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Is this a coincidence? The role of examples in fostering a need for proof

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Abstract

It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term “Is this a coincidence?”. In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for “proof” was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups.

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Notes

  1. We also created and examined algebraic versions of this type of task, but we do not discuss them here.

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Authors and Affiliations

  1. Technion-Israel Institute of Technology, Haifa, Israel

    Orly Buchbinder

  2. New York University, New York, USA

    Orit Zaslavsky

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  1. Orly Buchbinder
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  2. Orit Zaslavsky
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Correspondence to Orly Buchbinder.

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Buchbinder, O., Zaslavsky, O. Is this a coincidence? The role of examples in fostering a need for proof. ZDM Mathematics Education 43, 269–281 (2011). https://doi.org/10.1007/s11858-011-0324-7

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