Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia

Zetra Hainul Putra

doi:10.29333/ejmste/105867

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Zetra Hainul Putra ¹ ² ^*

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¹ University of Copenhagen, Department of Science Education, Copenhagen, DENMARK² University of Riau, Faculty of Teacher Training and Education, Pekanbaru, INDONESIA^* Corresponding Author

Abstract

The present study aims to introduce the notion of praxeological change, developed based on the Anthropological Theory of the Didactic, to describe a necessity of changing mathematical praxeologies when passing from natural to rational numbers. It is applied to study and compare Danish and Indonesian pre-service teachers’ (PSTs) knowledge of the density of rational numbers. They work in pairs to solve and discuss a hypothetical teacher task, which involves both mathematical and didactical tasks, related to the density of rational numbers. The findings highlight significant differences of the mathematical and didactical knowledge which are shared by the Danish and Indonesian PSTs. In particular, the Danish PSTs are more successful than the Indonesian PSTs in proposing didactical praxeologies to support pupils’ praxeological change. They use the mathematical idea of converting fractions into decimals or vice versa and representing fractions and decimals on the same number line, while the Indonesian pairs tend to suggest pupils to order fractions and decimals based on the ordering properties of natural numbers.

Keywords

praxeological change
anthropological theory of the didactic
hypothetical teacher task
density of rational numbers

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2019, Volume 15, Issue 5, Article No: em1711

