微积分教学中严格证明与直观理解的平衡——以两个教学实例为镜

Balancing Rigorous Proof and Intuitive Understanding in Calculus Teaching—Reflected in Two Teaching Examples

摘要: 工科微积分教学中，许多核心定理常以几何直观代替严格证明，这虽有助于节省课时，却可能使学生缺失对数学逻辑根基的理解。本文以闭区间上连续函数的可积性与最值定理、介值定理为教学实例，首先回顾现行工科教材中这些定理的直观化处理方式及其潜在的逻辑缺口，并简要介绍了它们的严密证明。在此基础上，提出一套嵌入正常教学进度的平衡策略：在不占用额外课时的前提下，通过关键概念的点拨，帮助学生从显然成立过渡到理解为何必然成立。该策略旨在实现避开证明，但不绕过思想的教学理想，为工科微积分课程中形式严格与直观理解的有机融合提供可操作的参考范式。

Abstract: In engineering calculus instruction, many core theorems are often presented through geometric intuition rather than rigorous proofs. While this approach helps save class time, it may prevent students from grasping the logical foundations of mathematics. Taking the integrability of continuous functions on a closed interval and the extreme value theorem and the intermediate value theorem as teaching examples, this paper first reviews the intuitive treatments of these theorems in current engineering textbooks and identifies their potential logical gaps, and then presents a concise outline of their rigorous proofs. On this basis, a balanced strategy embedded in the regular teaching schedule is proposed: without occupying extra class time, through key conceptual prompts, students are guided from obviously true to understanding why it must be true. This strategy aims to realize the pedagogical ideal of bypassing the full proof without circumventing the ideas, providing a practical reference for the organic integration of formal rigor and intuitive understanding in engineering calculus courses.