ZDM

, Volume 40, Issue 3, pp 401–412 | Cite as

Indirect proof: what is specific to this way of proving?

Original article

Abstract

The study presented in this paper is part of a wide research project concerning indirect proofs. Starting from the notion of mathematical theorem as the unity of a statement, a proof and a theory, a structural analysis of indirect proofs has been carried out. Such analysis leads to the production of a model to be used in the observation, analysis and interpretation of cognitive and didactical issues related to indirect proofs and indirect argumentations. Through the analysis of exemplar protocols, the paper discusses cognitive processes, outlining cognitive and didactical aspects of students’ difficulties with this way of proving.

Keywords

Proof Argumentation Indirect proof Proof by contradiction Proof by contraposition 

References

  1. Antonini, S. (2001). Negation in mathematics: obstacles emerging from an exploratory study. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 49–56). The Netherlands: Utrecht.Google Scholar
  2. Antonini, S. (2003a). Dimostrare per assurdo: analisi cognitiva in una prospettiva didattica. Tesi di Dottorato, Dipartimento di Matematica, Università di Pisa.Google Scholar
  3. Antonini, S. (2003b). Non-examples and proof by contradiction. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 2, pp. 49–55). Honolulu.Google Scholar
  4. Antonini, S. (2004). A statement, the contrapositive and the inverse: intuition and argumentation. In M. Johnsen Høines, & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 47–54). Norway: Bergen.Google Scholar
  5. Antonini, S., & Mariotti, M. A. (2007). Indirect proof: an interpreting model. In D. Pitta-Pantazi, & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 541–550). Cyprus: Larnaca.Google Scholar
  6. Balacheff, N. (1991). Treatment of refutations: aspects of the complexity of a contructivist approach to mathematics learning. In E. von Glasersfeld (Ed.), Radical Constructivism in Mathematics Education (pp. 89–110). The Netherlands: Kluwer.Google Scholar
  7. Barbin, E. (1988). La démonstration mathématique: significations épistémologiques et questions didactiques. Bulletin APMEP, 366, 591–620.Google Scholar
  8. Bernardi, C. (2002). Ricerche in Didattica della Matematica e in Matematiche Elementari. Bollettino Unione Matematica Italiana, Serie VIII, V-A, 193–213.Google Scholar
  9. Bellissima, F., & Pagli, P. (1993). La verità trasmessa. La logica attraverso le dimostrazioni matematiche. Firenze: Sansoni.Google Scholar
  10. Dummett, M. (1977). Elements of Intuitionism. New York: Oxford University Press.Google Scholar
  11. Durand-Guerrier, V. (2003). Which notion of implication is the right one? From logical considerations to a didactic perspective. Educational Studies in Mathematics, 53(1), 5–34.CrossRefGoogle Scholar
  12. Duval, R. (1992–93). Argumenter, demontrer, expliquer: coninuité ou rupture cognitive? Petit x, 31, 37–61.Google Scholar
  13. Duval, R. (1995). Sémiosis et Pensée Humain. Bern: Peter Lang.Google Scholar
  14. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht: Kluwer.Google Scholar
  15. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  16. Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive Unity of Theorems and Difficulties of Proof. In A. Olivier, & K. Newstead (Eds.), Proceedings of the 22th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 345–352). Stellenbosch: South Africa.Google Scholar
  17. Harel, G. (2007). Students’ proof schemes revisited. In P. Boero (Eds.), Theorems in school: from history, epistemology and cognition to classroom practice (pp. 65–78). Rotterdam: Sense Publishers.Google Scholar
  18. Leron, U. (1985). A Direct approach to indirect proofs. Educational Studies in Mathematics, 16(3), 321–325.CrossRefGoogle Scholar
  19. Mancosu, P. (1996). Philosophy of mathematical practice in the 17th century. New York: Oxford University Press.Google Scholar
  20. Mariotti, M. A., Bartolini Bussi, M., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorems in contexts: from history and epistemology to cognition. In E. Pehkonen (Ed.), Proceedings of the 21th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 180–195). Finland: Lathi.Google Scholar
  21. Mariotti, M. A., & Antonini, S. (2006). Reasoning in an absurd world: difficulties with proof by contradiction. In J. Novotnà, H. Moarovà, M. Kràtkà & N. Stelìchovà (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 65–72). Prague, Czech Republic.Google Scholar
  22. Pedemonte, B. (2002). Etude didactique et cognitive des rapports de l’argumentation et de la démonstration dans l’apprentissage des mathématiques. Thèse, Université Joseph Fourier, Grenoble.Google Scholar
  23. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.CrossRefGoogle Scholar
  24. Piaget, J. (1974). Recherches sur la contradiction. Paris: Presses Universitaires de France.Google Scholar
  25. Polya, G. (1945). How to solve it. Princeton University Press.Google Scholar
  26. Prawitz, D. (1971). Ideas and results in proof theory. In J.E. Fenstad (Eds.), Proceedings of the second Scandinavian Logic Symposium (pp. 235–307). Amsterdam.Google Scholar
  27. Reid, D., & Dobbin, J. (1998). Why is proof by contradiction difficult? In A. Olivier & K. Newstead (Eds.), Proceedings of the 22th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 41–48). Stellenbosch, South Africa.Google Scholar
  28. Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2004). Undergraduate students’ understanding of the contraposition equivalence rule in symbolic and verbal contexts. Educational Studies in Mathematics, 55(1–3), 133–162.CrossRefGoogle Scholar
  29. Szabó, A. (1978). The beginnings of Greek mathematics. Dordrecht: Reidel.Google Scholar
  30. Thompson, D. R. (1996). Learning and teaching indirect proof. The Mathematics Teacher, 89(6), 474–82.Google Scholar
  31. Wu Yu, J., Lin, F., & Lee, Y. (2003). Students’ understanding of proof by contradiction. In N.A. Pateman, B.J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 4, pp. 443–449). Honolulu.Google Scholar

Copyright information

© FIZ Karlsruhe 2008

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PaviaPaviaItaly
  2. 2.Dipartimento di Scienze Matematiche e InformaticheUniversità di SienaSienaItaly

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