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International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education
Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers
Zetra Hainul Putra 1 2 *
1 Department of Science Education, University of Copenhagen, DENMARK2 Faculty of Teacher Training and Education, University of Riau, INDONESIA
* Corresponding Author
APA
In-text citation: (Putra, 2019)
Reference: Putra, Z. H. (2019). Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers. International Electronic Journal of Mathematics Education, 14(3), 619-632. https://doi.org/10.29333/iejme/5775
AMA
In-text citation: (1), (2), (3), etc.
Reference: Putra ZH. Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers. INT ELECT J MATH ED. 2019;14(3), 619-632. https://doi.org/10.29333/iejme/5775
Chicago
In-text citation: (Putra, 2019)
Reference: Putra, Zetra Hainul. "Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers". International Electronic Journal of Mathematics Education 2019 14 no. 3 (2019): 619-632. https://doi.org/10.29333/iejme/5775
Harvard
In-text citation: (Putra, 2019)
Reference: Putra, Z. H. (2019). Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers. International Electronic Journal of Mathematics Education, 14(3), pp. 619-632. https://doi.org/10.29333/iejme/5775
MLA
In-text citation: (Putra, 2019)
Reference: Putra, Zetra Hainul "Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers". International Electronic Journal of Mathematics Education, vol. 14, no. 3, 2019, pp. 619-632. https://doi.org/10.29333/iejme/5775
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Putra ZH. Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers. INT ELECT J MATH ED. 2019;14(3):619-32. https://doi.org/10.29333/iejme/5775

Abstract

The aim of this study is to investigate Danish pre-service teachers’ (PSTs) mathematical and didactical knowledge of operations with rational numbers. This knowledge is studied through their collaborative activities to certain tasks related to the teaching of operations with rational numbers. An explicit model of the teachers’ mathematical and didactical knowledge is designed based on the anthropological theory of the didactic (ATD) and used to analyse five-groups of PSTs’ collaborative work. The findings show that the Danish PSTs prefer to use contextual or real-life situations in their teaching, but they encounter various challenges to realise general ideas in the context at hand.

Keywords

operations with rational numbers mathematical and didactical knowledge anthropological theory of the didactic collaborative work

