Generally speaking, mathematics learning can be divided into two levels: one is to master basic knowledge, and the other is to deepen knowledge.

The first level is that every student who studies advanced mathematics must do well in the exam; The second level is obviously necessary for students who want to learn advanced mathematics well, especially for science and engineering students who need to take the postgraduate entrance examination.

Now let's talk about the specific learning methods:

1. Understand knowledge points.

The knowledge points involved in advanced mathematics are: definition, theorem and formula.

1) What do you need to know about the definition?

A) First of all, we should grasp the basic meaning of the definition from the text.

B) Secondly, understand what knowledge the definition involves (what we have learned). For example, when we say "region", then this definition is closely related to interval and set. We can learn by comparison. We should not only analyze the similarities or connections of related concepts, but also pay attention to the differences or differences.

C) Define matters needing attention, or define the elements involved. If you define a set, you need to pay attention to the certainty of the elements in the set. Like tall students, it is hard to say how tall they are in this set, so they are not certain.

D) What is the nature of the definition? Fully understanding these properties can often help us better grasp the true connotation of the definition.

2) Theorem. A), b) and c) have the same definitions as attention.

D) conditions involved in the theorem. This is very important. Many students do not pay attention to the conditions for the existence of theorems, and as a result, theorems are used everywhere in solving problems, and the results often lead to wrong conclusions.

E) If you want to master the theorem well, you must do some related topics. Only in this way can we truly grasp its connotation. If we want to understand theorems in depth, we often have to do some topics involving multiple theorems or formulas. It needs to be understood in practice. If you learn theorems, but you can't do problems, then the knowledge you have learned is dead, and such knowledge is of little use.

3) Formula.

Some formulas are simple, such as derivative formula. As long as you understand the definition of derivative clearly, then using derivative formula is similar to applying multiplication formula.

But some formulas are more complicated, such as Gauss formula in multivariate calculus. These formulas are not so much formulas as theorems. For such a formula, when learning, you can refer to the learning method of the theorem introduced above.

2. Digest and consolidate knowledge points.

In this regard, in addition to doing the above-mentioned 1. The best way is to do exercises. Now we might as well introduce the problem-solving aspect.

Step 3 solve the problem.

Whether learning elementary mathematics or advanced mathematics, it is inseparable from solving problems. But in fact, many students feel that they have done a lot of problems and the effect is not good. Why?

We believe that,

1) First of all, do a good job in the topics in the textbook. These topics are often specially designed to digest and understand definitions, theorems and formulas, and are basic topics. So you must pass every question. These topics are often not difficult, but the digestion and understanding of basic knowledge points can not be underestimated. Some students are not sure about this. Typical negative examples are:

A) Because of the short time, or some problems can't be done, I copied my classmates' homework;

B) No matter whether he is right or wrong in the topic, deal with it first and give the homework to the teacher, which means that he has finished his usual homework, so the teacher will not deduct my usual score.

C) Don't do detailed argumentation and analysis, and calculate the answers to some questions; For some topics, let the wind out first, say what is obvious (actually not obvious), and then declare the original proposition established.

These are irresponsible practices. Some students may say, alas, there will be a meeting in the student affairs office today, or the fellow villagers are here today. Anyway, I really don't have time today. I'll make it up tomorrow. In fact, if you can't finish today's task, don't imagine that you can not only finish tomorrow's work, but also make up for what you left behind today. In the long run, more and more tasks will be left behind, and the more difficult it will be to study in the future.

2) solving problems cannot solve problems for the sake of solving problems.

Some students may be able to solve a problem when they encounter the same problem in the future, but they don't know how to do it if this problem is properly transformed. This kind of situation belongs to the situation of learning without thinking and solving problems in order to solve them. To solve a problem well, we should not only solve a problem, but also summarize the solutions to all similar problems. By analogy, you are not afraid that the questioner will change his moves. We hope that students must think more when solving problems. Every time they do a problem, they should think about it. What kind of problem can this problem boil down to? In this way, doing a problem is equivalent to solving one or more problems.

3) broaden your horizons.

Some students study well and often come up with all kinds of strange problems, which can often be solved. Why? Is that they have accumulated a lot of problem-solving skills. As mentioned in the martial arts novels, some people have created a new kind of martial arts and think that no one in the world can beat them. But such a martial arts expert, who had never seen any scenes, sealed all the portals with magical powers, secretly observed his martial arts routines, and finally found out the details of the other side's martial arts, so he succeeded in one blow. When it comes to solving math problems, the students are familiar with all kinds of problem-solving skills, so they tried various methods and finally found the answer.

