Learning mathematics requires not only a strong desire and enthusiasm for learning, but also scientific learning methods to learn mathematics well. From the analysis of mathematics learning activities, we can see that learning methods are not only restricted by classroom teaching, but also have some characteristics of their own. Therefore, on the one hand, we put forward learning methods suitable for classroom teaching, on the other hand, we summarized some special learning methods according to the characteristics of mathematics learning.

Methods of preview, lecture, review and homework

Learning methods suitable for mathematics classroom teaching are the basic methods of previewing, attending classes, reviewing and doing homework.

1, preview method

Preview is to read the upcoming mathematics content before class, understand its outline, and be aware of it, so as to grasp the initiative in class. Preview is an attempt of autonomous learning. Whether you understand the learning content correctly, whether you can grasp the key points and hidden thinking methods, etc. It can be tested, strengthened or corrected in time in class, which is conducive to improving learning ability and forming the habit of self-study, so it is an important part of mathematics learning.

Mathematics has a strong logic and coherence, and new knowledge is often based on old knowledge. Therefore, when previewing, you should find out what you need to learn new knowledge, and then recall or review it again. Once we find that the old knowledge is not well mastered or even understood, we should take timely measures to make up for it, overcome the learning obstacles caused by not mastering or forgetting, and create conditions for learning new content smoothly.

The method of preview, besides recalling or reviewing the old knowledge (or preparatory knowledge) needed to learn new content, should also understand the basic content, that is, know what to say, what problems to solve, what methods to adopt, where the focus is, and so on. In preview, reading, thinking and writing are generally used to draw out or mark the main points, levels and connections of the content, write down your own views or places and problems that you can't understand, and finally determine the main problems or solutions to be solved in class to improve the efficiency of class. In the arrangement of time, the preview is generally carried out after review and homework, that is, after finishing homework, read the content to be learned in the next class, which requires flexibility according to the specific situation at that time. If time permits, you can think more about some problems, study deeply, and even do exercises or exercises; Time does not allow, we can have fewer questions and leave more questions for lectures to solve. There is no need to force unity.

2, the method of listening to lectures

Listening to classes is the main form of learning mathematics. With the guidance, inspiration and help of teachers, we can avoid detours and reduce difficulties, and gain a lot of systematic mathematics knowledge in a short time, otherwise we will get twice the result with half the effort and it is difficult to improve efficiency. So attending classes is the key to learning math well.

The method of class, in addition to clarifying the tasks in the preview and solving the problems suitable for you, should also concentrate on keeping up with the teacher's lectures and use your brains to think about how the teacher asks questions, analyzes and solves problems, especially learning mathematical thinking methods, such as observation, comparison, analysis, synthesis, induction, deduction, generalization and specialization, that is, how to use formulas and theorems.

When listening to a class, on the one hand, we should understand what the teacher said, think or answer the questions raised by the teacher, on the other hand, we should think independently, identify what knowledge we have understood, what questions or new questions we have, and dare to put forward our own opinions. If you can't solve it in class for a while, you should write down the problems or problems you want to solve yourself or consult the teacher and continue to listen attentively. Don't stay here because you don't understand one thing, which will affect the later lectures. In general, in class, we should write down the main points, supplementary contents and methods of the teacher's lecture for review.

Step 3 review methods

Review is to learn the learned mathematical knowledge again, so as to achieve the purpose of in-depth understanding, mastery, refinement and generalization, and firm grasp. Review should be closely linked with lectures, and the contents of lectures should be recalled while reading textbooks or checking class notes, so as to solve the existing knowledge defects and problems in time. Try to understand the content of learning and really understand and master it. If you can't solve some problems for a long time, you can discuss them with your classmates or find a teacher to solve them.

On the basis of understanding the textbook, review should communicate the internal relations between knowledge, find out its key points, and then refine and summarize them to form a knowledge system, thus forming or developing and expanding the mathematical cognitive structure.

Review is a process of deepening, refining and summarizing knowledge, which can only be realized through the active activities of hands and brains. Therefore, in this process, it provides an excellent opportunity to develop and improve their abilities. The review of mathematics can't just stop at the requirements of reviewing and memorizing what we have learned, but we should try our best to think about how new knowledge is produced, how it is developed or proved, what its essence is and how it is applied.

