HPM视角下基于认知结构学习理论的教学设计——以“数系的扩充与复数的概念”为例

Teaching Design Based on Cognitive Structure Learning Theory from the Perspective of HPM—Taking "Expansion of Number Systems and the Concept of Complex Numbers" as an Example

摘要: 数系的扩充与人类历史进程紧密相连，每次突破都是原有认知的颠覆。本文以“数系的扩充与复数的概念”为例，从HPM视角出发，结合认知结构学习理论进行教学设计。通过融入数学历史发展脉络，揭示数系扩充的逻辑必然性，引导学生主动构建复数概念的认知框架。教学采用情境模拟、合作探究等方法，激发学生兴趣，促进知识内化。结果显示，学生在理解复数概念的同时，深化了对数学发展进程的认识，有效提升了数学思维能力。在此基础上，提出三点建议：注重学生认知发展，完善知识网络结构；数学史要抛弃表面化，深入融合教学内容；立足数学教育实际，加强案例实证研究。

Abstract: The expansion of the number system is closely related to the process of human history, and every breakthrough is a subversion of existing cognition. This article takes the expansion of number systems and the concept of complex numbers as an example, starting from the HPM perspective and combining cognitive structure learning theory for instructional design. By integrating the historical development of mathematics, revealing the logical inevitability of the expansion of the number system, and guiding students to actively construct a cognitive framework for the concept of complex numbers. Teaching adopts methods such as situational simulation and collaborative exploration to stimulate students’ interest and promote knowledge internalization. The results showed that while students understood the concept of complex numbers, they deepened their understanding of the development process of mathematics and effectively improved their mathematical thinking ability. On this basis, three suggestions are proposed: focus on students’ cognitive development and improve the knowledge network structure; The history of mathematics should abandon superficiality and deeply integrate teaching content; Based on the reality of mathematics education, strengthen case empirical research.