Educational Studies in Mathematics

, Volume 66, Issue 1, pp 23–41

How can the relationship between argumentation and proof be analysed?

Article

Abstract

The paper presents a characterisation about argumentation and proof in mathematics. On the basis of contemporary linguistic theories, the hypothesis that proof is a special case of argumentation is put forward and Toulmin’s model is proposed as a methodological tool to compare them. This model can be used to detect and analyse the structure of an argumentation supporting a conjecture (abduction, induction, etc.) and the structure of its proof. The aim of the paper is to highlight the importance of structural analysis between argumentation and proof. This analysis shows that although there are clear cases of continuity between argumentation supporting a conjecture and its proof, there is often a structural distance between the two (from an abductive argumentation to a deductive proof, from an inductive argumentation to a mathematical inductive proof).

Keywords

Argumentation Proof Structural continuity Toulmin Cognitive unity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anscombre, J. C., & Ducrot, O. (1983). L’argumentation dans la langue. Bruxelles: Mardaga.Google Scholar
  2. Aristotle (1974). Les topiques: Livres I–VIII (French translation Tricot J. Paris: Libraire philosophique J. Vrin.)Google Scholar
  3. Aristotle (1991). Rhetorique (French translation Dufour M., Wartelle A. Les belles lettres).Google Scholar
  4. Arsac, G., Germain, G., & Mante, M. (1991). Problème ouvert et situation-problème. Lyon: IREM.Google Scholar
  5. 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
  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). cK¢ 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. 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. Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational studies in mathematics, 24, 359–387.Google Scholar
  10. De Villiers, M. (1990). The role and function of proof in mathematics, adapted version of paper proof in the mathematics curriculum. Presented at the National Subject Didactics Symposium, University of Stellenbosh.Google Scholar
  11. Douek, N. (1998). Some remarks about argumentation and mathematical proof and their education implications. In I. Schwank (Ed.), First European Conference of the Research in Mathematics Education CERME 1, vol. 1, (pp. 125–139). Osnabrück, Germany.Google Scholar
  12. Ducrot, O., et al. (1980). Les mots du discours. Paris: Ed. de Minuit.Google Scholar
  13. Duval, R. (1992–1993). Argumenter démontrer expliquer : Continuité ou rupture cognitive? Petit X, 31, 37–61. Grenoble: IREM (Ed.).Google Scholar
  14. Duval, R. (1995). Sémiosis et pensée humaine. Edition: Peter Lang, Suisse.Google Scholar
  15. Fann, K. T. (1970). Peirce’s theory of abduction. The Hague, Holland: Martinus Nijhoff.Google Scholar
  16. Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive Unity of Theorems and Difficulty of Proof. Proceedings of the international group for the psychology of mathematics education PME-XXII, vol. 2, (pp. 345–352). Stellenbosch.Google Scholar
  17. Garuti, R., Boero, P., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems. Proceedings of the international group for the psychology of mathematics education PME-XX, vol. 2, (pp. 113–120). Valencia.Google Scholar
  18. Hanna, G. (1989). Proofs that prove and proofs that explain. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the international group for the psychology of mathematics education, PMEXIII, vol. 2, (pp. 45–51). Paris.Google Scholar
  19. Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking, 54–61. Dordrecht, The Netherlands: Kluwer.Google Scholar
  20. Hanna, G. (1995). Challenges to the Importance of Proof. For the Learning of Mathematics, 15(3), 42–49.Google Scholar
  21. Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zazkis (Eds.), Learning and teaching Number Theory. Journal of Mathematical Behavior (pp. 185–212). New Jersey, Ablex Publishing Corporation. Google Scholar
  22. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education, vol. 3, (pp. 234–283). American Mathematical Society.Google Scholar
  23. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.CrossRefGoogle Scholar
  24. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb, & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–269). Hillsdale, New Jersey: Lawrence Erlbaum Associates.Google Scholar
  25. Lakatos, I. (1976). Proofs and refutations. The logic of mathematical discovery. Cambridge University Press, Cambridge; (Italian translation Benelli D. (1979). Dimostrazioni e confutazioni la logica della scoperta matematica. Milano: Feltrinelli).Google Scholar
  26. Magnani, L. (2001). Abduction, reason and science. Processes of discovery and explanation. Dordrecht, The Netherlands: Kluwer.Google Scholar
  27. Mariotti, M.A. (2001). Introduction to proof : The mediation of a dynamic software environment. Educational studies in mathematics, 44, Issues 1 & 2, 25–53.Google Scholar
  28. 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. Proceeding of the international group for the psychology of mathematics education PME-21, vol. 1, (pp. 180–195). Lahti, Finland.Google Scholar
  29. Pedemonte, B. (2001). Some cognitive aspects of the relationship between argumentation and proof in mathematics. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education PME-25, vol. 4, (pp. 33–40). Utrecht (Olanda).Google Scholar
  30. Pedemonte, B. (2002). Etude didactique et cognitive des rapports de largumentation et de la démonstration dans lapprentissage des mathématiques. Thèse de doctorat. Grenoble I: Université Joseph Fourier.Google Scholar
  31. 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
  32. Peirce, C. S. (1960). Collected papers. Cambridge, Massachusetts: Harvard University Press.Google Scholar
  33. 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
  34. Plantin, C. (1990). Essais sur l’argumentation, Kimé (Ed.), Paris.Google Scholar
  35. Polya, G. (1954). Mathematics and plausible reasoning. Princeton University Press, London. (French translation Vallée R. (1958), Les mathématiques et le raisonnement « plausible ». Gauthier – Villars (Ed.), Paris).Google Scholar
  36. Polya, G. (1962). How to solve it? Princeton University Press, New York (French translation Mesnage C. Comment poser et résoudre un problème. Dunod (Ed.), Paris).Google Scholar
  37. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.Google Scholar
  38. Toulmin, S. E. (1993). The use of arguments. Cambridge: University Press (French translation De Brabanter P. (1958). Les usages de l’argumentation, Presse Universitaire de France).Google Scholar
  39. Yackel, E. (2001). Explanation, Justification and argumentation in mathematics classrooms. In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education PME-25, vol. 1, (pp. 1–9). Utrecht (Olanda).Google Scholar
  40. Yackel, E., & Rasmussen, C. (2002). Beliefs and norms in the mathematics classroom. In G. Toerner, E. Pehkonen, & G. Leder (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 313–320). Dordrecht, The Netherlands: Kluwer.Google Scholar
  41. Wood, T. (1999). Creating a Context for Argument in Mathematics Class Young Children’s Concepts of Shape. Journal for Research in Mathematics Education, 30(2), 171–191.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.ITD-CNR GenovaGenovaItaly

Personalised recommendations