Abstract
Studying the relationships between argumentation and proof could help teachers and students deal with the tension between the process by which a student develops a proof and the requirements the teacher places on the final product. This tension results from the need for students to experience freedom and flexibility during an initial exploratory phase whilst ultimately producing a proof that conforms to specific cultural constraints involving both logical and communicative norms. The chapter explores the issue of whether the activity of developing proofs under a teacher’s guidance can be used to introduce students to meta-mathematical concepts such as proof, definition, theorem, axiom, and theory. It also tries to clarify the perspectives underlying its specific proposals, to give insight into their richness, and to provide a basis for further research and innovation.
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Notes
- 1.
For a discussion about possible meanings of the terms axiom and postulate according to the Greeks, see Jahnke (2010).
- 2.
An énoncé-tiers is a statement already known to be true – namely an axiom, a theorem, or a definition – and generally is (or could be put) in the form “if p, then q”. In Toulmin’s (2008) model (see below), an énoncé-tiers corresponds to a warrant in an inference step.
- 3.
- 4.
This agrees with Mariotti’s definition of “theorem” (Mariotti et al. 1997): a system consisting of a statement, a proof – derived from axioms and other theorems according to shared inference rules – and a reference theory.
- 5.
Notice that if we consider decimal numbers with a given fixed number of digits following the decimal point, then these two questions are answered in the affirmative.
- 6.
The activity was also given to French students at a more advanced mathematical level, where it gave rise to very similar results.
- 7.
See Hsu et al. (2009) for another example of an instructional experiment concerning validity.
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Acknowledgements
We wish to thank the members of Working Group 2: Thomas Barrier, Thomas Blossier, Paolo Boero, Nadia Douek, Viviane Durand-Guerrier, Susanna Epp, Hui-yu Hsu, Kosze Lee, Juan Pablo Mejia-Ramos, Shintaro Otsuku, Cristina Sabena, Carmen Samper, Denis Tanguay, Yosuke Tsujiyama, Stefan Ufer, and Michelle Wilkerson-Jerde.
We are grateful for the support of the Institut de Mathématiques et de Modélisation de Montpellier, Université Montpellier 2 (France), IUFM C. Freinet, Université de Nice (France), Università di Genova (Italy), DePaul University (USA), and the Fonds québécois de recherche sur la société et la culture (FQRSC, Grant #2007-NP-116155 and Grant #2007-SE-118696).
We also thank the editors and the reviewers for their helpful feedback on earlier versions of this chapter.
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NB: References marked with * are in F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.) (2009). ICMI Study 19: Proof and proving in mathematics education. Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
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Durand-Guerrier, V., Boero, P., Douek, N., Epp, S.S., Tanguay, D. (2012). Argumentation and Proof in the Mathematics Classroom. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_15
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