As a math educator for many years, I want to write it here and share it with you, hoping to help you. The article is mainly divided into four parts. The first part is written for students with poor grades. The second part is the necessary condition for learning mathematics well; The third part is some learning methods that are easily overlooked, which is why they can't really become masters of mathematics; The fourth part is about exams and mentality.

For students with poor grades.

Many students have poor grades, mainly for the following reasons:

Do not study hard, not diligent enough. Diligence is a good quality, so is learning. As the saying goes, heaven rewards diligence is the truth. Generally speaking, in junior high school, there are relatively few knowledge points, and diligence is the main factor that determines grades. For high school, the influence of diligence will be relatively weak, and learning methods will be replaced. Natural question: What causes students to be not diligent? In other words, how can students learn actively and become diligent? In fact, it is very simple, that is, interest and motivation. So how can we make students interested and motivated? There are two main aspects: first, family education varies from family to family, with guiding principles, correct methods and moderation; Second, on the teacher's side, this is more difficult After all, now teachers mainly instill knowledge and communicate with students less, so it is difficult for students to have respect and lose interest. Parents can communicate with their children and are willing to give everything for them.

Passive learning

Many students have a strong dependence on teachers and parents, follow the inertia of teachers, study under the supervision of parents, and have no initiative in learning. Such as: not making plans, waiting for classes; Do not preview before class, do not understand what the teacher wants to do in class; Busy taking notes in class, not listening to the "doorway", not really understanding what you have learned, and so on. As a result, students' learning efficiency is low and their grades are naturally poor!

You can't learn law.

Teachers usually explain the ins and outs of knowledge in class, analyze the connotation of concepts, analyze key and difficult points, and highlight thinking methods. However, some students didn't concentrate in class, didn't hear the main points clearly or didn't listen completely, took a lot of notes and had many problems. After class, I can't consolidate, summarize and find the connection between knowledge in time, just write my homework in a hurry, do problems in disorder, have a little knowledge of concepts, laws, formulas and theorems, mechanically imitate and memorize; Some people work overtime at night, are listless during the day, or don't listen at all in class, so they have another set. The result is half the effort, with little effect. Learning blindness is the most troublesome problem for middle school students. What is the fundamental difference between top students and middle students? The foundation is not bad, mainly the learning method. Top students said, "Play before the quiz, play before the quiz". Secondary school students didn't get the first place in the exam because there is not much difference between learning methods and basic knowledge.

4. Don't pay attention to the foundation

Some students who "feel good about themselves" often despise the study and training of basic knowledge, basic skills and basic methods, and often only know how to do it, but they are interested in difficult problems to show their "level", and they are ambitious, either making mistakes in calculation or giving up halfway in formal homework or exams.

2. Necessary conditions for learning mathematics well

1. Mathematical operation

Operation is the basic skill to learn mathematics well. Junior high school is the golden age to cultivate mathematical operation ability. The main contents of junior high school algebra are related to operations, such as rational number operation, algebraic operation, factorization, fractional operation, radical operation, equation solving and so on. Junior high school students' poor operation ability will directly affect their future mathematics learning: from the current mathematics evaluation, accurate operation is still a very important aspect, and repeated operation errors will undermine students' confidence in learning mathematics. As far as personality quality is concerned, students with poor computing ability are often careless, with low requirements and low thoughts, which hinders the further development of mathematical thinking. From the self-analysis of students' test papers, there are not a few mistakes, most of which are operational errors, and they are extremely simple small operations, such as 7 1- 19=68, 7-9=2, etc. Although the mistake is small, it must not be taken lightly, and the real reason behind it must not be concealed by a "sloppy". It is one of the effective means to improve students' computing ability to help students carefully analyze the specific reasons for errors in operation. In the face of operation, we often pay attention to the following two points:

① Emotional stability, clear arithmetic, reasonable process, even speed and accurate results;

Have confidence and try to do it right once; Slow down and think carefully before writing; No verbal calculation, no mental calculation, no skipping steps. Write clearly on the draft paper and finally scan it with your eyes to see if there are any low-level mistakes.

Step 2 solve math problems

There is no shortcut to learning mathematics, and ensuring the quantity and quality of doing problems is the only way to learn mathematics well.

(1), how to ensure the quantity?

① Choose a tutorial book or workbook.