https://doi.org/10.29333/ejmste/105867

Publication date: 21 Mar 2019

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Article Downloads: 1146

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References

An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in China and the US. Journal of Mathematics Teacher Education, 7(2), 145-172. https://doi.org/10.1023/B:JMTE.0000021943.35739.1c.
BAN-PT. (2017). Directorate result of study programme accreditation Direktori hasil akreditasi program studi. Retrieved from https://banpt.or.id/direktori/prodi/pencarian_prodi.
Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: The case of limits of functions in Spanish high schools. In Beyond the apparent banality of the mathematics classroom (pp. 235-268). Springer US. https://doi.org/10.1007/0-387-30451-7_9.
Behr, M. J., Lesh, R., Post, T., & Silver, E. A. (1983). Rational number concepts. Acquisition of mathematics concepts and processes, 91-126.
Bosch, M., & Gascón, J. (2014). Introduction to the anthropological theory of the didactic (ATD). In A. Bikner-Ahsbahs & S. Prediger (Eds.), Networking of Theories as a Research Practice in Mathematics Education (pp. 67-83). Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-05389-9_5.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer Academic Publishers.
Chevallard, Y. (1992). Fundamental concepts in didactics: Perspectives provided by an anthropological approach. In R. Douady & A. Mercier (Eds.), Research in Didactique of Mathematics. Selected papers (pp. 131–167). Grenoble, France: La Pensée sauvage.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221-266.
Chevallard, Y. (2006). Readjusting didactics to a changing epistemology. European Educational Research Journal, 6(2), 131-134. https://doi.org/10.2304/eerj.2007.6.2.131.
Chevallard, Y. (2007). Steps towards a new epistemology in mathematics education. In M. Bosch (Eds.), Proceedings of the IV congress of the European society for research in mathematics education (pp. 21–30). Barcelona, Spain: FUNDEMI-IQS.
Christou, K. P. (2015). Natural number bias in operations with missing numbers. ZDM Mathematics Education, 47(5), 747-758. https://doi.org/10.1007/s11858-015-0675-6.
Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssen, R., Kelchtermans, G., Verschaffel, L., & Van Dooren, W. (2015). Teachers’ content and pedagogical content knowledge on rational numbers: A comparison of prospective elementary and lower secondary school teachers. Teaching and Teacher Education, 47, 82-92. https://doi.org/10.1016/j.tate.2014.12.009.
Durand-Guerrier, V., Winsløw, C., & Yoshida, H. (2010). A model of mathematics teacher knowledge and a comparative study in Denmark, France and Japan. Annales de Didactiques et de Sciences Cognitives, 15, 141-166.
Durkin, K., & Rittle-Johnson, B. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21-29. https://doi.org/10.1016/j.learninstruc.2014.08.003.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. https://doi.org/10.3102/00028312042002371.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.
McMullen, J., Laakkonen, E., Hannula-Sormunen, M., & Lehtinen, E. (2015). Modeling the development trajectories of rational number concept(s). Learning and Instruction, 37, 14-20. https://doi.org/10.1016/j.learninstruc.2013.12.004.
Ministry of Higher Education and Science. (2015). The Danish teacher education programme: B.Ed. programme for primary and lower secondary schools. Copenhagen: Danish Agency for Higher Education. Retrieved 04 October 2017 from https://ufm.dk/en/education-and-institutions/higher-education/university-colleges/.
university-college-educations/bachelor-of-education/b-ed-programme-for-primary-and-lower-secondary-schools.pdf.
Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. Retrieved from Boston College, TIMSS & PIRLS International Study. http://timssandpirls.bc.edu/timss2015/international-results/.
OECD. (2016), PISA 2015 Results (Volume I): Excellence and Equity in Education, PISA, OECD Publishing, Paris. http://dx.doi.org/10.1787/9789264266490-en.
Prediger, S. (2008). The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions. Learning and instruction, 18(1), 3-17. https://doi.org/10.1016/j.learninstruc.2006.08.001.
Prodi PGSD UR. (2017). The structure of curriculum for elementary school teacher education program Kurikulum berbasis KKNI tahun 2014: Struktur kurikulum program studi pendidikan guru sekolah dasar. Retrieved from http://pgsd.fkip.unri.ac.id/kurikulum/.
Putra, Z. H. (2016). Collective aspects of pre-service lower secondary teachers’ knowledge on density of rational numbers. In M. Achiam & C. Winsløw (Eds.), Educational Design in Math and Science: the Collective Aspect (pp. 53-61). Copenhagen, Denmark: University of Copenhagen.
Putra, Z. H., & C. Winsløw (in press). A framework for a comparative study of pre-service elementary teachers’ knowledge of rational numbers. Educaçao Matemática Pesquissa.
Rasmussen, J., & Bayer, M. (2014). Comparative study of teaching content n teacher education programmes in Canada, Denmark, Finland, and Singapore. Journal of Curriculum Studies, 46(6), 798-818. https://doi.org/10.1080/00220272.2014.927530.
Revina, S., & Leung, F. K. S. (2018). How the Same Flowers Grow in Different Soils? The Implementation of Realistic Mathematics Education in Utrecht and Jakarta Classrooms. International Journal of Science and Mathematics Education, 1-25. https://doi.org/10.1007/s10763-018-9883-1.
RISTEKDIKTI. (2016). A guideline for preparing a curriculum of higher education Panduan penyusunan kurikulum pendidikan tinggi. Jakarta: Direktorat Jenderal Pembelajaran dan Kemahasiswaan Kementerian Riset Teknologi dan Pendidikan Tinggi.
Rodríguez, E., Bosch, M., & Gascón, J. (2008). A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic. ZDM Mathematics Education, 40(2), 287-301. https://doi.org/10.1007/s11858-008-0094-z.
Stacey, K., Helme, S., Archer, S., & Condon, C. (2001). The effect of epistemic fidelity and accessibility on teaching with physical materials: A comparison of two models for teaching decimal numeration. Educational Studies in Mathematics, 47(2), 199-221. https://doi.org/10.1023/A:1014590319667.
Vamvakoussi, X., Christou, K. P., Mertens, L., & Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density. Learning and Instruction, 21(5), 676-685. https://doi.org/10.1016/j.learninstruc.2011.03.005.
Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453-467. https://doi.org/10.1016/j.learninstruc.2004.06.013.
Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181-209. https://doi.org/10.1080/07370001003676603.
Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2015). Inappropriately applying natural number properties in rational number tasks: Characterizing the development of the natural number bias through primary and secondary education. Educational Studies in Mathematics, 90(1), 39-56. https://doi.org/10.1007/s10649-015-9613-3.
Widjaja, W., Stacey, K., & Steinle, V. (2008). Misconceptions about density of decimals: Insights from Indonesian pre-service teachers’ work. Journal for Science and Mathematics Education in Southeast Asia, 31(2), 117-131.
Winsløw, C., & Durand-Guerrier, V. (2007). Education of lower secondary mathematics teachers in Denmark and France: A comparative study of characteristics of the systems and their products. Nordisk Matematikkdidaktikk, 12(2), 5-32.
Wu, H. (2014). Fractions, decimals, and rational numbers. Retrieved 05 December 2017 from https://math.berkeley.edu/~wu/NMPfractions.pdf.
Zhou, Z., Peverly, S. T., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers. Contemporary Educational Psychology, 31(4), 438-457. https://doi.org/10.1016/j.cedpsych.2006.02.001.

How to cite this article

APA

Putra, Z. H. (2019). Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia. Eurasia Journal of Mathematics, Science and Technology Education, 15(5), em1711. https://doi.org/10.29333/ejmste/105867

Vancouver

Putra ZH. Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia. EURASIA J Math Sci Tech Ed. 2019;15(5):em1711. https://doi.org/10.29333/ejmste/105867

AMA

Putra ZH. Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia. EURASIA J Math Sci Tech Ed. 2019;15(5), em1711. https://doi.org/10.29333/ejmste/105867

Chicago

Putra, Zetra Hainul. "Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia". Eurasia Journal of Mathematics, Science and Technology Education 2019 15 no. 5 (2019): em1711. https://doi.org/10.29333/ejmste/105867

Harvard

Putra, Z. H. (2019). Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia. Eurasia Journal of Mathematics, Science and Technology Education, 15(5), em1711. https://doi.org/10.29333/ejmste/105867

MLA

Putra, Zetra Hainul "Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia". Eurasia Journal of Mathematics, Science and Technology Education, vol. 15, no. 5, 2019, em1711. https://doi.org/10.29333/ejmste/105867