4, the method of homework

Mathematics learning is often to consolidate knowledge, deepen understanding and learn to use it by doing homework, thus forming skills and developing intelligence and mathematical ability. Because homework is done independently on the basis of review, it can check out the mastery and ability level of the learned mathematical knowledge, so when it finds many problems, difficulties or wrong questions, it often indicates that there are defects or problems in the understanding and mastery of knowledge, which should arouse vigilance and need to find out the reasons and solve them as soon as possible.

Usually, math homework is represented by problem solving, which requires the knowledge and methods learned. Therefore, you need to review before doing your homework, and then do it on the basis of basically understanding and mastering the textbooks you have learned. Otherwise, it will get twice the result with half the effort, take time and get the desired result.

Problems should be solved according to certain procedures and steps. First of all, we should make clear the meaning of the question, read it carefully and understand it carefully. For example, what are the known data and conditions, what are the unknown conclusions, what operations are involved in the problem, how they are related, and whether they can be represented by charts. We should carefully scrutinize and thoroughly understand them.

Secondly, on the basis of understanding the meaning of the problem, explore the way to solve the problem and find out the relationship between the known and the unknown, the condition and the conclusion. Recall related knowledge and methods, examples learned, problems solved, etc. And consider whether they can introduce appropriate auxiliary elements from form to content, from known numbers and conditions to unknown quantities and conclusions and can be used to find out a special problem or similar problems related to the problem, and whether solving them can enlighten the current problem; Whether we can separate, check or change them part by part, and then recombine them to achieve the expected results, and so on. That is to say, in the process of exploring and solving problems, we need to use a series of methods such as association, comparison, introduction of auxiliary elements, analogy, specialization, generalization, analysis and synthesis to learn from solving problems.

Thirdly, according to the explored solution, according to the required writing format and specification, describe the process of the solution, and strive to be simple, clear and complete. Finally, we should review the solution and check whether the solution is correct, whether each step of reasoning or operation is well-founded and whether the answer is detailed; Think about whether the problem-solving method can be improved or whether there is a new solution, whether the result of this problem can be popularized (in fact, many topics in middle school textbooks can be popularized) and so on. And sum up the experience of solving problems, and then develop and improve the thinking method of solving problems, and sum up some regular things.

Two Learning Methods from Thin to Thick and from Thick to Thin

"From thin to thick" and "from thick to thin" are the research methods mentioned by mathematician Hua many times. He believes that learning should go through the process of "from thin to thick" and "from thick to thin". "From thin to thick" means to understand and know the mathematical knowledge you have learned and know why. Learning should not only understand and memorize concepts, theorems, formulas and laws. We should also think about how they were obtained, what is the connection with the previous knowledge, what is missing in the expression, what is the key, whether we have a new understanding of knowledge, whether we have thought of other solutions, and so on. After careful analysis and thinking in this way, some notes will be added to the content, some solutions will be added or a new understanding will be generated. "The more books you read, the thicker you will be."

However, learning can't stop here. We need to integrate the knowledge we have learned, refine its spiritual essence, grasp the key points, clues and basic thinking methods, and organize it into refined content. This is a "from thick to thin" process. In this process, it is not the reduction of quantity, but the improvement of quality, so it plays a more important role. Usually, when summarizing the contents of a chapter, chapters or a book, we should have this requirement and use this method. At this time, due to the high generalization of knowledge, it can promote the transfer of knowledge and is more conducive to further learning.

"From thin to thick" and "from thick to thin" are a spiral rising process, with different levels and requirements, which need to be used many times from low to high in learning to achieve the desired results. This learning method embodies the dialectical unity of "analysis" and "synthesis", "divergence" and "convergence", that is to say, mathematics learning needs the unity of the two.

Third, the method of combining acceptance learning with discovery learning.

Mathematics learning should be meaningful acceptance learning and meaningful discovery learning. How to make them cooperate with each other, organically combine and give full play to their respective and comprehensive functions is an important aspect of learning methods.

