In primary and secondary schools, we will encounter such a situation. When students ask questions to teachers, some teachers often say: Please read the questions twice; Please talk about the problem; Please copy the question again; Wait a minute. What these teachers want to express is a meaning. Please read it again and understand it again.

We tell a true story. In universities, a "Mathematical Modeling Competition" is held every year. The problems in the competition are all practical problems. It takes three people to work for three days to complete a "paper" to solve the problem. You can use all kinds of books, network resources and tools (including computers and software). 1993 or 1994 Capital Normal University formed a team for the first time and asked us to be the instructor. We are very embarrassed. Students from Capital Normal University have to take the exam together with students from Peking University and Tsinghua. The gap is obvious and diverse. Our analysis shows that the biggest gap is the habit and ability to learn and understand mathematics independently. We changed the way of tutoring, let students choose the content, students say, we listen. At the beginning, we always said: Sorry, we didn't understand, please prepare again. Some students have said it four or five times, and when we think he really understands it, we will learn something else. This method is very good. After most students go through this process once, it will be easier to report other content. These students got good grades in the competition.

When learning a foreign language, you have a basic ability: reading comprehension. We think "math reading" is also very basic in math learning. In recent years, we have come into contact with the teaching practice of some primary and secondary schools, and there are fewer and fewer requirements and opportunities for primary and secondary school students to read mathematics independently. Teachers have good intentions. In order to improve students' test scores as soon as possible, to "talk more" and to "save time", teachers have done too much instead of students. We hope students realize that improving math reading ability is one of the basic skills to learn math well. We have done a survey, and in the papers of geological subjects, the number of mathematical formulas appears as many as 6 times per page on average. There are similar situations in other disciplines. In order to better illustrate the importance of mathematics reading in primary and secondary schools, we take mathematics "application problems" as an example to illustrate.

In mathematics teaching in primary and secondary schools, "application problem" is often a difficult point. Why is it difficult? There are two main reasons. One reason is that the background is rich, and they are all quadratic equations with one variable. But it can be displayed in various backgrounds, and it is difficult to define it as a question. If it is classified as "application problem of quadratic equation in one variable", it seems that it is not classified. It will be very complicated if it is classified from the background.

The second reason is that questions and conditions are not as standardized as traditional mathematical exercises, and sometimes it is necessary to define "required conclusions and conclusions to be proved" from narration. The relationship between "conditions" and "conclusions" is not as "decisive" as traditional mathematical exercises, that is, conditions are indispensable. In this way, it is necessary to analyze and judge which conditions are useful and which conditions are not, and the basis for analysis and judgment varies from topic to topic. These are not in line with the tone of primary and secondary school teaching-question type.

The application problem is "difficult" because it requires the ability of "mathematical reading comprehension", "difficult" because this ability can not be cultivated suddenly and can not be easily modeled, and "difficult" because teachers can not replace it.

Application problems, including mathematical modeling, have two educational functions. On the one hand, we can understand the relationship between mathematics and daily life, the relationship between mathematics and other disciplines, the role of mathematics in social development and the value of mathematics. On the other hand, from another angle, we can understand the process of doing mathematics. Mathematics is not only from concept to concept, from theorem to theorem, from some results to a new result; Mathematics has a background, which contains profound mathematical connotation and plays an important role in mathematical thinking. There will be a process in doing mathematics, which is a very interesting process. We need to find problems, guess, analyze and seek conditions, and we will constantly correct them, even repeatedly, and so on.

"Mathematics reading comprehension" ability is a basic ability, and both teachers and students should attach importance to it. It takes a long time to improve this ability, and teachers should provide different suggestions for different students.

There is a cognitive obstacle in mathematics teaching in primary and secondary schools. Some people think that "learning mathematics is doing mathematical exercises", while others think that "the ability to do problems is real, and everything else is empty." This view is reasonable, especially when dealing with exams. The ability to do mathematical exercises is an important aspect of reflecting mathematical ability. Doing exercises helps to understand some math skills and methods. However, the study of mathematics contains more contents, which we have talked about a lot before.

It is suggested that teachers should give students more opportunities to improve their "mathematical reading comprehension ability" for students with different levels and characteristics. Many teachers have accumulated some good experience in this field, for example, let students read textbooks and collect reference materials. In reading, let students think about the formation process of "some important concepts", the knowledge structure of some chapters, the internal relations of different concepts (such as functions and series), etc., and encourage students to write their own ideas into reports.

I hope students can broaden their thinking. Besides doing problems, they can also ask some questions worth thinking about and form the habit of thinking. When we were studying in the Department of Mathematics of Peking University, we asked Mr. Ding a question, which roughly means: What kind of students are good students? Teacher Ding's answer will never be forgotten. "Students who have no problems may not be good students." For many students, there is probably nothing worth thinking about except the exercises that they can't do. In the reading of mathematics, we should constantly ask questions to deepen our understanding of mathematics.

(2) Develop good math study habits.

In this curriculum reform, a three-dimensional goal is put forward, in which "process" is also regarded as a goal. "Study habit" embodies this process well.

What is a study habit?

