Knowledge and thinking are interrelated. Before teaching some thinking activities, we should first consider the existing knowledge structure of students.

What is knowledge structure? It is generally believed that mathematics includes definitions, axioms, theorems, formulas and methods. There is a relationship between them. People describe this relationship and function from a certain angle, sum up the laws and summarize them into a system, which is the knowledge structure. Only by understanding the students' knowledge structure in teaching can we further understand the level of thinking, consider whether it is enough to teach new knowledge and what kind of teaching methods to complete the teaching of mathematics activities.

For example, when explaining the quadratic equation of one variable and discussing its solution, it is necessary to use collocation method or factorization method, so before class, the teacher should know whether the students have mastered these methods and to what extent, so that the activity teaching can proceed smoothly.

Second, consider the thinking structure of students.

Mathematics teaching is the teaching of mathematical thinking activities, and it is natural for mathematics teaching to consider students' existing level of thinking activities.

Psychology has long proved that the thinking ability and intellectual quality develop with the growth of teenagers' age, and the thinking level of students at different ages is different. Stolyar introduced five different levels of children's learning geometry and algebra in mathematical pedagogy. In these five stages, there are obvious differences in students' mastery of knowledge, thinking methods and thinking level. Therefore, in order to make mathematics teaching become the teaching of mathematics activities, we must understand the students' thinking level. Here are two questions related to students' thinking level.

1. Characteristics of middle school students' thinking ability

We know that middle school students' operational thinking ability is in the stage of logical abstract thinking. Although several aspects of thinking ability have developed successively, the general trend is the same. The computing ability of the first-grade students is similar to that of the fourth and fifth grades of primary schools, and they are at the level of abstract thinking in images; The calculation ability of senior two and senior three students belongs to the abstract logical thinking of experience. Judging from the four indicators of generalization ability, spatial imagination ability, proposition ability and reasoning ability, the second grade is a new starting point of logical abstract thinking, a qualitative change period of operational thinking in middle school, and a critical period at this stage.

2. Several thinking forms of learning mathematics

(1) reverse thinking. Contrary to the thinking process of inferring conclusions from conditions, giving a conclusion or answer first requires various conditions to make it hold. For example, given a concentration problem, we list an equation; Conversely, given an equation, a concentration problem can be worked out. The latter belongs to reverse thinking.

(2) Case-making thinking. Some conditions or conclusions are often illustrated by examples, and their irrationality is often proved by counterexamples. Constructing examples according to needs is often a thinking process of returning from abstraction to concreteness and comprehensively applying all kinds of knowledge. For example, try to find a function whose inverse function is equal to itself.

(3) inductive thinking. Through observation and experiment, the general law is put forward in several examples.

(4) Open mind. That is, only the object or certain conditions of the research question are given, and the problems or conclusions that can be inferred from it are explored by the students themselves. For example, let students observe the image of y = sinx, tell its main properties and explain them one by one.

Understand the characteristics of students' thinking and several main forms of mathematical thinking. In teaching, combining with the characteristics of teaching materials and adopting effective teaching methods, the teaching of thinking activities will certainly receive good results.

Third, consider the logical structure of teaching materials.

Some of our existing middle school mathematics textbooks are arranged in a straight line or spiral line.

If the teaching of mathematics activities is carried out, the logical structure of the teaching materials will also change accordingly. For example, exponents, logarithms and roots can all be expressed as follows: the relationship between A, B and N is that the power of A equals N, can they be arranged to study together? For another example, there are problems of concentration, travel, engineering and equal product in middle school textbooks. When explaining, you can use an equation to express different problems, which is unified, but the forms of the problems are different. There is no essential difference in the form of the equation, so we can complete several problems at once.

In mathematics activity teaching, we should not only consider the characteristics of elementary mathematics and the logical structure of teaching materials, but also seriously study a specific knowledge block, and different contents should be handled in different ways. This is the problem of active teaching method to be discussed below.

Fourth, think about positive teaching methods.

Adopting active teaching methods varies from class to class, from person to person and from time to place. For example, for mathematical definitions and axioms whose contents are mostly logically dispersed, it is more appropriate to adopt the method of self-study counseling; The general formulas and theorems in teaching materials are best explored by questions; For theoretical difficulties in textbooks, it is generally better to explain them. Teachers should be flexible.

The teaching of mathematical activities is essentially the teaching of positive thinking activities, so it is extremely important to arouse students' enthusiasm in teaching. Generally speaking, the vividness of teaching content, the intuitiveness and interest of methods, the good evaluation of teachers and parents, and the quality of academic performance can all promote students' learning and improve their enthusiasm. In addition, extracurricular activities, such as visiting factories and computer rooms, introducing the application of mathematics in all walks of life, especially when the application of mathematics has made great achievements in various fields, can promote teenagers to broaden their horizons, enrich their knowledge and enhance their skills, thus developing their thinking ability and improving their initiative in learning. You can also talk a little about the history of mathematics, such as the great contribution of ancient scientists in China and their influence in the world, which can also stimulate students' enthusiasm.