利用AI大模型研究直角三角形中量的关系的案例

A Case Study on the Relationship among Quantities in Right Triangles Using an AI Large Language Model

摘要: 在人工智能重塑知识生产范式的背景下，本研究聚焦于数学教育中关系推理能力的培养，以直角三角形为载体，探究AI大模型对几何量关系网络的认知边界。针对现有研究多局限于公式验证而忽视系统性关系推理的现状，本文基于构建的四层认知水平模型，设计多阶段实验任务，以DeepSeek-R1模型为测试对象，系统评估其对边、角、高、面积等量关系的识别、推导与迁移能力。明确了当前AI在数学关系推理中的能力边界，更凸显了人类教师在几何本质引导与高阶思维培养中的不可替代性。这一发现为评估AI在数学关系推理中的认知边界提供了实证依据，也为后续相关研究提供了方法参照。

Abstract: Against the backdrop of artificial intelligence reshaping the paradigm of knowledge production, this study focuses on the cultivation of relational reasoning ability in mathematics education. Taking the right triangle as the research vehicle, it explores the cognitive boundaries of AI large language models in understanding networks of geometric quantitative relationships. In response to the current state of research, which largely confines itself to formula verification while neglecting systematic relational reasoning, this paper constructs a four-level cognitive model and designs multi-stage experimental tasks. Using the DeepSeek-R1 model as the test subject, it systematically evaluates its abilities in identifying, deriving, and transferring relationships among sides, angles, altitude, and area. The study delineates the current capacity boundaries of AI in mathematical relational reasoning. This finding provides empirical evidence for assessing the structural competence of AI in geometric quantitative cognition and offers a methodological reference for subsequent related research.