Some students do their homework (usually exercises) when they come home from school, and even if they have finished, they have completed their learning tasks.

Some students, after returning home, first carefully read the textbooks taught by the teacher, then do their homework and think about what they learned today and what they learned before.

Some students have the habit of summing up. When learning a paragraph, they must sort it out and write it down.

Some students don't like writing and like thinking. They often sit there in a daze, repeating the memories they have learned.

……

Different students have different study habits. Developing good study habits suitable for your own situation will improve your study efficiency, and this study habit will naturally be maintained and will benefit you for life.

Mathematics learning has its own characteristics. For example, many people like to draw pictures when explaining mathematics, and always use the most intuitive and vivid language to explain the essential content; Some people always like to choose some familiar examples when explaining abstract mathematical concepts, and they will express abstract concepts clearly at once; Some people always give people a sense of wholeness when teaching mathematics. Naturally, source, process, result and application are indispensable. Express abstract concepts with intuitive images; Use concrete examples to understand general things; Continuously form an overall knowledge framework; Wait a minute. These are very good habits.

The formation of these good habits takes a long time to accumulate, and teachers are consciously or unconsciously influencing students with their own habits. I hope teachers can do this more consciously and actively. I also hope that students will become conscientious people, form some effective good habits suitable for their own conditions, change some bad habits and improve learning efficiency.

(3) Learn to "take"-active learning

From the teacher's point of view, I always hope to do everything possible to give my things to students. Some students don't know how to accept these things; Some students accept everything, good or bad; Some will pick and choose, the good ones will stay, and the important ones will be put away; Wait a minute. However, generally speaking, teachers like students who will take the initiative to "take".

We often say, "It is better to teach people to fish than to teach them to fish." Teachers generally think more about how to teach fish, which is very challenging. The "good study habits" mentioned above are the category of "catching fish".

"Teach them to fish" has two aspects, one is the method, and "good study habits" is the method; The other is motivation, such as curiosity, interest, ambition and understanding the value of mathematics. The two are inseparable, and "confidence" reflects the connection between them. It takes some effort to learn math well. When you encounter difficulties, you should stick to it. Some of our masters or doctors often encounter some obstacles when doing their papers. In addition to analyzing and discussing together, we have been asking for "persistence". This process can not only help them build confidence, but also "force" them to sum up "methods". Many excellent teachers are resourceful in this respect.

From the perspective of students, the main task of students is to learn, not only to learn "knowledge", but also to turn other people's knowledge into their own; We should also learn to "seek knowledge" and keep getting what we need. The two also complement each other. Need to think. For example, when doing problems, some students have a good habit. When they are finished, they should often think about this problem and make an evaluation of it. Is that a good question? What did you leave me? These ideas make their study "get twice the result with half the effort", which is how they seek knowledge.

We hope to combine "teaching and learning" and establish teacher-student interaction in this respect, which will be our glory. Teachers should try their best to provide students with more opportunities to improve their initiative, help students develop their potential, give different suggestions to non-students and let more students get started as soon as possible. Turn passivity into initiative.

(4) independent thinking and discussion.

Learning mathematics requires independent thinking. We need to think about the background, problems, concepts, theorems, applications and their relationships, so that they can naturally stay in our minds. We also need to do the problems and exercises independently, even if we ask others, and finally, we need to do them ourselves.

At present, various forms of seminars have become a basic working mode of learning mathematics. In the teaching of graduate students and some undergraduates, more and more seminars are used. The discussion forms are different, the levels are different and the number of people is different. But the basic form is the same, and there are clear discussion questions. Members attending the seminar should think carefully, prepare in advance, have special reports, and fully discuss and communicate.

This form can also be used for reference in primary and secondary schools. Teachers and students organize together, and everyone benefits.

With the help of the Internet, a number of special discussion platforms have emerged, especially some "famous teacher studios". It would be better if there were more discussions in this form. This is the greatest convenience brought by information technology, and we should make full use of it.

★ How can I learn math well?

The answer to this question seems simple: just remember theorems and formulas, think hard and ask questions, and do more questions.

Actually, it's not. For example, some students can recite the bold words in the book word by word, but they can't use them. Some students do not attach importance to the process of knowledge and methods, and memorize conclusions mechanically; Some students are too arrogant to think and speak, but when it comes to writing and calculation, they are full of loopholes and mistakes. Some students are too lazy to do the problem, thinking that it is too difficult, too boring and too heavy a burden; Some students did a lot of exercises and read a lot of counseling books, but their grades just couldn't get up. Some students failed to review, learned a paragraph and lost a paragraph.

There are two reasons: First, the problem of learning attitude: some students are ambiguous in learning, unable to tell whether they are enterprising or retreating, insisting or giving up, maintaining or improving, their determination to study hard is often shaken, their learning energy is also very limited, their thinking is usually passive, shallow and extensive, and their academic performance is always stagnant. On the contrary, some students have clear learning goals and strong learning motivation. They have indomitable will, the spirit of hard study and the consciousness of independent study. They always try their best to solve the difficulties encountered in their studies and take the initiative to consult their classmates and teachers. They have good self-awareness and the ability to create learning conditions. Second, the problem of learning methods: some students don't ponder the learning methods at all, passively follow the teacher, take notes in class, do homework after class, cope mechanically, and have average grades; Some students try this method today, and try that method tomorrow. They are "in a hurry to see a doctor", and they never seriously understand the essence of learning methods, nor will they integrate various learning methods into their daily learning links to develop good study habits. More students have a one-sided or even wrong understanding of learning methods, such as what is "knowing"? Is it "understandable" or "able to write" or "able to speak" This kind of evaluative experience is very different for different students, which affects their learning behavior and its effect.

