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

Elise Lockwood, John S. Caughman, Keith Weber

Research output: Contribution to journalArticle


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.

Original languageEnglish (US)
Pages (from-to)173-189
Number of pages17
JournalEducational Studies in Mathematics
Issue number2
StatePublished - Feb 1 2020


All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Education


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

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