Student teachers’ work on instructional explanations in multiplication – representations and conversions between them

Authors

  • Anita Valenta
  • Ole Enge

Abstract

In this study we are analysing student teachers’ instructional explanations. The study is based on student teachers’ written work on two different tasks about different strategies and properties in multiplication and explaining these. Our research questions concern the type of representation registers student teachers use in their explanations. In explanations where several representation registers are used, we analyse what can be challenges in conversions between representations. Data is analysed using the framework of Duval’s cognitive analysis, and analyses and discussions are related to development of mathematical knowledge for teaching.

References

Alseth, B., Breiteig, T. & Brekke, G. (2003). Endring og utvikling ved R97 som bakgrunn for videre planlegging og justering: matematikkfaget som kasus. Notodden: Telemarksforskning.

Ball, D. L. & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In E. Simmt & B. Davis (Eds.), Proceedings of the 2002 annual meeting of the Canadian Mathematics Education Study Group (pp. 3-14). Edmonton: CMESG/GCEDM.

Ball, D. L., Lubienski, S. T. & Mewborn, D. S. (2001). Research on teaching mathematics: the unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.) (pp. 433- 456). Washington: American Educational Research Association.

Ball, D., 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

Berge, K. L. (2005). Skriving som grunnleggende ferdighet og som nasjonal prøve - ideologi og strategier. In A. J. Aasen & S. Nome (Eds.), Det nye norskfaget (pp. 161-188). Oslo: Fagbokforlaget.

Charalambos, Y. C., Hill, H. C. & Ball, D. L. (2011). Prospective teachers' learning to provide instructional explanations: how does it look and what might it take? Journal of Mathematics Teachers Education, 14, 441-463. https://doi.org/10.1007/s10857-011-9182-z

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 de sciences cognitives, 15, 141-166.

Duval, R. (2000). Basic issues for research in mathematics education. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th conference of PME (pp. 55-69). Hiroshima: I. Nishiki Print Co. Ltd.

Duval, R. (2002). The cognitive analysis of problems of comprehension in the learning of mathematics. Mediterranean Journal for Research in Mathematics Education, 1 (2), 1-16.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131. https://doi.org/10.1007/s10649-006-0400-z

Duval, R. (2008). Eight problems for a semiotic approach in mathematics education. In L. Radford, G. Schubring & F. Seeger (Eds.), Semiotics in mathematics education (pp. 39-62). Rotterdam: Sense Publishers. https://doi.org/10.1163/9789087905972_004

Empson, S. B. & Jacobs, V. R. (2008). Learning to listen to children's mathematics. I. D. Tirosh & T. Wood (Eds.), The international handbook of teacher education (Vol. 2, pp. 267-282). Rotterdam: Sense Publishers.

Evensen, L. S. (2010). En gyldig vurdering av elevers skrivekompetanse? In J. Smidt, I. Folkvord & A. J. Aasen (Eds.), Rammer for skriving: om skriveutvikling i skole og yrkesliv (pp. 13-31). Trondheim: Tapir Academic Press.

Fosnot, C. & Dolk, M. (2001). Young mathematicians at work. Constructing multiplication and division. Portsmouth: Heinemann.

Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25 (5), 443-471. https://doi.org/10.2307/749485

Grevholm, B. (1998). Teacher students' development of concepts in mathematics and mathematics education. In T. Breiteig & G. Brekke (Eds.), Theory into practices in mathematics education. Proceedings of Norma 98 (pp. 139-146). Kristiansand: Høgskolen i Agder.

Halliday, M. A. K. (1978). Language as social semiotic. London: Edward Arnold.

Jonsmoen, K. M. & Greek, M. (2012). "Hodet blir tungt - og tomt" - om det å skrive seg til profesjonsutøvelse. Norsk pedagogisk tidsskrift, 96 (1), 15-26. https://doi.org/10.18261/ISSN1504-2987-2012-01-03

Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). (2001). Adding it up: helping children learn mathematics. Washington: National Academies Press.