Learning, whether listening to systematic lectures or teaching materials given in the form of conclusions, does not involve any independent discovery. But in the process of learning, students are in a proactive state, not just accepting. They always ask themselves some questions, such as how the theorem was discovered or produced, how the idea of proof was worked out, and what key places need to be broken. Many mathematicians emphasize "not only to write, but also to read what is behind the book." In the process of acceptance and learning, we should also add some extreme points of discovery and learning, and learn ideas and methods of invention from them, rather than just staying in the acceptance of knowledge.

Discovery learning is to solve a problem independently by observing, comparing, analyzing and synthesizing the provided materials or problems, so as to acquire new knowledge. When solving a problem, we should really understand the essentials, principles, formulas, theorems and laws involved in the problem, understand the significance of each step of operation, and put forward and test the purpose of the hypothesis. When solving problems, we always need to connect the knowledge and methods we have learned in the past. If we can't recall them for a while, we must review them again to further understand the application. Some people encounter problems and even consult reference books or teachers to solve them. It can be seen that this period is also interspersed with learning.

Mathematics learning not only needs to accept learning, but also needs to find learning, which is conducive to thinking and cultivating creativity. Therefore, learning should be based on their own age, learning ability characteristics and teaching content requirements, so that the two can be closely combined.

Compared with other subjects, what are the characteristics of the second mathematics? What is its corresponding way of thinking? What kind of subjective conditions and learning methods does it require us to have? This lecture will briefly explain the characteristics, ideas and learning methods of mathematics.

I. Characteristics of Mathematics (1)

The three characteristics of mathematics are preciseness, abstraction and wide application. The so-called rigor of mathematics refers to the strong logic and high proficiency of mathematics, which is generally reflected by axiomatic system.

What is the axiomatic system? It refers to selecting a few undefined concepts and propositions without logical proof, and deducing some theorems to make them a mathematical system. In this respect, the ancient Greek mathematician Euclid is a model, and his Elements of Geometry studies most problems in plane geometry on the basis of several axioms. Here, even the most basic and commonly used original concepts cannot be described intuitively, but must be confirmed or proved by axioms.

There are still some differences in rigor between middle school mathematics and mathematics science. For example, the continuous expansion of several sets in middle school mathematics, the expansion operation law of several sets is not strictly deduced, but obtained by default. From this point of view, middle school mathematics is still far from rigorous, but to learn mathematics well, we must not relax the requirements for rigor and ensure the scientific content.

For example, arithmetic progression's general term is summed up through the recursion of the previous items, but it needs to be strictly proved by mathematical induction to be confirmed.

The abstraction of mathematics is manifested in the abstraction of spatial form and quantitative relationship. In the process of abstraction, it abandons the specific characteristics of more things, so it has a very abstract form. It shows a high degree of generality and symbolizes the concrete process. Of course, abstraction must be based on concreteness.

As for the wide application of mathematics, it is well known. Only in the past teaching and learning, we often paid too much attention to the abstract meaning of theorems and concepts, but sometimes gave up their wide application. If abstract concepts and theorems are compared to bones, then the extensive application of mathematics is like flesh and blood, and the lack of any one will affect the integrity of mathematics. The purpose of increasing the application space of mathematics knowledge and research-based learning in the new high school mathematics textbook is to cultivate students' ability to solve practical problems by applying mathematics.

Second, the characteristics of high school mathematics often lead to students' inability to adapt to mathematics learning after entering high school, which in turn affects their enthusiasm for learning and even their grades plummet. Why is this happening? Let's take a look at the changes in high school mathematics and junior high school mathematics.

1, theory strengthening 2, course increasing 3, difficulty increasing 4, requiring improvement.

Third, master mathematical thinking. High school mathematics is closer to advanced mathematics in learning methods and thinking methods. Learning it well requires us to master it from the height of methodology. When we study mathematical problems, we should always use materialist dialectical thinking to solve mathematical problems. Mathematical thought is essentially a reflection of the application of materialist dialectics in mathematics. The mathematical thoughts that should be mastered in middle school mathematics learning are: set and correspondence, initial axiom, combination of numbers and shapes, movement, transformation and transformation.

For example, the concepts of sequence, linear function and straight line in analytic geometry can be unified with the concept of function (special correspondence). For another example, the concepts of number, equation, inequality and sequence can also be unified into the concept of function.

Let's look at the following example of solving problems with a "contradictory" point of view.

