数学分析启发式教学的探索与实践——以级数的敛散性为例

Exploration and Practice of Heuristic Teaching in Mathematical Analysis—A Case Study on the Convergence and Divergence of Series

摘要: 作为连接初等数学与高等数学的桥梁，数学分析是数学思维训练的基石，变化与逼近的哲学，具有高度抽象性，对初学者并不友好。启发式教学强调以学生为主体，通过创设情境、设置障碍、类比引导等方式激发学生的学习动机。启发式教学思想在我国具有悠久的历史。文章以级数的敛散性为例，探讨在数学分析中应用启发式教学方法。通过还原知识的生成过程，设计富有层次的问题链，将知识的历史发展脉络、逻辑推演的过程融入到教学环节中。实践表明，启发式教学不仅能降低学生的认知困难，更能引导其洞察知识的本质，实现逻辑思维与创新能力的协同发展。

Abstract: As a bridge connecting elementary mathematics and advanced mathematics, mathematical analysis is a cornerstone for training mathematical thinking, and a philosophy for change and approximation. It is highly abstract and thus unfriendly to beginners. Heuristic teaching is a kind of special teaching model, which is characterized by student-centered. This approach is focus on stimulating students’ motivation by creating situations, setting question circumstances, and using analogies as guidance. The philosophy of heuristic teaching has a long history in China. Taking the convergence and divergence of series as an example, this paper explores the application of heuristic teaching method in mathematical analysis. By constructing a four-level heuristic framework “cognitive conflict, intuitive perception, logical construction, and transfer-creation”, the purpose of heuristic teaching is to guide students’ autonomous learning and to reconstruct the process of knowledge generation. A well-designed, hierarchical chain of questions is developed, integrating the historical development and logical deduction of knowledge into the teaching process. The teaching practice of several years shows that the heuristic teaching not only reduces students’ cognitive difficulties but also helps them gain insight into the essence of knowledge, thereby achieving the synergistic development of logical thinking and innovative ability.