The application of construction method in solving senior high school mathematics problems
构造法在高中数学解题中的应用

摘要

构造法是数学化归与转化的核心思维工具，但在高中解题教学中，学生常停留在机械模仿，缺乏对构造策略的元认知反思。本文基于“问题结构识别—构造路径选择—等价性验证”的三维框架，系统梳理函数构造、方程构造、向量构造和图形构造四类典型路径，并结合近年高考及模拟题，逐类剖析各类方法的触发条件、操作逻辑与适用边界。研究发现，构造法的有效运用更依赖于学生对问题深层结构的敏感度，而非知识量的积累；四类构造方法并非孤立，在综合问题中可形成策略联动。本文旨在为高中数学教学提供可操作的方法论，促进学生解题思维从经验型转向结构型。

Abstract

The constructive method is a core thinking tool for mathematical reduction and transformation. However, in senior high school problem-solving instruction, students often apply it through mechanical imitation, lacking metacognitive awareness of construction strategies. Based on a three dimensional analytical framework—“problem structure identification, construction path selection, and equivalence verification”—this paper systematically examines four typical construction pathways: function construction, equation construction, vector construction, and graphical construction. Using recent Gaokao and mock examination questions, we analyze the triggering conditions, operational logic, and applicable boundaries of each approach. The findings indicate that effective use of the constructive method depends more on students’ sensitivity to the deep structure of problems than on the sheer volume of knowledge acquired. Moreover, the four construction approaches are not isolated; they can form strategic synergies in complex composite problems. This study aims to provide actionable methodological guidance for senior high school mathematics teaching, facilitating the transformation of students’ problem solving thinking from experience based to structure oriented reasoning.