First, run self-confidence through the process of solving problems.

In the usual learning process, many students feel that they have a good grasp, but once they do a problem, they often can't do it. The teacher pulled it and suddenly it became clear. In other words, these problems are not absolutely impossible. As long as you think carefully, analyze and synthesize, use various mathematical ideas and methods, compare drawing, writing and calculation, and through tortuous reasoning or calculus, you can gradually find the essential relationship between the conditions and conclusions of the topic. Self-confidence is the secret of success, not empty talk. Be confident in the face of slightly complicated problems. You know, these problems are generally not beyond our own knowledge, and we can solve them with what we have learned. Dare to think and be good at thinking, which is a very important thinking quality. When solving specific problems, we must carefully examine the questions, correctly distinguish conditions from conclusions, and grasp two main links: First, we must firmly grasp the * * * relationship between this problem and a class of problems, and think about the general ideas and general solutions of such problems; The second is to firmly grasp the particularity of this topic and the difference between this topic and this kind of topic. Choose one or several conditions as the breakthrough point to solve the problem, and see what transitional conclusions can be drawn from these conditions, the more the better, and then filter out useful conclusions for further reasoning or calculus. This is what teachers often tell students: "Smart students study together, and unintelligent students study together". You know, the ocean of problems is endless. Only by drawing inferences from others can we jump out of the ocean of problems and understand the mystery of mathematics learning.

Second, remembering the necessary basic knowledge is the key to solving problems skillfully.

Some students think that only Chinese, English, politics, history, geography, biology and other subjects need to be memorized, while mathematics depends on operation, reasoning and analysis, and no memory is needed. This kind of understanding is all wet. "Learning from others" is the only way to learn. If you don't remember the necessary basic knowledge of mathematics, your mathematical thinking space will become narrower and narrower, which will inevitably lead to a dead end in your mathematics study. For example, I can't remember the "99 multiplication table" in primary school. Can I multiply smoothly? Although you understand that multiplication is the operation of the sum of the same addend, when you do 9×9, it is not cost-effective to add 9 9s to get 8 1. It is much more convenient to use "998 1" to find the results. For another example, when you solve the equation 2x2+3x- 1=0, if you don't remember the root formula of the quadratic equation with one variable, you can only use the complicated collocation method to reason step by step. In addition, this formula is the basis of learning the relationship between roots and coefficients of quadratic equations, quadratic functions, quadratic inequalities and other knowledge. Without this formula as the basis, the learning of these knowledge can only be in a dilemma. In fact, math learning is more like a game, such as playing China chess. If you don't remember the rules of the game, such as walking on the court and shooting every other player, how can you play China chess well? These rules of the game are like the basic knowledge in mathematics learning.

The New Curriculum Standard for Junior High School Mathematics in Nine-year Compulsory Education describes the basic knowledge in junior high school mathematics as follows: "The basic knowledge in junior high school mathematics includes the concepts, laws, properties, formulas, axioms and theorems in junior high school algebra and geometry, as well as the mathematical ideas and methods reflected in their contents."

Mathematical definitions, rules, properties, formulas, axioms, theorems, etc. Be sure to memorize it. It's catchy. We often say that we should remember on the basis of understanding. But some basic knowledge, such as definition, is unreasonable. For example, the definition of a linear equation with one unknown quantity, whose highest degree is 1 and whose coefficient cannot be 0, is called a linear equation with one unknown quantity. In this definition, why there is only one unknown instead of two or three, why the highest number of unknowns is 1 instead of 2 or 3, why the coefficient of unknowns cannot be 0, and so on. These questions are of little value, or the definition is just a prescribed or inherent meaning for a certain thing or phenomenon. And some basic knowledge, such as laws, formulas, theorems, etc. Know not only why, but also why. For example, the nature of parallel lines: two parallel lines are parallel, the same angle is equal, the inner angle is equal, and the inner angles on the same side are complementary. We should not only remember, but also be able to use what we have learned to explain why two parallel lines have such properties. This is what we call memory based on understanding. In the process of learning, it is inevitable that you will not understand some basic knowledge for a while. In this case, remember even by rote. After remembering it, you will gradually understand it in the learning process of the post-thread. In addition, some important mathematical methods and ideas also need to be remembered. Only in this way can you solve mathematical problems with ease, so as to experience the aesthetic value of mathematics and cultivate confidence in learning mathematics well.

Third, talk about "method" and "thought", and guide "method" with "thought". The two complement each other.

The so-called mathematical thought is an essential understanding of mathematical knowledge and methods, a rational understanding of mathematical laws, and an abstract thing belonging to mathematical concepts. The so-called mathematical method is the fundamental procedure to solve mathematical problems, the concrete embodiment of mathematical thought and the means to implement mathematical thought. Mathematical thought is the soul of mathematics, and mathematical method is the behavior of mathematics. The process of solving problems by mathematical methods is the process of accumulating perceptual knowledge. When the accumulation of this quantity reaches a certain procedure, it produces a qualitative leap and thus rises to mathematical thought. If mathematical knowledge is regarded as a magnificent building built by a clever blueprint, then mathematical methods are equivalent to architectural means, and this blueprint is equivalent to mathematical thoughts.

