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ZDM

, Volume 40, Issue 3, pp 385–400 | Cite as

Argumentation and algebraic proof

  • Bettina Pedemonte
Original article

Abstract

This paper concerns a study analysing cognitive continuities and distances between argumentation supporting a conjecture and its algebraic proof, when solving open problems involving properties of numbers. The aim of this paper is to show that, unlike the geometrical case, the structural distance between argumentation and proof (from an abductive argumentation to a deductive proof) is not one of the possible difficulties met by students in solving such problems. On the contrary, since algebraic proof is characterized by a strong deductive structure, abductive steps in the argumentation activity can be useful in linking the meaning of the letters used in the algebraic proof with numbers used in the argumentation. The analysis of continuities and distances between argumentation and proof is based on the use of Toulmin’s model combined with ck¢ model.

Keywords

Structurant Argumentation Structural Continuity Argumentation Activity Algebraic Proof Deductive Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Anscombre, J. C., & Ducrot, O. (1983). L’argumentation dans la langue. Bruxelles: Mardaga.Google Scholar
  2. Arsac, G., Germain, G., & Mante, M. (1991). Problème ouvert et situation-problème. Lyon: IREM.Google Scholar
  3. Arzarello, F., Bazzini, L., & Chiappini, G. (2001). A model for analyzing algebraic process of thinking. In R. Sutherland, T. Rojano, & A. Bell (Eds.), Perspectives on School Algebra (pp. 61–82). Dordrecht, The Netherlands: Kluwer.Google Scholar
  4. Balacheff, N. (1988). Une étude des processus de preuve en mathématiques chez les élèves de Collège. Thèse d’état. Grenoble: Université Joseph Fourier.Google Scholar
  5. Balacheff, N., & Gaudin, N. (2002). Students conceptions: an introduction to a formal characterization. Le Cahiers du Laboratoire Leibniz 65.Google Scholar
  6. Balacheff, N. (2000). A modelling challenge: untangling learners’ knowing. Journées Internationales d’Orsay sur les Sciences Cognitives: L’apprentissage, JIOSC2000, Paris.Google Scholar
  7. Balacheff, N., & Margolinas, C. (2005). Ckc modèle de connaissances pour le calcul des situations didactiques. In A. Mercier, & C. Margolinas (Eds.), Balises pour la didactique des mathématiques (pp. 75–106). Grenoble: La pensée sauvage.Google Scholar
  8. Boero, P., Garuti, R., & Mariotti, M. A. (1996). Some dynamic mental processes underlying producing and proving conjectures. In Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education PME-XX, vol. 2, (pp. 121–128). Valencia.Google Scholar
  9. Cerulli, M., & Mariotti, M. (2003). Building theories: working in a microworld and writing the mathematical notebook. In Proceedings of the 2003 Joint Meeting of PME and PMENA, CRDG, College of Education, University of Hawai’i, Honolulu, vol. 2, (pp. 181–188). HI, USA.Google Scholar
  10. Chevallard, Y. (1989). Arithmétique, Algèbre, Modélisation. Aix-Marseille: IREM.Google Scholar
  11. Duval, R. (1995). Sémiosis et pensée humaine, Edition: Peter Lang, Suisse.Google Scholar
  12. Duval, R. (2002). L’apprentissage de l’algèbre et le problème cognitif de la désignation des objets. Actes des Séminaires SFIDA-13 à SFIDA-16 vol. IV. Nice: IREM.Google Scholar
  13. Drohuard, J. P. (1992). Shells, shellettes and free shells, a framework for algebraic skills. Psychology for Mathematics Education PMEXVII, Tsukuba, Japan.Google Scholar
  14. Garuti, R., Boero, P., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems. In Proceedings of the International Group for the Psychology of Mathematics Education PME-XX, vol. 2 (pp. 113–120). Valencia.Google Scholar
  15. Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulties of proof. In Proceedings of the International Group for the Psychology of Mathematics Education PME-XXII, vol. 2, (pp. 345–352). Stellenbosh.Google Scholar
