In my opinion, mathematics is actually a very wonderful and interesting science. As long as you have a pair of eyes that are good at discovering and dare to discover, you can find the charm of mathematics and become interested in it. Interest is the best teacher. If you are interested in mathematics and are determined to learn it well, how can you not learn it well?
Textbooks are very important for mathematics. Many of the problems we do are examples in textbooks or their "variants". As long as we spend a little time reading textbooks, we will easily get these questions; On the contrary, if some basic concepts and theorems are vague, not only will the basic problems lose points, but it is even more impossible to do the difficult problems well. Mathematics is very logical and analytical, which can be said to be a purely rational science, requiring clear thinking, so basic knowledge is very important, especially for students who are not particularly good at mathematics.
The following are some points that I personally think are very necessary in the process of mathematics learning:
1, step by step. Mathematics is an interlocking subject, and any link will affect the whole learning process. Therefore, don't be greedy when studying. You should pass the exam chapter by chapter, and don't leave questions that you don't understand or understand deeply easily.
2. Emphasize understanding. Concepts, theorems and formulas should be memorized on the basis of understanding. My experience is that every time I learn a new theorem, I try to do an example first, and see if I can use the new theorem correctly without looking at the answer. If not, compare the answers to deepen the understanding of the theorem.
3. Basic training. You can't learn mathematics without training. Usually do more exercises with moderate difficulty. Of course, don't fall into the misunderstanding of dead drilling. Familiar with the questions that are often tested, and the training should be targeted.
4. Mark the key points. When reading textbooks, you can use bright-colored strokes when you encounter good problem-solving methods or key contents, so that it will be clear at a glance in future review.
Finally, I want to talk about the exam-taking skills of mathematics. To sum up, it is "easy first, then difficult". We often have the experience that when we are sober-minded, some difficult problems will be easily solved; On the contrary, when the mind is confused, some simple questions will waste a lot of time. It is inevitable to encounter obstacles in the exam. There are two possibilities to stop. First, it took me a lot of effort to finally figure it out, but because I spent a lot of time, I either didn't have enough time to finish the problem, or I was worried that I didn't have enough time, so I couldn't even do the simple problem for a while. Second, it still hasn't been done. The result is not only a waste of time, but also the following questions are not finished. The easier it is, the more confident you are, the clearer your mind will always be, or you will eventually solve the problem, or at least ensure that you will not lose points on the questions you can do.
June 5438 +2002 10, my advanced mathematics textbook only got 75 points. I looked at the foot-high draft paper and recalled the days and nights of these three months. I can't help sighing! So I wrote some of my experiences, but I haven't sorted them out yet. I hope that friends who are interested in reading will criticize and communicate.
In August, 2002, I decided to teach myself computer application major. My wife didn't object, supported or attacked, but only paid for it. I signed up for the high number and C++ that month. I chose difficulties and hopes. Most self-taught candidates have jobs at the same time. I am a policeman. Not only do I have to work, but I also have to work at night. I don't have fixed study time, I can't attend classes, and I don't have time to attend classes. It has been more than nine years since I failed in the college entrance examination in July. 1993. Mathematics retaking is difficult and takes a long time, accounting for 10 credits, comparable to the college entrance examination.
Experience: After you finish all the exercises in the book, even if you can't do them, you should copy down the answers.
Otherwise, how can we use up that foot-high draft paper! I spend a lot of time doing problems. When I am not on duty, I often do calculations until late at night and early the next morning. If you encounter a question that you can't answer, just look at the reference answer and calculate it again.
A lesson: only do exercises, not examples.
In fact, my first experience was the heaviest failure! Near the exam, I began to do the questions of the past years, and after doing it, I had an epiphany. The first is an example, the second is an example, and the third is an example! Do you remember the last question of this self-study exam? It's an example! There are examples or "variants" of big questions over the years! In fact, the "total exercises" in our textbooks are very difficult, and each question takes a lot of time! Generally speaking, it is not that difficult for us to take the self-taught exam. I usually spend the most time on "exercises, self-test questions and general remarks". In order to finish it, I have to reduce the time I spend reading books and examples. Get twice the result with half the effort! (pig! So I suggest latecomers: pay attention to examples and do it yourself. In the exercise, important chapters should be done, and a few parts should not be done. The self-test questions should be done after a chapter, and the general exercises should not be done.
Lesson 2: Full attack, no focus.