In the study of junior high school mathematics, we need to understand the mathematical ideas: equation function, combination of numbers and shapes, reduction, classified discussion, implicit condition, whole substitution, analogy and so on. The methods of "understanding" are: classification, analogy and reduction to absurdity; The methods that require "understanding" or "being able to use" include: undetermined coefficient method, elimination method, reduction method, collocation method, method of substitution method, image method and special value method. In fact, thoughts and methods cannot be completely separated. All kinds of methods used in junior high school mathematics reflect certain thoughts, and mathematical thoughts are rational understanding of methods. Therefore, it is an effective way to understand mathematical thoughts through the understanding and application of mathematical methods.

In the process of mathematics learning, we must fully infiltrate mathematical thinking methods, learn a knowledge point or do a problem, and seriously think about what mathematical thinking methods are used. Although the mathematical thinking method is different, it is limited after all. Correct use of mathematical ideas and methods to learn mathematics or solve problems is conducive to the comparison and classification of knowledge. Only in this way can we learn what we have learned systematically and flexibly, and truly integrate what we have learned into your knowledge structure and become your own wealth.

In addition, due to the abstraction of mathematical thought, although the mathematical method is more specific, the method itself is a science, a more important knowledge, and it is still more difficult. Therefore, when you first come into contact, it is inevitable that you can't sort out the clue. This is normal, you don't have to be afraid. In particular, mathematical thought is a gradual infiltration process, which should be understood in combination with specific mathematical knowledge or topics in the gradual learning process.

For example, when learning rational numbers, triangles, quadrilaterals, the proof of the theorem of circle angle and tangent angle, and the derivation of the root formula of quadratic equation in one variable, the idea of classified discussion will be involved. The principle of classified discussion is: unify the standard, and don't weigh or leak. Its advantage is that it has obvious logical characteristics and can train a person's thinking order and generality.

The idea of equation has realized the transformation from the arithmetic method in primary school to the algebra method in junior high school, which is a substantial leap in mathematical thought. The idea of equation refers to the relationship between unknown quantity and known quantity in mathematical problems, which can be solved by constructing equations. We will find that many problems can be solved easily if they are solved by the method of column equations.

The idea of combining numbers and shapes is conducive to visualizing abstract knowledge. In junior high school mathematics learning, "number" and "shape" are inseparable. For example, the concept and operation of rational numbers can be well understood with the help of the number axis. Many problems in solving application problems of series equations can easily find out the equal relationship between quantities by drawing the meaning of the questions, and function problems can not be separated from graphs. Often with the help of images, the problem can be clearly explained, and it is easy to find the key to the problem, thus solving the problem.

The idea of transformation is embodied in the transformation from unknown to known, from general to special and so on.

These mathematical ideas and methods will also run through the teaching process of teachers. Pay attention to the lectures, learn from the teachers and learn from the classroom. Bruner pointed out that mastering mathematical thinking methods can make mathematics easier to understand and remember. It fully illustrates the importance of mathematical thinking methods.

Fourthly, forming good thinking quality is the basis of understanding mathematical problems.

Mathematics, as a discipline to cultivate people's thinking ability, is fascinating with its rational thinking. Unlike sightseeing in the mountains, it is pleasing to the eye because of its charming scenery and lingering. Mathematics learning is to study the spatial form and quantitative relationship of things through thinking and reflection, so that the spatial form and quantitative relationship of things can be presented. Only by forming a good thinking quality and pulling away the appearance of things with the sharp blade of good thinking quality can we "see" the essence of things.

So what is a good thinking quality? Let's take the phenomenon of "visiting" in our life as an example to illustrate. Many people have this kind of life experience, let others take it to others' homes once, twice, maybe many times. One day you have to go to someone else's house by yourself. When you walk near someone's house, facing the same building, you are at a loss and don't know where someone is.

In the process of learning, we often have such a phenomenon. In class, the teacher made it very clear, and the students just nodded, which made me feel very clear. And let the students do the questions themselves, and they don't know where to start. The main reason is that students do not think deeply about what they have learned and do not understand the essence of what they have learned. Just like passing by, every time we go to other people's homes, we should remember the geographical environment around them, especially the special signs. To understand the characteristics of what you have learned and what you need to remember, especially what mathematical ideas and methods are involved in this part of knowledge, you need to master it in time. The content of this kind of memory should be carefully remembered, and only by remembering the necessary knowledge can thinking be based. In addition, pay attention to taking notes. Bacon said in On Knowledge: "Taking notes can make knowledge accurate. If a person is unwilling to take notes, his memory must be strong and reliable. " Pay attention to the key points the teacher said, especially some empirical and regular knowledge summarized by the teacher, so as to review in time after class. After-class review, we should think about which problems have been passed and which problems have not been passed, and do a good job of checking and filling gaps in time.

The above talks about how to learn junior high school mathematics well from four aspects. In order to learn junior high school mathematics well, in addition to the above, the key to learning mathematics well is hard study spirit, serious and careful study attitude and good study habits. In the classroom, we should not only learn new knowledge, but also subtly learn the teacher's way of thinking to solve problems. In the face of a problem, we should think ahead, find out our own way of thinking, and then compare our own way of thinking with the way of thinking of teachers, learn from each other's strengths and form our own way of thinking. Change "I want to learn" into "I want to learn", cultivate the initiative of learning and overcome the situation of passive learning. Really master the essentials of mathematics learning. The test of whether you can learn math well is whether you can solve problems. Understanding and memorizing the basic knowledge of mathematics, mastering the ideas and methods of learning mathematics is only the premise of learning mathematics well, and the ability to solve problems independently and correctly is the symbol of learning mathematics well.