The Reconstruction of Functions Using Didactic Transposition: A Focus on the Textbooks Following the Malaysian and Indonesian Mathematics Curricula

Nadya  Syifa Utami; Sufyani Prabawanto; Phoong Seuk Yen; Arrold Rynix Anak Rambee

doi:10.32890/mjli2025.22.2.9

Authors

Nadya Syifa Utami Department of Mathematics Education, Universitas Pendidikan Indonesia, Indonesia
Sufyani Prabawanto Department of Mathematics Education, Universitas Pendidikan Indonesia, Indonesia
Phoong Seuk Yen Department of Mathematics, Universiti Pendidikan Sultan Idris, Malaysia
Arrold Rynix Anak Rambee Department of Mathematics, Universiti Pendidikan Sultan Idris, Malaysia

DOI:

https://doi.org/10.32890/mjli2025.22.2.9

Keywords:

Functions, didactic transposition, Malaysian curriculum, Indonesian curriculum, comparative study

Abstract

Purpose – This study is aimed at exploring how the topic of functions was organised into knowledge to be delivered in textbooks based on the Malaysian and Indonesian mathematics curriculum, and how the topic reconstruction in both countries differed.

Methodology – A documentary research method was used in this study. Study data were collected through the examination of school mathematics textbooks, specifically Grade 8 textbooks in Malaysia and in Indonesia. The data used was the sequence of tasks presented in the chapter on functions in each country’s mathematics textbooks. This study analysed data by identifying the learning objectives of functions in both countries' curricula; categorizing task sequences in textbooks using the didactic co-determinacy framework—from domain level to sector level, theme level, and task level; and examining the similarities and differences on the topic of functions and the categorized task sequences in textbooks following the Malaysian and Indonesian mathematics curricula respectively.

Findings – The findings indicated that the presentation of functions in Malaysian and Indonesian textbooks followed a similar sequence, beginning with an introduction to functions, progressing to their representation, and concluding with their application in solving real-world problems. Malaysian and Indonesian textbooks both introduced functions as a subset of relations and demonstrate various representations, including arrow diagrams, ordered pairs, tables, formulas, and graphs. While both curricula shared a common foundation in the set-theoretical definition of functions, they differed in their approach to problem-solving. Malaysian textbooks emphasized the use of function graphs more extensively, whereas Indonesian textbooks frequently utilized function formulas as problem-solving tools.

Significance – This comparative study provides a practical application of the didactic transposition process of knowledge organised in the mathematics curriculum of two different countries. The results also contribute to the potential pathway for learning functions by drawing lessons from two different systems of education.

References

Abdullah, S. A. S. (2010). Comprehending the concept of functions. Procedia-Social and Behavioral Sciences, 8, 281–287. https://doi.org/10.1016/j.sbspro.2010.12.038

Akkoç, H., & Tall, D. (2005). A mismatch between curriculum design and student learning: The case of the function concept. Proceedings of the Sixth British Congress of Mathematics Education Held at the University of Warwick, 1–8.

Artigue, M., & Winsløw, C. (2010). International comparative studies on mathematics education: A viewpoint from the anthropological theory of didactics. Recherches En Didactique Des Mathématiques, 30(1), 47–82.

Ayalon, M., & Wilkie, K. J. (2019). Exploring secondary students’ conceptualization of functions in three curriculum contexts. The Journal of Mathematical Behavior, 56, 100718. https://doi.org/10.1016/j.jmathb.2019.100718

Bahariah, Baharizah, Jannah, N., Nurazreen, & Nazri M. (2017). Mathematics Form 2. Ministry of Education Malaysia.

Bailey, K. (2008). Methods of social research. Simon and Schuster.

Bakar, M. N. (1991). What do students learn about functions?: A cross cultural study in England and Malaysia. University of Warwick.

Bardini, C., Pierce, R., Vincent, J., & King, D. (2014). Undergraduate mathematics students' understanding of the concept of function. Indonesian Mathematical Society Journal on Mathematics Education, 5(2), 1-24. https://doi.org/10.22342/jme.5.2.1495.85-107

Biehler, R. (2005). Reconstruction of meaning as a didactical task: The concept of function as an example. Meaning in Mathematics Education, 61–81. https://doi.org/10.1007/0-387-24040-3_5

Borke, M. (2021). Student teachers' knowledge of students' difficulties with the concept of function. LUMAT: International Journal on Math, Science and Technology Education, 9(1), 670-695. https://doi.org/10.31129/LUMAT.9.1.1661

Bosch, M., & Gascón, J. (2006). Twenty five years of the didactic transposition. ICMI Bulletin 58, 51–62.

