构建学习进阶 培育核心素养——以“函数的单调性与导数”为例

Constructing Learning Progressions and Cultivating Key Mathematical Competencies—Taking “Monotonicity of Functions and Derivatives” as an Example

摘要: 本研究旨在解决高中数学“函数的单调性与导数”教学中普遍存在的知识点割裂、学生认知跃迁困难等问题。本研究选取人教A版高中数学教材中的对应教学内容作为研究载体，结合学习进阶理论与SOLO分类理论，搭建起针对该内容的分层进阶教学框架，这一框架从学生的经验直观出发(水平一)，通过探究引导建立概念(水平二)，再通过辨析与基础应用建立知识联系(水平三)，然后迁移到含参函数的复杂情境中(水平四)，最终整合体系并应用于解决实际问题和不等式证明(水平五)。本研究明确划定了教学的起点与终点，同时设计了“关系构建”与“导数工具应用”两个维度的进阶路径，以此推动学生完成从静态比较到动态分析的认知转变，在分层推进教学的过程中，逐步落地数学抽象、逻辑推理等数学核心素养的培养目标，帮助学生搭建起完整、系统的知识网络。

Abstract: This study aims to address the prevalent issues in the teaching of “Monotonicity of Functions and Derivatives” in high school mathematics, such as the fragmentation of knowledge points and students’ difficulties in cognitive leap. Taking the corresponding teaching content from the PEP (People’s Education Press) Version A high school mathematics textbook as the research carrier, this study combines the Learning Progression Theory and the SOLO (Structure of Observed Learning Outcomes) Taxonomy to construct a hierarchical progressive teaching framework tailored for this content. This framework starts from students’ empirical intuition (Level 1), guides concept construction through exploratory guidance (Level 2), then establishes knowledge connections through concept discrimination and basic application (Level 3), further transfers learning to the complex context of functions with parameters (Level 4), and finally integrates the knowledge system to solve practical problems and conduct inequality proofs (Level 5). This study clearly defines the starting point and end point of the teaching, and designs a two-dimensional progressive path consisting of “relationship construction” and “application of derivatives as a tool”. In this way, it promotes students’ cognitive transformation from static comparison to dynamic analysis. In the process of hierarchically advancing teaching, it gradually implements the cultivation goals of key mathematical competencies such as mathematical abstraction and logical reasoning, helping students build a complete and systematic knowledge network.