defined reflective learning as activities in facilitating learners’ reflection upon their learning experiences. the findings of their study recommend the significance of reflective learning to enhance students’ recognition of asking and assessing their learning in the class. furthermore, berger (2017) asserted that students are able to elevate the courage to encounter educational and life challenges. he cited one of the secondary schools, which has advanced academic courage in the students. as a result, 98% of their students graduated on time, and all of the students have been admitted to university. the previous studies were beneficial in conceptualising the present study. compared to the previous studies, the present study integrates growth mindset principles in the supplementary materials, which become the motivational activities, reflection activities, and instructional videos, that are not specifically implemented in the studies cited. moreover, based on the literature and studies displayed, there is a crucial need to conduct this study to enhance supplementary materials in accommodating growth mindset principles to enhance students' mindset and procedural fluency in mathematics. the study was anchored on the following theories. the primary theory of this study is the (1) mindset theory (2006) and was corroborated by (2) constructivists theory (mcleod, 2019) and (3) scaffolding theory of lev vygotsky (mcleod, 2019). carol s. dweck is the author of the growth mindset theory. she explained that a mindset believes in oneself, and one’s basic qualities. an individual with a fixed mindset believes that traits like intelligence, creativity, and ability are predefined and finite. these attributes are so established in persons with fixed mindsets that whatever they lack, they remain to be inadequate. a person with a growth mindset, on the other hand, believes that their natural talents were further strengthened via hard work and dedication. these intrinsic traits are merely the beginning; the results of success from hard work, study, and perseverance. dweck employed a school setting as an example who explained that fixed and developing mindsets emerge differently. students with a fixed mindset regulate image above all and, as a result, disregard the essential learning opportunities if performing poorly or admitting mistakes is required. in the classroom, cognitivist teaching approaches strive to help students assimilate new material into their existing knowledge while also make them produce the required modifications to their current intellectual structure in accommodating the new knowledge. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 14 according to the scaffolding theory, scaffolding involves adjusting the amount of assistance to the intellectual abilities of students. during a teaching session, the extent of coaching was modified to fulfil the student's potential level of performance. when a student experiences difficulties with a task, further assistance is provided, and as the student progresses, less guidance is performed. the objective of this study is to enhance students’ mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials in one of the junior high schools in the philippines. in particular, it examined the supplementary materials in ninthgrade mathematics which may be enhanced to integrate growth mindset principles. moreover, it was also performed in determining the effects of employing the ninth-grade supplementary materials with growth mindset principles integration in enhancing students’ procedural fluency in mathematics and fostering their growth mindset in mathematics. this study is deemed essential to students because the findings would convey messages that mathematics can be learned by all individuals who are able to learn and appreciate mathematics and that their ability is not fixed. for teachers, the findings of the study would assist them appreciate the implementation of supplementary materials promoting a growth mindset through motivational activities, reflective learning activities, and utilisation of instructional videos that make students encouraged, responsible, and possess a robust growth mindset in mathematics. in improving growth mindset and procedural fluency among students in mathematics, the school administrators, curriculum designers, and textbook writers may consider the study's findings in formulating programs, designing curriculums, and selecting a framework for writing textbooks respectively. methods research design the study employed the descriptive-developmental method using a quasi-experimental design. the study also utilised the mixed-method approach of research inquiry. the implementation group applies the innovation to determine the effect of utilising the ninthgrade mathematics supplementary materials with growth mindset integration on students’ procedural fluency and mindset in mathematics. the comparison group was implemented for the comparison of results which incorporated the textbook and lesson plans provided by the department of education without the supplemental materials. below is the study’s design diagram, where and are the pretest and posttest of the implementation group, respectively. x is the intervention (developed supplementary materials integrating growth mindset principles). meanwhile, and are the pretest and posttest of the comparison group, respectively, without the intervention: x respondents the respondents of the study were purposively selected two classes of one of the public secondary high schools in the philippines, sy 2020-2021. purposive sampling is a technique christian r. repuya & jedh esterninos 15 in which the researcher arranges criteria to select members of the population to be in charge as respondents of the study (dudovskiy, 2017). the researcher in this study selected the grade 9 students as it was the first level after the middle school, which is appropriate for the study, and they became the next group who conducted the nat. it is also because, during midschool classes, there is a rise in an imbalance of mathematics teaching (sun, 2015). in selecting the two groups, the students’ previous mathematics grades were examined to ensure no significant difference between the student’s initial performances in mathematics and were comparable. the implementation group and comparison group were established through a toss coin. each class comprised of 30 students, 17 male and 13 female in the implementation group, and eight male and 22 female in the comparison group. the researchers have explained the respondents that they were purposively selected, and informed consent was secured. they also discussed the role of the respondents and other ethical details. moreover, the confidentiality of respondents’ scores in the procedural test, mindset questionnaire, and other responses was assured. research procedure a letter of asking permission to conduct the study was created and proposed to the department of education’s schools division superintendent and the school supervisors for approval. the researchers began preparing, developing, and validating the data collection instruments after receiving the approval. the group of students in (implementation) and (comparison) was provided a pre-test and post-test in 30-item researcher-made test in mathematics to identify their procedural fluency. before the intervention, a pre-test (o1) was administered to the group, then a post-test (o2) was conducted after the intervention. the difference between the pre-test and post-test (q1 q2) was employed to ascertain whether there is a change or gain due to supplementary materials of integrating growth mindset principles (intervention, x). quantitative data encompasses students’ responses to the mindset questionnaire and feedback on their journals, and informal interviews regarding their experiences during the supplementary materials implementation. the students’ mindset scores from the mathematics interview schedule questionnaire were investigated by utilising a quantitative method. furthermore, journal and informal interviews were administered to demonstrate the improvement of a growth mindset among students in applying the supplementary materials. the data gathered were then accommodated and organised for presentations and analysis. the researchers prepared the research report after the data were arranged, analysed, and interpreted. research tool and analysis the study utilised the adopted validation guide from the deped as the instructional materials evaluation tool in validating the developed supplementary materials along with their features (deped naga city division order no. 441, series of 2019). the researcher demanded the panel of experts to portray their opinion on the statements and item regarding the features of the supplementary materials by incorporating comments and suggestions for improvement. