常微分方程问题导向教学设计探索与实践

Exploration and Practice of Problem-Oriented Instructional Design for Ordinary Differential Equations

摘要: 针对常微分方程教学中普遍存在的概念抽象、学生理解困难、学习动力不足等现实问题，本研究设计并实施了一套基于问题导向理念的教学模式。该模式以“一阶微分方程”核心教学内容为载体，通过精心设计与生活实际、工程背景及科学前沿紧密联系的问题情境，引导学生经历“问题识别–模型建立–数学求解–结果解释–拓展迁移”的完整探究过程，并辅以小组协作、课堂辩论与多模态成果展示等教学组织形式。在一轮完整的教学实践后，综合运用定量测试、问卷调查、深度访谈与课堂观察等多种方法进行评估。结果表明：问题导向教学模式能显著提升学生的课堂参与度、概念理解深度以及数学建模与解决实际问题的综合能力；同时，该模式有助于改善学生的学习体验，增强其内在学习动机。本研究为常微分方程乃至其他理论数学课程的教学改革提供了经过实践检验的设计方案与实施路径，具有一定的借鉴与推广价值。

Abstract: In response to common challenges in teaching Ordinary Differential Equations (ODEs), such as the abstract nature of concepts, difficulties in student comprehension, and lack of learning motivation, this study designed and implemented a teaching model based on the problem-oriented learning philosophy. Taking the core content of “first-order differential equations” as the vehicle, this model guides students through a complete inquiry cycle of “problem identification, model establishment, mathematical solution, result interpretation, extension and transfer” by designing problem scenarios closely linked to real-life situations, engineering contexts, and scientific frontiers. This process is supported by teaching organizational forms such as group collaboration, classroom debate, and multimodal presentation of outcomes. After a complete round of teaching practice, a comprehensive evaluation was conducted using quantitative tests, questionnaires, in-depth interviews, and classroom observations. The results indicate that the problem-oriented teaching model can significantly enhance students’ classroom engagement, depth of conceptual understanding, and comprehensive abilities in mathematical modeling and solving practical problems. Simultaneously, this model helps improve students’ learning experience and strengthen their intrinsic motivation. This research provides a practically tested design scheme and implementation path for the teaching reform of ODEs and even other theoretical mathematics courses, offering certain reference and promote value.