Proof is central to mathematics and has drawn substantial attention from the mathematics education community. Yet, valid and reliable measures of proof comprehension remain rare. In this article, we present a study investigating proof comprehension via students’ summaries of a given proof. These summaries were evaluated by expert judges making pairwise comparisons, which were used to generate a score for each summary. This approach, known as comparative judgement, has been demonstrated to generate reliable and valid scores when assessing other mathematical constructs. Our findings suggest that comparative judgement can produce valid and reliable assessments of the quality of student-produced proof summaries. We also explored which features of students’ proof summaries were most valued by the expert judges, and found that high-scoring summaries referenced a proof’s logical structure and the mechanism by which it reached a contradiction.

中文翻译：

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比较判断、证明概要和证明理解

证明是数学的核心，并引起了数学教育界的极大关注。然而，有效和可靠的证明理解措施仍然很少见。在本文中，我们展示了一项研究，通过学生对给定证明的总结来调查证明理解。这些摘要由专家评委进行成对比较进行评估，用于为每个摘要生成分数。这种被称为比较判断的方法已被证明可以在评估其他数学结构时产生可靠和有效的分数。我们的研究结果表明，比较判断可以对学生制作的证明摘要的质量进行有效和可靠的评估。我们还探讨了专家评委最看重学生证明摘要的哪些特征，