I did the textbook from beginning to end, because there were too many contents and formulas, so I forgot the front after doing it. Finally, the skull is still a paste. In fact, advanced mathematics is a very rigorous knowledge (say it again! ), from beginning to end! Several key points: limit, derivative, indefinite integral, empty solution, differential equation, there are a lot of exercises at the end of the book, and a small topic has 20 to 30 sub-questions, which is the key point.
Lesson 3: It's too difficult to see a dead end.
For example, to solve differential equations, I spent some time in the section of "second-order non-homogeneous equations with constant coefficients", but I didn't understand it first. I did a lot of problems and saw many examples before I understood what was going on! As a result, if you look at the test questions over the years, it is impossible for someone to make such complicated questions! There are many such examples, and there are various physical applications, so we will not test them at all! As for Fourier series, as long as we know the formula and the formulas on three boundaries, we can make it. As for how to come and how to apply it, leave him alone. So I have come to a conclusion: if you don't understand, you won't take the exam at all. If you can understand specious things, you should read more and practice more.
Freshmen —— Learning Methods of Advanced Mathematics
At present, when the one-year college entrance examination is over, millions of high school students stand out from their peers through their own efforts, enter their dream colleges and start studying in the new environment, the major media in society will keep repeating a topic: how can a high school student integrate into the new environment psychologically and physically as soon as possible and become a qualified freshman? Moreover, from time to time, TV news or newspapers have examples of freshmen falling asleep in the network or video games in the new environment and dropping out of school because they can't keep up with the learning progress of the university. The author believes that a senior high school student should not only adapt to the new study life from the environment and psychology, but also change his study methods. I have been engaged in advanced mathematics teaching in engineering colleges for more than 30 years. Advanced mathematics is a basic theoretical course in the teaching plan of engineering colleges and a compulsory course for freshmen. It plays a fundamental role in the study of follow-up courses of various majors and the working status of such engineers and technicians after graduation from university. For example, only by mastering the knowledge of advanced mathematics can we learn other basic professional courses, such as physics, engineering mechanics, electrotechnics and electronics, and learn our own professional courses well. For another example, when you graduate and go to work, you should always apply mathematical knowledge to solve engineering and technical problems well. Because today, with the continuous development of science and technology, mathematical methods have widely penetrated into all fields of science and technology. Therefore, it is a clear task for engineering freshmen to learn advanced mathematics well and lay a good foundation for their future study and work.
So, how do freshmen learn advanced mathematics well? The author talks about some superficial views on his own experience in teaching this course for many years for students' reference.
First, abandon the learning methods in middle schools.
After entering the university from middle school, there will be a big turning point in learning methods. First of all, they feel very uncomfortable with the teaching methods of universities, which is particularly obvious in the teaching of advanced mathematics courses, because it is a major theoretical basic course, and students are used to the imitative and unitary learning style, which has been cultivated from primary school to middle school for a long time and is difficult to change for a while.
The teaching methods in middle schools are qualitatively different from those in universities. Outstanding performance: middle school learning, students learn through imitation and singleness under the direct guidance of teachers, while universities require students to learn creatively under the guidance of teachers. For example, the teaching of mathematics in middle schools is completely in accordance with textbooks. Only the teacher is required to speak and the students are required to listen in class, and no notes are required. Teachers teach slowly and seriously, and there are many examples of calculation methods. After class, students only need to imitate what the teacher said and do some exercises. There is no need to delve into textbooks and other reference books (choosing some other reference books in the college entrance examination to enhance candidates' problem-solving ability is only the need to train their problem-solving ability), but advanced mathematics courses in universities are necessary. Students are required to take the key and difficult points in class as clues, read a lot of textbooks and similar reference books, fully digest and master the contents taught in class, and then do exercises after class to consolidate their knowledge. This is repeated creative learning. This is a hard mental work, which requires students not only to study actively and consciously, but also to restrain themselves in a relaxed environment, master better learning methods, learn what they want to learn solidly, and lay a good foundation for the study of professional courses. (To be continued)
Second, do three links well.
What is the best way to learn advanced mathematics? This varies according to everyone's study habits and ability to understand problems, but in general, we should do the following three links well. First, preview before class. This process is very important, because only by previewing before class can you know what the teacher says is difficult to understand and what is the key point, so that when you listen to the teacher with some questions, the effect will be obvious, and at the same time, you will cultivate your self-learning ability in the preview process, which will benefit you for life. It doesn't take much time to preview. Generally, a class takes about 30 to 40 minutes. You don't have to understand all the questions in the preview, just take these questions you don't understand to class. Second, listen carefully in class and take notes in class.
