基于差异分析的方程构建策略研究

Research on Equation Construction Strategies Based on Differential Analysis

摘要: 高中数学有关解三角形、立体几何及圆锥曲线的问题，通常具有条件隐含且逻辑路径非显性的特征，解答者面临如何从条件中抽离出代数关系，进而构建有效目标方程的逻辑困境。如果方程构建策略不科学，解答过程便会陷入低效的经验主义试错，缺乏逻辑导向的思维困境。针对这一困境，文章提出了一种“基于差异、建立等式”的方程构建策略。该策略突破了依赖大量重复训练以形成“解题直觉”的传统，致力于建立一套具有可操作性与可迁移性的认知模型。核心逻辑在于确定“桥梁”对象(即长度、角度、面积、体积)，用不同路径求解同一个“桥梁”对象，最终基于同一“桥梁”对象的不变性建立等量关系。文章分别从解三角形、立体几何、圆锥曲线各选取一个典型例题，对所提出的基于差异分析的方程构建策略进行阐释。基于差异分析的方程构建策略可高效构建目标方程，规避无效试探过程，同时能引导学习者快速挖掘问题的多种求解路径，并实现最优解法的筛选。基于差异分析的方程构建策略有效突破了传统解题中的思维局限，将零散的解题经验提炼、升华为结构化的理论方法。基于差异分析的方程构建策略不仅对上述板块的目标方程构建具有重要价值，其思想还可迁移至函数、数列等其他知识板块，对培养学习者的数学抽象、逻辑推理、数学建模等核心素养具有重要且深远的意义。

Abstract: Problems related to triangle solving, solid geometry, and conic sections in senior high school mathematics are usually characterized by implicit conditions and non-explicit logical paths. Solvers are confronted with the logical dilemma of how to extract algebraic relations from the conditions and then construct effective target equations. If the strategy of equation construction is unscientific, the solving process will fall into the inefficient empiricism of trial and error, lacking a logic-oriented thinking framework. To address this dilemma, this paper proposes an equation construction strategy of “identifying differences and establishing equalities”. Breaking away from the traditional approach that relies on extensive repetitive training to form “problem-solving intuition”, this strategy aims to establish an operational and transferable cognitive model. The core logic lies in identifying the “bridge” objects (i.e., length, angle, area, volume), calculating the same “bridge” object through different paths, and finally establishing an equivalence relation based on the invariance of this “bridge” object. In this paper, one typical example is selected respectively from triangle solving, solid geometry and conic sections to illustrate the proposed equation construction strategy based on difference analysis. This strategy can efficiently construct target equations, avoid the process of invalid trial and error, and at the same time guide learners to quickly explore multiple solution paths for a problem and select the optimal solution. The equation construction strategy based on difference analysis effectively breaks through the thinking limitations in traditional problem-solving, and refines and elevates scattered problem-solving experiences into a structured theoretical method. This strategy is not only of great value for constructing target equations in the above-mentioned modules, but also its ideas can be transferred to other knowledge modules such as functions and sequences. It has important and far-reaching significance for cultivating learners’ core mathematical competencies, including mathematical abstraction, logical reasoning, and mathematical modeling.