Skip to main content Skip to main navigation menu Skip to site footer
Beta: Jurnal Tadris Matematika
Articles
DOI: 10.20414/betajtm.v18i1.690
Submitted: 2025-02-18
Published: 2025-05-28

Didactical design of the proportionality concept based on anthropological theory of the didactic

Universitas Singaperbangsa Karawang
Anthropological theory of the didactic Mathematical praxeology Proportionality concept

Galleys

Abstract

[English]: This study aims to develop a didactic design grounded in the Anthropological Theory of the Didactic (ATD), with a focus on mathematical praxeology, to enhance students’ understanding of proportionality. The study uses the Didactical Design Research (DDR) methodology, which consists of three primary stages: retrospective analysis, metadidactical analysis, and prospective didactical situation analysis. Classroom observations, teacher and student interviews, and the examination of diagnostic tasks completed by thirty seventh-grade students at a junior high school in West Java, Indonesia, were used to gather data. The findings, which are interpreted using the Brousseau's framework, classify learning obstacles into three categories: (1) Ontogenic obstacles, which are associated with students' developmental readiness and prior knowledge; (2) Epistemological obstacles, which include misusing additive reasoning in multiplicative contexts and misinterpreting unit rates; and (3) Didactic obstacles, which are caused by unfamiliar task structures, implicit information, and insufficient instructional representations. The study has developed five learning tasks based on the praxeological framework, which included task, technique, technology, and theory. Despite not yet being used in classrooms, the didactic design's development is supported by both theoretical and empirical evidence. This study offers a different strategy for teachers dealing with proportionality learning obstacles and helps create more meaningful and contextually relevant math instruction. It also creates opportunities for further study to examine how the suggested design is implemented and how it affects students' mathematical thinking.

[Bahasa]: Penelitian ini bertujuan untuk mengembangkan desain didaktik berbasis Teori Antropologi Didaktik (ATD) dengan fokus pada praksiologi matematika untuk membantu siswa agar lebih memahami konsep proporsionalitas. Metode yang digunakan adalah Didactical Design Research (DDR), yang terdiri dari tiga tahap utama: analisis retrospektif, analisis metadidaktik, dan analisis situasi didaktik prospektif. Data dikumpulkan melalui observasi di kelas, wawancara dengan guru dan siswa, serta analisis tugas diagnostik yang dikerjakan oleh 30 siswa kelas VII di sebuah SMP di Jawa Barat. Menggunakan kerangka Brousseau hasil penelitian ini menunjukkan bahwa hambatan belajar siswa terbagi menjadi tiga jenis: (1) hambatan ontogenik, yaitu terkait kesiapan dan pengetahuan awal siswa; (2) hambatan epistemologis, seperti penggunaan penalaran aditif yang salah dalam konteks yang seharusnya menggunakan penalaran multiplikatif, serta kesalahan dalam memahami laju satuan; dan (3) hambatan didaktik, yang muncul karena struktur tugas yang asing, informasi yang tidak jelas, serta kurangnya representasi yang memadai dalam pembelajaran. Berdasarkan temuan tersebut, dirancang lima tugas pembelajaran melalui pendekatan praksiologi. Studi ini memberikan alternatif strategi bagi guru untuk mengatasi kesulitan belajar proporsionalitas dan mendukung terciptanya pembelajaran matematika yang lebih bermakna dan relevan dengan konteks siswa. Selain itu, penelitian ini juga membuka peluang untuk penelitian lebih lanjut terkait penerapan desain ini dan dampaknya terhadap cara berpikir matematika siswa.

Downloads

Download data is not yet available.

