基于数学建模理论的导数概念教学研究——以“平均变化率到瞬时变化率”课例为例

A Study on the Teaching of Derivative Concept Based on Mathematical Modeling Theory—Teaching Research on Derivative Concept Based on Mathematical Modeling: A Case Study of Average and Instantaneous Rate of Change

摘要: 导数作为衔接初、高等数学的关键概念，是高中微积分学习的核心难点。文章以数学建模教学理论为指导，将问题分析–模型假设–模型构建–模型求解–模型验证–模型应用流程嵌入导数概念生成全程。以一辆正在刹车的汽车为真实情境，从平均速度入手，通过缩小时间间隔完成从平均变化率到瞬时变化率的认知跃迁。有利于提升了学生对导数本质的理解与运用能力，增强了建模素养、逻辑推理能力及学习动机，改善传统概念课“重形式、轻本质”的现状，可为建模视角下的抽象概念课教学提供参考的案例。

Abstract: As a pivotal concept bridging elementary and advanced mathematics, the derivative represents a core challenge in senior high school calculus instruction. Grounded in the theory of mathematical modeling pedagogy, this study integrates the six-stage modeling cycle—problem analysis, model assumption, model construction, model solution, model verification, and model application—into the entire process of derivative concept formation. Using a braking automobile as an authentic contextual scenario, the instructional design begins with average velocity and, by progressively narrowing the time interval, facilitates students’ cognitive transition from the average rate of change to the instantaneous rate of change. This approach is expected to deepen students’ conceptual understanding and procedural fluency regarding derivatives, foster their modeling competence and logical reasoning, and enhance learning motivation. It also addresses the persistent limitation of traditional concept lessons that prioritize formal procedures over conceptual essence, thereby offering a referential case for teaching abstract mathematical concepts through a modeling lens.