This is the most effective way to learn

Chapter 18 Are You Afraid of Math? ——Mathematics learning method for top students (1)
Mathematics is the crystallization of human wisdom for thousands of years. Mathematics learning can cultivate and train thinking: through learning geometry, we learn how to use deductive reasoning to prove and think; through learning probability and statistics, we can learn how to avoid dead ends in thinking and how to maximize own chance.

Mathematics is the foundation of many science and engineering subjects.Learning mathematics, the most important thing is to get knowledge and ways of thinking from learning.

Top students should thoroughly study the problems in mathematics

A top student in the college entrance examination from the Northwest said that she basically didn't buy any teaching materials when she was reviewing, she just did the questions in the textbook honestly, and thought through the sample questions and exercises in the textbook.

Researchers have found from the college entrance examination and middle school mathematics test questions in recent years that a considerable part is slightly modified from textbook exercises, or even directly derived from textbook exercises.

It seems that the experience of top students is correct.Textbooks are the foundation. When studying, we must pay attention to digging deep into the functions of textbook exercises and give full play to certain functions.When solving a problem, don’t talk about the topic, and don’t stop thinking after solving the problem. Instead, you should extend your thinking and make analogies, associations, and promotions from all aspects of the exercises.

However, there are only a few questions in mathematics, what is there to think about?
A teacher thinks that the examples and exercises in the book can be considered at least from the following angles:

First, read the sample questions easily before class
Before class, you should carefully preview the content you will learn, and read through the examples word by word, and try to leave an impression on your mind.If you come across simple and easy-to-understand content, you can pass it by. Don’t worry if you encounter complicated and difficult things that you can’t understand. Mark them with a pen and leave them to be resolved during class.Don't stop reading when you encounter difficulties, think repeatedly, deliberate, and deliberate on difficulties, and you will become dizzy and unclear.We must learn to let go of problems in a timely manner and move forward lightly, otherwise we will waste time and increase fear of difficulties, and may also set up psychological obstacles for future learning.This also pays attention to the following points:

First of all, after grasping the methods in the textbook, pay attention to one method with multiple uses.

Secondly, ponder whether the example questions in the book can be changed in one question (note that it is not a question with multiple solutions).

Modifying the topics in the textbook can not only stimulate the interest of exploration, but also improve the ability of seeking common ground and different thinking, and enjoy the endless joy of creating the fruits of mental labor.

Third, ponder whether the questions in the book can be solved with one question and multiple solutions. In addition to using one question with multiple solutions for some of the more important exercises in the textbook, or more comprehensive questions, you can also solve some seemingly unremarkable exercises. Also make a certain effort to see if there is another solution,

Second, carefully select examples in class
It is impossible to concentrate a person's attention for a long time, and what the teacher said in class is difficult and easy for different students.In class, we must have relaxation and relaxation, combine "work" and "leisure" ("work" means high concentration of attention, "leisure" means relative concentration of attention), and grasp the rhythm of the teacher's lecture.Simple and easy to understand can be easily skipped (if necessary, you can also sneak away, think about a certain problem or memorize the theorems and formulas required to master this lesson), and focus on the difficulties marked during your preview.When talking about important and difficult points, the teacher often emphasizes the tone, slows down the speech speed, and repeats at the right time. We can grasp the characteristics of how the teacher lectures, "focus" attention in time, and never miss a word. Watch the teacher analyze, compare, and summarize , Synthesis, how to connect the knowledge learned before, how to master it, and draw inferences from one instance.The finishing touches should be done with great care, such as places that are prone to errors, the use of plus and minus signs, etc.Only by carefully digging out the example questions can we really clarify the problem-solving ideas of the example questions, grasp the key points, grasp the difficulties, and lay the foundation for the improvement of problem-solving ability.

Third, after-class analysis to see examples
Understanding the example questions in class does not mean that you have the ability to understand the questions and transfer knowledge.After class, it is necessary to re-examine and analyze the sample questions from a new perspective.Due to the mastery of new knowledge, the expansion of knowledge, and the guidance and advice of the teacher, when I look at the sample questions again, I have a different understanding of the difficulties and enter a higher level.You will have a deeper understanding of the application of basic knowledge in the questions, the choice of analysis and reasoning methods.For example: when doing a geometry proof question, by looking at the sample questions, you can analyze what knowledge you have learned before that the question involves, and see if there are other methods (using other auxiliary lines and theorems) to solve the problem.If you don't look at the example questions after class, your thinking will stay at a shallow level, and you will not be able to complete the transformation process from the shallow to the deep, from the surface to the inside.

Fourth, homework reasoning knowledge examples
Doing exercises is the most important and effective way to use knowledge to solve problems and improve ability, and it is also the key to learning mathematics well.When doing homework, you must first identify the type of question (that is, which type of example questions this chapter belongs to); secondly, recall how to solve the problem in the book, analyze several problem-solving methods again, and finally clarify which one The easiest way.It should be pointed out that when identifying the question type, you should carefully recall the specific problem-solving steps. If you can't remember clearly or forget the sample questions you have learned before, you should spare no time to read, analyze, and memorize.By doing exercises, you can synthesize the knowledge you have learned, analyze the types of questions you have seen, and keep in mind the steps and methods of solving the questions.

