APOS理论下等差数列的教学研究

Teaching Research on Arithmetic Sequences under the APOS Theory

摘要: 数列作为刻画离散数学模型的重要工具，是连接初等数学与高等数学的关键桥梁。在新高考改革全面推进与《普通高中数学课程标准》深入实施的背景下：数列，特别是等差数列的教学被赋予了新的内涵与要求。本研究旨在深入剖析等差数列在高中数学课程体系中的核心地位与教学价值。通过对人教A版教材的结构化分析，厘清等差数列内容的知识脉络与编排逻辑。进而，以杜宾斯基等人提出的APOS理论(操作–过程–对象–图式)为框架，构建等差数列概念、通项公式及前n项和公式的渐进式教学模式。将等差数列置于新高考强调的“突出本质、体现综合、重视应用、彰显素养”的命题导向下审视，并结合建构主义的APOS理论进行教学设计，能有效促进学生从具体操作到抽象思维、从孤立知识点到系统知识网络的认知建构，从而不仅掌握等差数列的知识技能，更能发展数学抽象、逻辑推理、数学建模等核心素养。

Abstract: As an important tool for characterizing discrete mathematical models, sequences serve as a key bridge connecting elementary mathematics and higher mathematics. Against the backdrop of the comprehensive promotion of the new college entrance examination reform and the in-depth implementation of the “General High School Mathematics Curriculum Standards”, the teaching of sequences, particularly arithmetic sequences, has been endowed with new connotations and requirements. This study aims to analyze in depth the core position and teaching value of arithmetic sequences within the high school mathematics curriculum system. Through a structured analysis of the People’s Education Press (PEP) version A textbooks, the study clarifies the knowledge framework and organizational logic of the content related to arithmetic sequences. Furthermore, using the APOS theory (Action-Process-Object-Schema) proposed by Dubinsky and others as a framework, it constructs a progressive teaching model for the concepts of arithmetic sequences, the general term formula, and the sum of the first n terms formula. By examining arithmetic sequences under the new college entrance examination’s emphasis on “highlighting essence, reflecting integration, valuing application, and demonstrating competence”, and integrating constructivist APOS-based instructional design, teaching can effectively promote students’ cognitive construction from concrete operations to abstract thinking, and from isolated knowledge points to a systematic knowledge network. This approach not only enables students to master the knowledge and skills of arithmetic sequences but also fosters core competencies such as mathematical abstraction, logical reasoning, and mathematical modeling.