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 16 the instructional material evaluation tool possesses a five-point scale with the following descriptive interpretation: 4.51-5.00 (outstanding), 3.51-4.50 (very satisfactory), 2.51-3.50 (moderately satisfactory), 1.51-2.50 (poorly satisfactory), and 1.00-1.50 (not satisfactory). the supplementary materials were assessed in three components in particular: mathematics content, features, and technical aspects. along with the mathematics content, the supplementary materials were examined concerning the consistency to learning competencies, objectives and accuracy of the content. along with the features, the evaluation process concentrated on the supplementary materials’ features and appropriateness to the objectives as reflective learning activities, motivational activities, and instructional videos. finally, along with technical aspects, the supplementary materials were evaluated in regarding the application of stimulating graphics, utilising colours pleasing to the eye, readability, and organisation of texts, consistency in the utilisation of font, colours, and design, presentation of the cover page, and the easiness to reproduce. the validated procedural fluency test was employed for pre-test and post-test in identifying students’ procedural fluency. it incorporates the lessons established for the 4 th quarter of the sy 2020-2021 based on the most essential learning competencies (melcs) under the k-12 curriculum of the deped. the procedural fluency test was designed with a table of specifications to ensure that the test items were equally distributed among the topics and learning competencies encompassed in the test. the researcher implemented the mindset questionnaire of dweck (2006) during interview schedules to collect students’ preand post-scores in mindset in mathematics. the questionnaire comprises 20 statements illustrating growth or fixed mindset (questions directed to mathematics performance). the respondents responded with the scales of strongly agreed, agreed, disagreed, or strongly disagreed with the statement. the statements were assigned point values. if the statement was a fixed mindset statement, the point values were: 0 points for strongly agree, 1 point for agree, 2 points for disagree, and 3 points for strongly disagree. if the statement demonstrated a growth mindset, the point values were: 3 points for strongly agree, 2 points for agree, 1 point for disagree, and 0 points for strongly disagree. the interpretation of the total mindset scores comprises 00-20 (strong fixed mindset), 21-33 (fixed mindset with some growth ideas), 34-44 (growth mindset with some fixed ideas), and 45-60 (strong growth mindset). the fixed mindset statements encompass, “some people are good in mathematics and kind (patience in mathematics), and some are not – it is not frequently that people change”. meanwhile, the growth mindset statements incorporated a statement like “all human beings are capable of learning mathematics”. for instance, supposing a student’s mindset survey score declines between 00-20. in that case, he possesses a strong fixed mindset which implies his belief that mathematics ability is fixed and cannot be learned through effort, as he responded to the questions in the mindset survey. furthermore, if a survey score of students’ mindsets descends between 21-33, he owns fixed ideas about mathematics ability but also believes in any other method that mathematics can be learned through exercise, as he responded to the mindset questions. unstructured interview, focus group discussion questionnaires, journals, and outputs were also considered to corroborate the findings of the study through thematic analysis. frequency christian r. repuya & jedh esterninos 17 count was employed to calculate the number of occurrences of respondents’ answers under the different questions and the number of students’ correct responses in the mathematics researcher-made pre-test and post-test in procedural fluency. moreover, it also administered to measure the number of students’ growth mindsets based on the scale and interpretation. the difference between the pre-test and post-test of the students on the procedural fluency exam was determining the application of the mean difference/mean gain method. the students’ procedural fluency and performance levels were identified by utilising the percentage technique. according to the national council of teachers of mathematics (nctm) (2014), the ability to implement procedures precisely, efficiently, and flexibly; to transfer procedures to new issues and contexts; to generate and adjust procedures from other procedures, and to comprehend the moment of one technique or method is more appropriate to implement than the other is procedural fluency. in this study, students’ performance level in procedural fluency concerns on the student's scores on a 60-item procedural fluency multiple-choice test formulated and validated by the teacher, and answered by students in the pre-test and post-test. the performance level of the students was interpreted by administering the scale: 35 and below (very low mastery), 36-65 (low mastery), 66-85 (average mastery), 86-95 (moving towards mastery), and 96-100 (mastered). results and discussion the primary concern of this study is to enhance supplementary materials for ninth-grade mathematics with growth mindset principles integration which was expected to help students encourage and enjoy learning and possess meaningful learning while studying at home. the study’s outcomes could serve as supplemental materials to corroborate mathematics teachers in teaching mathematics to enhance students’ procedural fluency and mindset in mathematics. the developed supplementary materials encompassed six lessons for ninth-grade mathematics, such as the six trigonometric ratios, the trigonometric ratios of special angles, angles of elevation and angles of depression, trigonometric ratios to solve real-life problems incorporating right triangles, laws of sines and cosines, and problems accommodating oblique triangles. each lesson was anchored to the most essential learning competencies of the ninthgrade mathematics curriculum and adhered the policy on lesson planning of the department of education. five jurors subjected the developed supplementary materials to critiquing. the jurors provided their comments, suggestions, and recommendations to enhance each part of the lessons and supplementary materials. the developed lessons possess specific features in which the developed supplementary materials incorporate growth mindsets. these are motivational activities, reflective learning activities, and integration instructional videos to assist students enhance a strong growth mindset in