融合APOS理论和RMI原则的函数概念教学研究——以“对数的概念”为例

A Study on Teaching the Concept of Functions by Integrating APOS Theory and RMI Principles—Taking the “Concept of Logarithm” as an Example

摘要: 对数的概念是高中数学的核心内容，传统教学偏重定义记忆与技能训练，学生难以经历概念的心理建构过程，导致对对数是指数逆运算的本质理解不清。本研究以APOS理论和RMI原则为双重理论框架，以对数的概念为例，设计并实施了四个环节：创设GDP情境，试算感知认知冲突；逆运算类比，提炼对数本质内涵；辨析定义性质，固化符号运算对象；强化概念迁移、构建完整运算体系。在此基础上，探究了二者在概念教学中的协同机制。研究表明，APOS理论提供纵向认知建构路径，RMI原则提供横向问题转化工具，二者融合能有效促进学生对对数概念的深度理解。该设计既落实数学抽象、逻辑推理、数学建模等核心素养，又自然融入课程思政，为高中抽象概念教学提供了可操作的理论融合范式。

Abstract: The concept of logarithm is a core component of high school mathematics. Traditional teaching lays excessive emphasis on definition memorization and skill training, making it difficult for students to go through the psychological construction process of the concept, which results in their vague understanding of the essence that logarithm is the inverse operation of exponentiation. Taking the APOS theory and the RMI principle as the dual theoretical framework, this study designs a teaching case for the logarithm concept and constructs four teaching links: creating a GDP context to trigger cognitive conflict through trial calculation and perception; extracting the essential connotation of logarithm by analogy with inverse operations; discriminating definitions and properties to solidify symbolic operation objects; and strengthening concept transfer to build a complete operation system, so as to explore the collaborative mechanism of the two theories in concept teaching. The research shows that the APOS theory provides a vertical path for cognitive construction, while the RMI principle offers a horizontal tool for problem transformation. The integration of the two can effectively promote students’ in-depth understanding of the logarithm concept. This design not only develops core mathematical competencies such as mathematical abstraction, logical reasoning and mathematical modeling, but also naturally integrates curriculum ideology and politics, providing an operable theoretical integration paradigm for the teaching of abstract concepts in senior high school.