Thinking mode is a psychological phenomenon that people consider, analyze and solve problems according to fixed thinking mode and customary methods. The thinking mode has duality, which enables people to solve problems quickly by using the methods they have mastered under the same environment; When the situation changes, it will prevent people from adopting new methods, and negative thinking mode is the shackles that bind creative thinking. In teaching, teachers should give full play to their strengths, give full play to their positive role, avoid their weaknesses and strive to overcome their negative influence.

When people consider and study problems, they often like to analyze and think with fixed patterns and ideas, which is the so-called thinking mode in psychological education. This formula has its positive side in solving mathematical problems, that is, in general, students can use the knowledge and methods they have learned to accumulate experience and solve similar problems correctly and effectively; However, it can not be ignored that it also has a negative side, because the thinking mode is often accompanied by the rigidity and narrowness of thinking, which leads students to apply mechanically when solving problems, which is very unfavorable for cultivating students' creative thinking. In view of the duality of thinking mode, teachers should make full use of their strengths and avoid their weaknesses in teaching, not only give full play to their positive role, but also strive to overcome their negative effects and improve students' mathematical thinking ability.

First, Lenovo analogy, play an active role in thinking mode

People's learning process is essentially a process of establishing various thinking modes, and a large number of mathematical standard problems can be solved by thinking modes. Under normal circumstances, most students can quickly associate and use the knowledge and methods they have mastered when solving problems, and put some new problems that need to be solved into the category of old problems that have been solved, showing the positive role of thinking mode. Association is the spark of thinking and the bridge between the known and the unknown. Strengthening association analogy is conducive to promoting the positive transfer of thinking and improving the ability to solve mathematical problems.

3. Strengthen method guidance and broaden Lenovo channels. In mathematics teaching, it is far from enough to grasp only two foundations and observe and think. Some students memorize theorems, rules and formulas, but it is common to think out of line when solving problems. The main reason is that the channels of association are not smooth enough, so teachers must strengthen the guidance of methods and broaden the channels of thinking association. In the process of theorem proving and formula derivation, many important mathematical thinking methods have appeared. If teachers can pay attention to these mathematical thinking methods in teaching and guide students to use these methods to solve mathematical problems, they can greatly broaden students' association channels. In addition, teachers can organically combine the contents of textbooks and guide students to master some thinking strategies to solve mathematical problems, thus broadening students' association channels and improving their mathematical thinking ability.

Second, divergent thinking, to overcome the negative impact of thinking mode.

Thinking mode has both positive and negative aspects. Because of the fixed thinking, people's thinking always follows the inherent track, which limits the creativity. Especially in the process of forming thinking mode, it is often accompanied by rigidity and narrowness of thinking, which leads students to copy the existing problem-solving experience and follow a certain problem-solving mode, only paying attention to the same points and ignoring the different points, thus leading to problems or mistakes. The negative influence of their thinking mode is largely related to classroom teaching. Some teachers pay attention to habitual thinking and ignore the cultivation of heterosexual thinking in classroom teaching; Some teachers are keen on the teaching mode of "type+method", which leads to students' thinking fixed in the framework set by the teacher and leads to students' negative thinking mode over time. In order to overcome the negative influence of thinking mode, it is very important to cultivate students' divergent thinking. Divergent thinking, also known as radial thinking, refers to thinking about known information in many directions and angles, so as to find a variety of solutions and results. It plays a very important role in broadening the thinking of solving problems and cultivating creative thinking. How to cultivate students' divergent thinking?

1. Break the routine and cultivate students' reverse thinking. Reverse thinking is an important form of divergent thinking. It is to think and analyze problems from the opposite direction of existing habits. It is manifested in the reverse use, reverse reasoning and reverse proof of definitions, theorems, rules and formulas. Reverse thinking reflects the discontinuity and mutation of the thinking process, and it is an important way of thinking to get rid of the mindset, break through the old thinking framework, produce new ideas and discover new knowledge.