mathematics. motivational activities table 1 presents the integration matrix of the growth mindset principle through motivational activities in the developed supplemental materials for teaching ninth-grade mathematics. motivational activities in every lesson were employed to make students more enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 18 prepared. each topic was associated with them, leading them to think about the growth mindset principles. for instance, in lesson 1, along with the “let’s start”, the students read the direction and employed the words in the box to complete the sentence. it came up with the phrases “i will do my best”, “i can perform more effort”, “i can learn from my mistakes”, “i can achieve my goals”, and “i can overcome challenges”. these phrases are concerning about the growth mindset that the students require to enhanced. according to dweck (2006), if the students think that their success is based on hard work, learning, and training, individual possesses a growth mindset. in this activity, students will identify what they require to remember to more succeed the learning on particularly topic. in that case, the student can improve their growth mindset. in lesson 2, along with the “let’s start” part, the activity assisted the students develop their growth mindset in a method that they require to realise that for them to enhance their performance, they need to learn new things, welcome constructive feedback, discover creative solutions to solve a problem, elevate with practice, and train their brain. the activities in lessons 3 and 4, “discovering a solution to a problem” and “converting negative to positive”, may enhance the students’ growth mindset. table 1 integration of growth mindset principles via motivational activities lessons activities description lesson 1: the six trigonometric ratios: sine, cosine, tangent, secant, cosecant, and cotangent challenge yourself to succeed direction: use the words in the box to fill in the blanks 1. i will do my ___________ 2. i can put in more _______ 3. i can learn from my _____ 4. i can achieve my _______ 5. i can overcome ________ this activity was anchored on the growth mindset principle. this activity allows students to understand that our belief about intellectual abilities can be enhanced for us to be successful. in this activity, students have discovered what they need to remember to be more successful in understanding the topic. hence, the students are able to improve their growth mindset. lesson 2: the six trigonometric ratios of special angles improve and develop direction: match the appropriate word column a column b i can learn ______________ a. solutions i welcome constructive ____ b. practice i can train my ___________ c. new things i can improve with _______ d. brain i can find creative _______ e. feedback this activity assists the students to be aware that there are principles that we need to follow to enhance our performance. the students matched the appropriate word from column b to complete the sentence to realise this. this part concerns on developing the students’ growth mindset that knowledge is not fixed but can be improved. challenges goals best mistakes effort christian r. repuya & jedh esterninos 19 lessons activities description lesson 3: angle of elevation and angle of depression finding a solution to a problem direction: match the correct solution to a problem from column a to column b the students matched column a to column b to get the correct solution to a problem or scenario. this activity focuses more on developing the students’ growth mindset that there is no problem if there is no solution. lesson 4: solving word problems involving right triangles negative to positive direction: convert the following negative thought to positive thinking. negative thoughts your positive thoughts 1. mathematics is a very hard subject for me. 1. 2. i cannot solve any problems in trigonometry concerning right triangles. 2. 3. i never understand the concept and process of solving problems in mathematics. 3. 4. i think that my teacher does not care about my learning. 4. 5. i could not complete high school because there was a lot of homework and projects. 5. 6. i cannot go to college because it is very hard. 6. 7. i cannot study without the help of my classmates or my friends 7. 8. i think that i am not good at anything. 8. 9. i never do anything to succeed. 9. 10. i cannot achieve my dream and be a successful person. 10. in this activity, “negative to positive”, the students convert the following negative thoughts to positive thinking. this activity is more on elevating the students' growth mindset that for us to be successful in life, we require to practice converting negative thoughts into positive thinking. the students answered the following questions: 1. how did you convert the following negative thoughts to positive thoughts? 2. how can this activity assist you to be optimistic and motivated to learn? enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 20 lessons activities description lesson 5: oblique triangles: sine law, cosine law, and the application you can do it. trust yourself direction: formulate your own motivational quotes that you can always remember to be a successful person someday. answer the following questions: 1. how can this motivational quote help you in your study about this topic in this module? 2. what are the things you need to perform now to achieve your dreams? 3. from the quotes you have encountered in the previous modules, what quote do you like the most? why and how will you apply those to grow and develop as an individual? in this activity, the students generated motivational quotes that they can always remember for them to be a successful person someday. hence, it helps the students to enhance their growth mindset. this section is able to develop the students’ growth mindset; hence, they need to realise that every problem, big or small, must possess a solution, and failure and mistakes are not bad things. the motivational activity in lesson 5 is “you can do it, trust yourself”. in this activity, the students created motivational quotes that they can always remember to be a successful person someday. this section enhances students' growth mindset; thus, the students will realise that there is no other way to enhance themselves but by trusting themselves. briggs (2015) stated in his article that learning takes time and do not anticipate learning everything in one session. it indicates that the students need to exert efforts to learn and realize that knowledge is not fixed but can be enhanced. when students make a mistake or own a problem, they have not failed but they have learned instead. he recommended replacing the word “failing” with “learning” which can improve students’ growth mindset. furthermore, students should stop finding approval from others because when students prioritise approval over learning, they sacrifice their growth potential. reflective learning activities reflective learning was one of the features of the developed supplementary materials accommodated on the growth mindset employed in this study. gray (2021) defined reflective learning as students’ thinking about what they have learned in the classroom and how it is implemented into their individual life. it is a method of learning in which students reflect upon their learning experiences. reflective learning also allows the students to reflect on their learning. in this study, reflective learning concerns on the students’ reflections in the lesson format which implement supplementary material attached on growth mindset principles and the approach employed in the development of the lessons. table 2 presents the promotion of reflective learning, another feature of the developed supplemental materials for teaching ninth-grade mathematics. christian r. repuya & jedh esterninos 21 table 2 promoting reflective learning lessons reflective learning theme description 1 “success is not an accident. success is a choice”. – stephen curry (n.d.) in this lesson, reflective learning was performed in the “let’s reflect” part, in which the students reflect on their learning. the students reflect on the provided quotation, that is about being successful. this activity will motivate the students to study hard for the successful outcome. 