(2) After finishing all the exercises in a section, correct the answers. Never do a pair of answers, because it will cause thinking interruption and dependence on answers; Easy first, then difficult. When you encounter a problem that you can't do, you must jump over it first, go through all the problems at a steady speed, and solve the problems that you can do first; Don't be impatient and discouraged when there are too many questions you can't answer. In fact, the questions you think are difficult are the same for others, but it takes some time and patience; There are two ways to deal with examples: "do it first, then look at it" and "look at it first, then take the exam".

(3) Choose questions with thinking value, communicate with classmates and teachers, and record your own experience in the self-study book.

④ Ensure about 1.5 hours of practice time every day.

(2) How to ensure the quality?

(1) There are not many topics, but they are good. Learn to dissect sparrows. Fully understand the meaning of the question, pay attention to the translation of the whole question, and deepen the understanding of a certain condition in the question; See what basic mathematical knowledge it is related to, and whether there are some new functions or uses? Reproduce the process of thinking activities, analyze the source of ideas and the causes of mistakes, and ask to describe your own problems and feelings in colloquial language, and write whatever comes to mind in order to dig out general mathematical thinking methods and mathematical thinking methods; One question has multiple solutions, one question is changeable and pluralistic.

② Execution: Not only the thinking process but also the solving process should be executed.

(3) Review: "Reviewing the past and learning the new", redoing some classic questions several times and reflecting on the wrong questions as a mirror is also an efficient and targeted learning method.

3. Make a series of personal mistakes. I give my classmates a formula: less mistakes = more pairs. If you make a mistake, no matter what mistakes you find, no matter how simple they are, they are included; I believe that once you really do it, you will be surprised to find that your mistakes can't be corrected once. On the contrary, many mistakes are made for the second, third or even more times! Looking at my wrong suit, alas, it's shocking. This is really a good place for self-reflection and a good way to improve your grades. The later you review, the less likely you are to break through knowledge, and the room for correcting mistakes is not small. If you don't have this habit, prepare a book, collect your mistakes, classify them, and then look through them when you have nothing to do to warn yourself, and you will certainly gain a lot.

4. A reference book is enough. I want to say, don't be superstitious about reference books. There are not many reference books, but one main one is enough. I found a very strange phenomenon. Nowadays, many reference books on the market sell well, all of which are branded by a famous teacher. How well they speak. As a result, many students took one book after another, confused. In fact, our time for study and review is limited, and the time we can leave for ourselves is even more limited. In these limited time, it's best not to read this reference book for a while and that reference book for a while. By memorizing the main points of the knowledge structure of the textbook, you can review all the knowledge in a short time. Doing this is much more important than reading some reference books called "Golden Keys and Silver Keys". In a word, grasp the most fundamental and important, don't read reference books blindly, especially don't read many reference books.

5. What should I do if I encounter difficulties? First of all, we should try our best to solve it through our own efforts. If it can't be solved, we should also find out why it won't and where the problem lies. I often say: never expect not to encounter problems, but never allow yourself not to understand where the problems are. When you can't solve it by yourself, you can solve those problems by discussing and asking the teacher. The solution is by no means that you can do it with the help of others, but after you can do it, look back and compare the reasons why you can't, and be sure to find out the reasons, otherwise you will lose an opportunity to improve and lose the meaning of doing the problem.

6. How to jump out of the sea of questions? I think everyone must be very concerned about this topic, because physics is difficult to understand, chemistry is difficult to remember and mathematics has endless problems. But the topic is the heart of mathematics, and it is absolutely impossible not to do it. And there are so many problems before us that it seems endless. Try the following methods. First, on the basis of completing the homework, analyze how each topic is investigated, which knowledge points are investigated, and whether there are other ways to investigate this knowledge point; Second, when you continue to do the problems, there is absolutely no need to work out every problem in detail. As long as you have read it, you can classify it into the problems we analyzed above, and you can skip if you know the way to solve the problem! In this way, for every knowledge point, we can master the examination method, which is the real improvement. If you don't realize this, doing the problem is just doing the problem, "topic", you can't jump out of the topic, you can't see the essence of the problem, and you can't do anything when you meet a new topic, which is slightly different. What else can we talk about? How can we get rid of the ocean of problems that plague you?

7. Diligence and perseverance are the necessary conditions for learning mathematics well. In any case, we should have a hard-working spirit in our study, but we should not only be hard-working, but also be good at learning and summing up, so as to get twice the result with half the effort. There are "seven bread principles" in mathematics: you won't feel anything after eating the first six breads, and you will suddenly feel full after eating the seventh bread. After doing a lot of problems, you may feel ineffective and get nothing, but if you do a few problems, your math level will make a qualitative leap. So don't give up, the revolution has not yet succeeded, comrades still need to work hard, and one day they will be pleasantly surprised.

