Abstract
In this chapter, we present some results from a project about mathematical argumentation and proving in the form of dialogues. Tasks were prepared in the form of written dialogues between two imaginary pupils discussing a mathematical problem, and pupils were invited to write their own dialogues continuing the mathematical discussion. An analysis of dialogues about fractions written by 33 pupils from two classrooms in Norway in Grades 5 and 6 (10–12-year-olds), working in small groups, revealed that many of the 5th grade pupils used forms of argumentation supported by visual representations of fractions, while the 6th graders used more rule-bound approaches based on conversion. The analysis showed that three of ten groups in 6th grade used both diagrammatic and narrative argumentation in contrast to 5th grade where half of the groups were able to use these two kinds of argumentation. Those groups who made use of both types of argumentation were most successful in their argumentation. We relate these findings to the theory of relational and instrumental understanding in mathematics .
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References
Adler, J. (1999). The dilemma of transparency: Seeing and seeing through talk in the mathematics classroom. Journal for Research in Mathematics Education, 30(1), 47–64.
Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna et al. (Ed.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 115–135). Springer Science + Business Media.
Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183–203.
de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
Dörfler, W. (2006). Diagramme und Mathematikunterricht [Diagrams and the mathematics classroom]. Journal für Mathematikdidaktik, 27(3/4), 200–219.
Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving in the classroom. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education. New ICMI study series (Vol. 15). Springer Science+Business Media.
Gholamazad, S. (2007). Pre-service elementary school teachers’ experiences with the process of creating proofs. In J. Woo, H. Lew, K. Park, & D. Seo (Eds.), Proceedings of the 31th Conference of the International Group for the Psychology of Mathematics Education (pp. 265–272). Seoul: PME.
Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Greenwich: CT: Information Age.
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
Hemmi, K. (2008). Students’ encounter with proof: The condition of transparency. ZDM—The International Journal on Mathematics Education, 40, 413–426.
Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153–170). New York: Routledge.
Krummheuer, G. (1999). The narrative character of argumentative mathematics classroom interaction in primary education. In Proceedings of the European Society for Research in Mathematics Education I (pp. 331–341). Osnabrück: Forschungsinstitut für Didaktik der Mathematik.
Krummheuer, G. (2013). The relationship between diagrammatic argumentation and narrative argumentation in the context of the development of mathematical thinking in the early years. Educational Studies in Mathematics, 84(2), 249–265.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.
Lekaus, S., & Askevold, G.-A. (2015). Dialogues as in instrument in mathematical reasoning. In B. Di Paola & C. Sabena (Eds.), Teaching and learning mathematics: Resources and obstacles, Proceedings of CIEAEM 67 (pp. 399–403). Aosta, Italy.
Mellin-Olsen, S. (1984). Eleven, matematikken og samfunnet. Oslo: NKI-forlag.
Mellin-Olsen, S. (1996). Oppgavediskursen i matematikk. Tangenten, 7(2), 9–15.
Ministry of Education and Research. (2013). Curriculum for the common core subject of mathematics (MAT1-04). Oslo: Norwegian Directorate for Education and Training. https://www.udir.no/kl06/MAT1-04?lplang=eng. Accessed: March 08, 2017.
Peirce, C. S. (1978). Collected papers of Charles Sanders Peirce. Cambridge, MA: Harvard University Press.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.
Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). Reston, VA: National Council of Teachers of Mathematics.
Wenger, E. (1998). Communities of practice. Cambridge: Cambridge University Press.
Wille, A. (2011). Activation of inner mathematical discourses of students about fractions with the help of imaginary dialogues: A case study. In Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (pp. 337–344). Ankara, Turkey: PME.
Wille, A. (2013). Mathematik beim Schreiben denken—Auseinandersetzung mit Mathematik in Form von selbst erdachten Dialogen. In M. Rathgeb, M. Helmerich, R. Krömer, K. Lengnink, & G. Nickel (Eds.), Mathematik im prozess (pp. 239–254). Wiesbaden: Springer Fachmedien.
Wille, A., & Boquet, M. (2009). Imaginary dialogues written by low-achieving students about origami: A case study. In Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematical Education (Vol. 1, pp. 337–344). Thessaloniki, Greece: PME.
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Askevold, GA., Lekaus, S. (2018). Mathematical Argumentation in Pupils’ Written Dialogues. In: Stylianides, A., Harel, G. (eds) Advances in Mathematics Education Research on Proof and Proving. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-70996-3_11
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