波利亚解题思想下可视化技术赋能高中圆锥曲线动点最值问题的探究——以GeoGebra辅助解题为例

Visualizing Moving-Point Extrema on Conic Sections: A Pólya-Inspired Approach in High School Mathematics—A Case Study of GeoGebra-Assisted Problem Solving

摘要: 《普通高中数学课程标准（2022版）》要求发展学生直观想象、逻辑推理、数学建模等核心素养。圆锥曲线是高考高频考点，试题难度系数0.3~0.4，区分度较好，动点最值问题是学生学习的主要难点。受静态习题呈现形式限制，学生难以想象点、线的动态变化，解题过程易出现思维卡顿与畏难情绪。本文以波利亚解题表的“理解题意、拟订方案、执行计划、回顾反思”四阶段为框架，结合GeoGebra可视化平台，选取椭圆双动点最值问题开展教学案例研究，落实“观察–猜想–验证”的探究过程。借助软件轨迹、滑动条、动画演示功能直观呈现几何关系，先通过可视化提出几何猜想，再完成严格数理证明，并通过变式探究提炼圆锥曲线与定圆联动最值问题的一般规律。教学实践表明，将GeoGebra融入波利亚解题流程，能够降低学生认知负荷，培养动态几何思维，落实数学核心素养。

Abstract: The General Senior High School Mathematics Curriculum Standards (2022 Edition) calls for developing students’ core competencies, including intuitive imagination, logical reasoning, and mathematical modeling. Conic sections are a frequent topic in the college entrance examination, with a difficulty coefficient ranging from 0.3 to 0.4 and a good degree of discrimination. The problem of finding the maximum or minimum values of moving points is the main difficulty for students. Due to the limitations of the static presentation form of exercises, students find it difficult to imagine the dynamic changes of points and lines, and the problem-solving process is prone to thinking blockages and feelings of difficulty. This article takes the four stages of “understanding the problem, formulating a plan, executing the plan, and reviewing and reflecting” from Polya’s problem-solving table as the framework, combines the GeoGebra visualization platform, selects the problem of the maximum or minimum values of the double moving points of an ellipse for teaching case research, and implements the “observation-conjecture-verification” exploration process. The article uses the software’s trajectory, slider, and animation demonstration functions to present geometric relationships intuitively. First, it proposes a geometric conjecture through visualization, then completes a strict mathematical proof, and through variations, refines the general rules of the linkage maximum or minimum value problem of conic sections and fixed circles. Teaching practice shows that integrating GeoGebra into Polya’s problem-solving process can reduce students’ cognitive load, cultivate dynamic geometric thinking, and implement core mathematical literacy.