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.

中文翻译：

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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：从实践中挖掘出一个微妙的问题

多年来，我们注意到我们的学生正在构建证明同构性保持可交换性的证据，而这些同构性并未明确使用同构是排斥的事实。这些证明通常在其他方面有效。但是，这样的证明是无效的，因为它们会证明错误的说法，即任何同构都可以保持可交换性。来自实践的观察提出了可研究的问题：这种现象有多普遍？这种现象的本质是什么，我们可以解释为什么学生提出这种论点吗？在本文中，我们报告了一个由两部分组成的小规模调查研究和一个初步访谈研究，旨在开始探索这些问题。我们的结果表明，这种现象很可能很普遍，并且超出了对证明细节的简单遗漏。借助研究文献和我们的后续采访，我们提出了对此现象的潜在解释。最后，我们讨论了两种方法来考虑支持犯错的学生，一种侧重于完善学生的证明，另一种涉及鼓励学生使用陈述的结论来构造证明。