The flexibility of thinking refers to the timeliness of being able to improvise with the changes of things, not being too affected by the mindset, and being good at getting rid of the old model or the usual constraints. Cultivate students' rigor, profundity and extensiveness in mathematical thinking, but without developing the flexibility of thinking, it is possible to make thinking tend to a specific method and way, unilaterally pursue the stylization or patterning of analyzing and solving problems, and produce thinking inertia.

Flexible thinking is characterized by using knowledge freely, being good at adapting and adjusting ideas, and being good at using differential thinking to analyze specific problems, which is an important manifestation of flexible thinking.

Second, cultivate the rigor of mathematical thinking.

The rigor of thinking refers to the rigor and justification of considering problems. To improve the rigor of students' thinking, we must be strict and strengthen training.

First of all, students are required to think step by step and clearly, that is, to think in a certain logical order. Especially when learning new knowledge and methods, we should start with the basic steps and deepen them step by step.

Secondly, students are required to think comprehensively and carefully, and there must be sufficient reasons for reasoning and argumentation. Use the power of intuition, but don't stop at intuitive understanding; Use analogy, but don't believe the result of analogy; When examining questions, we should not only pay attention to obvious conditions, but also pay attention to discovering those hidden conditions; When applying the conclusion, pay attention to the conditions under which the conclusion is established; Carefully distinguish the differences between concepts, understand the connotation and extension of concepts, and use concepts correctly; Give the answers to all the questions, don't leave them out.

Third, cultivate the depth of mathematical thinking.

Thinking depth refers to the abstraction and logical level of thinking activities, as well as the depth and difficulty of thinking activities. In mathematics learning, students often can't understand the conclusion well. When doing exercises, they can't understand the essence of problem-solving methods at all. Without books and teachers, they can't solve problems independently. This phenomenon is a manifestation of students' lack of deep thinking in long-term study. To overcome this phenomenon, we must consciously and often carry out profound thinking training.

1. Whether we can see the essence of mathematics through phenomena and the essence and connection of mathematical objects through superficial phenomena is the main manifestation of profound thinking. In many mathematical problems, the conditional relationship is relatively hidden. If we only look at the surface of the problem, we can't start. Therefore, in mathematics learning, we should think from the outside to the inside and grasp the essence and law of the problem.

2. Pay attention to the careful examination of questions to prevent students from forming a mindset after solving similar problems many times with a certain thinking mode. When encountering similar new problems, they often show a tendency to mechanically apply the previous thinking mode. The more times the same method is used, the more obvious this tendency is.

Fourth, cultivate the broadness of thinking.

The broadness of thinking means that a problem can be considered from many aspects. Specifically, it can explain a fact in many ways, express an object in many ways, and put forward many solutions to a topic. In mathematics learning, paying attention to multi-directional and multi-angle thinking and broadening the thinking of solving problems can promote students' thinking broadening.

5. Cultivating the critical thinking of thinking means being good at strictly estimating thinking materials and carefully examining the thinking process in thinking activities. In mathematics teaching, the critical performance of students' thinking is that they are willing to test and reflect in various ways, and can put forward their own views on existing mathematical expressions or arguments, instead of blindly following them. They have completely accepted things ideologically, and they should also seek improvement and put forward new ideas and viewpoints.