Kinach, B. M. (2002). Understanding and learning to explain by representing mathematics: epistemological dilemmas facing teacher educators in the secondary mathematics "methods" course. Journal of Mathematics Teacher Education, 5, 153-186. https://doi.org/10.1023/A:1015822104536

Krogh, E. & Jensen, M. J. (2008). Portfolioevaluering og portfoliodidaktik. Fredriksberg: Dansklærerforeningens forlag.

Leinhardt, G. (1987). Development of an expert explanation: an analysis of a sequence of subtraction lessons. Cognition and Instruction, 4, 225-282. https://doi.org/10.1207/s1532690xci0404_2

Leinhardt, G., Putnam, R. T., Stein, M. K. & Baxter, J. (1991). Where subject knowledge matters. In J. Brophy (Ed.), Advances in research on teaching (Vol. 2, pp. 87-113). London: JAI Press inc.

Leinhardt, G. (2001). Instructional explanations: a commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook for research in teaching (4th ed.) (pp. 333-357). Washington: American Educational Research Association.

Leinhardt, G., Steele, M. D. (2005). Seeing the complexity of standing to the side: Instructional dialoges. Cognition and Instruction, 23, 87-163. https://doi.org/10.1207/s1532690xci2301_4

Lo, J., Grant, T. & Flowers, J. (2008). Challenges in deepening prospective teachers' understanding of multiplication through justification. Journal of Mathematics Teacher Education, 11, 5-22. https://doi.org/10.1007/s10857-007-9056-6

Ma, L. (1999). Knowing and teaching elementary mathematics: teachers' understanding of fundamental mathematics in China and the United States. Hillsdale: Erlbaum. https://doi.org/10.4324/9781410602589

Måsøval, H. S. (2005). Student reasoning constrained by the didactical contract. Nordic Studies in Mathematics Education, 10 (3-4), 83-99.

Måsøval, H. S. (2011). Factors constraining students' establishment of algebraic generality in shape patterns: a case study of didactical situations in mathematics at a university college (Unpublished doctoral dissertation). Kristiansand: University of Agder.

Nicol, C. (1999). Learning to teach mathematics: questioning, listening, and responding. Educational Studies in Mathematics, 37, 45-66. https://doi.org/10.1023/A:1003451423219

Niss, M. & Højgaard T. (Eds.). (2011). Competencies and mathematical learning ideas and inspiration for the development of mathematics teaching and learning in Denmark (English edition, October 2011. IMFUFA tekst no. 485). Roskilde University. (Published in Danish in 2002). Retrieved April 11, 2013 from http://milne.ruc.dk/ImfufaTekster/pdf/485web_b.pdf

Säljö, R.(1999). Læring i praksis. Et sosiokulturelt perspektiv. Oslo: J. W. Cappelens forlag.

Skott, J., Larsen, D. M. & Østergaard, C. H. (2011). From beliefs to patterns of participation - shifting the research perspective on teachers. Nordic Studies in Mathematics Education, 16 (1-2), 29-55.

Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.2307/1175860

Strauss, A. & Corbin, J. (1998). Basics of qualitative research: techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks: Sage.

Stylianou, D. (2010). Teachers' conceptions of representation in middle school mathematics. Journal of Mathematics Teacher Education, 13, 325-343. https://doi.org/10.1007/s10857-010-9143-y

Winsløw, C. & Durand-Guerrier, V. (2007). Education of lower secondary mathematics teachers. Nordic Studies in Mathematics Education, 12 (2), 5-32.

Zaslavsky, O., Chapman, O. & Leikin, R. (2003). Professional development in mathematics education: trends and tasks. In A. J. Bishop, M. A. Clemens, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 877-917). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-010-0273-8_28

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Published

2024-11-19

How to Cite

Valenta, A., & Enge, O. (2024). Student teachers’ work on instructional explanations in multiplication – representations and conversions between them. NOMAD Nordic Studies in Mathematics Education, 18(1), 31–59. Retrieved from https://tidsskrift.dk/NOMAD/article/view/148497

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