Logical Reasoning beyond Classical Logic: An Illustration with Pythagoras Theorem

• Autores: Francisco Vargas M., Keith Stenning
• Localización: International Electronic Journal of Mathematics Education, ISSN-e 1306-3030, Vol. 15, Nº. 1, 2020
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • We report on a study conceived with the idea that the use of logic in regard to mathematical reasoning as it actually occurs in practice is not limited to its prominent role in formal deductions and proofs. Interpretation of different mathematical situations elicits in fact the use of mostly unconscious forms of reasoning, close to those of narrative processing, which do not coincide with the expectations of traditional logic. This is pervasive, in particular, in educational situations at different levels, as we illustrate with interpretations which can emerge alongside an apparently obvious mathematical statement, namely, Pythagoras Theorem. We defend the position that analyses of “errors”, should start by understanding their prevalence and non arbitrariness. Accordingly, we use a nonclassical logics whose features may give new insights to the kind of learning obstacles often found in the literature, as well as in our results.

• Referencias bibliográficas
  • Brousseau, G. (1997). Theory of Didactical Situations in Mathematics. Netherlands: Kluwer.
  • D’Amore, B. (2005). Secondary school students’ mathematical argumentation and Indian Logic (Nyaya). For the learning of mathematics, 25(2),...
  • Doets, K. (1994). From Logic to Logic Programming. Cambridge, MA: MIT Press. https://doi.org/10.7551/mitpress/3133.001.0001
  • Durand-Guerrier, V. (2003). Which notion of implication is the right one? From logical considerations to a didactic perspective. Educational...
  • Durand-Guerrier, V., Boero, P., Douek, N., Epp, S. S., & Tanguay, D. (2011). Examining the role of logic in teaching proof. In Proof and...
  • Evans, J. S. B., Newstead, S. E., & Byrne, R. M. (1993). Human reasoning: The psychology of deduction. Psychology Press.
  • Fischbein, H. (1987). Intuition in science and mathematics: An educational approach (Vol. 5). Springer Science & Business Media.
  • Grice H. P. (1975). Logic and conversation. In P. Cole & J. Morgan (Eds.), Syntax and Semantics: Speech Acts, volume 3. London: Academic...
  • Hoyles, C., & Küchemann D. (2002). Students’ understanding of logical implication. Educational Studies in Mathematics, 51, 193-223. https://doi.org/10.1023/A:1023629608614
  • Kahneman, D. (2011). Thinking, fast and slow. Macmillan.
  • Kowalski, R. (2011) Computational Logic and Human Thinking: How to be Artificially Intelligent. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511984747
  • O’Brien, T. C., Shapiro, B. J., & Reali, N. C. (1971), Logical thinking – language and context, Educational Studies in Mathematics, 4,...
  • Stenning, K., & van Lambalgen, M. (2008). Human Reasoning and Cognitive Science. Cambridge, MA: MIT Press. https://doi.org/10.7551/mitpress/7964.001.0001
  • Van Lambalgen, M., & Hamm, F. (2005). The proper treatment of events. New Jersey, USA: Blackwell Publishing. https://doi.org/10.1002/9780470759257
  • Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20, 273–281. https://doi.org/10.1080/14640746808400161