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Proof constructions and evaluations

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Abstract

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.

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Notes

  1. Morris (2002) also examined prospective elementary teachers’ understanding of the distinction between proofs and empirical arguments, but her sample included additionally prospective middle school teachers and the results were not reported separately for the two groups.

  2. There was also a mathematics pedagogy course but its focus was on teaching methods.

  3. As a starting point for the development of this list of criteria, the class used Beckmann’s (2005, p. 7) characteristics of “good explanations” in mathematics.

  4. This phenomenon is not particular to prospective elementary teachers. For examples with other populations, see Knuth (2002), Smith (2006), and Sowder and Harel (1998).

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Acknowledgements

The two authors contributed equally to the preparation of this article. The work reported herein received support from the Spencer Foundation (Grant numbers: 200700100 and 200800104) and, during the preparation of the article, the first author received support from the UK’s Economic Social and Research Council (Grant number: RES-000-22-2536). The opinions expressed in the article are those of the authors and do not necessarily reflect the position, policy, or endorsement of either organization. The authors wish to thank Heinz Steinbring and anonymous reviewers for useful comments on earlier versions of the article. Part of an earlier version of the article will be presented at, and published in the proceedings of, the 19th Study Conference of the International Commission on Mathematical Instruction (Taipei, Taiwan, 2009).

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Authors and Affiliations

  1. Faculty of Education, University of Cambridge, 184 Hills Road, Cambridge, CB2 8PQ, UK

    Andreas J. Stylianides

  2. University of Pittsburgh, Posvar Hall, Room 5517, 230 S. Bouquet Street, Pittsburgh, PA, 15260, USA

    Gabriel J. Stylianides

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  1. Andreas J. Stylianides
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  2. Gabriel J. Stylianides
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Correspondence to Andreas J. Stylianides.

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Stylianides, A.J., Stylianides, G.J. Proof constructions and evaluations. Educ Stud Math 72, 237–253 (2009). https://doi.org/10.1007/s10649-009-9191-3

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