Progressive change refers to the change of topics from special to general, while the basic knowledge needed for solving problems remains unchanged. First, the topic conditions from special to general, from simple to complex, can form a progressive variant problem group. Progressive variant problem set refers to a set of mathematical problems designed reasonably and effectively according to students' cognitive rules in order to achieve a certain teaching purpose in classroom teaching, and this set of mathematical problems has certain internal logical connection, that is, the former problem is a special case of the latter problem, and the latter problem is a general case of the former problem, so the combination of special to general problems is called progressive variant problem set. This kind of progressive variant problem set is progressive step by step, from shallow to deep, from simple to complex, step by step, spiraling up, which is conducive to students' in-depth understanding of the essence of the problem, and then master the law of solving problems and break through the teaching difficulties. Second, under the condition that the general law of solving problems remains unchanged, it is helpful for students to separate the general law from it by changing the non-essential attributes. Third, it is beneficial to students at different levels. Because the problem is from simple to complex, students of different levels can climb the stairs step by step and master the general rules. For example, in the teaching of "fractions", the following assignments are designed.

Case 1:

In the above three questions, the numerator of the fraction starts from x-3, and the variant is

The condition that the molecule is 0 is increasing. The denominator is changed from 2x- 1 to x-3, so when the numerator is 0, the denominator is also 0. The three questions are different and different, but the essence of solving the problem is the condition that the fractional value is 0, the numerator is 0 and the denominator is not 0. Through these three levels of questions, students can not only find the essence of solving problems, but also find their own breakthrough points, thus helping students at different levels to sum up the laws of solving problems and form a complete mathematical cognitive structure for such problems.

Second, discuss variation.

Discussion variation refers to the variation of topics in the direction of classified discussion. First, mathematical concepts are the cells and basic units of thinking. Mathematics concept is the core of mathematics teaching, and it is the element of judgment and reasoning. Clear concept is the basic requirement of logical thinking. Therefore, the understanding of concepts directly affects students' mathematical thinking ability. Second, the mathematical concept itself is declarative knowledge, but if the concept is used to solve problems, it belongs to procedural knowledge. Only in problem-solving teaching can students understand the essence of concepts. Third, through classified discussion, let students understand the essence of the concept. For example, understand the concept of linear equation of one variable and design the following assignments.

Case 2:

Third, the background changes.

Background change refers to the background change of the problem, while the method of solving the problem remains unchanged. For similar problems, the background has changed, and the solutions have not changed. First, it helps students to find the general law of solving problems. Second, it helps students to improve their generalization ability. Students need certain thinking operation, and sum up general rules from different background topics. Third, it is helpful for students to expand the scope of analogy transfer. For example, in the problem of finding rules, the following homework is designed by using background changes.

Case 3:

(1) As shown in figure 1, observe the graph and fill in the form 1.

Problem variation means that the problems are different, but the mathematical methods used to solve them are the same. Mathematics comes from life, and many problems in life can be transformed into equation problems. The famous mathematician Descartes once conceived a general method to solve all problems. First turn any problem into a mathematical problem, then turn the mathematical problem into an algebraic problem, and finally reduce any algebraic problem to an equation problem. Make students master the solution of a kind of problem, and generalize the basic mathematical methods to solve the problem through the differences of the specific problems to be solved. This can not only improve students' generalization ability, but also enable students to form a problem domain to solve a class of problems. At the same time, due to the constant changes of problems, students can clearly understand the relationship with other types of problems, which will make the mathematical cognitive structure formed by students scientific and reasonable, that is, form the CPFS structure. China scholars have proved that the CPFS structure is a. For example, in the teaching of the application of binary linear equations, the following mutation homework is designed.

Verb (abbreviation of verb) graphic change

Graphic variation means that the graphics are different, but the basic graphics separated from the graphics are the same. First, mathematics is a science that studies the relationship between spatial form and quantity in the real world. Due to the complexity of spatial form, it is impossible for human beings to clearly understand the essence of all graphics. However, it is found that many complex graphics are composed of basic graphics, and it is the crystallization of human wisdom to solve the problem of complex graphics into basic graphics. Therefore, the nature of basic graphics and how to separate basic graphics from complex graphics need students to focus on characterization. There are many basic figures in junior high school geometry knowledge, such as A-shape, 8-shape, first-line third triangle, mother-daughter triangle and so on. Understanding and flexible application of these basic graphics is the thinking carrier for students to solve complex problems. Through the variation of graphics, but the basic graphics that make up graphics remain unchanged, it is beneficial to improve students' ability to separate basic graphics from complex graphics. It is beneficial to improve students' ability to solve problems by using basic graphics. For example, in the teaching of similar triangles's Judgment, there is a special problem with one line and three angles, and the following variational homework is designed.

Sixth, some ideas.

1. Variational teaching is based on the variational theory, and the variation of the topic should be carried out around the unchangeable essence. The purpose of variation is to let students discover and summarize the general principles (laws) of solving problems through several examples. Therefore, when doing variation, we should first clarify the nature of the problem, and then change the non-essential attributes around the nature of the problem to highlight the essential attributes of the problem and highlight the general principles of this kind of problem.

Second, repetition helps to improve the memory strength of students' mathematical knowledge. Variations are carried out under the condition that the essence is unchanged, that is to say, students use the same thinking method to solve such problems. Therefore, it is a repetitive thinking activity for students to use the same principle to solve problems repeatedly. The research of cognitive psychology shows that repetition can enhance students' memory of knowledge and increase the memory intensity in long-term memory, that is, the traces of memory are large, which makes it easy for students to extract the information that needs to be transferred from long-term memory when answering other questions, thus improving their ability to analyze and solve problems.

Thirdly, variation helps students at different levels to discover and summarize the general laws of mastering problems. Differences among students exist objectively, and different students have different abilities to solve problems and summarize. Therefore, when changing the topic, we should make the topic have a certain gradient, that is, gradually change from simple to complex, so that students at different levels can analyze and discover the general laws.