AI错例辨析驱动的数学分析证明思维培养

Cultivating Undergraduates’ Mathematical Proof Reasoning through AI-Generated Error Analysis

摘要: 生成式人工智能给出的数学证明往往语言顺畅、形式完整，但其中可能存在量词错误、定理条件缺失或推理依据不足等问题。本科生在数学分析学习中也容易重视结论和模仿，忽视检查证明条件。针对这两个课堂难点，本文将经过教师审定的AI错误证明用作证明评估材料。教学设计以数学证明理论和支架理论为依据，从定义与条件、逻辑审查、反例证伪和证明重构四个方面提出学习要求。课堂活动分为六个连续步骤，分别是呈现错误证明、独立辨识、说明依据、修正证明、迁移练习和反思总结。相关活动安排在极限、连续性、中值定理、级数与积分等八个教学单元中。本文以一致连续性和Rolle定理为例，说明教师如何运用追问、反例和逐步减少提示，引导学生从判断答案对错转向检查定义、条件与推理依据。在这一过程中，AI只提供可供检验的材料。教师仍负责筛选错例、组织课堂和把握数学标准。该教学方案可以直接用于数学分析常规课堂。

Abstract: Generative artificial intelligence can produce proofs that read smoothly and appear complete, but errors may remain in quantifiers, theorem conditions, or logical reasoning. Undergraduates in mathematical analysis courses may also focus on conclusions and imitate familiar forms without checking why each step is valid. To address these classroom difficulties, this article uses teacher-vetted erroneous AI proofs as materials for proof evaluation. The design draws on studies of mathematical proof and instructional scaffolding. It sets four learning targets concerning definitions and conditions, logical scrutiny, counterexample-based refutation, and proof reconstruction. Classroom work includes six connected tasks. Students read an erroneous proof, make an independent judgment, explain the diagnosis, repair the proof, apply the checking method to a new problem, and reflect on the process. Eight activity units cover limits, continuity, mean value theorems, series, and integration. Cases involving uniform continuity on a compact interval and Rolle’s theorem show how questions, counterexamples, and gradually reduced prompts can move students from checking an answer to examining definitions, hypotheses, and reasons. AI only provides claims for examination. Teachers remain responsible for selecting examples, organizing discussion, and maintaining mathematical standards. The resulting approach can be used in regular mathematical analysis lessons. Its classroom effects remain to be examined empirically.