References

  1. Artigue, M. (2009). Didactical design in mathematics education. In Nordic research in mathematics education (pp. 5–16). Brill.
  2. Artigue, M., & Trouche, L. (2021). Revisiting the French didactic tradition through technological lenses. Mathematics, 9(6), 629. https://doi.org/10.3390/math9060629
  3. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). Macmillan.
  4. Bofferding, L., & Wessman-Enzinger, N. M. (2018). Nuances of prospective teachers’ interpretations of integer word problems. In Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 191–212). https://doi.org/10.1007/978-3-319-90692-8_8
  5. Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition. ICMI Bulletin, 58(58), 51–65.
  6. Boyer, T. W., & Levine, S. C. (2015). Prompting children to reason proportionally: Processing discrete units as continuous amounts. Developmental Psychology, 51(5), 615–620. https://doi.org/10.1037/a0039010
  7. Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer Academic Publishers. https://doi/10.1007/0-306-47211-2
  8. Charalambous, C. Y. (2010). Mathematical knowledge for teaching and task unfolding: An exploratory study. The Elementary School Journal, 110(3), 247–278.
  9. Chevallard, Y. (1989). On didactic transposition theory: Some introductory notes. In Proceedings of the international symposium on selected domains of research and development in mathematics education (Vol. 1, pp. 51–62). University of Bielefeld, Germany, and University of Bratislava, Slovakia.
  10. Chevallard, Y. (1991). La transposición didáctica. Del saber sabio al saber enseñado (3rd ed.).
  11. Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In Proceedings of the IV congress of the European society for research in mathematics education (pp. 21–30).
  12. Clarke, D. (2021). Calling a spade a spade: The impact of within-class ability grouping on opportunity to learn mathematics in the primary school. Australian Primary Mathematics Classroom, 26(1), 3–8. https://search.informit.org/doi/10.3316/informit.759554458530056
  13. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. D. Reidel Publishing Company.
  14. González-Forte, J. M., Fernández, C., Van Hoof, J., & Van Dooren, W. (2020). Various ways to determine rational number size: An exploration across primary and secondary education. European Journal of Psychology of Education, 35(3), 549–565. https://doi.org/10.1007/s10212-019-00440-w
  15. Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students' fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196
  16. Hartnett, J. E. (2012). Mathematics for machine technology. Cengage Learning.
  17. Hohensee, C. (2016). Student noticing in classroom settings: A process underlying influences on prior ways of reasoning. The Journal of Mathematical Behavior, 42, 69–91. https://doi.org/10.1016/j.jmathb.2016.03.002
  18. Jitendra, A. K., Harwell, M. R., Karl, S. R., Slater, S. C., Simonson, G. R., & Nelson, G. (2016). A replication study to evaluate the effects of schema-based instruction on middle school students' proportional problem-solving performance. Society for Research on Educational Effectiveness.
  19. Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235–287). State University of New York Press.
  20. Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age Publishing.
  21. Lobato, J., Ellis, A., & Zbiek, R. M. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6–8. National Council of Teachers of Mathematics.
  22. Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. The Journal of Mathematical Behavior, 21(1), 87–116. https://doi.org/10.1016/S0732-3123(02)00099-8
  23. Maudy, S. Y. (2023). Transposisi didaktik berpikir aljabar awal [Doctoral dissertation, Universitas Pendidikan Indonesia]. UPI Repository. https://repository.upi.edu/90055/
  24. Maudy, S. Y., & Ruli, R. M. (2024). Learning obstacle instrument analysis of proportion concept with praxiology framework. Journal of Didactic Mathematics, 5(3), 194–210. https://doi.org/10.34007/jdm.v5i3.2435
  25. Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children's proportional reasoning. The Journal of Mathematical Behavior, 22(3), 335–368. https://doi.org/10.1016/S0732-3123(03)00025-7
  26. Oehrtman, M., Carlson, M., & Thompson, P. (2008). Foundational understandings for undergraduate mathematics majors: A framework for quantitative reasoning. Mathematical Association of America.
  27. Paun, E. (2006). Transposition didactique: Un processus de construction du savoir scolaire. Carrefours de l’éducation(2), 3–13.
  28. Prediger, S., & Zindel, C. (2017). School academic language demands for understanding functional relationships: A design research project on the role of language in reading and learning. Eurasia Journal of Mathematics, Science and Technology Education, 13(7b), 4157–4188. https://doi.org/10.12973/eurasia.2017.00804a
  29. Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes: An overview on achievements and challenges. ZDM, 47, 877–891. https://doi.org/10.1007/s11858-015-0722-3
  30. Radford, L. (2014). Phenomenology, praxis, and the question of mathematical objects. Educación Matemática, 26(1), 124–145.
  31. Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Lawrence Erlbaum Associates.
  32. Sensevy, G., Tiberghien, A., Santini, J., Laubé, S., & Griggs, P. (2008). An epistemological approach to modeling: Case studies and implications for science teaching. Science Education, 92(3), 424–446. https://doi.org/10.1002/sce.20268
  33. Thompson, P. W. (1994). The role of quantitative reasoning in developing algebraic concepts. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 289–296). University of Lisbon.
  34. Thompson, P. W., & Thompson, A. G. (1994). Talking about rates conceptually, Part I: Teacher’s struggle. Journal for Research in Mathematics Education, 25(3), 279–303. https://doi.org/10.5951/jresematheduc.25.3.0279
  35. Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9–35. https://doi.org/10.1023/B:EDUC.0000005212.03219.dc
  36. Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2022). Elementary and middle school mathematics: Teaching developmentally. Pearson.
  37. Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication… and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360–381. https://doi.org/10.1080/07370008.2010.488306
  38. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). Academic Press.
  39. Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83–94.
  40. Wijaya, A., van den Heuvel-Panhuizen, M., & Doorman, M. (2015). Opportunity-to-learn context-based tasks provided by mathematics textbooks. Educational Studies in Mathematics, 89, 41–65. https://doi.org/10.1007/s10649-015-9595-1

How to Cite

Yugni Maudy, S., & Ruli, R. M. (2025). Didactical design of the proportionality concept based on anthropological theory of the didactic. Beta: Jurnal Tadris Matematika, 18(1), 21–48. https://doi.org/10.20414/betajtm.v18i1.690

License

Copyright (c) 2025 Septiani Yugni Maudy, Redo Martila Ruli

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.