基于APOS理论的“三角函数的概念”教学设计研究

Teaching Design Study of “Trigonometric Function Concept” Based on APOS Theory

摘要: 数学概念是构建知识体系的核心，也是学生数学学习的基础。针对现有研究中APOS理论在三角函数概念教学中偏重宏观阶段划分、忽视微观认知活动设计的问题，本文提出“四阶段、八环节”的“三角函数的概念”教学设计。“四阶段”是指APOS理论中的活动、过程、对象、图式四阶段，“八环节”是指观察(生活现象)、思考(数学抽象)、探究(关系建构)、定义(形式化)、深化(符号认知)、拓展(一般化)、应用(问题解决)、总结(图式整合)的阶梯式设计。“四阶段、八环节”旨在让学生在概念学习过程中完成从现实世界到数学知识的抽象，实现模型的构建，对三角函数的概念形成深层次的理解，了解概念本质，提升数学核心素养。为APOS理论的实践提供可操作的课堂实施框架，对抽象数学概念教学具有方法论启示。

Abstract: Mathematical concepts are the core of the knowledge system and the foundation of students’ mathematical learning. Aiming at the problem that the APOS theory in the teaching of trigonometric function concepts favors the division of macroscopic stages and neglects the design of microscopic cognitive activities, this paper proposes the teaching design of “concepts of trigonometric function” with “four stages and eight links”. The “four stages” refers to the four stages of activity, process, object and schema in the APOS theory, and the “eight links” refers to the observation of (life phenomenon), thinking (mathematical abstraction), inquiry (relational construction), definition (formalization), deepening (symbolic cognition), extension (generalization), application (problem solving), and conclusion (schematic integration) in a stepwise design. The “four stages and eight links” are designed to enable students to complete the abstraction from the real world to mathematical knowledge in the process of conceptual learning, to realize the construction of models, to form a deep understanding of the concept of trigonometric functions, to understand the essence of the concepts, and to improve the core mathematical literacy of mathematics. It provides a practical framework for the implementation of APOS theory in the classroom, and provides methodological insights into the teaching of abstract mathematical concepts.