Given that the moving point Q moves on the circle x2+y2= 1, fix the point P (2 2,0) and find the locus of the midpoint of the straight line PQ.

Analyzing this problem, P, Q and M are mutually restricted, and the movement of Q drives the movement of M; The main contradiction is the movement of point Q, whose trajectory follows the equation x02+y02 =1①; Secondary contradiction: m is the midpoint of the straight line PQ, and the coordinates (x, y) of m can be expressed by the midpoint formula with the coordinates of point Q.

X = (x0+2)/22y = y0/2③ Obviously, the trajectory can be obtained by substituting x0 and y0 in the elimination problem.

Mathematical thinking method is different from problem-solving skills. In proving or solving, it can be said that solving problems by induction, deduction and method of substitution is a technical problem, and mathematical thinking is a guiding general thinking method. When solving a problem, from the overall consideration, how to start, what are the methods? It is a common problem under the guidance of mathematical thinking methods.

With mathematical ideas, we should master specific methods, such as method of substitution, undetermined coefficient method, mathematical induction, analysis, synthesis and induction. Only under the guidance of problem-solving thought and flexible use of specific problem-solving methods can we really learn mathematics well. It is often difficult to make mathematics learning enter a higher level by mastering specific operation methods without considering problems from the perspective of problem-solving thinking, which will bring great trouble for further study in universities in the future.

In terms of specific methods, commonly used are: observation and experiment, association and analogy, comparison and classification, analysis and synthesis, induction and deduction, general and special, finite and infinite, abstraction and generalization.

If you want to win the battle, you can't just fight bravely, not afraid of death or suffering. You must formulate tactics and strategies that have a bearing on the overall situation. When solving mathematical problems, we should also pay attention to solving the problem of thinking strategy, and often think about what angle to choose and what principles to follow. Generally speaking, the general idea adopted to solve problems is a principled thinking method, a macro guidance and a general solution.

Mathematical thinking strategies commonly used in middle school mathematics are:

Simple control of complexity, combination of numbers and shapes, advance and retreat freely, and turn life into familiarity, difficulty, harmony and dynamic and static transformation, which complement each other. If you have correct mathematical thinking methods, appropriate mathematical thinking strategies, rich experience and solid basic skills, you will certainly learn high school mathematics well.

Fourth, the improvement of learning methods is in the strange circle of exam-oriented education. Every teacher and student can't help falling into the ocean of problems. Teachers pay attention to a certain kind of questions, and the college entrance examination can't do it. Students are afraid to do one less question. In case the loss is too heavy, in such an atmosphere, the cultivation of learning methods is often ignored. Every student has his own method, but what kind of learning method is correct? Is it necessary to "read a lot of questions" to improve your level?

Reality tells us that it is a very important issue to boldly improve learning methods.

( 1)

Learn to listen and read. We listen to teachers and read textbooks or materials at school every day, but do we listen and read correctly?

Let's talk about listening (listening, classroom learning) and reading (reading textbooks and related materials).

What students learn is often indirect knowledge, abstract and formal knowledge, which is refined on the basis of previous exploration and practice, and generally does not include the process of exploration and thinking. So be sure to listen to the teacher, concentrate and think positively. Find out what the content is. How to analyze it? What is the reason? In what way? Is there a problem? Only in this way can we understand the teaching content.

The process of attending classes is not a process of passive participation. On the premise of attending class, we need to analyze: what thinking method is used here, and what is the purpose of doing so? Why can the teacher think of the shortest way? Is there a more direct way to solve this problem?

"Learning without thinking is useless, thinking without learning is dangerous", so we must have positive thinking and participation in the process of listening to classes, so as to achieve the highest learning efficiency.

Reading mathematics textbooks is also a very important way to master mathematics knowledge. Only by reading and reading mathematics textbooks can we master mathematics language well and improve our self-study ability. We must change the bad tendency of using textbooks as dictionaries to look up formulas without reading books. When reading textbooks, we should also strive for the guidance of teachers. When reading the content of the day or the content of a unit or a chapter, we should consider comprehensively and have a goal.

For example, learning the arcsine function, in terms of knowledge, through reading, should ask the following questions:

(1) Does every function have an inverse function? If not, when does a function have an inverse function?

