人工智能前沿融入《数学分析》的教学重构——“一图三层五环”模式及实施路径

Instructional Reconstruction of Mathematical Analysis by Integrating Frontiers of Artificial Intelligence—The “One-Map, Three-Level, Five-Cycle” Model and Its Implementation Path

摘要: 面向人工智能应用对数学基础、算法解释与工程迁移能力提出的新要求，针对《数学分析》课程中理论知识与算法场景割裂、抽象概念呈现单一、实践任务碎片化以及生成式人工智能使用规范缺位等问题，采用文献研究、设计研究、行动研究与专家评议相结合的方法，构建“一图三层五环”教学重构模式。其中，“一图”是数学分析知识与人工智能技术的双向映射图谱；“三层”依次为概念锚定、算法阐释和工程迁移；“五环”包括问题情境、数学推导、代码实现、可视诊断与迁移反思。研究进一步设计梯度下降、链式法则与误差反向传播、管网仿真模型参数调优等案例，并建立与课程目标一致的多元评价量规和“可用–需核–须述–可追”的人工智能使用规则。该模式把人工智能由外置教学工具转化为检验数学概念、定理条件和知识迁移的应用语境，在不削弱证明训练的前提下形成可实施、可评价、可迭代的课程改革路径。

Abstract: To respond to the growing demand for mathematical foundations, algorithmic interpretation, and engineering transfer in artificial-intelligence applications, this study addresses four persistent problems in Mathematical Analysis: the separation of theory from algorithmic contexts, limited representations of abstract concepts, fragmented practical tasks, and the absence of explicit rules for the use of generative artificial intelligence. This study employs a combination of methods including literature review, design-based research, action research, and expert review. The resulting “one-map, three-level, five-cycle” model consists of a bidirectional knowledge map linking Mathematical Analysis to artificial-intelligence techniques; three progressive levels—concept anchoring, algorithm interpretation, and engineering transfer; and five learning cycles—problem context, mathematical derivation, code implementation, visual diagnosis, and transfer reflection. Representative cases are developed around gradient descent, the chain rule and backpropagation, and parameter tuning for a pipeline-network simulation model. A multidimensional assessment rubric and a four-part governance rule—permitted use, required verification, mandatory disclosure, and traceability—are also proposed. This model transforms artificial intelligence from an external teaching tool into an application context for testing mathematical concepts, theorems, and knowledge transfer, forming an implementable, evaluable, and iterative curriculum reform path without weakening proof training.