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How can we learn math better?
The methods of preview, lecture, review and homework are the basic methods of preview, lecture, review and homework. 1, preview method preview is to read the upcoming mathematics content before class, understand its outline, and be aware of it, so as to grasp the initiative of listening to the class. Preview is an attempt of autonomous learning. Whether you understand the learning content correctly, whether you can grasp the key points and hidden thinking methods, etc. It can be tested, strengthened or corrected in time in class, which is conducive to improving learning ability and forming the habit of self-study, so it is an important part of mathematics learning. Mathematics has a strong logic and coherence, and new knowledge is often based on old knowledge. Therefore, when previewing, you should find out what you need to learn new knowledge, and then recall or review it again. Once we find that the old knowledge is not well mastered or even understood, we should take timely measures to make up for it, overcome the learning obstacles caused by not mastering or forgetting, and create conditions for the smooth learning of new content. The method of preview, besides recalling or reviewing the old knowledge (or preparatory knowledge) needed to learn new content, should also understand the basic content, that is, know what to say, what problems to solve, what methods to adopt, where the focus is, and so on. In preview, reading, thinking and writing are generally used to draw out or mark the main points, levels and connections of the content, write down your own views or places and problems that you can't understand, and finally determine the main problems or solutions to be solved in class to improve the efficiency of class. In the arrangement of time, the preview is generally carried out after review and homework, that is, after finishing homework, read the content to be learned in the next class, which requires flexibility according to the specific situation at that time. If time permits, you can think more about some problems, study deeply, and even do exercises or exercises; Time does not allow, we can have fewer questions and leave more questions for lectures to solve. There is no need to force unity. 2, the method of attending classes Listening to classes is the main form of learning mathematics. With the guidance, inspiration and help of teachers, we can make fewer detours, reduce difficulties and acquire a large number of systematic mathematical knowledge in a short time, otherwise we will get twice the result with half the effort and it is difficult to improve efficiency. So attending classes is the key to learning math well. The method of class, in addition to clarifying the tasks in the preview and solving the problems suitable for you, should also concentrate on keeping up with the teacher's lectures and use your brains to think about how the teacher asks questions, analyzes and solves problems, especially learning mathematical thinking methods, such as observation, comparison, analysis, synthesis, induction, deduction, generalization and specialization, that is, how to use formulas and theorems. When listening to a class, on the one hand, we should understand what the teacher said, think or answer the questions raised by the teacher, on the other hand, we should think independently, identify what knowledge we have understood, what questions or new questions we have, and dare to put forward our own opinions. If you can't solve it in class for a while, you should write down the problems or problems you want to solve yourself or consult the teacher and continue to listen attentively. Don't stay here because you don't understand one thing, which will affect the later lectures. In general, in class, we should write down the main points, supplementary contents and methods of the teacher's lecture for review. 3. The method of reviewing is to learn the learned mathematical knowledge again, so as to achieve the purpose of in-depth understanding, mastery, refinement and generalization, and firm mastery. Review should be closely linked with lectures, and the contents of lectures should be recalled while reading textbooks or checking class notes, so as to solve the existing knowledge defects and problems in time. Try to understand the content of learning and really understand and master it. If you can't solve some problems for a long time, you can discuss them with your classmates or find a teacher to solve them. On the basis of understanding the teaching materials, review should also communicate the internal relations between knowledge, find out its key points and keys, and then refine and summarize them to form a knowledge system, thus forming or developing and expanding the mathematical cognitive structure. Review is a process of deepening, refining and summarizing knowledge, which can only be realized through the active activities of hands and brains. Therefore, in this process, it provides an excellent opportunity to develop and improve their abilities. The review of mathematics can't just stop at the requirements of reviewing and memorizing what we have learned, but we should try our best to think about how new knowledge is produced, how it is developed or proved, what its essence is and how it is applied. 