(2) Under what circumstances does a sine function have an inverse function? If so, how to express its inverse function?

(3) What is the relationship between the image of sine function and the image of arcsine function?

(4) What are the properties of the arcsine function?

(5) How to find the value of the arcsine function?

(2)

Learn to Think Einstein once said, "Cultivating the general ability of independent thinking and independent judgment should always be the first." Being diligent and good at thinking is the most basic requirement for us to learn mathematics. Generally speaking, we should try our best to do the following two things.

1, good at finding and asking questions 2, good at reflection and reverse seeking.

In the process of communicating with students, Teacher Xie Dahong feels that the problems that students often encounter in their study have a great influence on their learning process.

As educators, the mentality of dealing with students' learning problems is different from that of parents. Because of family relations, parents are easily impatient. However, impatience cannot solve the problems in learning and growth. There must be scientific ways, methods and educational means to guide students to solve these learning problems.

One of the characteristics of mathematics is that it is important and boring. The importance is obvious. Mathematics, as a basic subject, must be tested in both the college entrance examination and the senior high school entrance examination. At the same time, it is boring, which seems to be a contradiction. To deal with this contradiction, we must solve a technical and psychological problem in mathematics learning. Of course, it is impossible for everyone to learn math well. Because of different sexual orientation, some people tend to be in humanities, some people tend to think logically, some people tend to think spatially, and some people tend to practice ... Everyone has different tendencies, different professional fields and different mathematical levels, but one thing is to master some basic requirements of mathematics, such as some basic principles and mathematical methods in mathematics. Because no matter what industry we are engaged in in in the future, mathematics is very important as a basic way to deal with things. The average child can achieve it through the right method and correct guidance. The following are several important contents in mathematics learning emphasized by Teacher Xie Dahong:

First, questions about mathematical concepts

The formation of the concept needs a process. Different from concepts such as philosophy of life, mathematical concepts are superimposed, and new concepts are recognized on the basis of the superposition of old concepts. Concept is a fundamental problem in mathematics, not backrest, but gradually formed through continuous application. It must go through the process of comparison, practice, exploration, summary and induction, and finally establish a complete concept. This process can even be said to be a painful and long stage.

This concept is long-term. Every concept has a process of failure-re-understanding-failure, which is deepened in your frustration of misunderstanding this concept.

This concept deepens with one's knowledge. Learning mathematics is very important for a person to establish a complete way of thinking. With the in-depth understanding of different mathematical concepts, the way people deal with problems can become more and more rigorous.

It is necessary to establish a mathematical concept network. Mathematics is a lattice of concepts, and all related and subordinate concepts must form a network in the mind. When learning concepts, we should make clear the concepts that cannot be included in them or related concepts. There should be a clear analysis of how the related concepts in the general concept develop.

Understand a mathematical concept from different levels. Only through comparison can we understand a mathematical concept, and we should be good at understanding it from four levels: front, side, top and bottom. Unclear understanding of similar or similar concepts or their internal relations is not conducive to understanding concepts, which shows that mathematics is in-depth at the end of learning.

2. Computational ability: Symbolization and patterning are a major feature of mathematics, which should be deeply understood.

Modeling. Some theorems, principles and axioms of mathematics have certain patterns. "Because ………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

Symbolization. Mathematical symbols are different from expressive symbols, and expressive symbols in literature and art need us to understand their meanings carefully; Symbol in mathematics is a substitute symbol. It doesn't need us to think about its meaning, its function lies in deduction. It's just a body pair, which helps us to think mathematically, so we can't spend too much energy on its meaning. Mathematics is a symbolic game, and you must be proficient in symbols to quickly transform.

There are several important theorems in middle school: three orthogonal theorems, sine and cosine theorems, the relationship between roots and coefficients, and quadratic trinomial theorems. We must master these theorems skillfully.

3. Problem solving skills.

From the way of doing the problem, the usual homework can be divided into hard homework and soft homework: hard homework refers to the homework that needs to be done seriously every day, which should be done meticulously according to formal steps, aiming at training your writing skills and writing ability; Soft homework refers to taking some time to browse some exercises every day. This kind of exercise is mainly used to exercise your thinking ability. The specific method is to look at the exercises without writing, and quickly go through the ideas and practices of this problem in your mind. The whole process is a bit like air to air. Therefore, in daily practice, the two methods should be used in combination, and serious exercises and browsing exercises should be used in combination.

