The so-called mathematical variant training means that in the process of mathematics teaching, concepts, properties, theorems, formulas and problems are effectively transformed from different angles, levels, situations and backgrounds, so that their conditions or forms change, while their essential characteristics remain unchanged. In mathematics teaching, it is only one aspect for students to understand knowledge. More importantly, it is necessary to cultivate students' thinking ability and master mathematical thinking methods.

Variance is actually innovation. Of course, variation is not blind. We should grasp the essential characteristics of the problem, follow the development of students' cognitive psychology and make variation according to actual needs. The implementation of variant training should grasp the main line of thinking training, appropriately change the problem situation or change the thinking angle, cultivate students' adaptability and guide students to seek solutions to problems from different ways. Stimulate the enthusiasm and profundity of students' thinking by asking more questions, thinking more and using more. Based on theoretical research and the practice of mathematics classroom teaching, I will talk about how to carry out variant training in mathematics teaching to cultivate students' thinking ability.

First, in the process of forming mathematical concepts, use variants to inspire students to actively participate in observation, analysis and induction, and cultivate students' thinking ability of correct generalization.

From the requirement of cultivating students' thinking ability, it is more important to form mathematical concepts and prompt their connotation and extension than the definition of mathematical concepts itself. In the process of concept formation, variants can be used to guide students to actively participate in the whole process of concept formation, so that students can "discover" and "create" themselves, improve students' learning enthusiasm through diversified variants, and cultivate students' ability of observation, analysis and generalization.

Through the deformation of the formula, we can gradually deepen our understanding of the concept and have a very clear understanding of the essential things in the concept. Therefore, in future practice, teachers should also make clear the examination direction of similar knowledge points to prevent teachers from blindly giving questions and students from blindly practicing, so as to maximize the benefits in a limited time.

Second, in the process of understanding theorems and formulas, students can use variants to deeply understand various relationships between concepts in theorems and formulas, thus cultivating students' multi-directional flexible thinking ability.

The development of mathematical thinking depends on mastering and applying theorems and formulas for reasoning, argumentation and calculus. Because the essence of theorems and formulas is also a generalization of the essential relationship between concepts, the key to mastering theorems and formulas lies in clearly understanding the relationship between concepts in theorems and formulas. Any form of mechanical understanding of this relationship is the root of the inability to skillfully apply theorems and formulas and the result of the lack of multi-directional flexible thinking ability. Therefore, in the teaching of theorems and formulas, variants can also be used to express the relationship between related theorems and formulas, as well as the conditions for the establishment and addition of theorems and formulas, so as to train students to distinguish and use the judgments related to theorems and formulas.

Through variant training, we can prevent the formalization and mechanical recitation and application of formulas and theorems, and improve students' ability to think flexibly and use concepts, formulas and theorems flexibly.

Thirdly, in problem-solving teaching, we can change the conditions or conclusions of the topic by using variants, reveal the relationship between conditions and goals, and the relationship and laws between methods in problem-solving thinking, and cultivate students' thinking ability of association, transformation, reasoning, induction and exploration.

(A) multiple solutions to a problem, appropriate variants, to cultivate students' thinking ability of seeking common ground while reserving differences.

Many mathematical exercises seem to be different, but their internal essence (or ideas and methods of solving problems) is the same, which requires teachers to pay attention to the collection and comparison of such topics in teaching, guide students to seek common solutions, let students feel the internal relationship between them and form mathematical thinking methods.

(B) multiple solutions to a problem, analogy, cultivate students' divergent thinking ability, cultivate students' thinking flexibility.

The essence of multiple solutions to a problem is to reflect the inevitable essential relationship between conditions and conclusions in different ways of argumentation. In teaching, teachers should actively guide students to think in various ways and methods. In this way, it can not only expose students' thinking process of solving problems, increase the transparency of teaching, but also enable students to broaden their thinking and master the internal relations of knowledge skillfully. There are many examples in this respect, especially geometric proof questions. By solving a problem with multiple solutions, students can think and solve problems from different angles, which can stimulate students' strong desire to seek differences and cultivate their thinking flexibility.

(c) Changing topics, summarizing laws and cultivating students' exploratory and profound thinking.

Through variant teaching, we can not solve a problem, but solve a class of problems, curb "sea tactics", develop students' problem-solving ideas, cultivate students' inquiry consciousness, and achieve "get twice the result with half the effort"

Galileo once said that "science is advancing in the exploration of constantly changing the angle of thinking". Therefore, classroom teaching should always be innovative and fickle, and more new questions with relevance, similarity and opposition can be derived through the original questions, so as to dig deep into the educational function of example exercises.

