Methods of learning advanced mathematics
Learning advanced mathematics needs a kind of spirit, in the words of the great mathematician Hua, that is, the spirit of "learning to think and never giving up". Because of the characteristics of advanced mathematics itself, students can't master it all at once. Some contents, such as the continuity and discontinuity of functions, the substitution method of integral, and the method of step-by-step integration, are difficult to master at the moment, and each student needs to ponder, think and train repeatedly. By comparing the positive and negative examples, we can learn some truth from them, and let us go from ignorance to a little knowledge to basic mastery. Here I only talk about some methods of learning advanced mathematics in combination with general learning methods for reference. 1. "Learning, thinking and learning" is a big model for learning advanced mathematics. The so-called learning includes learning and asking questions, that is, learning and asking questions from teachers, classmates and themselves. Only by asking questions in study and study can we digest the concepts and theories of mathematics. Method. The so-called thinking is to learn the content, get the essence through thinking and processing, and grasp the essence and essence. Hua Qin's thinking and thinking method of learning mathematics from coarse to fine is worth learning from. The so-called learning, as far as advanced mathematics is concerned, is to do problems. Mathematics has its own characteristics, and exercises are generally divided into two categories. First of all, every chapter and section is accompanied by basic training exercises. This kind of problem is relatively simple and not difficult, but it is very important and is the basic part. Knowledge is broader, not limited to this section of this chapter, and a variety of mathematical tools are used in solving problems. Mathematics practice is an extremely important link to digest and consolidate knowledge, otherwise it will not achieve the goal. Second, pay close attention to the foundation and proceed step by step. In any subject, the basic content is often the most important part, which is related to the success or failure of learning. Advanced mathematics itself is the foundation of mathematics and other disciplines, and it has some important basic contents, which are related to the overall situation. Taking differential product division as an example, the limit runs through the whole calculus, and the continuity and properties of the function run through a series of theorem conclusions. Derivation and integration of elementary functions are related to the latter three subjects. Therefore, we should work hard from the beginning and firmly grasp these basic contents. To learn advanced mathematics, you should study and practice step by step, and the door to success will certainly open to you. Third, classify and summarize, from coarse to fine. The general principle of memory is to grasp the outline and remember it in use. Classified summary is an important method. The classification method of higher mathematics can be summarized into two parts: content and method, and illustrated by taking representative problems as examples. When classifying chapters, we should pay special attention to some conclusions drawn from the basic content, that is, some so-called intermediate results, which often appear in some typical examples and exercises. If you can master more intermediate results, you will feel relaxed when solving general problems and comprehensive training problems. Fourth, read a reference book intensively. Practice has proved that under the guidance of teachers, we can accurately grasp a reference book and read it intensively. If you can read a representative reference book well, you can easily read other reference books. Fifth, pay attention to learning efficiency. According to practical experience, the mastery of mathematical methods and theories often needs to be greater than 4, otherwise practice makes perfect, leading to analogy. It is impossible for a person to master what he has learned through one study, and it needs to be repeated many times. The so-called "learning from time to time" and "reviewing the old and learning the new" all mean that learning has to be repeated many times. The memory of advanced mathematics must be based on understanding and skillfully doing problems, and rote memorization is useless. There is no royal road to science, but "there are obstacles in science, and efforts can pass." "How many beats can life have?" "Life can always hit a few times!" Every college student should be able to "try his luck" in advanced mathematics. First of all, the knowledge of analytic geometry is necessary. Only the establishment of knowledge system can make you know more about this knowledge. Second, you should learn to make full use of the knowledge of plane geometry in junior high school. Analytic geometry is, in the final analysis, a kind of calculation, which itself is a system established to solve plane geometry problems. The test is who can calculate accurately and quickly, so you should try to reduce the steps and time of calculation to be faster and more accurate, which requires knowledge of plane geometry. Sometimes when used, the topic will become very simple. Third, it is a familiar method, and several common methods for solving point trajectories must be familiar. Also, sometimes when doing problems, don't pursue some ideas too much. The definition and essence of regression is also a good method, and the simplest is the best. Fourth, doing more questions is the quickest way to get familiar with these methods and skills, and you don't have to practice a lot of calculations, but more skills.
I believe that if you find a way to learn, you will get good grades! Personally, I think that learning mathematics should actually include two parts, namely, mathematical discovery+mathematical proof. But unfortunately, most of the current textbooks throw away the mathematical findings on the grounds of rigor. In this way, the textbook is likely to be written like this: definition 1, definition 2, proof 1, proof 2, example 1, definition 3, definition 4, ... very strict. However, the object of writing books is people, mostly beginners. The consequences of formal dictionary writing are mostly chaotic. After reading it for a long time, I don't know what to say. In this way, I will probably have fear, disgust and even disgust for mathematics. As we all know, when I was studying mathematics in college, it was almost impossible for a person to learn mathematics well if he was not interested in it or even rejected it. Only then did I realize that I could only do the problems designed by others. When I study math by myself, I will find that there is no problem and I feel very confused. I have no idea, no direction, no inspiration and so on. As a result, I mostly lamented my poor mathematical talent and low IQ.
To tell the truth, except for a few geniuses, is the IQ gap between people really that big? The same family, they are closely related, and their IQ should be similar. But the gap in mathematics level is not an order of magnitude. As far as SCIbird is concerned, he is not the smartest in the family now. But my relatives on my father's side and my mother's side are not as good at math as me. And I have established a far-ahead advantage in mathematics since junior high school. I never thought this mathematical advantage was innate.
I summed up my own experience: diligence+attitude+method.
The first is diligence. If genius is born, we can't change it. So hard work can change it.
The second is attitude, low-key, open-minded and enterprising. Don't be proud of speaking fast. If you want to improve your math, you have to put on airs. Instead of trying to get a bargain with your mouth, sit down and read more books.
Method, that may be a long story. I only say one thing: learning mathematics should include two parts: mathematical discovery and mathematical proof.