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Proof, Mathematical Problem-Solving, and Explanation in Mathematics Teaching

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Explanation and Proof in Mathematics

Abstract

We expect mathematical proofs to explain why the propositions in question are true or why certain mathematical phenomena occur in certain situations. In this paper, I reexamine explanation-building processes by taking them as problem-solvers’ understanding processes and by referring to research that has analyzed the relationships between explorations, understandings, and explanations in mathematical problem-solving. I discuss some interactive features among these components during problem-solving processes by introducing some examples and referring to that research. I then use those features to offer an elaborated conception of explanation-building processes that takes into consideration local explanations, full explanations, and the direct and indirect relationships between local and full explanations.

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Notes

  1. 1.

    Corfield (1998, pp. 280–281) proposed that the hard core of mathematical research programmes includes aims of developing good understandings of targeted objects and its positive heuristics are composed of “favoured means” to achieve these aims. Similarly, in mathematical problem-solving processes discussed here, it is usually expected to develop good understandings through favored means.

  2. 2.

    As discussed next, this solution was backed by a certain operational image: opening the folded net which consisted of acute-triangles (Fig. 15.6). This proof may be a kind of picture proof (Brown 1997) with dynamic components.

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Authors and Affiliations

  1. Department of Learning Support, Joetsu University of Education, Joetsu, Japan

    Kazuhiko Nunokawa

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  1. Kazuhiko Nunokawa
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Correspondence to Kazuhiko Nunokawa .

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Editors and Affiliations

  1. Ontario Institute for Studies in, University of Toronto, Bloor Street W. 252, Toronto, M5S 1V6, Canada

    Gila Hanna

  2. FB Mathematik, Universität Duisburg-Essen, Essen, 45117, Germany

    Hans Niels Jahnke

  3. Inst. Philosophie, Universität Bochum, Universitatsstr. 150, Bochum, 44801, Germany

    Helmut Pulte

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Nunokawa, K. (2010). Proof, Mathematical Problem-Solving, and Explanation in Mathematics Teaching. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_15

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