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Systematic Exploration of Examples as Proof: Analysis with Four Theoretical Frameworks

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Advances in Mathematics Education Research on Proof and Proving

Part of the book series: ICME-13 Monographs ((ICME13Mo))

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Abstract

This chapter offers a multi-layered analysis of one specific category of students’ example-based reasoning , which has received little attention in research literature so far: systematic exploration of examples. It involves dividing a conjecture’s domain into disjoint sub-domains and testing a single example in each sub-domain. I apply four theoretical frameworks to analyze student data: The Mathematical-logical framework for the interplay between examples and proof, Proof schemes framework, Transfer-in-pieces framework, and the Theory of instructional situations . Taken together, these frameworks allow to examine the data from mathematical, cognitive and social perspectives, thus broadening and deepening the insights into students thinking about the relationship between examples and proving. Implications for teaching and learning of proof in school mathematics are discussed.

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Notes

  1. 1.

    See Nelsen (1993) for several elegant proofs without words of the mediant property.

  2. 2.

    All names are pseudonyms.

  3. 3.

    According to the framework, understanding is operationalized as consistent application of inferences that are aligned with conventional mathematical knowledge.

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Authors and Affiliations

  1. University of New Hampshire, Durham, USA

    Orly Buchbinder

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  1. Orly Buchbinder
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Correspondence to Orly Buchbinder .

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Editors and Affiliations

  1. Faculty of Education, University of Cambridge, Cambridge, United Kingdom

    Andreas J. Stylianides

  2. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    Guershon Harel

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Buchbinder, O. (2018). Systematic Exploration of Examples as Proof: Analysis with Four Theoretical Frameworks. In: Stylianides, A., Harel, G. (eds) Advances in Mathematics Education Research on Proof and Proving. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-70996-3_18

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  • DOI: https://doi.org/10.1007/978-3-319-70996-3_18

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-70996-3

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