2 “learn everything you can, learn anytime you can from anyone you can – there will always come a time when you will be grateful for what you did” – sarah caldwell (n.d.) the quotation about learning by sarah caldwell was employed to promote reflective learning. in the “let's reflect” part, the students reflect on their learning and the provided quotation about the importance of learning. the students will realise that learning is essential. it also helps to improve the growth mindset of the students. 3 “work to find solutions instead of always highlighting the problems”. – pictureqoutes.com (n.d.) the promotion of reflective learning in this lesson is on the “let’s reflect” part, in which the students reflect on the quotation about discovering a solution for a problem. it will also assist the students to enhance their problem-solving skills, accommodated to the growth mindset principles. 4 “once you replace negative thoughts with positive ones, you will start having positive results” – willie nelson (n.d.) in this lesson, reflective learning was introduced in the “let’s reflect” part, in which the students reflect on their learning. the students also reflect on the provided quotation, which is about being positive. this activity will encourage students to study hard and avoid any negative thoughts. 5 “trust yourself. you know more than you think you do”. – benjamin spock the quotation about trusting oneself by benjamin spock was administered to promote reflective learning. in the “let's reflect” part, the students reflect on their learning and the provided quotation about the importance of self-confidence. the students will realise that trusting oneself is crucial in learning and achieving goals. in the “let’s reflect” part, the last part of every developed lesson, the students reflect on the provided quotations. for instance, in lesson 1, the quotation is about success by curry (n.d) “success is not an accident, success is a choice”. the student reflected on this quote to encourage them to study hard for being successful. this part also allows the students to review and reflect on their learning and new understanding. in this part, the students were also asked about what they have learned, the difficulties they experienced, what they learned from these quotes at each start of the lesson, and the way to apply the things they have learned to acquire their goals. as another part of their reflections, they were also asked to write their significant learning experiences to capture their significant learning experience during the conduct of the study. elias (2010) explained that students’ reflections were identified as one of the top six activities to assist students to be successful. he asserted that students should formulate their goals, record the achievements, and show and share the learning with others. promoting reflective learning in lesson 2 was on the “let's reflect” part. the students reflected on the quote by caldwell (n.d), which is “learn everything you can, learn anytime you can, learn from anyone you can – there will always come a time when you will be grateful for what you did”. in this activity, the students realised that learning was essential and help them enhance their growth mindset. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 22 malec (2021) elaborated that continuous learning is vital because it assists students feel joy in life and elevate their productivity. when students realise the importance of learning, they will be encouraged and eager to learn. the students also reflected on what they did and learned from activity two and how it helped them prepare for the topic. the difficulties they have encountered in the lesson and how they solved it, how the concepts they have learned are applied in real life, and how it may assist them to achieve their goals. the theme in lesson 3 was “finding a solution to a problem”. problems in learning are frequently present, hence, the students sometimes do not understand what to perform. in this part of the promotion of reflective learning, the students reflected on the quotation about finding a solution to a problem: “work to find solutions instead of always highlighting the problems” (pictureqoutes.com). it helps the students improve their problem-solving skills and promote growth mindset principles. in lessons 4 and 5, the students reflected on the quotations about possessing positive thoughts and trusting themselves. one of methods for them to succeed is to be positive and trust themselves. as cited by williams (2017), dewey asserted that reflection was a tremendous significant part of learning. he advocates that reflective learning is not merely reading and obtaining knowledge. it is an active examination of learning from the information in accordance with the students’ learning experiences. the students need to think deeply based on what they have read and learned from the lesson. then, they can implement the knowledge to their context. the implementation of supplementary material with its features makes learning meaningful to the students since they can reflect and freely express their selves about what they have studied. it also encourages learning transfer because students would easily identify the concepts as they reflect on their personal experiences. furthermore, the deped promoted reflective learning in the k to 12 basic education to provide a meaningful learning experience. reflective learning allows learners to employ what they have learned and why they need to learn (department of education order no. 21, series of 2019). additionally, briggs (2015) stated that if the students are allowed to reflect on their understanding at least regularly, they can enhance their growth mindset. opportunities for reflection encourage the students to be motivated to learn and elevate their interest in learning. integration of instructional videos integrating instructional videos was one of the features of the developed supplementary materials encompassed on the growth mindset employed in this study. any video displaying a procedure, transmitting knowledge, explaining a concept, or presenting someone how to accomplish something are classified as an instructional video. it means that it is a form of instruction utilising technology to convey knowledge to the viewers (simon, 2012). according to mendoza et al. (2015), employing instructional materials and integrating instructional video obtains the student's interest. it also helps the students to concern on the lessons. table 3 presents the developed supplemental materials featuring the integration instructional video teaching ninth-grade mathematics. videos in each lesson were utilised as instructions that assisted students to comprehend each topic in mathematics and were christian r. repuya & jedh esterninos 23 associated with them, improving their understanding and interest in mathematics. in this study in lesson 1, the instructional video is all about the concept of the six trigonometric ratios, portraying the parts of a right triangle and illustrating the six trigonometric ratios. it also displays the method to identify the missing parts of the triangle and its application in real life. in lesson 2, the instructional video discusses about the “trigonometric ratios of special angles”. the students watched the video which portrays the concept of trigonometric ratios of special angles, discovering the missing parts of the tringle utilising special angles, and its application in real life. after watching the instructional video, they were writing what they had learned in their notebook and answered some questions associated with the discussion. table 3 integrating instructional videos lessons instructional videos descriptions lesson 1: the six trigonometr ic ratios: sine, cosine, tangent, secant, cosecant, and cotangent “the parts of the right triangle” “the six trigonometric ratios” “trigonometry – application” this instructional video in this section integrated into the “let’s start” part, activity number 2, in which the students watched the video segment to assist them to recall the different concepts about a triangle and answered the following questions: 1. what are the parts of the right triangle? 