This is the most effective way to learn

Chapter 19 Are You Afraid of Math? ——Mathematics learning method for top students (2)
The so-called important exercises are examples and exercises in a unit that closely follow the key points and difficulties of the textbook and have certain mathematical methods.Proficient in the knowledge points and problem-solving methods involved in this type of exercises, you can control a type of exercises, and you can adapt to all changes with the advantage of winning more with less.Therefore, this kind of exercises must be repeated continuously to achieve the purpose of memorizing, memorizing, and being able to use.A girl who was admitted to Peking University once introduced that her method was to memorize all the examples in the textbook.

So what are some good math memory methods for top students for your reference?

Method [-], understanding memory method
Memory is based on understanding, and only things that are understood can be easily remembered and accepted by people.Conversely, things that you don't understand or don't understand well are hard to remember and often error-prone.

For example, for the algebraic formula (a+b) 2=a2+2ab+b2, some students memorize it by heart, one by one, and memorize the three items in order.This is mechanical memory, not only laborious, but also easy to forget.

Some top students understand it from the perspective of graphics: isn't the so-called (a+b)2 the area of ​​a square whose side length is a+b?If you draw this plot, you can see that the entire square is composed of four pieces, one is a2, one is b2, and two are ab. When added together, it becomes a2+2ab+b2.This is easier to remember.Some students also learned from the perspective of understanding that after the expansion of (a+b)2, there will be quadratic terms of a and b anyway, so there is no need to remember them. Just remember that there is a primary term "2ab", put In the middle is fine.

Method [-], comparative memory method

The method of comparative memory is the method of memorizing knowledge by comparing the content, form and characteristics of knowledge in mathematics learning.This method is implanted into the human brain with distinct features, coupled with rich connotations and deep impressions, and can often achieve twice the result with half the effort.

Method [-], formula memorization method

Some concepts, rules, theorems and formulas in mathematics have certain rules, and we can use popular language to compile formulas for memory.There are 54 most common induced formulas like triangles.It is very inconvenient to remember in isolation.It is easy to remember if it is compiled according to the situation of the angle as "odd changes to even and remains unchanged, and the symbols look at the quadrant".

Method four, image memory method

Intuitive image is the basis of memory, and it can be associated and abstracted with the help of certain specific graphics and function images, which is an effective memory method.

As a special teacher said, middle school students who are poor in mathematics are generally poor in understanding, retention, recall, and recognition of mathematical knowledge.That is to say, I can't remember it when I read it, I can't remember it when I use it, and I don't know it when I see it.Therefore, grasping the link of memory may be a shortcut to rewrite math scores.

take notes to learn math
Insisting on taking notes is an important means to learn mathematics well. In addition to memorizing the content of lectures, mathematics notes mainly include the following aspects:
Record problems in homework

The quality of homework can truly reflect the effect of learning and expose the defects in learning.Errors in homework can be divided into general errors and individual errors, frequent errors and occasional errors.Remembering the problems in the homework, analyzing the reasons for the mistakes, and re-establishing the correct answers is actually a process of re-learning, re-experience, and re-recognition, and it is also a process of further understanding the key points and difficulties of the textbook.By recording the problems in the homework, you can understand which knowledge you are easily confused and error-prone, and take appropriate remedial measures in time to understand and grasp the knowledge thoroughly, without leaving any tails and future troubles.Persisting in taking such notes is conducive to discovering the weak links in one's own learning in time; it is conducive to a deep understanding of the knowledge learned; it is conducive to cultivating the ability to think independently and solve problems.

The method of memorizing multiple solutions to one question
Sometimes the solution to math problems is not unique. With the continuous expansion of knowledge, there are more and more ways to solve problems.Frequently discussing multiple solutions to a problem, looking for the optimal solution, can integrate the knowledge learned and receive the effect of excellence; it can also promote the firm grasp of basic knowledge and basic skills; it can also accumulate problem-solving experience, improve the ability to analyze problems and solve problems. ability to problem.For example, the proof of the proposition "In a right triangle, the midline on the hypotenuse is equal to half of the hypotenuse" can be proved by means of the median line theorem, congruent triangles, properties of rectangles, equal inner radii of circles, and complex numbers.Practice has proved that for a mathematical proposition, analyzing it from different angles, using different basis and different methods to solve it can broaden one's thinking and cultivate thinking ability.