On the Learning Methods of Higher Mathematics
Advanced mathematics is an important basic course in colleges and universities, and it is extremely important for every college student to learn it well.
Here, I put forward some suggestions on how to learn this course well for students' reference:
First, grasp the three links to improve learning efficiency
Preview before class: know what the teacher is going to say and review the relevant content accordingly.
Second, listen carefully: pay attention to the teacher's explanation methods and ideas, as well as the process of analyzing and solving problems.
Taking notes in class is a process of listening, taking notes and thinking.
Review after class three: Be sure to recall what the teacher said that day and see how much you remember;
Then open notes and teaching materials, improve notes and communicate; Finally finish the homework.
Second, understand on the basis of memory, deepen in the completion of homework, and build a framework of knowledge structure in comparison.
Third, we should understand and deepen the learning knowledge according to the idea of "new = old+poor".
Four, "among the three, there must be my teacher", participate in the teacher's counseling, ask questions to students and discuss with each other.
Five, the basic methods to deal with mathematical problems:
A division and summation method;
The second is the method of directly finding the curve;
Three identical deformation methods:
① arithmetic addition and subtraction; ② multiplication and division factor method; ③ Integral derivative method;
④ Triangular substitution method; ⑤ Number-shape combination method; ⑥ Relational iteration method;
⑦ Recursive formula method; (8) methods of mutual communication; Pet-name ruby attack before and after;
Attending the method of reflection and verification; ⑾ builder's method; ⑿ stepwise decomposition method.
Sixth, stage review and comprehensive consolidation.
How to learn advanced mathematics? 2006-6- 13 20:26:47
On the Learning Methods of Higher Mathematics
Advanced mathematics is an important basic course in colleges and universities, and it is extremely important for every college student to learn it well.
Here, I put forward some suggestions on how to learn this course well for students' reference:
First, grasp the three links to improve learning efficiency
Preview before class: know what the teacher is going to say and review the relevant content accordingly.
Second, listen carefully: pay attention to the teacher's explanation methods and ideas, as well as the process of analyzing and solving problems.
Taking notes in class is a process of listening, taking notes and thinking.
Review after class three: Be sure to recall what the teacher said that day and see how much you remember;
Then open notes and teaching materials, improve notes and communicate; Finally finish the homework.
Second, understand on the basis of memory, deepen in the completion of homework, and build a framework of knowledge structure in comparison.
Third, we should understand and deepen the learning knowledge according to the idea of "new = old+poor".
Four, "among the three, there must be my teacher", participate in the teacher's counseling, ask questions to students and discuss with each other.
Five, the basic methods to deal with mathematical problems:
A division and summation method;
The second is the method of directly finding the curve;
Three identical deformation methods:
① arithmetic addition and subtraction; ② multiplication and division factor method; ③ Integral derivative method;
④ Triangular substitution method; ⑤ Number-shape combination method; ⑥ Relational iteration method;
⑦ Recursive formula method; (8) methods of mutual communication; Pet-name ruby attack before and after;
Attending the method of reflection and verification; ⑾ builder's method; ⑿ stepwise decomposition method.
Sixth, stage review and comprehensive consolidation.
Five principles of learning methods
Learning methods are closely related to the process, stages and psychological conditions of learning, which not only includes the understanding of learning rules, but also reflects the understanding of learning content. In a certain sense, it is also a learning method with personal characteristics. Learning methods vary from person to person, but the correct learning methods should follow the following principles: step by step, careful reading, self-satisfaction, combination of knowledge and practice, and unity of knowledge and practice.
1. "Step by step"-that is, people learn systematically and step by step according to the knowledge system of the subject and their own intellectual conditions. It requires people to attach importance to the foundation, avoid aiming too high and be eager for success. The principle of gradual progress is embodied in: first, we must lay a good foundation. Second, from easy to difficult. Third, we should do what we can.
2. "Read carefully and think carefully"-that is, according to the dialectical relationship between memory and understanding, they should be closely combined and not neglected. We know that memory and understanding are closely related and complement each other. On the one hand, only by understanding on the basis of memory can we understand thoroughly; On the other hand, only with the participation of understanding can memory be strong. To "read well", we must achieve "three": heart, eyes and mouth. To "seriously think", we should be good at asking and solving problems, and use "self-questioning method" and "people's questioning method" to ask and ask questions.