2. what are the definitions of the parts of a right triangle? 3. is it easy to identify the parts of a right triangle? why? or why not? another instructional video in this section also accommodated “let’s find out” part, and they wrote the things they had learned from the video. the concepts of the six trigonometric ratios were further discussed. in the “let’s apply” part, the students watched the instructional video and solve the provided worded problem about the six trigonometric ratios and answered the following questions: 1. did you obtain the correct answer? how? 2. is the problem easy to solve? write the difficulties you've encountered if you have one? 3. if you were asked to make your trigonometric ratio problem, how would you make it? lesson 2: the six trigonometr ic ratios of special angles “exact trigonometric ratios by employing your hands”. “trigonometric ratios of special angles” “trigonometry – application” the instructional video in the “let’s start” part was on activity number 2. the students watched the video segment to help them remember the exact trigonometric values employing their hands. then, they answered the following questions: 1. how did you feel after you watched the video? 2. did employing the hand trick in finding exact trigonometric values make you easily compute the values of trigonometric ratio? 3. how could you utilise the trigonometric ratios hand trick in finding the unknown angles or sides of a special right triangle? in the “let’s find out” part, the instructional video in this part talks about the concept of trigonometric ratios of special angles. the students watched the video, and they wrote the things that they had learned. another instructional video incorporates on “let’s apply” part. the students watched the video about solving problems involving trigonometric ratios of special angles and solve the provided worded problems. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 24 lessons instructional videos descriptions lesson 3: angle of elevation and angle of depression “angle of elevation and angle of depression” clinometer (how to make and use) the instructional video in this lesson was on the “let’s find out” part, and it discussed about the concept of the angle of elevation and the angle of depression. this content discussed all the content. the students watched the video segment and wrote what they had learned. questions: 1. based on the video you’ve watched, what is the definition of angle of elevation? 2. what is angle depression? 3. what are the differences between the angle of elevation and depression? the instructional video in this part was the video containing the way to create a clinometer. the students followed the steps in making a clinometer based on the instructions from the video, and then the students performed activity 4. in activity 4, “your shadow”, students calculate the height of their friend and the length of his/her shadow. applying the concept of trigonometric ratios, the students determined the angle of elevation from the ground to the sun. lesson 4: solving word problems involving right triangles “the scientific calculator” and “example problem” the instructional video in this lesson was about utilising the scientific calculator and solving problems incorporating the right triangle. the students watched the video segment about the example of problems involving right triangles applying the trigonometric ratios and wrote what they had learned. lesson 5: oblique triangles: sine law, cosine law, and its application “sine and cosine law: finding the missing parts of an oblique triangle”. “sine and cosine law – application” the instructional video in this lesson was all about oblique triangles. the students watched the video segment entitled “oblique triangles” sins and cosine law: finding the missing part of an oblique triangle, and afterward, they wrote the things they had learned. from the video, the concepts were further discussed. another video on this part discussed about the application of sine law and cosine law. the students watched the video about sine law and cosine law–applications and solved the provided worded problems. there is an instructional video in activity 2, in which the students were watching the video segment entitled “trigonometric ratios shortcuts” to assist them remember the exact trigonometric values by employing their hands. the short video presents the way to memorise exact trigonometric values by employing the hand trick. the video helps the students easily compute trigonometric ratios’ values, and employ the trigonometric ratios hand trick in identifying the unknown angles or sides of a special right triangle. there is also a video about the implementation and how to solve problems regarding trigonometric ratios of special angles. this video displays the students how to solve problems involving trigonometric ratios of special angles. in lesson 3, the instructional video in this lesson is on the “let’s find out” part. it discusses about the “angle of elevation and angle of depression”. the students were watching the video segment about the “angle of elevation and angle of depression”. the video displays the way the angle of elevation and angle of depression is employed in solving word problems christian r. repuya & jedh esterninos 25 incorporating right triangles and application in real life. they were also writing about what they had learned from the video for their reflection and evaluation purposes. there is also a video about how to make a clinometer. the video teaches the students on how to make and utilise the clinometer to calculate the angle of objects. after they have watched the video, an activity follows, which is activity 4. in activity 4, entitled “your shadow”, students measured the height of their friend and the length of his/her shadow. employing the concept of trigonometric ratios, the students examined the angle of elevation from the ground to the sun. employing the clinometer, they could calculate the angles. after the activity, there are also questions that they require to answer as a part of the evaluation, in lesson 4, the instructional video is on the “let’s find out” part. it is all about solving problems encompassing the right triangle, utilising the scientific calculator, and employing concepts in the previous lesson to solve problems incorporating the right triangle. the students watched the video segment about example problems concerning right triangles utilising the trigonometric ratios. in lesson 5, the instructional video is on the “let’s find out” part. it was about oblique triangles. the students watched the video segment entitled “oblique triangles” sins and cosine law: discovering the missing part of an oblique triangle. there was also a video on the “let’s apply” part. the video on this part discussed about implementing sine and cosine law. the students watched this video and solved the