Abstract Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’ conceptions about the validity of a direct argument after the students received instruction on the ECE framework. We describe student activities in response to the intervention, and we identify students’ conceptions that are inconsistent with canonical notions of mathematical proving and appear to be barriers to using the ECE framework.

中文翻译：

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消除反例：提高青少年批判直接论点能力的案例研究干预

摘要 学生在论证、证明方面的困难以及反例在证明中的作用都有很好的记录。本研究中的学生经历了一种干预，以改善他们的论证和证明实践。干预包括消除反例 (ECE) 框架作为构建和批评一般主张的可行论点的一种手段。该框架涉及构建对条件声明的所有可能反例的描述，并确定直接论证是否消除了反例的可能性。本案例研究调查了美国八年级（13 岁）数学学生在接受 ECE 框架指导后关于直接论证有效性的概念。我们描述了学生响应干预的活动，