生成式人工智能融入工科数学课堂的效果与边界——基于《数值计算方法》课程两轮准实验的实证分析

Effects and Limits of Integrating Generative AI into Engineering Mathematics Classrooms—An Empirical Analysis Based on Two Rounds of Quasi-Experiments in a Numerical Methods Course

摘要: 《数值计算方法》的教与学长期卡在一个地方：学生在公式层面“看得懂”，却说不清公式的适用条件，写不出对应的程序，也解释不了误差从何而来。生成式人工智能进入课堂以后，教师可以在讲解现场生成代码、绘制图形、构造反例，学生也能围绕同一算法反复追问。但它究竟只是提高了课堂的新鲜感，还是真的帮助学生跨过了“会套公式”到“能解释算法过程”的那道坎，需要放在一门真实课程、两届自然班和可量化的学习结果里检验。本文以《数值计算方法》连续两年的教学改革为现场，比较两种AI课堂组织方式与传统教学的成绩差异。2024~2025学年，23级一个自然班(n = 24)采用教师AI演示，同年级三个平行班为对照(n = 97)；2025~2026学年，24级一个自然班(n = 21)改为学生AI对抗式学习，同年级三个平行班为对照(n = 84)。两年均采用覆盖10个知识点的期末分项考核。结果显示，两轮实验班总分均高于对照班：23级高14.55分(p < 0.001, Hedges g = 0.78)，24级高12.00分(p = 0.009, Hedges g = 0.62)，及格率与优秀率同步提升。差异中的差异分析表明，从教师演示转向学生对抗式学习并未带来显著的总分额外增益(−2.55分，p = 0.645)。逐知识点分析揭示明显异质性：曲线拟合、代数精度、数值积分、数值微分受益稳定，Newton法和部分复杂迭代内容提升有限。研究说明，生成式AI能够改善工科数学课程的学习表现，但其效果取决于知识点性质、教师支架与课堂任务结构，不能简单理解为学生使用AI越多越好。

Abstract: Students in Numerical Methods courses routinely reproduce formulas on paper without understanding when those formulas apply, how to implement them in code, or where errors originate. With generative AI, instructors can now generate code, plot functions, and construct counterexamples in real time, while students can probe the same algorithm from multiple angles. Whether these tools actually deepen understanding, rather than adding surface novelty, can only be judged through controlled trials with intact cohorts and measurable outcomes. This study reports two years of such trials in a Numerical Methods course. In 2024~2025, one intact class from the 2023 cohort (n = 24) received instructor-led AI demonstrations, with three parallel classes serving as controls (n = 97). In 2025~2026, one intact class from the 2024 cohort (n = 21) shifted to student-driven adversarial AI learning, again with three parallel controls (n = 84). Both years concluded with a final itemized assessment covering 10 knowledge points. The experimental classes outperformed their controls in both rounds, by 14.55 points for the 2023 cohort (p < 0.001, Hedges’ g = 0.78) and by 12.00 points for the 2024 cohort (p = 0.009, Hedges’ g = 0.62), and both pass rates and excellence rates improved. A difference-in-differences analysis showed that shifting from instructor demonstration to student adversarial learning yielded no significant additional gain in total scores (−2.55 points, p = 0.645). Item-level analysis also showed uneven gains across topics: curve fitting, algebraic precision, numerical integration, and numerical differentiation improved consistently, whereas Newton’s method and portions of the complex iterative content showed only limited benefits. Generative AI can improve learning outcomes in engineering mathematics, but its value depends on what is being taught, how instructors scaffold the work, and how classroom tasks are designed. The shift from instructor-led demonstration to student-driven adversarial learning brought no extra benefit, and the gains varied sharply by topic. These patterns suggest that effectiveness is not a simple function of how much students interact with AI.