First of all, the question is raised. Educators in ancient China have always emphasized that learners must pay attention to the unity of learning and thinking. For example, Confucius thought that "learning without thinking is useless, and thinking without learning is dangerous"; Cheng Yi, an educator in the Song Dynasty, thought that "the way to learn must be based on thinking, and if you think, you will get it, and if you don't think, it will be easy", which further highlighted the supreme position of thinking in learning. These words of wisdom have had an important influence on the study of later generations. Throughout the syllabus of all countries in the world, the cultivation of students' independent thinking ability is put in a more prominent position. In China's "Middle School Mathematics Syllabus", in addition to the specific requirements for basic knowledge, skills, thinking methods and four abilities, it is also clearly stated that "we should pay attention to cultivating students' independent thinking and self-learning ability". Therefore, cultivating students' independent thinking ability is one of the goals of middle school mathematics education. In a broader sense, "the goal of a school should be to cultivate people who think and work independently" (Einstein's language). Independent thinking is the premise of discovery, breakthrough and innovation. 2. Understanding of independent thinking ability Independent thinking ability is a comprehensive ability, which shows that individuals can face different situations and use different ways of thinking, methods and skills to solve the problems they face. To cultivate this ability, we should first let students participate in specific activities and improve their participation as much as possible. Secondly, it is necessary to help students gradually master the methods of thinking and analyzing problems. Finally, we should pay attention to cultivating students' thinking quality and form the habit and ability of independent thinking. The age characteristics and cognitive level of middle school students determine that the degree of independent thinking is relative. Generally speaking, with the growth of age, students' cognitive level and activity ability are constantly improved, and the independence of thinking is also constantly enhanced. In other words, students' independent thinking ability must go through a long process before it can be gradually cultivated, constructed and developed. Independent thinking does not exclude cooperation and mutual assistance among students, but cooperative learning must be based on individual independent thinking. For a specific problem, if you don't form your own unique point of view and are eager to cooperate and talk with others, it will definitely affect the initiative of thinking, thus affecting the improvement of thinking ability. It can be said that without independent thinking, there is no essential content of cooperative learning, and cooperative discussion becomes passive water and rootless wood, and cooperation can only become a mere formality. 3. Teaching approaches to cultivate students' independent thinking ability 3. 1, which puts forward the requirements of independent thinking; Educate students and strengthen the consciousness of independent thinking. Through questionnaire survey, student discussion, statistics, analysis and judgment, it is found that independent thinking is positively related to learning effect. Generally speaking, the better the students are, the better the habit of independent thinking, and good study habits are gradually transformed into a kind of ability, thus providing support and guarantee for independent thinking activities. In order to form this virtuous circle, only by adopting different teaching strategies for all kinds of students in teaching can students' independent thinking ability be gradually improved. The policy we adopt is: for students with higher level, we should adopt the way of "letting go" to provide them with broader time and space for independent thinking; For middle school students, we should take the "exciting" way, provide them with appropriate questions, and gradually develop the habit of independent thinking; For poor students, we should adopt the way of "induction", give more encouragement and inspiration, and form the consciousness of independent thinking. In teaching, some famous experts and scholars at home and abroad are introduced to students in time in combination with teaching materials, and their achievements in independent thinking, hard study and study are illustrated with examples to illustrate the importance of "independent thinking" from a historical perspective. For example, in the teaching of the chapter "Sequence and Limit", short stories such as "Gauss summation method" and "Zeno paradox and its decoding" are interspersed to let students understand the scientific value of independent thinking in interesting situations; In the teaching of the chapter "Complex Numbers", combined with the development history of mathematics, such as the bumpy course of irrational numbers being discovered, students can feel the personality charm of independent thinking. At the class meeting, let the students and alumni who have made continuous progress and achieved excellent results introduce their experiences in learning mathematics well, and explain in detail the specific methods of independent thinking and daring to explore. Explain the necessity and feasibility of independent thinking from the perspective of learning. 3.2 Guide students and cultivate the habit of independent thinking; Create situations and teach students to think independently. Starting from the five links of students' learning (preview, lecture, review, homework and feedback), independent thinking is put in a more prominent position. Taking preview as an example, students are required to change browsing into thinking, so that preview can become meaningful learning. Our approach is to induce students to think independently through the form of question strings, and then communicate with each other in class to refine and summarize. Independent thinking is not whimsical thinking, but must follow correct laws and methods. The scientific thinking method is not divorced from the process of acquiring and using knowledge, but permeates and permeates in this process. In this process, teachers should not only tell students the conclusion, but also let students know the process and method of drawing the conclusion, and know the ins and outs of knowledge and their relationship. Through the process of learning knowledge, learn to think correctly at the same time, and gradually build a systematic thinking method to prepare for independent thinking in the true sense. Taking the teaching process of "complex related concepts" as an example, with the deepening of students' classroom discussion, teachers and students have jointly constructed the knowledge structure of complex concepts, and in the process of solving this problem, some thinking methods have been refined. 3.3 Leave room to stimulate students to think independently; Postpone judgment and encourage students to think independently. Teachers should not explain in detail, leaving room for students to think, explore and develop themselves. Otherwise, it seems thorough, but it is difficult to internalize it into students' views, and students' independent thinking ability cannot be formed. Therefore, in the teaching process, we should make the most basic and important things clear, so as to facilitate knowledge transfer; For some unfolding problems, simple deduction and demonstration, comparison, difference and connection of knowledge before and after, induction and summary of knowledge and methods, etc. It can leave room for students and stimulate them to study and think. Taking the teaching of "Examples of the Application of Mathematical Induction" as an example, the key point is to explore the conclusion of the problem. In this process, there is a contradiction between finite and infinite, thus strengthening students' understanding of the application function of mathematical induction. Postponement of judgment is a principle of Osborne's intellectual incentive law. The main point of this principle is that it is limited to the stage of thinking and discussing problems, and it is not appropriate to make judgments and criticisms too early. We should create a relaxed atmosphere, make the discussants feel safe and free psychologically, constantly induce creative ideas, and finally make their thinking flexible, profound and comprehensive. 