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When Students Prove a Theorem without Explicitly Using a Necessary Condition: Digging into a Subtle Problem from Practice

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Abstract

Over the years, we have noticed our students constructing proofs that commutativity is preserved by isomorphism that do not explicitly use the fact that the isomorphism is surjective. These proofs are typically valid otherwise. However, such proofs are invalid because they would prove the false claim that commutativity is preserved by any homomorphism. This observation from practice raises researchable questions: How common is this phenomenon? What is the nature of this phenomenon and can we explain why students produce this type of argument? In this paper, we report a small-scale two-part survey study and a preliminary interview study designed to begin exploring these questions. Our results suggest that this phenomenon is likely quite common and goes beyond a simple omission of a proof detail. Drawing on the research literature and our follow-up interviews, we propose potential explanations for this phenomenon. Finally, we discuss two different ways to think about supporting students who make this error, one that focuses on refining the students’ proofs and one that involves encouraging students to use the conclusion of a statement to structure a proof.

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Notes

  1. Survey A and Survey B were shuffled together and handed out to both classes. While we are not claiming this is a fully randomized approach, we can reasonably assume survey distribution does not account for our later results.

  2. Let G and H be isomorphic groups. If a,b in G, then ab = ba implies if c,d in H, then cd = dc.

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Authors and Affiliations

  1. Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX, 78666, USA

    Kathleen Melhuish

  2. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, 724 SW Harrison Street, Portland, OR, 97201, USA

    Sean Larsen

  3. Wheelock College of Education and Human Development, Boston University, 2 Silber Way, Boston, MA, 02215, USA

    Samuel Cook

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  1. Kathleen Melhuish
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  2. Sean Larsen
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  3. Samuel Cook
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Correspondence to Kathleen Melhuish.

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Melhuish, K., Larsen, S. & Cook, S. When Students Prove a Theorem without Explicitly Using a Necessary Condition: Digging into a Subtle Problem from Practice. Int. J. Res. Undergrad. Math. Ed. 5, 205–227 (2019). https://doi.org/10.1007/s40753-019-00090-9

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