provided worded problems. there are six lessons developed with instructional videos, that are the six trigonometric ratios, the six trigonometric ratios of special angles, the angle of elevation and angle of depression, solving word problems incorporating right triangles, and oblique triangles: sine law, cosine law, and its application. effects on the students’ procedural fluency in mathematics procedural fluency is the ability to implement procedures accurately, efficiently, and flexibly, to correlate procedures to different problems and contexts, build and modify procedures from the other ones, and recognise when one strategy or procedure is more appropriate. in developing procedural fluency, students should experience rationalising both informal approaches and general employed procedures mathematically, should establish and defend their choices of appropriate procedures and reinforce their knowledge and skills through the distributed practice (nctm, 2014). employing supplementary materials anchored on growth mindset principles in the development of the lessons in ninth-grade mathematics allows students to enhance procedural fluency. table 4 presents the pre-test and post-test results provided for students in the implementation group. it indicates that the mean scores of the students in the implementation group during the pre-test were all interpreted as very low mastery except for the fourth competency on solving real-life problems which encompass right triangles classified as low mastery. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 26 table 4 result of pre-test and post-test in the procedural fluency implementation group comparison group topics no. of items pre-test post-test mean gain pre-test post-test mean gain mean interpret ation mean interpret ation mean interpret ation mean interpret ation the six trigonometric ratios 4 0.97 (24.3%) vlm 3.1 (77.5%) am 2.13 1.2 (30%) vl m 2.2 (55%) lm 1 trigonometric ratios of special triangles 4 0.93 (23.3%) vlm 1.9 (47.5%) lm 1.27 1.1 (27.5%) vl m 1.5 (37.5%) lm 0.4 angles of elevation and depression 7 1.37 (19.6%) vl m 3.7 (52.9%) lm 2.33 2.6 (37.1%) lm 3.1 (44.3%) lm 0.5 solving real-life problems involving right triangles 3 1.4 (46.67%) lm 2.3 (76.7%) am 0.9 1.4 (46.7%) lm 1.9 (63.3%) lm 0.5 the laws of sines and cosines. 9 2.37 (26.3%) vlm 4.6 (51.1%) lm 2.23 3.5 (38.9%) lm 4 (44.4%) lm 0.5 solving problems involving an oblique triangle 3 0.6 (20%) vlm 1.4 (46.7%) lm 0.8 0.8 (26.7%) vl m 1.2 (40%) lm 0.4 total 30 7.63 (25.5%) vlm 16.87 (56.2%) lm 9.24 10.67 (35.6%) vl m 13.93 (46.4%) lm 3.23 legend: vlm – very low mastery, lm – low mastery, am – average mastery, hm – high mastery, vhm – very high mastery, p* tested at a .05 level of significance looking at the data, the learning competency “solving real-life problems involving right triangles” (46.67%) owned the highest performance level, while the lowest was on “illustrates angles of elevation and angles of depression” (19.6%). during the post-test, the students acquire the highest performance level on “the six trigonometric ratios” (77.5%), while the lowest performance was on “solving problems involving oblique triangle” (46.7%). overall, the total mean score for the pre-test was 7.66 (25.5%), interpreted as very low mastery, and 16.87 (56.2%), construed as a low mastery for the post-test, which displayed an increase of 9.24. results presented a difference in the students' performance in the group from pre-test to post-test. the table revealed an increase in the mean of post-test of 3.1, 1.9, 3.7, 2.3, 4.6, and 1.4 on competencies 1 to 6, respectively. it indicates that the students in the implementation group who employed the instructional materials enhanced the students’ procedural fluency in terms of the lessons incorporated in the study. with the result, it is implied that the students who were taught utilising the supplemental materials obtained a high mean score in the post-test since there was an increase in the mean score of the pre-test. the possible reason for this accomplishment and improvement in performance level may be attributed to the supplemental materials, which enable students to comprehend well and exert effort in learning. one of the students expressed her thoughts about learning with the materials which is the way she applied what she has learnt and the way to achieve the goal. christian r. repuya & jedh esterninos 27 the students explained: student 3: “in achieving our goal, we require to possess hardship or work through step by step, hence, it will be easy for us to perform such things”. student 7: “i have some difficulties in finding the best way to solve the problem, particularly of which trigonometric ratios will be utilised. however, by trying to understand the problem, again and again, i can solve the problem”. based on the student’s entry, they learned to solve trigonometric ratios based on their growth mindset. even though solving worded problems was complicated, they attempted their best to discover the right answer. the student explained that learning step by step to solve the problem was significant for her to better understand the problem and to succeed identifying solutions. in conclusion, the lessons and activities in the supplementary material significantly enhanced the students’ procedural fluency. the data collected corroborated the students’ reflection, and it revealed that employing activities (such as studying with the instructional videos that allow them to think, reflect and correlate their lives to the lessons) enables the students to understand the mathematics concepts better. moreover, the students manifested a growth mindset in mathematics, leading them to strive and learn the topics more. table 4 portrays that the mean scores of the students in the comparison group during the pre-test were interpreted as low mastery of “angles of elevation and depression”, “solving real-life problems involving right triangles”, and "the laws of sines and cosines, and very low mastery for “the six trigonometric ratios”, “trigonometric ratios of special triangles” and “solving problems involving oblique triangle”. observing the data, the learning competency “solving real-life problems involving right triangles” (46.7%) possessed the highest performance level while the lowest was on “solving problems involving oblique triangle” (26.7%). based on the same table, in the post-test of the comparison group, the students acquired the highest performance level on “real-life solving problems involving right triangles” (63.3%), while the lowest performance was on “trigonometric ratios of special triangles” (37.5%). data also unveiled that the students owned low mastery of all six competencies. overall, the total mean score for the pre-test was 10.67 (35.6%), construed as very low mastery, and 13.93 (46.4%), also portrayed as a low mastery for the post-test. based on the table, it could be inferred that the utilisation of the traditional method of teaching and reference materials also enhanced the students’ procedural fluency in terms of the lessons encompassed in the study, with the mean scores of 10.67 and 13.93 in the pre-test and posttest, respectively, which displayed an increase of 3.23. the implementation group possesses a higher increase with an increase of 9.24 for the implementation group than the comparison group with an increase of 3.23. a t-test for an independent sample was administered to determine if there was a significant difference in the mean gain scores of the students between the implementation and comparison groups in procedural fluency. the results revealed a significant difference between the mean gain scores of the two groups, conveyed by the t-value of 4.890 and p-value of 0.000, which was significant at a 95% confidence level. