Combinatorial proof is an important topic both for combinatorics education and proof education researchers, but relatively little has been studied about the teaching and learning of combinatorial proof. In this paper, we focus on one specific phenomenon that emerged during interviews with mathematicians and students who were experienced provers as they discussed and engaged in combinatorial proof. In particular, participants used a wide variety of cognitive models to interpret multiplication by a constant when reasoning about binomial identities, some of which seemed to be more (or less) effective in helping produce a combinatorial proof. We present these cognitive models and describe episodes that illustrate implications of these cognitive models for our participants’ work on proving binomial identities. Our findings both inform research on combinatorial proof and highlight the importance of understanding subtleties of the familiar operation of multiplication.

中文翻译：

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调查组合证明者关于乘法的推理

组合证明是组合教育和证明教育研究者的重要课题，但是关于组合证明的教与学的研究相对较少。在本文中，我们关注于在与数学家和学生的访谈中出现的一种特定现象，这些学生在讨论和从事组合证明时经验丰富。特别是，在对二项式身份进行推理时，参与者使用各种各样的认知模型通过常数来解释乘法，其中一些似乎在帮助产生组合证据方面更为有效（或不太有效）。我们介绍了这些认知模型，并描述了说明这些认知模型对我们参与者证明二项式身份的工作的意义的情节。