Fifth, the comprehensive series of exam questions

Review before the exam to summarize and compress knowledge, and read the book thinly.To achieve this, in addition to the commonly used "knowledge string" method, "example string" is also an essential and effective means of learning mathematics.The so-called "example series" is to find out how many chapters there are in the book, how many sections are in each chapter, how many example questions are in each section (in fact, each example question is a type of question), and how to answer each type of question. After the sample questions in the book are well known, it enters the last process: copying the sample questions and solving the example questions.Copy down a sample question neatly in the homework book, and get familiar with the question type.Close the book (don't read the solution first and then solve the problem), and solve the problem carefully according to the problem-solving steps and methods in the book (never sloppy deletion or omission).After answering, open the book and compare them with examples one by one to see if the problem-solving methods and steps are consistent with those in the book. If there are differences, analyze the reasons, find the existing knowledge blind spots and the pros and cons of doing so, and finally correct and memorize them.If the closed-book exam answers for the example questions are exactly the same as those in the book (methods, steps), then you will be able to easily complete all the practice questions in the book without losing step points.This fully shows that you have not only mastered the content of the whole book, but also standardized the problem-solving process.

This method can be used not only when studying mathematics, but also when studying physics, chemistry and other courses.Example questions are one of the main parts of science textbooks (just imagine deleting example questions, what is left in a textbook).When the book editors choose sample questions, they have spent a lot of effort: what kind of questions to choose?How many to choose?How to choose?Therefore, textbooks are the best reference books, and you must ponder deeply and thoroughly.

Do what you can't do, don't do what you can do
Do you have such doubts: I have done many math problems, and some problems have been done more than once. Why can’t my grades improve?
The experience of Feng Yu, the number one student in the college entrance examination of the High School Attached to Shanghai Normal University, is: Do what you can’t do, and don’t do what you can do.It means that there is no need to repeat the questions that can be done.While this method of learning may sound a bit "speculative", it works very well.When implementing it in his usual study, he said, "For example, in class, I can't guarantee my full attention all the time, but I will listen carefully to those new knowledge points, and I will pay special attention when the teacher explains some topics that I don't know how to do." The same is true. , Zhang Fengyu said that he is also "selective" when doing exercises, of course, this is mainly based on what he said "generally you will not make mistakes on the same topic twice".

Another champion, Zhou Hong, said: "My favorite subject is mathematics. I benefited from a teacher who won the first place in a single subject in mathematics. This teacher once said such a famous saying: If you can't do it, you can do it If you don’t do it, don’t do it. This sentence made me ponder for a long time, and I think it is very reasonable. First of all, we have to ask, why do the questions? What is the purpose of doing the questions? If you have not mastered, what things you have mastered, the focus should be on what you have not mastered, instead of following the questions aimlessly, then you will lose your initiative. Everyone's situation is different, and some people are stronger in this aspect. Some people are weak in that aspect, if you can find out which aspects you are weak in, it is half the battle, and then you can work on your weaknesses. In this way, you will continue to improve."

Classmate Zhou Hong said that he personally thinks that it is better to do one topic that you are not familiar with than doing ten repetitive questions.Make a set of papers, and after finishing, you must summarize and find problems from it.You must not let go of any questions that you have done wrong, otherwise you will make mistakes again and again and mess up your position.With such an attitude, after a long time, you will feel that there are not many questions to do.When you feel that there are no questions to do, you have almost done the review.

"Do what you don't know, and don't do what you know." This is really a simple truth.You might say: What kind of experience is this?Don't jump to conclusions yet.In fact, how many students just forget this simple truth, they don’t do what they don’t know, and they still do what they know, wasting precious time.

Since it is said that "do what you don't know", first of all, you must figure out what you don't know, and secondly, you must figure out "what to do" for it.

Since it is said that "if you know it, you will not do it", then you must also find out whether you really know it or fake it?Will it be 50.00% or only [-]%?If it's true, don't do it; if it's just a fake meeting, then you have to do it.

If you know it 50.00%, then you don’t need to do it; if you only know [-]%, then you have to figure out what is the half that you don’t know, and you have to practice this part!

The "literature of science" approach to mathematics
You may think that mathematics mainly depends on doing questions, and there is not much to remember.Indeed, compared with liberal arts, and even compared with chemistry, which is also a science, the amount of memory in mathematics is indeed not large.But this does not mean that learning mathematics does not require memorization.At least some of the following should be kept in mind:
Memorize axioms, theorems, and definitions
Solid Geometry Chapter 1 is the foundation of solid geometry. It has many axioms, theorems, and definitions. Whether you can accurately memorize these axioms and theorems is the key to learning solid geometry smoothly.Therefore, classroom memory, classroom questions, classroom tests, homework and other methods should be used to repeatedly strengthen memory, so that most students can accurately describe every axiom, theorem, and definition, laying a solid foundation for further study. Foundation.

Remember how to apply axioms and theorems
The purpose of accurately memorizing axioms and theorems is to apply these axioms and theorems to solve practical problems, but many students often do not know where to start for specific geometric problems.Therefore, how to apply the conditions for the application of each axiom and theorem may be listed one by one, and it is required to be memorized, so as to form a pattern in thinking, that is, how to deal with any problems encountered.For example, "the property theorem of parallel planes" can be memorized backwards and forwards, but they are still at a loss when encountering specific problems. Therefore, for the content of the theorem itself, it is required to remember "planes are parallel planes".In this way, through repeated practice, the handling of such problems has become very proficient, and the mathematical ability is gradually formed in this repeated memory training.

Memorize commonly used mathematical methods

The mastery of commonly used mathematical methods is an important content of the "double base" of mathematics, and it is a skill that students must master.Mathematics is inseparable from methods. During review, pay attention to repeated training of commonly used mathematical methods such as analysis, synthesis, proof, formula, combination of numbers and shapes, etc., so as to achieve the purpose of memory through repeated training.At the same time, pay attention to reproducing these important mathematical methods at regular intervals to prevent forgetting.

Remember important exercises

(End of this chapter)