When Students Prove a Theorem without Explicitly Using a Necessary Condition: Digging into a Subtle Problem from Practice
When Students Prove a Theorem without Explicitly Using a Necessary Condition: Digging into a Subtle Problem from Practice
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当学生在没有明确使用必要条件的情况下证明定理时：从实践中挖掘微妙的问题
DOI：
10.1007/s40753-019-00090-9
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发表时间：
2019
影响因子：
1.5
通讯作者：
Samuel A. Cook
中科院分区：
文献类型：
--
作者：
K. Melhuish;Sean Larsen;Samuel A. Cook
多年来，我们已经注意到我们的学生构造的证明，交换性是由同构，没有明确使用的事实，同构是满射。这些证明通常是有效的，否则。然而，这样的证明是无效的，因为它们会证明交换性被任何同态保持的错误主张。这一来自实践的观察提出了一些值得研究的问题：这种现象有多普遍？这种现象的本质是什么？我们能否解释为什么学生会产生这种类型的论点？在本文中，我们报告了一个小规模的两部分调查研究和初步的访谈研究，旨在开始探索这些问题。我们的研究结果表明，这种现象可能是相当普遍的，超越了一个简单的遗漏证明细节。根据研究文献和我们的后续访谈，我们对这一现象提出了潜在的解释。最后，我们讨论了两种不同的方式来思考支持学生谁犯这个错误，一个侧重于完善学生的证明，一个涉及到鼓励学生使用一个声明的结论，以结构的证明。
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.