An essay on proof, conviction, and explanation: multiple representation systems in combinatorics
An essay on proof, conviction, and explanation: multiple representation systems in combinatorics
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一篇关于证明、信念和解释的文章：组合学中的多重表示系统
DOI：
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发表时间：
2020
影响因子：
3.2
通讯作者：
Keith Weber
中科院分区：
文献类型：
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作者：
E. Lockwood;John S. Caughman;Keith Weber
在数学教育文献中有一个关于解释性证明与仅说服性证明的长期对话。在这篇文章中，我们提供了一个特征的解释性证明，有三个目标。我们首先提出了一个理论的解释证明数学教育的表示系统。然后，我们说明了这些想法的组合证明，集中在二项式身份。最后，我们利用我们的理论来解释解释性证据的观众依赖性和观众不变方面。在整个过程中，我们使用组合数学的背景来强调要点，并提供可以解释或仅令人信服的证明示例，具体取决于人们如何理解所提出的主张。
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.