Introduction-to-proof courses are becoming more prevalent in mathematics departments as more recognize the need to support students while they transition from courses focused on computation (such as calculus) to proof-intensive courses (such as real analysis). In such introduction courses, there are some common proving techniques to teach (induction, contradiction, and contraposition to name a few), but the content varies from institution to institution. This note adds to the discussion on content in such courses, by analyzing two prior studies, one using a coding scheme designed to illuminate step-by-step justifications in a proof, and the other focused on interviews with course instructors. Our analysis of the literature shows that there may be reason to believe that content-based introduction-to-proof courses inadvertently overemphasize specific mathematical-area reasoning, which may not translate effectively to subsequent proof-based courses in different content areas. Simply put, while some mathematicians may be convinced of this, a real analysis, number theory, or abstract algebra course may not be the most effective introduction-to-proof course for students to transition to other proof-based courses.

中文翻译：

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入门课程中内容是否重要？

证明入门课程在数学系中变得越来越普遍，因为人们越来越认识到在学生从专注于计算（例如微积分）的课程过渡到证明密集型课程（例如实际分析）的过程中需要支持学生的需求。在此类入门课程中，有一些通用的证明技术可以教（归纳法，矛盾法和矛盾法等），但内容因机构而异。通过分析两项先前的研究，本注释增加了对此类课程内容的讨论，一种是使用旨在阐明证据中逐步依据的编码方案，另一种则是针对与课程讲师的访谈。我们对文献的分析表明，可能有理由相信基于内容的证明课程无意中过分强调了特定的数学领域推理，这可能无法有效地转化为不同内容领域的后续基于证明的课程。简而言之，虽然一些数学家可能对此深信不疑，但对于学生来说，过渡到其他基于证明的课程时，真正的分析，数论或抽象代数课程可能不是最有效的证明入门课程。