8. Write more, think more, and combine the two. This is the most important thing, and it is also a magic weapon to learn mathematics and even science well. Mathematics is responsible for cultivating students' computing ability, logical thinking ability, spatial imagination ability, and the ability to analyze and solve problems by using what they have learned. It is characterized by a high degree of abstraction, logic and wide applicability, and requires high ability. Learning mathematics must pay attention to "living", not only reading books without doing problems, but also burying oneself in doing problems without summarizing and accumulating. We must be able to access textbook knowledge and find the best learning method according to our own characteristics. This is why Mr. Hua advocates the learning process of "from thin to thick" and "from thick to thin". Methods vary from person to person, but four steps (preview, class, arrangement and homework) and one step (review and summary) are indispensable. Think more when you do the problem, draw inferences from others, and don't do it by death. There are many new questions in the college entrance examination, which are suitable for students who like thinking rather than doing it.

3. Easy-to-ignore but vital learning methods

1. The principle of doing a problem is: to do a problem, you must know ten or dozens of problems. Because every math problem (mother problem) will cover several math problems (sub-problems). Do a motif, and every sub-topic is understood, so you don't have to do all the topics that belong to the motif, wasting limited precious time.

I have my own way of learning mathematics. I always understand the connotation of various knowledge points in class, then deepen my understanding by doing some representative questions after class, and then read books to understand the basic knowledge points again. In mathematics learning, doing problems is not an end, but a means: doing problems is to achieve a deeper understanding. Don't do problems for the sake of doing them, but at the same time, do some representative exercises in moderation. Usually, after every exam, I always take a correcting book to copy down the wrong questions, correct them carefully, and indicate the methods used next to the key steps, then write comments after the wrong questions and summarize the reasons for the mistakes. Before every math exam, I always read this book carefully and remember why I made mistakes, so as to avoid making similar mistakes again.

3. Thoroughly understand the textbook method. In the process of learning, everyone has had such an experience. Sometimes when I see a topic, I can't find the idea for a while, so I can't wait to see the answer. When reading the answer, they often feel that every step of the answer is logical. It is easy to use any theorem and method. They think they have understood the topic thoroughly. Do this problem again in a few days, and there is still no way to start. I think this situation is mainly because my acceptance of this issue is a passive process. In this process, I only saw the specific problem-solving process mechanically, but I didn't really understand the problem-solving idea.

Actively seeking ways to solve problems is just the opposite of this passive learning method. This method emphasizes starting with simple exercises, because simple exercises will be easier to do, and then go from shallow to deep after finishing. When you encounter a difficult problem in practice, consciously force yourself not to look at the answer, the formula and ask others for help (these are passive methods), but to calm down, actively mobilize your brain knowledge base and actively seek ideas to solve the problem. In this way, you can train yourself from the shallow to the deep, plus the classification analysis of common problems. When you see the math exercises again, you will reflect the knowledge points and ways of thinking examined in the topic at the first time, and you will feel handy. Many students think that the topics in math textbooks are very simple, and they are all said by the teacher in class. After class, they often put their textbooks aside and do other exercises that they think are more difficult. But when it comes to exams, there are often problems, but simple questions are easy to lose points. Therefore, we should pay special attention to learning textbooks and do every question well in textbooks, which is also the first point I want to say. The second point is the basic concepts and ideas in textbooks. Textbooks are not only important for exercises, but also for basic concepts and ideas. There are many big concepts in bold in math textbooks, which we usually pay attention to, but in some small words, there are often some very subtle concepts and principles that are easy to be ignored, and when we take the exam, we often take out those problems that we ignore. And as soon as the exam is taken, everyone will "pour a big chunk." Therefore, when reading textbooks, we must see every word, sentence and even tiny truth clearly. There are many important conclusions in the exercises of trigonometric function, solid geometry and analytic geometry, which should be remembered. You can't overemphasize the importance of a textbook.

