The Application of the Function and Equation Thought in Middle School Mathematics Problem-Solving

Abstract

The idea of functions and equations is a core approach running through algebra, geometry, probability and other knowledge modules in middle school mathematics. Its value in cross-module problem-solving and connection with curriculum standards has become increasingly prominent. Taking sequences, trigonometric functions and solid geometry as research objects, this study systematically analyzes the problem-solving logic of the function and equation idea, so as to provide empirical references for front-line teaching to implement curriculum standards and improve students’ key mathematical competencies. This paper adopts literature review, case analysis and classification research. First, it defines the connotation and relationship between function thought and equation thought, clarifies their core feature of “interconversion”, and meets the curriculum standard requirement of “understanding the essence of mathematical thought”. Second, based on recent college entrance examination questions and typical examples, it analyzes the application paths of these thoughts by module. Finally, it summarizes their general problem-solving logic and puts forward teaching and problem-solving suggestions in line with curriculum standards. The research shows that the function and equation idea permeates middle school mathematics and is crucial for developing core competencies such as mathematical abstraction and logical reasoning. It is often tested with the combination of numbers and shapes, classified discussion in the college entrance examination, reflecting the orientation of "emphasizing literacy and application". It simplifies complex problems and improves students’ problem-solving efficiency and thinking quality.