Thus, the correct learning attitude and scientific learning methods are the two cornerstones of learning mathematics well. The formation of these two cornerstones can not be separated from the usual mathematics learning practice. Let's talk about how to learn mathematics well on some specific problems in mathematics learning practice.

First, mathematical operations.

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, solving equations and so on. The poor operation ability of junior high school will directly affect the learning of senior high school mathematics: judging from the current mathematical evaluation, accurate operation is still a very important aspect, and repeated mistakes in operation will undermine students' confidence in learning mathematics. From the perspective of personality quality, students with poor computing ability are often careless, unsophisticated and low-minded, which hinders the further development of mathematical thinking. From the self-analysis of students' test papers, there are not a few questions that will be wrong, and most of them are operational errors, and they are extremely simple small operations, such as 71-kloc-0/9 = 68, (3+3)2=8 1 and so on. Although mistakes are small, they must not be taken lightly, let alone left unchecked. 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 complex operations, 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; Less mental arithmetic, less skipping rope, and clear draft paper.

Second, the basic knowledge of mathematics

Understanding and memorizing the basic knowledge of mathematics is the premise of learning mathematics well.

★ What is understanding?

According to constructivism, understanding is to explain the meaning of things in your own words. The same mathematical concept exists in different forms in the minds of different students. Therefore, understanding is an individual's active reprocessing process of external or internal information and a creative "labor".

The standards of understanding are "accuracy", "simplicity" and "comprehensiveness". "Accuracy" means grasping the essence of things; "Jane" means simple and concise; "All-round" means "seeing both trees and forests", with no emphasis or omission. The understanding of the basic knowledge of mathematics can be divided into two levels: first, the formation process and expression of knowledge; The second is the extension of knowledge and its implied mathematical thinking method and mathematical thinking method.

★ What is memory?

Generally speaking, memory is an individual's memory, maintenance and reproduction of his experience, and it is the input, coding, storage and extraction of information. It is an effective memory method to try to recall with the help of keywords or hints. For example, when you see the word "parabola", you will think: What is the definition of parabola? What is the standard equation? How many properties does a parabola have? What are the typical mathematical problems about parabola? You might as well write down your thoughts first, and then consult and compare them, so that you will be more impressed. In addition, in mathematics learning, memory and reasoning should be closely combined. For example, in the chapter of trigonometric function, all formulas are based on the definition and addition theorem of trigonometric function. If we can master the method of deducing the formula while reciting it, we can effectively prevent forgetting.

In a word, sorting out the basic knowledge of mathematics in stages and memorizing it on the basis of understanding will greatly promote the learning of mathematics.

Third, solve mathematical 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) Select a tutorial or workbook that is synchronized with the textbook.

(2) After completing 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) guarantee the practice time of about 1 hour 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.

Fourth, mathematical thinking.

The integration of mathematical thinking and philosophical thinking is a high-level requirement for learning mathematics well. For example, mathematical thinking methods do not exist alone, but all have their opposites, which can be transformed and supplemented each other in the process of solving problems, such as intuition and logic, divergence and orientation, macro and micro, forward and reverse. If we can consciously turn to the opposite method when one method fails, there may be a feeling that "there is no way to doubt the mountains and rivers, and there is another village." For example, in some series problems, in addition to deductive reasoning, inductive reasoning can also be used to find the sum formula of general formula and the first n terms. It should be said that understanding the philosophical thinking in mathematical thinking and carrying out mathematical thinking under the guidance of philosophical thinking are important methods to improve students' mathematical literacy and cultivate their mathematical ability.

In short, as long as we attach importance to the cultivation of computing ability, grasp the basic knowledge of mathematics in a down-to-earth manner, learn to do problems intelligently, and reflect on our own mathematical thinking activities from a philosophical point of view, we will certainly enter the free kingdom of mathematics learning as soon as possible.

I remember someone once said. ...

For people majoring in mathematics, Chinese is often better, so it is better to regard mathematics as a language and literature ... almost.

First, think about how you learn Chinese well. Is it because of your own interest or because you have read widely since you were a child?

Then, for mathematics, whether you understand it or not, memorize the formulas and common methods first. When you can't do the problem, write down the formula first. See if the formula can be used. Not yet. Try more.

In fact, mathematics can only be integrated on the basis of rote memorization. Personally, I feel that no matter what you do, you should pretend or force yourself to be interested. If you master the initiative, you will naturally have no problem.

Besides, you haven't learned much as a primary school student now. You can rely on more than 70 to show that the foundation has not stopped. It is easy to improve your grades.

Trust you ~ ~ ~ ~