In the process of solving problems, many students often can't write because they can't find ideas. There are only two kinds of math problems: solving problems and proving problems. Solving a problem allows you to seek a result, and proving a problem allows you to prove a conclusion. Personally, I prefer these two methods: list the known conditions and see what conclusions can be drawn, which are also conditions, and then see what new conclusions can be drawn from these new conditions, layer by layer, like branches of the trunk, more and more. Since it can be deduced forward, it can also be deduced backward. Starting from the results you require or the problems you need to prove, you can see what conditions you need for the desired results and what conditions you need to get these conditions, step by step and think backwards. When the branches stretch out more and more, eventually two will be intertwined, and the problem will be solved. When you start using this method, it is really time-consuming, but it is still quite effective. After you are more proficient, you can often see the key to the problem at a glance and find a breakthrough quickly. Then there is the network summary method of knowledge points, that is, when you usually do a problem, if you encounter a problem in the solution, you will list the solution methods related to this problem and the knowledge points you have tested next to the problem, and then summarize them in your notebook. In this way, after a period of training, you can see the topic in the exam and associate it with the relevant knowledge points, and quickly find the corresponding solution. On the one hand, this method can improve the speed of solving problems and save a lot of time for candidates, on the other hand, the correct rate of doing problems is high and the hit rate of solving problems is improved.

Some people say that mathematical thought is the soul of mathematics, and I think it makes sense. True masters of mathematics often compete for mathematical ideas, which requires their own accumulation and cultivation. Below I just want to say two very rustic views: First, the consciousness of "the answer lies in the topic". This awareness is crucial. Doing a problem is nothing more than this: the problem-what have you learned-the answer. We can't find the answer because we can't find any knowledge to solve it, and the questioner often uses the topic to give you hints or directions. Whether an expert is good or not depends on experience. Look at the topic carefully and see through the conditions that can be excavated hidden in the topic, and the general answer will be naked in front of you. Therefore, I am slow to do my own problems mainly by watching. Sometimes I have passed the test word by word, but I have to read it again if I don't understand it. After reading the question, the idea basically came out and the answer was made. So I do geometry, whether the topic is vivid or not, I have to draw it myself, which is very helpful to understand the meaning of the topic. Especially the big topic, there are several small topics, all of which are interlocking. The following questions are basically based on the answers to the previous small questions. Second, simplify thinking. Most people who have learned a certain degree will consciously complicate some simple topics, which is a big taboo. Many problems seem complicated and difficult to understand. In fact, everyone thought it was very simple after reading the answer, but they didn't think of it at that time. The more unconventional the problem, the simpler it is. Don't complicate it yourself

4. About exams and mentality

1. The magic weapon to win in the examination room. The first is to get rid of psychological fear. You can remind yourself, "What are you afraid of? No matter how difficult it is, everyone is like me. " This kind of self-psychological suggestion has calmed my heart a lot after a period of time. In fact, the most important thing in studying and taking exams is not how to learn or how to take exams, but how to play one's own level, which is also the premise of playing beyond the level. You might as well have a try, maybe it will work well! Secondly, we should have correct learning and examination strategies, so as to be "humiliated". Don't be nervous, especially when you encounter difficulties. There is such a phenomenon in the exam that once you encounter a problem that you can't solve for a long time, you will fidget, which will seriously affect the following problems and then affect the exam results. In my opinion, in this case, we should give up this problem for the time being and continue to do it to ensure that other problems are not affected. Believe this: difficult topics are difficult for everyone, and there is nothing wrong with not doing them; In the end, when all the other questions have been answered, I will calmly look back and maybe I will succeed. There are always two or three difficult problems in the college entrance examination paper, but I hope everyone will pay attention to the fact that it is not those difficult problems that really distance you from others, but those problems that can be solved by everyone's efforts.

2. Understand the exam correctly. Actually, here, I just remind you of a fact. That is to say, if it is not a competition, then more than 80% of the contents in the test paper are copied from what we have practiced in our usual study. In other words, more than 80% of the questions are very basic, and more than 80% of the scores can be obtained by each of us through hard work. If you don't believe me, you can go and see for yourself. Suppose you have mastered these basic topics. What level is it? So every student should see this fact and make himself confident.

3. The relationship between "knowing how to do" and "scoring": To turn one's problem-solving strategy into a scoring point, it mainly depends on accurate and complete mathematical language expression, which is often ignored by some candidates, so there are a lot of cases of "meeting but not right" and "being right but not complete" on the test paper, and the candidates' own evaluation scores are far from the actual scores. If the words and figures in the probability question are unclear, there will be many points deducted. Only by paying attention to the language expression of the problem-solving process can we grade the "can do" questions.