How can I learn problem-solving skills? The first is to sum up yourself. In solving problems, thinking more and comparing with the knowledge learned in the past can often form a family and acquire problem-solving skills that are difficult to see in other books. The second is to obtain problem-solving skills through books or network resources.

The more problem-solving skills you master, the more you can cope with all kinds of problems.

On our website, we have collected tens of thousands of exercises, many of which are classics. Some topics also summarize the problem-solving skills in particular. May wish to find some topics to do and exercise through the home page to the columns where various courses are located.

Answer a classmate's question:

1. I can often solve difficult problems, but I often lose points on basic problems. Is there any way?

This is mainly because the basic skills are not solid. We can imagine a high-rise building whose building materials are all high-quality steel, but it may still collapse. Why? Because some places have a lot of floating soil on the ground and soft geology. It is difficult to build a high-rise building in this place, no matter how good the building materials above you are.

Of course, some students think that the basic skills are solid, but they are slightly careless. Actually, it is not. Imagine if a college student is asked to calculate 1+2, will he make mistakes because of carelessness? The answer is no, of course, because he has a solid basic skill in this field.

2. I like some technical topics, which are very enjoyable and fulfilling. The topics in those textbooks are so corny that I can know the result at a glance. Is this view appropriate?

The answer is: no!

It's like a person never goes out, doesn't do any exercise and takes tonic every day. Will there be good health like this? Although some topics in the tutorial are generally not difficult, doing them is our homework. Even if you can do it easily, you might as well do it. When we think we are doing this, some topics may have been proved completely wrong. Even if you can be sure that you can do it, you might as well do more analysis and summary, and even draw up a related topic for yourself on the basis of this topic. Let's imagine that literati in the past liked to write couplets in order to show their talents. Are those who are good at couplets just making couplets for others? That's not true. These people often want to find out what is good at home, and once they find something good, they will try to match it at home.

3. Is learning advanced mathematics similar to learning elementary mathematics?

There are many similarities in learning methods. But there are also many differences. Specifically, there are the following points:

A) Elementary mathematics focuses on solving practical problems, such as calculation; In addition to calculation, advanced mathematics needs to be understood in theory. A thorough understanding of a theorem is often directly related to learning advanced mathematics well.

B) Advanced mathematics involves a lot of contents, and it often takes a semester to learn a completely different subject of advanced mathematics. Therefore, the repeated excavation of a course in the past will not be able to schedule in time. So we should be able to digest and understand knowledge as soon as possible.

C) The teacher-led model should be transformed into the student-led model as soon as possible.

In middle school, teachers have arrangements for what to learn and how much to learn every day. Basically, just finish the task assigned by the teacher. It will be very dangerous if we continue to study like this in college. It's hard to even guarantee that all the doors will pass. When studying advanced mathematics, everyone should take the initiative to study. In addition to completing the tasks assigned by the teacher, we should ponder and think about the knowledge in the book repeatedly after class, so that our understanding will be profound. Moreover, just doing the questions in the textbook is not enough in terms of the amount of questions, and some extracurricular questions need to be supplemented appropriately.

D) The thinking of elementary mathematics research is completely different from that of advanced mathematics. The problem solved by elementary mathematics is mainly poverty; The focus of advanced mathematics is infinite problems. For example, when we are in one-dimensional calculus, we will soon come into contact with the basic concept of limit. The introduction of this concept indicates that our learning ideas need to be transformed into infinite problems immediately. Many problems, when there is poverty, may be wrong from an infinite perspective. For example, we generally think that the set {1, 2, 3, ..., n, ...} is obviously smaller than the set formed by all rational numbers; But when we study with advanced mathematics theory, the number of these two groups of numbers is the same. Why? I'll leave this question for you to think about after learning advanced mathematics.

4. Is the learning method of college mathematics the same as other subjects?

In terms of student-centered learning methods, all college courses are the same.

But specifically, mathematics still has its own characteristics. We have talked a lot about this. Add it here. Mathematics has a strong continuity, so we must not stop learning a part in the middle, or put the contents of a part in the middle for a while and make up later. If we are unfortunate enough to miss some courses, we will find it painful that we can't make up for the tasks that have been missed for a month.

Are you satisfied with the above answers?