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 28 comparing the increase of the mean score of the implementation group to the comparison group, the implementation group owns a higher increment with an increase of 9.24 for the implementation group compared to the comparison group with an increase of 3.23. the study uncovered that the students in the implementation group taught by applying the supplementary materials enhanced procedural fluency in mathematics more than those in the comparison group. the latter was taught by the traditional method and did not employ supplementary materials. bautista (2013), in his study, revealed that to remodel students’ performance on procedural fluency and mathematical explanation toward problem-solving, the instructional activities, mindset, and contents have to be complementary to one another. furthermore, the study by arthur, dogbe, and asiedo-addo (2021) discovered that motivation in learning mathematics and quality of mathematics teaching obtained significant positive effects on students’ mathematics performance. in the present study, the improvement in the implementation group could be attributed to the student's application of the supplementary materials in which they are able to learn the materials at home. the contribution of the study in elevating the students’ procedural fluency were administering the motivational activity with a growth mindset that helps the students be motivated to learn and possess a high interest in learning the topics in mathematics. reflective learning activities provide a clear goal for attaining the learning objectives in which students are encouraged to perform reflection activities. the instructional videos help students learn and comprehend more of the concept at home and be able to employ technology such as phones and computers as an additional tool for researching instructional resources. the study by sharma (2018) discovered that students receiving steady experience to instructional videos and real-life activities performed better than students receiving some of the exclusive instructional treatments. the students, during the interviews, manifested their belief that instructional videos and real-life activities were able to enhance their understanding on mathematical concepts. effects on the students’ mindset in mathematics the effects of the motivational activities on students’ mindsets were also the manifestations of the development of a growth mindset among students in performing the activity. dweck’s (2006) mindset questionnaire was administered to calculate it by utilising an interview schedule. students answered with a growth mindset or a fixed mindset for each statement. a student with a growth mindset and some fixed ideas experienced that his or her mathematical skill could be improved, but he or she also believed that some aspects of his or her mathematical skills could not be modified. a student with a fixed mindset, on the other hand, acknowledged that his or her mathematical skills could not be changed or elevated; nevertheless, he or she might believe that it could be enhanced in certain ways. the data was analysed through frequency count after determining the students’ mindsets. table 5 presents the summary of the pre-test and post-test frequency counts of the students’ mindsets in the implementation group and comparison group. christian r. repuya & jedh esterninos 29 in the implementation group, 17 students owned a “fixed mindset with some growth ideas”, while 13 possessed a “growth mindset with some fixed ideas”. 13 students owned a fixed mindset with some growth ideas during the post-test, 14 students possessed a growth mindset with some fixed ideas, and three students owned a strong growth mindset. table 5 summary of the pre-test and post-test frequency counts of two groups on growth mindset in mathematics mindsets of the students in mathematics pre-test post-test implementation group comparison group implementation group comparison group fixed mindset 0 0 0 0 the fixed mindset with some growth ideas 17 10 13 9 growth mindset with some fixed ideas 13 19 14 19 strong growth mindset 0 1 3 2 total 30 30 30 30 while 19 students in the comparison group possessed a growth mindset with some fixed ideas, ten students owned a fixed mindset with some growth ideas, and one student possessed a strong growth mindset during the pre-test. nine students acquired a fixed mindset with some growth ideas during the post-test, 19 students obtained a growth mindset with some fixed ideas, and two students gained a strong growth mindset. based on the results, the implementation and comparison groups had students in the pretest with a “fixed mindset with some growth ideas”. the students in the implementation group all exhibited a growth mindset in the post-test, with no students possessing a robust fixed mindset. however, the post-test presented that one student was moved from the students with a “fixed mindset with some growth ideas” to the students with a “growth mindset with some fixed ideas”, and nothing was added to the students with a “growth mindset with some fixed ideas”, elevating the strong growth mindset from one (1) to two (2). specifically, during the post-test in the implementation group, three students from growth mindset with some fixed ideas were involved to the students with a strong growth mindset, and four students from fixed mindset with some growth ideas were joined into a growth mindset with some fixed ideas. meanwhile, the one student in the comparison who was transferred to possess a growth mindset with fixed ideas in the post-test obtained a fixed mindset with some growth ideas in the pre-test. furthermore, one from a growth mindset with fixed ideas was switched to strong growth mindset. during the post-test in the implementation group, the three students who leaped to students with a strong growth mindset enhanced more than the other students. the other students have also developed their growth mindset, although just a little. based on the findings, students in the implementation group cultivated a stronger growth mindset than students in the comparison group. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 30 the students in the implementation group acknowledge the significance of a growth mindset in their education. during the interview, a student (student 21) moving from a growth mindset with some fixed ideas to a strong growth mindset stated the following: “nakatabang po talaga sya (motivational activities) sako kasi, sa kada lesson, ugwa pong inspiring quotes na nagmomotivate samo, pati si mga activities about motivation nakatabang sya na marealize me na kaipuhan ming mag adal para makatapos saka makagraduate” (the motivational activities tremendously assisted me a lot because in every lesson, there are quotes which encourage us to possess a growth mindset, and the activities about motivation help us realise that it is crucial for us to learn diligently to accomplish our study, to graduate). students’ development of a growth mindset might be associated with utilising supplemental materials, which enhances a growth mindset through self-learning that emphasises efforts in learning mathematics, in accordance with the students’ statements, as previously outlined in their learning experiences. sun (2019) argued that current school mechanisms, such as tracking regulations, instructional methodologies, and standardised assessment, may postpone teaching which consistently initiates growth mindset messages concerning mathematics ability. it indicates that supplementary materials which utilise activities to promote a growth mindset in mathematics could help the students develop a growth mindset despite different external factors that convey fixed mindsets. the supplementary materials were formulated to enhance students’ growth mindset in mathematics, as