Remember "one method with multiple uses" and "one question with multiple changes"

To learn knowledge, you must develop a good habit of exploring the law.String together mathematical propositions with intrinsic connections to form a chain of questions, and write them down in notes according to their classification, so as to achieve the effect of "drawing inferences about other cases from one instance".If you study a question, you will be able to solve a series of questions, and if you do a question, you will be able to solve a series of questions.When encountering a new proposition in the future, I will not think about whether I have done it before, but consider what simple form it came from, and what method to use to turn it into a standard proposition, which can eliminate negative thinking patterns in the long run influence, so that the basic knowledge learned is clear, and the problem-solving is lively and not chaotic.

"One method with multiple uses" means the divergence of the proposition angle and the convergence of the solution angle, while "one problem with multiple changes" means that both the proposition angle and the solution angle diverge at the same time.Therefore, it is an effective way to cultivate creativity.

Write down life phenomena or examples related to the textbook
Mathematics is distilled from real life.It is the gymnastics of abstract thinking. To keep it youthful, life must be used as a "fulcrum".For this reason, pay attention to the phenomena in life and connect them with the content of mathematics textbooks to obtain vivid and appropriate metaphors, which can arouse learning interest, strengthen the understanding of mathematical concepts, and enhance the memory of knowledge.For example, the number of photos obtained from the changes in the positional relationship of each person in the group photo deepens the understanding of the concept of arrangement and combination; the stacking of steel pipes on the construction site strengthens the memory of the summation formula of the number sequence; The difficulty is "necessary and sufficient conditions"... Persisting in this work over time will keep the learning mood full and make learning mathematics more fun.It further improves the abstract thinking ability and improves the mathematical literacy to a higher level.

Remember the lessons learned from successful experiences and failures
In a long learning career, everyone has a lot of successful experiences and lessons from failures, which are important personal wealth.Recording them in time can help you continuously improve your learning effect and enhance your talents.

The summary of experience and lessons should be written in a simple and simple way, and talk about one aspect and experience that you feel the most deeply. You don't have to be exhaustive, but you must correctly describe (grasp) the facts and the process of their occurrence in the learning practice, and reveal the relationship between the facts. The cause-and-effect relationship of the law, put forward the understanding of the regularity, so that it can guide the practice in the future.Persevere, keep it down for a long time, and experience will naturally be transformed into ability.Lessons also translate into experience, and over time an individual's mathematical literacy will reach new heights.

Practice shows that insisting on taking good mathematics notes and organically infiltrating them into the process of mathematics learning can simplify complex problems and reduce the abstraction of concepts.At the same time, it can enrich the teaching materials, enrich the experience, and enhance the fun of learning.Thus changing the boring and abstract current situation of mathematics textbooks, improving the interest in learning mathematics, enhancing mathematics literacy, and completing the transformation from "knowledge-type" to "ability-type"; from "closed" to "open"; from " Transformation from "experiential" to "scientific research".

