Abstract
Enhancing the teaching of mathematics in ways that support the development of students’ competence in argumentation and proof calls for increasing teachers’ awareness of the crucial role played by logical reasoning in proof. The chapter examines the relevance of, and interest in, teaching logic in order to foster competence with proof in the mathematics classroom. It reviews various positions on the role of logic in argumentation and proof, taking psychological studies into account, discusses these positions from an educational perspective, and offers some suggestions about how to modify curriculum to help students develop their logical reasoning abilities. A challenge for the future is to develop and implement the suggestions in research programmes.
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Notes
- 1.
Ufer et al. (2009) propose a cognitive model based on mental models.
- 2.
A; and “If A, then B”; hence B.
- 3.
Not B; and “If A, then B”; hence not A.
- 4.
If A then B; If B then C; hence If A then C.
- 5.
A or B; not B; hence A.
- 6.
If not A, then both B and not B; hence A.
- 7.
In the academic year 1998–1999, 22 mathematics teachers from various French scientific universities were asked to comment on an invalid proof written by an undergraduate (see Durand-Guerrier and Arsac 2005, pp.159–163).
- 8.
Mathematicians with broad knowledge of mathematical subjects can similarly recall mathematical facts without having to re-derive them, which could be one of the reasons they may underrate the extent to which logic plays a role in their work.
- 9.
In addition, the logical analysis of statements may vary from one natural language to another. For example, in French statements of the form “tous les A ne sont pas B” (“All A are not B”) are ambiguous. Depending on the context, they may mean either “There is an A that is not-B” (existential statement) or “For all A, A is non-B” (universal statement). In written Arabic, however, this ambiguity does not exist (Durand-Guerrier and Ben Kilani 2004). Such issues were discussed in a paper presented at ICMI Study 21: Mathematics Education and Language Diversity, São Paolo, Brazil, September 16–21, 2011.
- 10.
This will be discussed in Sect. 3.4.
- 11.
A theorem-in-action is a mathematical property that a person may not be consciously aware of but may use in certain situations, such as to find an answer to a mathematical question. However, because the property may not apply to all the situations in which the person might try to use it, it could lead to an invalid deduction.
- 12.
These stacked quantifiers may also be described as instances of the dependence rule for variables: in a statement of the form “For all x, there exists y such that F(x, y),” y is dependent on x.
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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 these chapters.
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NB: References marked with * are in Lin, F. L., Hsieh, F. J., Hanna, G. & de Villiers M. (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). Examining the Role of Logic in Teaching Proof. 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_16
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