4, the method of homework Mathematics learning is often done by doing homework, in order to consolidate knowledge, deepen understanding, learn to use, thus forming skills and skills, and developing intelligence and mathematical ability. Because homework is done independently on the basis of review, it can check out the mastery and ability level of the learned mathematical knowledge, so when it finds many problems, difficulties or wrong questions, it often indicates that there are defects or problems in the understanding and mastery of knowledge, which should arouse vigilance and need to find out the reasons and solve them as soon as possible. Usually, math homework is represented by problem solving, which requires the knowledge and methods learned. Therefore, you need to review before doing your homework, and then do it on the basis of basically understanding and mastering the textbooks you have learned. Otherwise, it will get twice the result with half the effort, take time and get the desired result. Problems should be solved according to certain procedures and steps. First of all, we should make clear the meaning of the question, read it carefully and understand it carefully. For example, what are the known data and conditions, what are the unknown conclusions, what operations are involved in the problem, how they are related, and whether they can be represented by charts. We should carefully scrutinize and thoroughly understand them. Secondly, on the basis of understanding the meaning of the problem, explore the way to solve the problem and find out the relationship between the known and the unknown, the condition and the conclusion. Recall related knowledge and methods, examples learned, problems solved, etc. And consider whether they can introduce appropriate auxiliary elements from form to content, from known numbers and conditions to unknown quantities and conclusions, and use them to find out a special problem or similar problem related to the problem, and whether solving them can enlighten the current problem; Whether we can separate, check or change them part by part, and then recombine them to achieve the expected results, and so on. That is to say, in the process of exploring and solving problems, we need to use a series of methods such as association, comparison, introduction of auxiliary elements, analogy, specialization, generalization, analysis and synthesis to learn from solving problems. Thirdly, according to the explored solution, according to the required writing format and specification, describe the process of the solution, and strive to be simple, clear and complete. Finally, we should review the solution and check whether the solution is correct, whether each step of reasoning or operation is well-founded and whether the answer is detailed; Think about whether the problem-solving method can be improved or whether there is a new solution, whether the result of this problem can be popularized (in fact, many topics in middle school textbooks can be popularized) and so on. And sum up the experience of solving problems, and then develop and improve the thinking method of solving problems, and sum up some regular things. The learning methods of "from thin to thick", "from thick to thin" and "from thin to thick to thin" are the research methods mentioned by mathematician Hua many times. He believes that learning should go through the process of "from thin to thick" and "from thick to thin". "From thin to thick" means to understand and know the mathematical knowledge you have learned and know why. Learning should not only understand and memorize concepts, theorems, formulas and laws. We should also think about how they were obtained, what is the connection with the previous knowledge, what is missing in the expression, what is the key, whether we have a new understanding of knowledge, whether we have thought of other solutions, and so on. After careful analysis and thinking in this way, some notes will be added to the content, some solutions will be added or a new understanding will be generated. "The more books you read, the thicker you will be." However, learning can't stop here. We need to integrate the knowledge we have learned, refine its spiritual essence, grasp the key points, clues and basic thinking methods, and organize it into refined content. This is a "from thick to thin" process. In this process, it is not the reduction of quantity, but the improvement of quality, so it plays a more important role. Usually, when summarizing the contents of a chapter, chapters or a book, we should have this requirement and use this method. At this time, due to the high generalization of knowledge, it can promote the transfer of knowledge and is more conducive to further learning. "From thin to thick" and "from thick to thin" are a spiral rising process, with different levels and requirements, which need to be used many times from low to high in learning to achieve the desired results. This learning method embodies the dialectical unity of "analysis" and "synthesis", "divergence" and "convergence", that is to say, mathematics learning needs the unity of the two. The method of combining reception learning with discovery learning Mathematics learning should be meaningful reception learning and meaningful discovery learning. How to make them cooperate with each other, organically combine and give full play to their respective and comprehensive functions is an important aspect of learning methods. Learning, whether listening to systematic lectures or teaching materials given in the form of conclusions, does not involve any independent discovery. But in the process of learning, students are in a proactive state, not just accepting. They always ask themselves some questions, such as how the theorem was discovered or produced, how the idea of proof was worked out, and what key places need to be broken. Many mathematicians emphasize "not only to write, but also to read what is behind the book." In the process of acceptance and learning, we should also add some extreme points of discovery and learning, and learn ideas and methods of invention from them, rather than just staying in the acceptance of knowledge. Discovery learning is to solve a problem independently by observing, comparing, analyzing and synthesizing the provided materials or problems, so as to acquire new knowledge. When solving a problem, we should really understand the essentials, principles, formulas, theorems and laws involved in the problem, understand the significance of each step of operation, and put forward and test the purpose of the hypothesis. When solving problems, we always need to connect the knowledge and methods we have learned in the past. If we can't recall them for a while, we must review them again to further understand the application. Some people encounter problems and even consult reference books or teachers to solve them. It can be seen that this period is also interspersed with learning.