Understand the system and start with the big picture

There are a lot of concepts and fractions in mathematics, and it is hard to remember, what should I do?
Zhao Jing, a classmate who was admitted to the Department of International Trade of Nanjing University with honors, said that her math performance has always been poor. Later, the teacher taught her a very simple method: find out how many chapters there are in the math textbook, and what each chapter is about. All of a sudden enlightened.In her college entrance examination summary, she wrote:
Mathematics has been a big problem for me since the beginning of the year. I suffered a lot during the senior high school entrance examination, but I really lack passion for the boring calculations of letters upside down, so my grades have not improved after high school.As a result, the head teacher of the first parents' meeting in the third year of high school severely "warned", saying that I would "succeed in mathematics and fail in mathematics" in the college entrance examination!Of course I don't want to lose, but after I have died a few brain cells for mathematics, I am still in the "superior" state where I can understand at a glance and make mistakes as soon as I do it.I desperately wanted to rush forward, but I always felt that I was standing still. In December, my math score was still struggling. Shameless to ask the math teacher.Her first question left me speechless. How many volumes are there in mathematics books from the first year of high school to the third year of high school, and how many chapters are there in each volume?Seeing my embarrassment, she pointed out my shortcomings with a smile: I can only talk about the topic, if I don’t understand the book system, I can’t integrate it well, and if I don’t understand the question types, I can’t draw inferences.If I realize something, under the guidance of the teacher, I will first return to the books and notes, outline the main points and basic question types of each chapter, and ponder over and over again.It is not like doing random questions like before, but when you get a question, analyze the stem of the question first, draw out the key words and conditions, identify which knowledge points it has tested, and then think about what thinking methods need to be used to solve it.I specially selected two reference books, and did the above questions three times. After each time, I summed up the same type of questions and wrote them down in the notebook.As a result, when I picked up the questions again, I was no longer at a loss like a headless chicken. There are many questions that I can know at a glance what knowledge points they are testing.My math grades began to rise, and I even got the first place in the class with a score of 12.This is exactly "another village with dark willows and bright flowers".Although my foundation is not good, I only scored 140 points in the college entrance examination, but fortunately, it did not hold me back in the end.

Apparently, the method taught by this brilliant teacher to Zhao Jing is:

The first step is to find out how many mathematics textbooks there are in the three years of high school?How many chapters does each book have?What is each chapter mainly about.When this problem is clarified, in fact, the knowledge framework of high school teaching is basically clear.

The second step is what are the basic question types in each chapter.

The third step is to outline the knowledge framework and basic question types and read them repeatedly.

The fourth step is to familiarize yourself with and supplement the above outline by doing the questions.

What should I do after I figure out how many books there are in mathematics and how many chapters are in each book?

Find out how many volumes and chapters there are in the textbook, which is similar to Hu Zhi, who was admitted to Peking University as the number one student in science in the college entrance examination.He believes that high school mathematics content seems to be quite a lot, but it is nothing more than a few "sections" when summed up.The first is the function block, the second is the triangle block, the third is the solid geometry block, the fourth is the analytic geometry block, the fifth is the sequence limit block, the sixth is the permutation and combination block, and the seventh is the complex number block.Among them, the first, second, and fourth sections are particularly important. Most of the more difficult questions come from these three sections, so you can spend more effort.When reviewing, you can first review according to the large sections, try to figure out the various types of questions in each section, and be able to deal with each type of questions proficiently.

It seems that the hero sees the same thing.

According to the new national curriculum standards and new textbooks, the content of high school mathematics has been slightly adjusted.But no matter how the content is adjusted, Hu Zhanzhi's method is still applicable.We can summarize and enrich his method into the following steps:

The first step is to divide mathematical knowledge into several "sections".Pay attention to the analysis of the college entrance examination questions and study the characteristics of each section.

In the second step, each "plate" can be divided into several small "plates".For example, under the big "plate" of function, it can be divided into several small "plates" such as quadratic function, monotone function, inverse function, and logarithmic function.You should also be aware of the characteristics of each small "plate".

In the third step, you should be aware of the basic question types under each small "section".You should see and do as many relevant question types as possible.

In the fourth step, you should not only be able to do the relevant question types, but also be familiar with them.

According to the principle of "from big to small", the content is connected with various knowledge points.Clarify the backbone, and the next step is to summarize and list specific knowledge points, because what you have learned is not scattered or messy, but orderly.It's like giving you a lot of colorful beads. You have to sort them by size and color, wipe them clean, and find the lost beads. Then, use your own aesthetic point of view to match and combine the beads properly. When you wear a strong thread, when you need it, you will not grab a handful of beads in a hurry, pick up this one and lose that one, but easily pick up a string of beads.

(End of this chapter)