Formulas in mathematics are two-way, but many students are used to thinking forward and applying formulas in solving problems, but they are not used to using formulas in reverse, especially using deformed formulas. For example, simplifying cos(π/4-α)cosα-sin(π/4-α)sinα, some students are influenced by the positive formula, and the process of simplifying cos(π/4-α) and sin(π/4-α) is very complicated. If the inverse formula can be completed in one step, the solution will be much simpler. In order to get rid of this mindset and form the habit of two-way thinking, after teaching a formula and its application, teachers should seize the opportunity to cite some examples of using the formula in reverse and strengthen the training of reverse thinking, so as to cultivate students' thinking flexibility and improve their mathematical problem-solving ability.

2. Contact all subjects to cultivate students' lateral thinking. Lateral thinking is another form of divergent thinking. It is based on the horizontal similarity between knowledge, that is, to examine the object from different angles of different branches of mathematics, such as algebra, geometry and trigonometric functions, or to simulate, imitate and analyze the thinking mode from different disciplines, such as mathematics, physics and biology. Lateral thinking uses the similarity between things, spans the knowledge and methods of different branches or disciplines, obtains hints and inspirations from lateral or lateral connections, and solves problems in this field with knowledge and methods in other fields.

Cultivating students' lateral thinking can not only communicate the internal relationship between the knowledge of various courses, but also deepen their understanding and mastery of the knowledge and methods they have learned from different aspects, and help overcome the rigidity and narrowness of thinking brought about by the thinking mode, cultivate the broadness of thinking and improve their ability to solve problems by comprehensively applying the knowledge of various disciplines.

For example, three identical squares are arranged as shown in the following figure, which proves that ∠α+∠β=π/4.

To solve this problem, we can analyze and think from the angles of geometry, algebra and trigonometric function. (1) From the geometric point of view, since ∠ EAC+∠ β = ∠ AED = 45, it is only necessary to prove ∠EAC=∠α, which can be obtained by △AEF∞△CEA. (2) From the perspective of trigonometric function, it is only necessary to prove that tg(α+β)= 1 and 0.

3. Variant teaching cultivates students' multi-directional thinking. Multi-directional thinking is a typical form of divergent thinking. It is to examine the same question from as many aspects as possible, so that thinking is not limited to one mode or one aspect, so as to obtain multiple answers or multiple results. "Multiple solutions to one problem", "one method is multi-purpose" and "one problem is changeable" are the basic forms of multi-directional thinking. Judging from the composition of thinking mode, "multiple solutions to one question" is the divergence of concentrated solutions of propositions, "multiple uses of one method" is the divergence of concentrated solutions of propositions, and "changeable questions" is the divergence of propositions and solutions. It can be seen that "one question is changeable" is relatively divergent. It is easier to induce and cultivate students' creative thinking by using creative thinking properly and timely in mathematics teaching.

In teaching, we can also adopt the teaching of changing questions and variants under the same conditions to guide students to think deeply and cultivate the profundity of thinking. For example, if the projection △ABC of the vertex S of a triangular pyramid on the bottom surface is a point O, then the necessary and sufficient condition for the point O to be △ABC is that the three slopes of the triangular pyramid are equal in height. After proving this question, guide the students to start the proposition. Extension 1: A necessary and sufficient condition for point O to be △ABC. What other equivalent statements are there? (1) The angles formed by the three sides of the triangular pyramid and the bottom surface are equal; (2) Each side of a triangular pyramid is equal to the angle formed by the bottom of its vertex. Extension 2: If the inner heart in the question is changed to the outer heart, what is the necessary and sufficient condition for O point to be the outer heart of △ABC? (1) Three sides of a triangular pyramid are equal; (2) The angles formed by the three sides of the triangular pyramid and the bottom surface are equal. Extension 3: If the outer center in the question is changed to △ABC, what is the necessary and sufficient condition for O point to be △ ABC? Three groups of opposite sides of a triangular pyramid are perpendicular to each other, and the projection of each vertex (as a tetrahedron) on the opposite side is the vertical center of the triangle. This variant not only deepens students' understanding of knowledge, but also improves students' ability to solve problems and cultivates students' creative thinking.

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