Bosch, M., & Gascón, J. (2014). Introduction to the Anthropological Theory of the Didactic (ATD). In A. Bikner-Ahsbahs & S. Predinger (Eds.), Networking of Theories as a Research Practice in Mathematics Education (Advances i, pp. 67–83). Springer International Publishing. https://doi.org/0.1007/978-3-319-05389-9_5

Bosch, M., & Winsløw, C. (2020). The external didactic transposition of mathematics at the university level: Dilemmas and challenges. Educação Matemática Pesquisa Revista do Programa de Estudos Pós-Graduados em Educação Matemática, 22(4), 373-386. https://doi.org/10.23925/ 1983-3156.2020v22i4p373-386

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically. Portsmouth, NH: Heinemann.

Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: An attempt at a principled approach. Hiroshima Journal of Mathematics Education, 12, 71–114. https://www.jasme.jp/ hjme/volumes.html

Chevallard, Y., Bosch, M., & Kim, S. (2015). What is a theory according to the anthropological theory of the didactic? CERME 9-Ninth Congress of the European Society for Research in Mathematics Education, 2614–2620.

Chevallard, Y., & Sensevy, G. (2014). Anthropological approaches in mathematics education, French perspectives. In S, Lerman (Eds.), Encyclopedia of Mathematics Education, (pp. 38–43).

Chinofunga, M., Chigeza, P., & Taylor, S. (2022). A framework for content sequencing from junior to senior mathematics curriculum. Eurasia Journal of Mathematics, Science and Technology Education, 18(4). https://doi.org/10.29333/ejmste/11930

Denbel, D. G. (2015). Functions in the secondary school mathematics curriculum. Journal of Education and Practice, 6(1), 77–81. https://doi.org/10.1080/10986065.2016.1107819

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131. https://doi.org/10.1007/s10649-0060400-z

Eames, C. L., Graf, E. A., van Rijn, P. W., Budzban, G., & Voepel, T. (2021). The finite-to-finite strand of a learning progression for the concept of function: A research synthesis and cognitive analysis. The Journal of Mathematical Behavior, 62, 100864. https://doi.org/10.1016/j.jmathb. 2021.100864

Eisenberg, T. (1991). Functions and associated learning difficulties. In Advanced Mathematical Thinking (pp. 140–152). Springer.

Gholami, H. (2023). Performance of Malaysian foundation level students in mathematical problem solving as well as gender comparison. Mathematics Teaching Research Journal, 15(2), 104120.

Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. In ZDM (Vol. 41, pp. 535–540). Springer. https://doi.org/10.1007/s11858-009-0214-4

Hong, D. S., & Choi, K. M. (2018). A comparative analysis of linear functions in Korean and American standards-based secondary textbooks. International Journal of Mathematical Education in

Science and Technology, 49(7), 1025–1051. https://doi.org/10.1080/0020739X.2018.1440327

Ismail, S. N., Muhammad, S., Omar, M. N., Shanmugam, S. K. S., & Rajoo, M. (2022). The practice of critical thinking skills in teaching mathematics: Teachers’ perception and readiness. Malaysian Journal of Learning and Instruction, 19(1), 1-30. https://doi.org/10.328 90/mjli2022.19.1.1

Jukic Matic, L., Kehler-Poljak, G., & Rukavina, S. (2022). The influence of curriculum on the concept of function: An empirical study of pre-service teachers. European Journal of Science and Mathematics Education, 10(3), 380–395. https://doi.org/10.30935/scimath/12042

Kang, W., & Kilpatrick, J. (1992). Didactic transposition in mathematics textbooks. For the Learning of Mathematics, 12(1), 2–7.

Kjeldsen, T. H., & Lützen, J. (2015). Interactions between mathematics and physics: The history of the concept of function—teaching with and about nature of mathematics. Science & Education, 24, 543–559. https://doi.org/10.1007/s11191-015-9746-x

Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282. https://doi.org/10.2307/2686848

Kleiner, I. (1993). Functions: Historical and pedagogical aspects. Science & Education, 2, 183–209. https://doi.org/10.1007/BF00592206

Lichti, M., & Roth, J. (2018). How to foster functional thinking in learning environments using computer-based simulations or real materials. Journal for STEM Education Research, 1(1–2), 148–172. https://doi.org/10.1007/s41979-018-0007-1

Lundberg, A. L. V., & Kilhamn, C. (2018). Transposition of knowledge: Encountering proportionality in an algebra task. International Journal of Science and Mathematics Education, 16(3), 559–579. https://doi.org/10.1007/s10763-016-9781-3

Michelsen, C. (2006). Functions: A modelling tool in mathematics and science. ZDM, 38, 269–280. https://doi.org/10.1007/BF02652810

Ministry of Education Indonesia. (2022). Capaian pembelajaran pada pendidikan anak usia dini, jenjang pendidikan dasar, dan jenjang pendidikan menengah pada kurikulum merdeka.