4. Relationship between quickness and accuracy: The word "accuracy" is particularly important in the current situation of large number of questions and tight time. Only "accuracy" can score, and only "accuracy" can save you the time of examination, while "quickness" is the result of usual training, not a problem that can be solved in the examination room. Quite a few candidates miscalculated even a function in their haste. Although the follow-up questions are correct in thinking and take time to calculate, they hardly get points, which is inconsistent with the actual level of candidates. Slow down and be more accurate, and you can get a little more points; On the contrary, if you hurry up and make mistakes, you will not get points if you spend time.

5 The relationship between difficult problems and easy problems

After you get the test paper, you should read the whole volume. Generally speaking, you should answer in the order from easy to difficult, from simple to complex. In recent years, the order of examination questions is not completely difficult. When answering questions, we should arrange the time reasonably. Don't fight a "protracted war" on a stuck question, which will take time and won't get points, and the questions that can be done will be delayed. In recent years, mathematics test questions have changed from "one question to many questions", so the answers to the questions have set clear "steps". Wide entrance, easy to start, but difficult to go deep and finally solve. Therefore, seemingly easy questions will also have the level of "biting hands", and seemingly difficult questions will also be divided. So don't take the "easy" questions lightly in the exam, and don't be timid when you see the "difficult" questions of new faces. Think calmly and analyze carefully, and you will get the due score.

In fact, there will be some people around you. They are top students in the eyes of teachers, and they are especially good at solving problems, especially difficult problems. He can always come up with some wonderful ways to solve these problems. However, in the senior high school entrance examination or college entrance examination, these so-called top students always fall into the average score of the whole city or the whole province, and do not show their advantages in this subject. Why? Because they only pay attention to the comprehensive application of some knowledge and thinking methods, but ignore the foundation and the most important quality of doing math problems-carefulness. It seems that everyone has such a misunderstanding that as long as the questions they usually practice are much more difficult than the senior high school entrance examination, they will certainly do well in the exam, which is all wet! In fact, everyone thinks that grasping the foundation means mastering the basic knowledge. I think it is more important to cultivate a mentality of attaching importance to the foundation. Usually, I despise those small questions from my heart, so I seldom do such questions, and I have no experience and feeling in doing such questions at all. Therefore, when you encounter minor problems in the big exam, you often capsize in the gutter. As we all know, math test is often an experience and a state of mind. This is the kind of problem you have done. You were more concerned at that time. Just be careful. Therefore, many awesome people, regardless of basic knowledge or difficult problems, have no problem, but are often squeezed out in the big exam. Why? Because they usually do less basic questions, they are full of contempt and don't pay attention to them at all, so in the big exam, those foundations always set some easy traps to let them jump down and then steal their scores. It's that simple. Therefore, especially those students who get good grades, they must stabilize their mentality, do those basic questions well, be kind to them and respect them, so that they will not have a hard time with you in the big exam.

6. About the mentality of studying at ordinary times: under the guidance of the correct method, study hard and be indifferent to scores and rankings. Just like the characters in the martial arts novels in TV dramas, they think that Laozi is the best in the world all day, so we are generally this way than that. There are only two endings, either possessed or killed by others, which is very tragic. We should be as indifferent to fame and fortune as sweeping monk in Tianlong Babu, and naturally achieve perfection. Take a student recommended by my colleague as an example. He took three math classes and got the first place in his class in the mid-term exam. There is a simple reason. He is correct, obedient and practical, and doesn't care about scores and rankings. After winning the first prize, he began to feel uneasy and began to doubt himself. He thinks this score is better than mine and that one is better than mine. How can I be the first in the exam? What if I can't get the first place in the next exam? Apply this principle to your life: when you go out into the society, the more you want to get promoted, the more you want to compete for fame and fortune, and the situation will only make you worse; Down-to-earth, do a good job, and help others after work, naturally there will be unexpected gains. Usually, when we study mathematics, just like watching martial arts novels, we just watch the excitement and don't pay attention to the doorway at all. Summarize this principle, and then summarize it with Tianlong Babu as the motif (typical example) and apply it to the sub-topic. This is the true meaning of mathematics!

7. In the whole examination process, we should maintain a mentality of "emphasizing the process and neglecting the results", and the wall will be pushed by everyone, and we will be rigid without desire.

It takes sweat to get good grades, but it is more important to master the correct learning methods. Too many diligent students failed in the exam, and I think blindly increasing the intensity of study is often counterproductive.

Of course, you will encounter many difficulties in your usual study. Everyone should have a healthy attitude to face these challenges, and the eagle that has experienced the storm can fly to a broader blue sky. Facing the study life with confidence, there is no unattainable ideal.

Finally, I wish everyone a dream come true and become the first! ! !