An essay on proof, conviction, and explanation: multiple representation systems in combinatorics


There is a longstanding conversation in the mathematics education literature about proofs that explain versus proofs that only convince. In this essay, we offer a characterization of explanatory proofs with three goals in mind. We first propose a theory of explanatory proofs for mathematics education in terms of representation systems. Then, we illustrate these ideas in terms of combinatorial proofs, focusing on binomial identities. Finally, we leverage our theory to explain audience-dependent and audience-invariant aspects of explanatory proof. Throughout, we use the context of combinatorics to emphasize points and to offer examples of proofs that can be explanatory or only convincing, depending on how one understands the claim being made.

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  1. 1.

    Providing such a definition was not necessarily their intent.

  2. 2.

    Balacheff (2009) also defined explanation as a psychological construct, proof as a social construct, and both characterized in terms of representation systems. However, at least in terms of student-generated proofs, he viewed a proof as a socially accepted explanation (i.e., all proofs are explanations). Our characterization says a student-generated proof might not be explanatory if it was generated with non-natural inferences or in an RS that a student did not personally value.

  3. 3.

    The theorem extends to other, non-integer values, but we focus on this version.


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We thank the anonymous reviewers for helpful comments.

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Correspondence to Elise Lockwood.

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Lockwood, E., Caughman, J.S. & Weber, K. An essay on proof, conviction, and explanation: multiple representation systems in combinatorics. Educ Stud Math 103, 173–189 (2020).

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  • Proof
  • Proofs that explain and convince
  • Combinatorics
  • Binomial identities