the motivational activities with the integration of the principles of growth mindset are able to make students more engaged in the learning process. harackiewicz et al. (2018) emphasised that interest is a robust motivational mean which elevates learning and is essential for the educational success. when students develop their interest in studying mathematics due to the motivation, their desire to learn escalates. it implies that teachers should determine both considerations in enhancing students’ interest and growth mindset in their instruction. boaler (2013) explained that through instructional and classroom activities, teachers and institutions regularly transfer messages to students regarding their skill and comprehension (i.e., instructional materials). she emphasised that a thorough review of all areas of education is required if one is genuinely committed to the transfer and training of a growth mindset. she added that, norm-setting, questioning, committing mistakes, providing activities, grouping, grading, and feedbacking were all employed to convey mindset messages. briggs (2015) asserted that a growth mindset can alter goals and perceptions of achievement. he emphasised that conducting growth mindset-related actions such as employing the word “yet”, learning from others’ failures, setting new goals for each goal achieved, taking opportunities in the company of others, and thinking realistically about time and effort can eventually help students enhance the growth mindset. the students were demanded to share their experiences by writing them in their journals to corroborate the study’s findings with the actual manifestations of the development of a growth mindset among the students in performing the activities. the students’ responses in the informal interviews and focus group discussions were also sources to examine the manifestations of the development of a growth mindset among the students. christian r. repuya & jedh esterninos 31 after thematic analysis of the unstructured interviews responses, focus group discussion notes, and journals entries of the student during the implementation and after the utilisation of the supplemental materials, five themes were generated (table 6): the students (1) acknowledging and embracing imperfections, (2) viewing challenges as opportunities, (3) replacing the word “failing” with the word “learning”, (4) providing regular opportunities for reflection, and (5) thinking realistically about time and effort. sample statements of the students in each theme were presented in the matrix in table 6. the students’ statements imply that the students developed their growth mindset by acknowledging and embracing imperfections. the students accepted that the activities were difficult, but it manifested that they possessed a growth mindset due to their acknowledgement of imperfections. another manifestation of the development of a growth mindset was the student’s response during the interview, which was by replacing the word “failing” with the word “learning”, and the students learned to perform their best even when they committed mistakes. it may be attributed to one of the elevated supplementary materials, reflective learning activities, which allow the students to reflect on their learning. it is a method of learning in which the students are able to reflect upon their learning encounters. based on the students’ reflection, they acknowledged that through the effort, he could learn to solve problems. according to le cunff (2022), another thing about enhancing a growth mindset is situating growth before speed. she asserted that fast learning does not necessarily mean equate to good education and learning well requires an investment of time. it means that students should also think about the importance of time and effort in acquiring new skills. based on the student’s response, it was identified that the student understands that it takes time to grow and improve. table 6 sample statements of the students in each theme themes description 1 acknowledge and embrace imperfections “today, i experienced that those activities are not easy to resolve, but if we perceive all things positively, we can achieve our goals”. 2 perceive challenges as opportunities “i learned to improve myself, particularly in solving skills, by studying in this module about matrices solving real-life problems and being positive”. 3 replace the word “failing” with the word “learning”. “today, i also learned to be positive. even though i committed mistakes, i attempted my best to learn from them. somehow solving involving right triangle is not easy, but i was willing to learn so that i tried”. 4 provide regular opportunities for reflection “i learned that learning could be everywhere. we could learn everything at any time from anyone or ourselves. i learned that to learn. we required the effort to accomplish something. today, i learned that learning is the best way to progress. we required to perform whatever we could do; we just needed to put some effort to do good and solve problems even in real-life experiences”. enhancing mathematics learning by integrating growth mindset principles in ninth-grade supplementary materials 32 themes description 5 think realistically about time and effort “learning fast does not mean learning well, and learning well requires allowing time for mistakes. thinking realistically about the time and effort is required to acquire a new skill”. “in our life, success is the very best thing to acquire but in order to succeed, you need to set your goals, put more efforts, learn from mistakes, try our best to solve the challenges we have, like in solving trigonometric, we need to effort to solve and identify the solution to attain the correct answers and try our best to determine the right value of it”. overall, the students’ mindsets shifting from a fixed mindset to a growth mindset is one of the impacts of utilising the developed supplementary materials such as motivational activities, reflection activities, or instructional videos. the study results are associated with the study of masterson and koch (2021). the research result discovered that moving from a fixed mindset to a growth mindset required adjustments to the mathematics methods. in this study, supplementary materials integrating growth mindset principles are able to help students develop a growth mindset in mathematics. conclusion the supplementary materials developed to integrate growth mindset principles were motivational activities, reflection activities, and instructional videos. the students possessed significant learning experiences by employing supplementary material, which developed procedural fluency and assisted students in shifting from fixed mindsets to growth mindsets in mathematics. the students learned to acknowledge and embrace imperfections, view challenges as opportunities, acknowledge failures as a key to learning, reflect on learning, and think realistically about time and effort. the implementation of the supplementary materials affects students’ procedural fluency and mindset in mathematics better than conventional teaching in accordance with the findings of this study. in strengthening students’ learning and growth mindset, researchers could employ the supplementary materials in various areas of learning and with larger sample sizes. acknowledgement the students, parents, educators, and school principal who participated in the study were acknowledged for their participation and cooperation during the implementation and data collection. the researchers also appreciate the department of science and technology for providing funding and the department of education for approving the study in the school. references arthur, y. d., dogbe, c. s., & asiedo-addo, s. k. 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