First, cultivate students' abstract generalization ability in the process of teaching mathematical concepts.
Mathematical concept is a form of mathematical thinking, which reflects the quantitative relationship of real objects and the essential characteristics of spatial form. In mathematics, judgment and reasoning, as general forms of thinking, are expressed in the form of theorems, rules and formulas, and mathematical concepts are their basis. Correctly understanding and flexibly using mathematical concepts is the premise of mastering basic mathematical knowledge and operational skills, and developing logical argumentation and spatial imagination. Mathematical concepts are highly generalized, and the teaching of concepts plays a great role in cultivating students' abstract generalization ability. The teaching of mathematical concepts should be a process problem, not a simple conclusion problem. First of all, we have a perceptual understanding of the concept and a concrete image through examples and graphics. Then we observe these examples and figures, analyze and compare them, and abstract the essential attributes of concepts. For example, when introducing the concept of angle and distance formed by straight lines in different planes, we should first review the knowledge that the position relationship of intersecting straight lines in plane geometry is determined by the size of the angle and the distance is determined by the length of the common vertical line of parallel straight lines. Then, by rotating and translating two bamboo needles or rulers, students can intuitively form an intuitive image of the change of angle and distance, and then transform the measurement of spatial angle into the measurement of the same plane angle, which will help students master this concept and help the teaching of dihedral angle measurement later.
Second, cultivate students' abstract generalization ability in the application of mathematical formulas.
The application of formulas is the application of students' abstractness and concreteness to solving problems, and it is also very important to the generalization ability of formulas. In teaching, it is inevitable that students can't remember the formula or the formula can't be applied. Therefore, it can help students summarize some methods and steps of applying formula theorem, and make students use formula theorem and principle more skillfully and accurately. For example, when learning trigonometric functions, it is difficult for many students to remember inductive formulas. There is a popular saying in high school mathematics education: "Singular variables remain unchanged, symbols look at quadrants", which highly summarizes the inductive formula. When finding the period, maximum and monotone interval of trigonometric function, the method of transforming the same name and angle is often used. The various skills of transforming the same name and the same angle can be summarized into four small skills: higher power reduction, sum and difference of multiplication, multiplication and reanalysis of multiplication.
Thirdly, cultivate students' abstract generalization ability in analogy and association.
The completeness and rigor of mathematics make mathematical conclusions and methods relevant and similar. In classroom teaching, teachers should make full use of these correlations and similarities, adopt analogy and association methods, and let students explore and discover many new conclusions or methods by themselves. In teaching, teachers often ask students to compare and guess unknown formulas and properties according to existing formulas and properties. First analogy, then questions, and finally proof.