基于生成式AI的数值迭代法教学设计与案例研究——以《数值分析》课程为例

Teaching Design and Case Study of Numerical Iteration Methods Based on Generative AI—A Case of the Numerical Analysis Course

摘要: 随着人工智能技术，尤其是生成式大语言模型在教育领域的快速发展，高等理工科课程教学迎来了创新变革的机遇。针对《数值分析》课程中非线性方程迭代法教学中存在的理解难点与编程实践障碍，本文设计并实施了一套基于生成式AI辅助的嵌入式教学方案。该方案涵盖课前预习材料的AI自动生成与可视化动画制作，课堂中的动态演示与师生互动，以及编程实践环节中的多层次代码模板和AI调试支持，最后辅以基于AI的个性化作业布置与自动反馈体系。通过具体案例详述每个环节的设计与实施流程，保障教学内容的系统性与操作的可复制性。实证结果表明，AI技术有效提升了学生对迭代法数学原理的理解深度与编程实现能力，促进了师生互动和个性化学习支持，显著优化了教学效果与学习体验。本文为理工科数值计算课程在新兴AI环境下的教学创新提供了理论基础与实践范式，具有重要的推广价值和参考意义。

Abstract: With the rapid development of artificial intelligence technologies, especially generative large language models (LLMs), higher education in science and engineering is undergoing significant instructional transformation. Focusing on the challenges in teaching nonlinear equation iterative methods within the Numerical Analysis curriculum—such as difficulty in understanding abstract mathematical concepts and lack of programming practice—this study designs and implements an embedded AI-assisted instructional scheme. The scheme integrates AI-generated pre-class learning materials and visual animations, dynamic in-class demonstrations with interactive guidance, multi-tiered coding templates with AI-powered debugging support, and AI-based personalized homework assignments with automatic feedback mechanisms. Each component of the instructional design is described in detail to ensure systematic structure and replicability. Empirical results show that the use of AI significantly enhances students’ comprehension of numerical iteration theory, improves their programming proficiency, and fosters personalized learning and classroom interaction. This study offers a theoretical foundation and practical framework for teaching innovation in computational mathematics courses under the emerging AI-embedded educational paradigm, with strong implications for future adoption.