GeoGebra环境下基于APOS理论的高中圆锥曲线概念教学设计研究——以椭圆的概念为例

Research on Teaching Design of High-School Conic Section Concept Based on APOS Theory in GeoGebra Environment—A Case Study of the Concept of Ellipse

摘要: 圆锥曲线概念教学长期存在学生机械记忆定义与方程以及“重结论、轻过程”的问题。本文依据高中数学课程标准对信息技术与数学课程深度融合的要求，以APOS理论的活动、过程、对象、图式四阶段为框架，结合GeoGebra图形、代数多重表征实时联动的优势，研究圆锥曲线概念教学设计。首先梳理国内外相关研究，分析GGB与APOS理论融合的理论逻辑与实践逻辑，构建“操作体验、反思内化、对象抽象、图式整合”的教学流程；进而以椭圆概念为例设计教学案例，借助航天情境、数学史与Dandelion双球模型，引导学生在动态探究中自主归纳椭圆定义、推导标准方程并整合知识体系。研究表明，该设计有助于突破机械记忆弊端，深化学生对椭圆几何本质的理解，也为信息技术环境下圆锥曲线概念教学提供了参考。

Abstract: Problems such as students’ rote memorization of definitions and equations, alongside the phenomenon of “emphasizing conclusions while neglecting processes”, have long existed in the teaching of conic section concepts. In accordance with the requirements of in‑depth integration of information technology and mathematics curriculum specified in the General High‑School Mathematics Curriculum Standards, this paper adopts the four‑stage framework of APOS Theory, namely Action, Process, Object and Schema, and combines the strengths of GeoGebra in real‑time linkage of multi‑representations between graphics and algebra to investigate the teaching design for conic section concepts. Firstly, relevant domestic and international studies are reviewed, and the theoretical and practical logics underlying the integration of GeoGebra and APOS Theory are analyzed. A teaching procedure consisting of “operational experience, reflective internalization, object abstraction and schema integration” is constructed. Furthermore, a teaching case is designed taking the concept of ellipse as an example. With the aid of aerospace contexts, history of mathematics and Dandelion’s two‑sphere model, students are guided to induce the definition of ellipse independently, derive its standard equation and integrate knowledge systems through dynamic inquiry. The research indicates that the proposed teaching design helps to overcome the drawbacks of rote learning, deepen students’ understanding of the geometric essence of ellipses, and provide references for conic‑concept teaching under information‑technology‑enhanced environments.