一道高考数学题的多种解法探究及思考——以2025年全国新课标II卷第17题为例

Research and Reflections on Multiple Solutions to a College Entrance Examination Mathematics Problem—A Case Study of Question 17 in the 2025 National New Curriculum Standard Volume II

摘要: 在高中数学知识体系里，立体几何是考查学生空间想象能力、逻辑推理能力及数学建模素养的关键内容，高考中的立体几何题常因综合性强、解法灵活而成为区分学生思维层次的典型题型。以2025年全国新课标II卷第17题为例，探究其多种解法。研究发现，不同解法在构建解题桥梁时存在依赖坐标系、辅助线构造要求高、延长线探索随机性强等局限，反映出工具应用与数学本质失衡的问题。由此得出教学中需平衡“代数运算”与“几何直观”训练，解题策略应注重“转化思想”渗透的启示。

Abstract: Within the knowledge system of senior high school mathematics, solid geometry serves as a core component for examining students’ spatial imagination ability, logical reasoning capability, and mathematical modeling literacy. Solid geometry questions in the National College Entrance Examination have always been typical items for differentiating students’ thinking levels due to their strong comprehensiveness and flexible solution approaches. Taking Question 17 of the 2025 National New Curriculum Standard Volume II as the research object, this paper explores its multiple solving methods. The research results indicate that different solutions present distinct limitations when building bridges for problem solving, including excessive reliance on coordinate systems, high requirements for auxiliary line construction, and strong randomness in extended line exploration, which reflect the imbalance between tool application and the essence of mathematics. Accordingly, this paper concludes that mathematics teaching should balance the training of algebraic operation and geometric intuition, and problem-solving strategies should emphasize the infiltration of transformation thinking.