According to the spirit of the syllabus, the basic ideas of junior high school mathematics mainly refer to the basic methods such as transformation, classification, combination of numbers and shapes, and mainly refer to the undetermined coefficient method, elimination method, matching method, method of substitution method and image method. Because most mathematical methods are specifically expressed in textbooks, mathematical thinking is implicit in the knowledge system, which brings certain difficulties to strengthen mathematical thinking methods. To this end, let's talk about the expressions of transformation, classified discussion and combination of numbers and shapes in junior high school mathematics. Transformational thinking The so-called transformational thinking refers to the way of thinking that one research object is transformed into another under certain conditions. Transforming thinking is the core of mathematical thinking method. Other mathematical thinking methods are all means or strategies of transformation. The application of transformation thought in junior high school mathematics is reflected in the following three aspects: (l) Transforming new problems into previously learned problems, such as rational number subtraction into addition, division into multiplication, etc. (It helps to turn complex problems into simple ones. When new problems cannot or are difficult to be solved by existing methods, establish new research methods, such as introducing negative numbers and establishing number axes; Change the nature of the inverse operation to the nature of the equation, and so on. ? 2. The idea of classified discussion The so-called classified discussion refers to the way of thinking that complex objects are divided into different types according to the similarities and differences of their essential attributes, and the nature of various objects is studied, so as to understand the essence of the whole. In the discussion of classification, we should pay attention to the identity of standards, that is, the division will always be the same standard, which must be scientific and reasonable; Domains are mutually exclusive, that is, the divided categories should not contain each other, that is, the sum of all categories should be equal to the complete set of discussions; The gradual nature of the field, some problems can be classified into various categories. Guiding mathematics teaching with the idea of classified discussion is helpful for students to summarize their own mathematics knowledge, make it systematic and orderly, and gradually form a complete network of knowledge structure, which is helpful for students to explore problem-solving ideas strictly, clearly and reasonably and improve their mathematical thinking ability. In junior high school mathematics, the problems that need to be classified are mainly manifested in many ways: (The proof of some mathematical concepts and theorems includes many situations. Such problems need to be discussed in categories. Such as the classification of dihedral angles, quadrangles, angles, the theorem of circle angle, the theorem of circle power, the theorem of tangent angle, etc. , all involve the classification of I-inch theory (solving the problem that numbers without parameters or absolute value symbols are one-way, inequalities, discussing arithmetic roots, coefficients of quadratic terms in proportional and inverse proportional numbers, and the direction of opening L: of images). Because the positions of these parameters are different or the absolute value signs are removed, there are different results. This kind of problem needs to be discussed in categories. (3) There are some math problems. Although the conclusion is unique, the premise of this conclusion is different. This kind of problem should also be discussed in categories. (3) The idea of the combination of numbers and shapes refers to the combination of abstract mathematical language and intuitive graphics, so as to realize a way of thinking that transforms from abstract to concrete. Hua, a famous mathematician, said: numbers are not intuitive when they are missing shapes, and it is difficult to understand that some numbers are most relevant when they are missing shapes. With the help of the nature of graphics, many abstract concepts and complex relationships can be visualized, visualized and simplified, while some properties of graphics can be rigorous with the help of quantitative calculation and analysis. In junior high school, the combination of number and shape can be number axis, image of function, geometric figure and so on. They all have visual characteristics. The idea of combining numbers and shapes in junior high school mathematics is mainly manifested in the following two aspects; (l) Using shapes to help students understand mathematical concepts deeply. For example, teachers can use the corresponding relationship between points on the number axis and real numbers to clarify the concepts of opposition and absolute value and the method of comparing the sizes of two numbers; Using the properties of function images, we can discuss the roots of a cubic equation with one variable and a quadratic equation with seven degrees. (2) Use numbers to help students simplify problem-solving methods. Mathematical thinking methods, such as analogy, induction and association, also permeate junior high school mathematics. They permeate and promote each other and should be organically combined in mathematics teaching.