3. "Self-satisfaction"-that is, give full play to the initiative and enthusiasm of learning, tap their inherent learning potential as much as possible, and cultivate and improve their autonomous learning ability. The principle of self-satisfaction requires you not to study for the sake of learning, but to digest and absorb what you have learned and turn it into your own use.
4. "Combination of Bo and Yue"-that is, combining the two according to the dialectical relationship between Bo and Yue. As we all know, the relationship between Bo and Yue is based on Bo. Under the guidance of Yue, Bo and Yue combine and promote each other. Insist on learning from others. First, read widely. The second is intensive reading.
5. "Unity of knowledge and practice"-that is, according to the dialectical relationship between knowledge and practice, combine learning with practice to avoid learning without using it. As the saying goes, "the knower begins to do what he knows, and the walker becomes what he knows", which is effective only under the guidance of knowledge, and is blind without knowledge. Similarly, the knowledge verified by lines is insightful, and the knowledge divorced from lines is empty. Therefore, the unity of knowing and doing should pay attention to practice: First, we should be good at learning, practicing, learning and accumulating in practice. The second is practice, that is, applying the knowledge learned to practical work and solving practical problems.
Mathematics learning method
● Comprehensive review, reading thin books.
It can be seen from the content distribution of examination papers over the years that all the contents mentioned in the examination syllabus may be tested, and even some unimportant contents may appear in the big questions of a certain year. For example, in Mathematics No.1 Middle School in 1998, not only the third question was pure analytic geometry, but also two questions were combined with linear algebra to test the content of analytic geometry. It can be seen that the review method of guessing questions is not reliable, but we should refer to the examination outline, integrate our interests, and leave no omissions.
A comprehensive review is not about memorizing all the knowledge. On the contrary, it is about grasping the essence of the problem and the essential connection between the content and the method, and minimizing the things to be memorized (try to make yourself understand what you have learned, grasp the connection of the problem more, and memorize less knowledge). Besides, it's reliable not to remember. Facts have proved that some memories will never be forgotten, while others can be the basis of memorizing basic knowledge.
● Highlight key points and strive for perfection.
In the requirements of the examination syllabus, there are three levels of requirements for the content: understanding, understanding and knowing; Generally speaking, the content to be understood and the methods to be mastered are the focus of examination. In previous years' exams, the probability of these problems is very high. The same test paper has many scores in this respect. People who "guess the questions" often have to work hard in this respect. Generally speaking, they can guess several points. But when they encounter comprehensive questions, these questions contain secondary content in the main content. At this time, "guessing questions" will not work. When we talk about highlighting the key points, we should not only work hard on the main contents and methods, but more importantly, find the key contents. Cover the whole content with key content. The main content is thoroughly understood, and other contents and methods will be readily solved. In other words, grasping the main content is not to abandon the secondary content and isolate the main content, but to naturally highlight the main content from the comparison by analyzing the relationship between the contents. Such as differential mean value theorem, Rolle theorem, Lagrange theorem, Cauchy theorem and Taylor formula. Because Rolle theorem is a special case of Lagrange theorem, Cauchy theorem and Taylor formula are the generalization of Lagrange theorem. Comparing these relations, we naturally take Lagrange's theorem as the core and have a thorough understanding, and several other theorems are also well grasped from the connection. In the examination syllabus, both Rolle's theorem and Lagrange's theorem are required to be understood and are the focus of examination, and we highlight Lagrange's theorem more, which can be said to be Excellence.
Basic training is repeated.
To learn mathematics, we should do a certain number of problems and practice the basic skills thoroughly, but we do not advocate the tactic of "sea of problems" and advocate simplicity, that is, we should do some typical problems repeatedly, and one problem is changeable. Train abstract thinking ability, prove some basic theorems, deduce basic formulas and practice some basic problems. You don't need to write, just do "blind chess". In other words, we can get the exact answer. This is what we mentioned in the preface. We can complete 10 objective questions in 20 minutes. Some questions can be answered at a glance without writing. This is called well-trained, "practice makes perfect", and people with solid basic skills have many ways to encounter problems and are not easily stumped. On the contrary, when doing exercises, they are always looking for problems. Many candidates misjudge the questions they can do, which is classified as carelessness. Indeed, people will be careless, but people with solid basic skills will find out immediately when they make mistakes, and rarely make "careless" mistakes.
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