For example, there is a question in the book that proves that the quadrilateral obtained by connecting the midpoints of each side of the quadrilateral in turn is a parallelogram. Teachers can lose no time to make variants to stimulate students' interest in thinking. Variant (1) connects the midpoints of each side of the rectangle in turn, which is the quadrilateral? Variant (2) What figure is the quadrilateral obtained by connecting the midpoints of the sides of the diamond in turn? Variant (3) What is the quadrilateral obtained by connecting the midpoints of the sides of a square in turn? After these four exercises, the teacher can further guide the students to conclude that it is the diagonal characteristics of the original quadrangle that affect the shape essence of the figure.

Another example is the application problem teaching, which is the difficulty of junior high school teaching. In teaching, the same type of questions can be presented to students in a variant way, so that students' thinking can be deepened gradually.

For example, when explaining the practice and exploration of the one-dimensional linear equation, the teacher made up an application question about catching up with the Olympic champion Meng in training. A speedboat was at the same starting point as Meng's kayak, and the speedboat advanced 20 meters at a speed of 5 meters per second. In order to catch the speedboat, Meng had to paddle hard. Students, please think about how many seconds it will take him to catch up with the speedboat if he rows at a speed of 6 meters per second. Then the teacher can make the following changes to this example.

Variant 1: A speedboat and Meng's kayak are at the same starting point. The speedboat is 20 seconds ahead at a speed of 5 meters per second. In order to catch the speedboat, Meng had to paddle hard. Students, please think about it. If he paddles at a speed of 6 meters per second, how many seconds can he catch up with the speedboat? (From 20 meters ahead to 20 seconds ahead)

Variant 2: Our school has a 300-meter track, which often involves the problems of meeting and catching up when running.

At present, A and B are racing. The speed of A is10m/s, and the speed of B is 8m/s, both of which start from the same place.

(1) Two people walk towards each other at the same time. A few seconds later, they met.

(2) Two people walk in the same direction at the same time and meet for the first time after a few seconds.

(3) B starts for 5 seconds, then A starts and asks A how many seconds it takes them to meet for the first time.

This problem should be the circular track on the playground that students are familiar with at ordinary times. Here, the three questions are also a set of variant questions. (1) and (2) are problems of meeting and catching up at the same time, and (3) are problems of meeting and catching up at different times. This problem also contains the idea of classified discussion.

Variant 3: A speedboat and Meng's kayak are at the same starting point. The speedboat advanced at a speed of 5 meters per second 10 seconds. The coach asked him to catch up with the speedboat in 45 seconds. In order to catch the speedboat, Meng had to paddle hard. He rowed at a speed of 6 meters per second. After rowing for 5 seconds, he found that he could not catch up with the speedboat within the specified time. 45 seconds. Is he right? If he wants to catch up, please calculate how long it will take Meng to catch up with the speedboat within the specified time.

This variant covers the basic types of travel problems, such as simultaneous satisfaction, different time satisfaction, simultaneous catch-up and different time catch-up. In this way, through the practice of a problem, we not only solved a class of problems, but also summed up the most essential thing between quantity and quantity. In the future, when students encounter similar problems, their thinking direction will be accurate, which will cultivate the profundity of their thinking. Students don't have to be trapped in the ocean of problems.

(3) Ask more questions, expand and develop the original function through variant extension, and cultivate students' innovative consciousness and ability to explore and summarize.

Newton said, "Without bold guesses, great discoveries are impossible." Middle school students are rich in imagination. They can encourage and guide students to guess boldly through the structural characteristics provided by examples, thus cultivating students' creative thinking and divergent thinking.

Special attention should be paid to the "modification" or extension of textbook examples and exercises in teaching. Mathematical thinking methods are hidden in textbook examples or exercises. In teaching, we should be good at digging this kind of exercises, that is, covering knowledge points as much as possible through a typical example and stringing scattered knowledge points into a line, which often has unexpected effects and is conducive to the construction of knowledge.

In a word, in mathematics classroom teaching, following the law of students' cognitive development and strengthening variant training according to the teaching content and objectives play an important role in laying a solid foundation, cultivating thinking and improving ability. In particular, variant training can cultivate students' qualities of dare to think, dare to associate and dare to doubt, and cultivate their independent inquiry ability and innovative spirit. Of course, the variant questions in classroom teaching should be textbook-based, student-oriented, reflect "from textbooks, higher than textbooks" and penetrate into students' learning in daily teaching. Let students learn to "change questions" and explore, analyze and synthesize by themselves, so as to improve students' mathematical quality.