Doing the problem should be rhythmic and difficult. Pay attention to the quality of the questions, and don't focus on partial questions, difficult questions and strange questions, because there are 20% difficult questions in the college entrance examination. If you usually focus on difficult questions, the basic knowledge will inevitably be lost, so you can usually do some questions with moderate difficulty. The key is to learn the basic knowledge well and step by step.

When you do the problem, you should leave experience and traces. Learning is divided into three processes: imitation, taste and transfer. Imitation is a way that often works in the early days, step by step with reference to teachers or textbooks. After repeated imitations, we have further processed and experienced these problems in our minds and formed our own understanding of the formation of such problems. After the accumulation of the first two stages, the original knowledge system is finally integrated with the existing knowledge to realize the latest experience of old and new knowledge.

4. Mathematical methods. Common mathematical methods are as follows:

Reduction, that is, substitution and elimination, is an idea that turns a complex problem into several simple problems. The idea of eliminating parameters in senior two and senior three mathematics is an example of this method.

Pay attention to frequent induction, arrangement and summary of knowledge, promote the knowledge learned to be more systematic and organized, and smooth out the internal relations when solving problems.

When you do the problem, you should establish an idea of order and relevance. The elements in a mathematical stem are generally arranged in a certain order and relationship. Before you do the questions, you should carefully examine the meaning of the questions, prioritize them and break them down one by one.

Thinking method of equation.

The method of classified discussion.

The fourth article mathematics learning methods

Review in an all-round way and read a book.

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 and review it comprehensively without leaving any omissions.

Comprehensive review is not about memorizing all the knowledge. On the contrary, it is about grasping the essence and content of the problem and the essential connection of various methods, 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). Moreover, if you don't remember, you will, and if you remember, you will be reliable. Facts have proved that some memories will never be forgotten, while others can be obtained by using the relationship between them on the basis of remembering the basic knowledge. This is the significance of comprehensive review.

Second, focus on 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; There are two levels of requirements for mastering methods, knowing (or 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 this aspect is relatively high; The same test paper, the test questions in this area also occupy more scores. People who "guess the questions" often have to work hard in this respect. Generally speaking, you can really guess a few points. But when it comes to 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 content and methods, but more importantly, we should find the connection between the key content and the secondary content, so that the main content is the secondary content and the key content covers all the content. The main content is thoroughly understood, and other contents and methods will be readily solved. We should grasp the main content, don't give up the secondary content and isolate the main content, but naturally highlight the main content by analyzing the relationship between the contents and comparing them. Such as differential mean value theorem, Rolle theorem, Lagrange theorem, Cauchy theorem, Taylor formula and so on. Because Rolle theorem is a special case of Lagrange theorem, Cauchy theorem and Taylor formula are the generalization of Lagrange theorem. By comparing these relations, we naturally come to the conclusion that Lagrange's theorem is the core, and we can thoroughly understand this theorem and grasp several other theorems from the connection. In the examination syllabus, Rolle's theorem and Lagrange's theorem are both required to be understood and are the focus of the examination. We highlight Lagrange's theorem more, which can be described as Excellence.

Third, the basic training is repeated.

To learn mathematics, we should do a certain number of problems and thoroughly practice the basic skills, but we do not advocate the tactics of "problems" and advocate refinement, that is, we should repeatedly do some typical problems, solve many problems for one problem and change one problem. Training the ability of abstract thinking, proving some basic theorems, deducing basic formulas and doing some basic exercises don't require writing, just like a chess player's "blind chess", you only need to meditate with your brain to get the exact answer. This is what we mentioned in the preface, 20 minutes to complete 10 objective questions. Some questions can be answered at a glance without writing. This is called well-trained, "practice makes perfect" people with solid basic skills have many ways to solve problems and are not easily stumped. On the contrary, when doing problems, I always find difficult problems, and as a result, I will encounter similar problems I have done before when I go to the examination room. Many candidates misjudge the questions they can do, which is classified as carelessness. It is true that people are careless, but people with solid basic skills will find out immediately when they make mistakes, and rarely make "careless" mistakes.