3.4 Advocate open teaching and improve the taste of independent thinking. In order to let students think independently, teaching methods should first advocate openness, and never "speak at once", otherwise this independence will soon be greatly limited. Secondly, try to cultivate students to be good at asking questions independently. Because you can ask questions independently, you can't do it without independent thinking; And good questions just illustrate the depth of thinking. For example, the focus chord of parabola is a common geometric model in analytic geometry. We asked the students to ask some questions, and as a result, they showed great enthusiasm. After finishing, * * * put forward more than 20 questions with thinking value. In addition, we should expand students' knowledge and broaden their thinking as much as possible. For example, the reading materials, practical assignments, research-based learning and other contents set in the new textbook of senior one are carefully guided and organized by the teachers in the experimental class, and some "open-ended" questions are put forward for students to explore and think. 4. Establish the object of thinking and improve the effectiveness of independent thinking. 4. 1 Think about the background, process and function of knowledge formation. Taking the teaching of "function periodicity" as an example, we listed the following background materials for students to think about: What is periodicity? What is the period of the earth's rotation? What is the period of the earth's revolution? How is the period defined in physics? On this basis, let the students review the practice of sine function image (moving pursuit method after dividing the unit circle equally). Through multimedia demonstration, let the students think about the physical meaning and mathematical representation of image repetition, and gradually abstract the definition of function periodicity. On this basis, the arbitrariness of constants T and X in the definition is questioned: What is the given constant T? Is it unique? Does it have to have the smallest positive value? In f(x+T)=f(x), why must x be any value in the domain? If a is a nonzero constant, and for any X, it satisfies (1) f(x+a) = f(x-a), (2) f(x+a) =-f(x), and (3) f(a-x)=f(x) respectively. Ask f(x). These "question strings" improve the pertinence and effectiveness of students' independent thinking about the concept of function periodicity. There are too many mathematical thinking methods to arouse students' independent thinking. You can see several teaching cases provided earlier. 4.2 Think about problem-solving strategies and review after solving problems. According to Paulia's "How to solve a problem table", we often ask students the following questions: (1) What are the conditions of this problem? What is the conclusion? What is the connection between them? What is the difference? (2) Do you know any questions related to this topic? What if you come up with a more special question? More general questions? A similar problem? (3) Can a solution be found and implemented? In teaching, we use these questions to ask students questions and induce them to think independently, so many students gradually learn how to seek solutions to problems. Through problem-solving review, students' metacognitive ability can be improved. After solving the problem, ask the students: (1) Can you check your results? Can you tell me about the detour you took to solve the problem? (2) Can we get the result by other methods? (3) Can this result or method be transferred to other problems? (4) Can the results or methods be popularized? Once students develop this habit of asking and answering questions, it will undoubtedly play a great role in promoting the formation of students' independent thinking ability. 4.3 Thinking about the mistakes in learning. The solution to thinking mistakes is not only to find the basis for correction, but also has a deeper function: First, it is the "mother machine" of correct thinking, and the exposure of the root causes of mistakes is often accompanied by correct understanding, which leads to the emergence of correct thinking; Secondly, the study of various possible ideas fully exposes the students' thinking process. In this process, guiding students to think from all directions and angles can not only cultivate divergent thinking, but also constantly optimize problem-solving methods and enhance the profundity and criticism of thinking; Thirdly, in the process of error correction, students must try their best to find loopholes, construct counterexamples and adjust strategies, that is, students must undergo complex psychological changes to achieve the purpose of error correction, so this process itself is a highly independent thinking activity. 4.4 Thinking about the vertical and horizontal connection between mathematical thinking methods and knowledge. For example, how to analyze and synthesize? How to resolve the induction? How to transform? How to classify? How to discuss? How to replace? How to make an analogy? How to master the combination of numbers and shapes? How to build a mathematical model? Wait a minute. Usually, teaching is always carried out step by step with a single knowledge, so students accept and store knowledge in a piecemeal way, which leads to a high forgetting rate and hinders the development of students' independent thinking ability. So at the end of each unit, we ask students to list their own review outlines and form a good outline under the guidance of teachers. On this basis, through independent thinking on a series of problems, the same and essential characteristics are extracted and controlled by mathematical thinking methods, so that students can master them from the height of methodology, thus improving their macro-thinking ability. In teaching practice, we realize the difference of independent thinking level. For example, "thinking aloud" is only the reappearance of thinking knowledge, and practical operation is thinking skills and methods, both of which are low-level thinking. The choice of problem-solving strategies needs to be constantly searched from mathematical concepts, ideas, methods and knowledge, which can be experienced repeatedly. Therefore, it needs a process, and its thinking form is often "silent", but its thinking activities and psychological activities are extremely rich and complex, which can be said that more is told in silence than in audio thinking. Reflective learning is actually the embodiment of students' metacognitive ability. Students should think independently from a broader perspective, so they have a longer time, a wider space and a higher level of thinking. In teaching, letting students "think aloud" and "do mathematics" are the most basic guarantee for independent thinking activities, but we should not just stay at the level of these operations, but should leave enough time for students to further "reflect" on the abstraction of their practical activities, so as to make thinking activities move to a higher level. Among them, it is an effective way to improve students' independent thinking ability to guide students to think about problem-solving strategies with mathematical thinking methods and let students constantly reflect on their learning activities.