References

  • Alenazi, A. (2016). Examining middle school pre-service teachers’ knowledge of fraction division interpretations. International Journal of Mathematical Education in Science and Technology, 47(5), 696-716. https://doi.org/10.1080/0020739X.2015.1083127
  • 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
  • Austin, P., Carbone, R. E., & Webb, P. (2011). Prospective primary school Teachers’ attempts to pose acceptable word problems on the addition of fractions: Some insights from South Africa and the United States of America. African Journal of Research in Mathematics, Science and Technology Education, 15(2), 168-178. https://doi.org/10.1080/10288457.2011.10740710
  • Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The elementary school journal, 90(4), 449-466. https://doi.org/10.1086/461626
  • Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
  • 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
  • Bradshaw, L., Izsák, A., Templin, J., & Jacobson, E. (2014). Diagnosing teachers’ understandings of rational numbers: Building a multidimensional test within the diagnostic classification framework. Educational Measurement: Issues and Practice, 33(1), 2-14. https://doi.org/10.1111/emip.12020
  • Castro-Rodríguez, E., Pitta-Pantazi, D., Rico, L., & Gómez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129-146. https://doi.org/10.1007/s10649-015-9673-4
  • Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational studies in mathematics, 64(3), 293-316. https://doi.org/10.1007/s10649-006-9036-2
  • Chevallard, Y. (1992). Fundamental concepts in didactics: perspectives provided by an anthropological approach. In R. Douady & A. Mercier (Eds.), Research in Didactique of Mathematics (pp. 131-167). Grenoble: La Pensée sauvage.
  • Chevallard, Y. (2006). 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). Grenoble: La Pensée Sauvage.
  • Chevallard, Y. (2007). Readjusting didactics to a changing epistemology. European Educational Research Journal, 6(2), 131 – 134. https://doi.org/10.2304/eerj.2007.6.2.131
  • Danish Ministry of Education. (2011). Moderne matematiske færdigheder fra skolestart til studiestart [Modern mathematics skills from school start to study start]. Retrieved from http://www2.mat.dtu.dk/people/S.Markvorsen/DISPLAY/ABOUT_MATH1_PROOPG/Faerdigrapport_Dec_16.pdf
  • Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssen, R., Kelchtermans, G., ... & 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
  • Dixon, J. K., Andreasen, J. B., Avila, C. L., Bawatneh, Z., Deichert, D. L., Howse, T. D., & Turner, M. S. (2014). Redefining the whole: Common errors in elementary preservice teachers’ self-authored word problems for fraction subtraction. Investigations in Mathematics Learning, 7(1), 1-22. https://doi.org/10.1080/24727466.2014.11790336
  • 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 didactique et des sciences cognitives, 15, 147-172.
  • Erdem, E., Gökkurt, B., Şahin, Ö., Başıbüyük, K., & Soylu, Y. (2015). Examining prospective middle school mathematics teachers’ modelling skills of multiplication and division in fractions. Croatian Journal of Education, 17(1), 11-36. https://doi.org/10.15516/cje.v17i1.830
  • Francisco, J. M. (2013). Learning in collaborative settings: Students building on each other’s ideas to promote their mathematical understanding. Educational Studies in Mathematics, 82(3), 417-438. https://doi.org/10.1007/s10649-012-9437-3
  • Freil, O, & Kaas, T. (2006). Kolorit matematik for sjette klasse-grundbog [Kolorit mathematics for the sixth grade-basics]. København: Gyldendalske Boghandel, Nordisk Forlag A/S.
  • Freil, O, & Kaas, T. (2007). Kolorit matematik for femte klasse-grundbog (2. Oplag) [Kolorit mathematics for the fifth grade-basics 2nd edition]. København: Gyldendalske Boghandel, Nordisk Forlag A/S.
  • Gabriel, F. C., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4, 715. https://doi.org/10.3389/fpsyg.2013.00715
  • Jacobson, E., & Izsák, A. (2015). Knowledge and motivation as mediators in mathematics teaching practice: the case of drawn models for fraction arithmetic. Journal of Mathematics Teacher Education, 18(5), 467-488. https://doi.org/10.1007/s10857-015-9320-0
  • Kar, T., & Isik, C. (2014). Analysis of problems posed by pre-service primary teachers about adding fractions in terms of semantic structures. International Electronic Journal of Mathematics Education, 9(2), 135-146.
  • Klahr, D., & Nigam, M. (2004). The equivalence of learning paths in early science instruction: Effects of direct instruction and discovery learning. Psychological Science, 15(10), 661-667. https://doi.org/10.1111/j.0956-7976.2004.00737.x
  • Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd Ed.). Mahwah, NJ: Lawrence Erlbaum Associates. https://doi.org/10.4324/9780203803165
  • Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ understanding of fundamental mathematics in China and the United State. New Jersey: LEA. https://doi.org/10.4324/9781410602589
  • Newton, K. J. (2008). An extensive analysis of pre-service elementary teachers: Knowledge of fractions. American Educational Research Journal, 45(4), 1080 – 1110. https://doi.org/10.3102/0002831208320851
  • Piaget, J. (1964). Part I: Cognitive development in children: Piaget development and learning. Journal of research in science teaching, 2(3), 176-186. https://doi.org/10.1002/tea.3660020306
  • Putra, Z. H. (2016). Pengetahuan mahasiswa pendidikan guru sekolah dasar dalam merepresentasikan operasi pecahan dengan model persegi panjang [Pre-service elementary teachers’ knowledge on representing the fraction operations with rectangle models]. Jurnal Elemen 2(1): 1 – 13. Jurnal Elemen. https://doi.org/10.29408/jel.v2i1.174
  • Putra, Z. H. (2018). A comparative study of Danish and Indonesian pre-service teachers’ knowledge of rational numbers (PhD thesis). University of Copenhagen, Denmark.
  • Putra, Z. H. (2019). Elementary teachers’ knowledge on fraction multiplication: An anthropological theory of the didactic approach. Journal of Teaching and Learning in Elementary Education,2(1), 47-52. https://doi.org/10.33578/jtlee.v2i1.6964
  • Putra, Z. H., & Winsløw, C. (2018). Teachers’ collective knowledge: the case of equivalent fractions. In R. Johar et al. (Eds.), Proceeding of the 6th South East Asia Design Research International Conference (6th SEA-DR IC) (vol. 1088, pp. 1-6), Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1088/1/012003
  • Putra, Z. H., & Winsløw, C. (2019). Prospective teachers’ knowledge of comparing decimals. International on Emerging Mathematics Education, 3(1), 57-68. http://dx.doi.org/10.12928/ijeme.v3i1.11314
  • 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, 40(2), 287-301. https://doi.org/10.1007/s11858-008-0094-z
  • Şahin, Ö., Gökkurt, B., & Soylu, Y. (2016). Examining prospective mathematics teachers’ pedagogical content knowledge on fractions in terms of students’ mistakes. International Journal of Mathematical Education in Science and Technology, 47(4), 531-551. https://doi.org/10.1080/0020739x.2015.1092178
  • Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational researcher, 15(2), 4-14. https://doi.org/10.3102/0013189x015002004
  • Siegler, R. S., & Lortie-Forgues, H. (2017). Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People. Current Directions in Psychological Science, 26(4), 346-351. https://doi.org/10.1177/0963721417700129
  • Slattery, J., & Fitzmaurice, O. (2014). ‘Ours is not to reason why, just invert and multiply’: an insight into Irish prospective secondary teachers’ conceptual understanding of the division of fractions. Irish Educational Studies, 33(4), 467-488. https://doi.org/10.1080/03323315.2014.982346
  • Tatto, M. T., Schwille, J., Senk, S. L., Ingvarso, L., Peck, R., & Rowley, G. (2008). Teacher education and development study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics conceptual framework. East Lansing, MI: Teacher Education and Development International Study Center, College of Education, Michigan State University.
  • Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 5-25. https://doi.org/10.2307/749817
  • Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behaviour, 31(3), 344-355. https://doi.org/10.1016/j.jmathb.2012.02.001
  • Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1-4. https://doi.org/10.1016/j.learninstruc.2015.01.001
  • Van Steenbrugge, H., Lesage, E., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge? Journal of Curriculum Studies, 46(1), 138-161. https://doi.org/10.1080/00220272.2013.839003
  • Winsløw, C., Barquero, B., De Vleeshower, M., & Hardy, N. (2015). An institutional approach to university mathematics education: from dual vector spaces to questioning the world. Research in Mathematics Education, 16(2), 95-111. https://doi.org/10.1080/14794802.2014.918345
  • Winsløw, C., & Durand-Guerrier, V. (2007). Education of lower secondary mathematics teachers in Denmark and France, Nordic Studies in Mathematics Education 12(2), 5-32.

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