依测度收敛学习中的认知难点与教学突破

Difficulties in Cognition and Teaching Breakthroughs in Learning Convergence in Measure

摘要: 依测度收敛是实变函数中极具深度的概念，是学生从数学分析进入现代分析(如概率论、泛函分析)时遇到的一个关键点。然而，该概念因其高度的抽象性、反直观性以及与逐点收敛的复杂关系，成为学生学习过程中面临的主要认知障碍。本文应用杜宾斯基APOS理论，分析了学生学习依测度收敛概念时所遇到的认知困难的根源，提出教学突破策略，帮助学生构建清晰的知识网络，克服学习难点，提升数学抽象思维与辨析能力。

Abstract: Convergence in measure is a highly profound concept in real variable functions, and it is a key point that students encounter when transitioning from mathematical analysis to modern analysis, such as probability theory and functional analysis. However, due to its high degree of abstraction, anti-intuitiveness, and complex relationship with point by point convergence, this concept has become the main cognitive barrier faced by students in the learning process. In the article, applying the Doubinsky’s APOS theory, we analyze the root of cognitive difficulties encountered by students in learning the convergence in measure. Some teaching breakthrough strategies are proposed to help students establish clear knowledge networks, overcome learning difficulties, and enhance mathematical abstract thinking and analytical abilities.