First, turn to thinking,
The so-called "transformation" refers to a way of thinking that through transformation, the problems to be solved or unsolved are reduced to problems that have been solved or are relatively easy to solve, and finally the problems are solved. We often call it "changing ideas". For example, solving fractional equations is transformed into solving integral equations, solving binary equations is transformed into solving univariate equations, and solving polygonal problems is transformed into solving triangular problems.
Second, the thinking method of combining numbers and shapes
The idea of combining numbers and shapes refers to a way of thinking that combines numbers and shapes to solve problems. Hua, a famous mathematician, once said: "The number of missing shapes is less intuitive, and the missing shapes are difficult to be nuanced." This is to emphasize the importance of the combination of numbers and shapes. Using points on the number axis to represent rational numbers in the textbook Rational Numbers is the simplest embodiment of the idea of combining numbers with shapes.
Third, the thinking method of classified discussion
In the process of infiltrating the idea of classification discussion, I think the first thing is classification. For example, in rational numbers, the research on the sign law of the multiplication of opposites and absolute values with rational numbers can be divided into three categories according to rational numbers: positive number, negative number and zero. In the chapter of understanding plane graphics, the classification of angles, the classification of the positional relationship between points and straight lines, and the classification of the positional relationship between two straight lines are all carried out with the idea of classification discussion. This way of thinking can mainly avoid missing and wrong solutions.
Fourth, the idea of equation.
The idea of equation refers to a problem-solving strategy to solve unknown quantities by solving equations. As we know, equation is an effective mathematical model to describe the real world. Therefore, the idea of equation is actually a mathematical modeling idea from the abstraction of practical problems to the process of equation. For example, many practical problems can be solved by using linear equation and quadratic equation.
Five, from the special to the general way of thinking
The mathematical thinking method from special to general is to observe some special situations first, then analyze their common characteristics and draw general conclusions.
If letters are used to represent numbers, students always think that "-A is negative" and "the sum of two numbers is greater than any addend" and so on, and can give different values, thus finding that these conclusions are incorrect. This is the mathematical thinking method from special to general infiltration.