Ministry of Education Malaysia. (2018). Secondary School Standard Curriculum: Mathematics: Standard Curriculum and Assessment Documents Form 4 and Form 5. Curriculum Development Division.

Muzaffer, O. (2013). Learning difficulties experienced by students and their misconceptions of the inverse function concept. Educational Research and Reviews, 8(12), 901. https://doi.org/10. 5897/ERR2013.1457

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. NCTM.

Oliveira, H., Polo-Blanco, I., & Henriques, A. (2021). Exploring prospective elementary mathematics teachers’ knowledge: A focus on functional thinking. Journal on Mathematics Education, 12(2), 257–278. https://doi.org/10.22342/jme.12.2.13745.257-278

Panaoura, A., Michael-Chrysanthou, P., & Philippou, A. (2015). Teaching the concept of function: Definition and problem solving. In CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (pp. 440-445).

Payne, J., & Payne, G. (2004). Key concepts in social research. Key Concepts in Social Research, 1–248.

Ramful, A. (2015). Reversible reasoning and the working backwards problem solving strategy. The Australian Mathematics Teacher, 71(4), 28–32.

Ronda, E. (2015). Growth points in linking representations of function: A research-based framework. Educational Studies in Mathematics, 90, 303-319. https://doi.org/10.1007/s10649015-9631-1

Rolfes, T., Roth, J., & Schnotz, W. (2018). Effects of tables, bar charts, and graphs on solving function tasks. Journal Fur Mathematik-Didaktik, 39(1), 97–125. https://doi.org/10.1007/s13138-0170124-x

Sajka, M., & Rosiek, R. (2019). Struggling with physics and mathematics curricula based on the notion of function in the context of the educational reform in Poland. AIP Conference Proceedings, 2152(1). https://doi.org/10.1063/1.5124773

Septyawan, S. R., & Suryadi, D. (2019). Learning obstacles on the concept of function: A hermeneutic phenomenological study. In Journal of Physics: Conference Series (Vol. 1280, No. 4, p. 042041). IOP Publishing. https://doi.org/10.1088/1742-6596/1280/4/042041

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715

Sierpinska, A. (1992). On understanding the notion of function. The Concept of Function: Aspects of Epistemology and Pedagogy, 25, 23–58.

Sproesser, U., Vogel, M., Dörfler, T., & Eichler, A. (2022). Changing between representations of elementary functions: Students’ competencies and differences with a specific perspective on school track and gender. International Journal of STEM Education, 9(1), 1–18. https://doi.org/10.1186/s40594-022-00350-2

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

Tesfamicael, S. A., & Lundeby, Ø. A. (2019). A comparative study of Norwegian and Ethiopian textbooks: The case of relations and functions using anthropological theory of didactics (ATD).

Tohir, M., As'ari, A. R., Anam, A. C., & Taufiq, I. (2022). Matematika SMP/MTs Kelas VIII. Ministry of Education Indonesia

Trujillo, M., Atarés, L., Canet, M. J., & Pérez-Pascual, M. A. (2023). Learning difficulties with the concept of function in maths: A literature review. Education Sciences, 13(5), 495. https://doi.org/10.3390/ educsci13050495

Utami, N. S., Prabawanto, S., & Suryadi, D. (2023). How students generate patterns in learning algebra? A focus on functional thinking in secondary school students. European Journal of Educational Research, 12(2), 913–925. https://doi.org/10.12973/eu-jer.12.2.913

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305. https://doi.org/10.1080/0020739830140305

Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366. https://doi.org/10.5951/jresematheduc.20. 4.0356

Walde, G. S. (2017). Difficulties of concept of function: The case of general secondary school students of Ethiopia. International Journal of Scientific & Engineering Research, 8(4), 1-10.

Widada, W., Herawati, A., Fata, R., Nurhasanah, S., Yanty, E. P., & Suharno, A. S. (2020). Students’ understanding of the concept of function and mapping. In Journal Of Physics: Conference Series (Vol. 1657, No. 1, p. 012072). IOP Publishing. https: //doi.org/10.1088/1742-6596/1657/ 1/012072

Winsløw, C. (2011). Anthropological theory of didactic phenomena: Some examples and principles of its use in the study of mathematics education. Un Panorama de TAD, CRM Docume, 117, 138.