The commonly used mathematical thinking methods in mathematics mainly include: the idea of using letters to represent numbers, the idea of combining numbers with shapes, the idea of transformation (reduction), the idea of classification, the idea of analogy, the idea of functions, the idea of equations, the idea of no approximation, and so on.

1. The idea of using letters to represent numbers: this is one of the basic mathematical ideas. This idea is mainly reflected in the second chapter of the first volume of algebra, "Elementary knowledge of algebra".

2. The combination of numbers and shapes is one of the most important and basic thinking methods in mathematics, and it is an effective idea to solve many mathematical problems. It is a famous saying of Professor Hua Luogeng, a famous mathematician in China, that "the number is less intuitive when it is missing, and it is difficult to be nuanced when it is countless", which highly summarizes the function of the combination of number and shape.

3. Transformation thought: In the whole junior middle school mathematics, transformation (conversion) thought has been running through it. Transforming thought is to solve an unknown (to be solved) problem into a solved or easy-to-solve problem, such as simplifying the complex, making the difficult easy, turning the unknown into the known, turning the high order into the low order, etc. It is one of the most basic ideas for solving problems, and it is one of the basic thinking methods of mathematics.

4. classification ideas: rational number classification, algebraic expression classification, real number classification, angle classification, triangle classification, quadrilateral classification, the position relationship between points and circles, straight lines and circles, etc. are all discussed through classification.

5. Analogy: Analogy reasoning is of great significance in people's activities to understand and transform the objective world. It can draw lessons from analogy and inspire thinking, which is not only the basis for solving a large number of problems in daily life, but also a powerful tool for scientific research and invention.

6. Thought of function: Dialectical materialism holds that everything in the world is in the process of movement, change and development, which requires us to attach importance to the thinking method of function in teaching.

7. Equation: it is the main content of algebra in junior high school. In junior high school, I mainly studied the solutions of several kinds of equations and equations, and in junior high school, I will form the idea of equations. The so-called idea of equations is to highlight the equal relationship between known quantities and unknown quantities, and to achieve the problem-solving ideas and strategies for the purpose of evaluation by setting unknowns, listing equations or equations, solving equations or equations, and so on.

Expand the data. The idea of equation is to start with the quantitative relationship of the problem, use mathematical language to transform the conditions in the problem into mathematical models (equations, inequalities, or mixed groups of equations and inequalities), and then solve the problem by solving equations (groups) or inequalities (groups).

Starting from the overall nature of the problem, we highlight the analysis and transformation of the overall structure of the problem, find the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with an "integrated" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment. The holistic thinking method is widely used in the simplification and evaluation of algebraic expressions, solving equations (groups), geometric solutions and so on.

Reference: Baidu Encyclopedia-Mathematical Thought