  16. Hanna, G., & Jahnke, N. (1993). Proof and application. Educational Studies in Mathematics, 24, 421–438.CrossRefGoogle Scholar
  17. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.CrossRefGoogle Scholar
  18. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59–78.CrossRefGoogle Scholar
  19. Lakatos, I. (1976). Proofs and refutations. The logic of Mathematical Discovery. Cambridge: Cambridge University Press (Italian translation Benelli D. (1979). Dimostrazioni e confutazioni La logica della scoperta matematica. Milano: Feltrinelli).Google Scholar
  20. Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 78, 39–65.Google Scholar
  21. Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173–196.CrossRefGoogle Scholar
  22. Mariotti, M. A., Bartolini Bussi, M. G., Boero, P., Ferri, F., & Garuti, M. R. (1997). Approaching Geometry theorems in contexts: from history and epistemology to cognition. In Proceedings of the International Group for the Psychology of Mathematics Education PME 21, vol. 1 (pp. 180–195). Lahti, Finland.Google Scholar
  23. Mariotti, M. A. (2001). La preuve en mathématique. La Revue canadienne de l’enseignement des sciences, des mathématiques et des technologies, pp. 437–458.Google Scholar
  24. Miyakawa, T. (2002). Relation between proof and conception: the case of proof for the sum of two even numbers. In Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education PME-26, (pp. 21–26). Norwich.Google Scholar
  25. Pedemonte, B. (2002). Etude didactique et cognitive des rapports de l’argumentation et de la démonstration en mathématiques. Thèse de Doctorat. Grenoble I: Université Joseph Fourier. Google Scholar
  26. Pedemonte, B. (2005). Quelques outils pour l’analyse cognitive du rapport entre argumentation et démonstration. Recherche en didactique des mathématiques, 25(3), 313–348.Google Scholar
  27. Pedemonte, B. (2007a). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66, 23–41.CrossRefGoogle Scholar
  28. Pedemonte, B. (2007b). Structural relationships between argumentation and proof in solving open problems in algebra. In Proceedings of the V Congress of the European Society for Research in Mathematics Education CERME 5, (pp. 643–652). Larnaca, Cyprus.Google Scholar
  29. Pedemonte, B., Chiappini, G. (2008) Algebra on Numerical sets: a system for teaching and learning algebra. International Journal Continuing Engineering Education and Life-Long Learning (in press).Google Scholar
  30. Peirce, C. S. (1960). Collected Papers. Cambridge, Massachusetts: Harvard University Press.Google Scholar
  31. Perelman, C., & Olbrechts-Tyteca, L. (1958). Traité de l’argumentation-La nouvelle rhétorique Editions de l’Université de Bruxelles, Bruxelles 1992 (5éme édition).Google Scholar
  32. Plantin, C. (1990). Essais sur l’argumentation. Paris: Kimé.Google Scholar
  33. Polya, G. (1962). How to solve it? New York: Princeton University Press (French translation Mesnage C. Comment poser et résoudre un problème. Paris: Dunod).Google Scholar
  34. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.CrossRefGoogle Scholar
  35. Toulmin S. E. (1993). The use of arguments. Cambridge: Cambridge University Press (French translation De Brabanter P. (1958). Les usages de l’argumentation, Presse Universitaire de France).Google Scholar
  36. Yerushalmy M., & Chazan, D. (2002). Flux in school algebra: curricular change, graphing technology, research on student learning and teacher knowledge. In L. English (Ed.) Handbook of International Research in Mathematics Education (pp. 725–755). Hillsdale, NJ: Erlbaum.Google Scholar

Copyright information

© FIZ Karlsruhe 2008

Authors and Affiliations

  1. 1.Istituto per le Tecnologie Didattiche, CNRGenovaItaly

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