The study of ascending to mathematical thinking method.

Simply put, thought is the method in method, and method is the concrete realization of thought. Internal unification of various methods is the embryonic stage of methods. Methods must be guided by ideas. Based on the dialectical unity of thinking methods, here I will discuss the learning of mathematical thinking methods and the learning of basic mathematical knowledge together.

The predecessors summarized and discussed many mathematical thinking methods for us. The question now is how to turn these other people's thinking methods into their own.

First, a large collection

Collect and sort out a large number of mathematical thinking methods, online and in books. The problem is that the way of thinking is endless. When will this collection and sorting stage end? One way to judge is repetition, and repetition to a certain extent is enough. We can also judge the universality and importance of mathematical thinking methods by the degree of repetition.

Second, the preliminary classification summary

According to certain standards, it is preliminarily classified and summarized to form a general system network framework. Explain it in detail below.

For example, according to the application fields, it can be divided into: mathematics research methods, mathematics learning methods and mathematics teaching methods. According to the degree of universality, it can be divided into philosophical methodology, general scientific methodology and specific scientific methodology. Mathematical methods include at least the above three fields and three levels. They are interrelated, showing a trend of mutual penetration and mutual transformation. We just want to grasp and reveal the relationship between them through preliminary induction, classification and summary.

Such as abstraction and generalization, induction and deduction, classification and classification, comparison and analogy, analysis and synthesis, can be regarded as both philosophical methodology and general scientific methodology, and there is only a very thin line between them. If we stand on the height of philosophy to reflect and demonstrate, that is philosophical methodology; If you focus on how to apply perfection in science, it is the general scientific methodology.

Abstraction and generalization are mainly manifested in the idealization and modeling methods in mathematics; In mathematics, induction and deduction are mainly manifested in mathematical induction, axiomatization and formalization; Comparative analogy is a very important mathematical conjecture method in mathematics; In fact, all kinds of mathematical methods are the combination of all kinds of philosophical methods, which are not as simple and linear as mentioned above. For example, axiomatic and formal methods mainly include deduction and abstraction; Mathematical model method also includes abstraction, classification, deduction and calculation.

The preliminary summary is as follows:

Basic thinking method of mathematics

1. Idealized method and modeling method.

2. Induction and deduction: mathematical induction, axiomatic method and formal method.

3. Comparison and analogy: mathematical conjecture method

4. Analysis and synthesis: analysis and synthesis.

5. Classification and classification: equivalent division method and classification discussion method.

Unique mathematical thinking method

1. thinking set method:

2. Mapping ideas and methods: correspondence, function, RMI (reflection principle of relational mapping).

3. Other ways of thinking: reduction, construction, recursion, iteration, combination of numbers and shapes, and equation method.

4. Mathematical problem solving methods: reduction to absurdity, method of substitution, undetermined coefficient method, matching method, elimination method and factorization method.

Although it is a leak, everything mentioned is very important.

Third, break the foundation of mathematics.

There are many attractive theories in modern mathematics. Every time I want to study deeply, I always feel that the foundation is weak and it is difficult to make progress. I really don't think I can move. We must concentrate on every stage of learning the basic knowledge of mathematics. Through the study of relatively simple basic knowledge, we can understand and master common and important mathematical thinking methods, and learn mathematics by doing mathematics. In the process of doing mathematics, we should deeply experience and understand the thinking method of mathematics. Only through this process can other people's mathematical methods become their own thinking methods.

Fourth, gradually improve and optimize.

To gradually form its own ideological methodology system, it is necessary to integrate various thinking methods and gradually systematize, network and enrich them. Therefore, we must strengthen our philosophical cultivation and mathematical cultivation. Through various channels, select some related master classics for research. "I've been thinking about it all day. It's better to learn it in an instant." "Listening to you is better than studying for ten years." Only by studying the master's classic original works can we play such a role and effect. In addition, we should continue to do mathematics in combination.

How to learn Chinese well in junior high school

Chinese is an important tool discipline and the foundation of human language. It is very important to learn Chinese well.

The most important thing to learn Chinese well is to cultivate interest. Students all think that Chinese is a very boring subject with poor flexibility and knowledge. In fact, Chinese has three main characteristics: first, openness. The extension of Chinese is equal to life, and any content of life can not be separated from Chinese. As long as you study consciously, you can learn Chinese in any life situation. Therefore, Chinese has the outstanding feature of being the most convenient for self-study. The second is emotion. "The article is not heartless." Not only articles, but also the words of our nation are full of emotions. Teachers can only teach well if they teach with emotion; It is possible for students to learn well with emotionalism. It is one of the important wishes for students who have no emotional factors to be tired of learning Chinese. The third is flexibility. The knowledge system of Chinese subject is not arranged in a linear or chain way like other subjects, but spirals up. Therefore, the gradual learning of Chinese is not very strict, but relative. Learning Chinese can't be self-enclosed and mechanical.

So, how can we better master the subject of Chinese?

First, learn textbook knowledge well.

Textbook knowledge is the foundation of Chinese. Only by laying a good foundation can we learn better. Many students think that their grades can be improved as long as they listen carefully, take notes diligently, finish their homework carefully after class and review consciously. In fact, this is not enough. The most important stage of learning is preview. In other words, you should teach yourself the text before the teacher attends class. In the preview, we should make full use of the acquired knowledge and methods, actively solve the problems we can solve, write down the problems we don't understand, and study and discuss with teachers and classmates in class. Read the textbook again and again until you see the problem thoroughly and understand it. In order to consolidate your knowledge, you'd better do some exercises after class, so that you can master your knowledge more firmly. This not only has a good learning effect, but also cultivates your own learning ability.

Second, pay attention to extracurricular accumulation.

Knowledge is like the ocean, and the knowledge in textbooks is just a spray on the sea, which is far from meeting our needs, so we should read properly after class. When we enter the intense study stage, we can't have a lot of spare time to read, so we should choose when reading. We should browse all kinds of books, newspapers and magazines extensively, get information from TV, radio and internet, and take notes in an orderly way. We should care about social life, understand social trends, and make our thoughts progress continuously. This will not only enable us to accumulate more knowledge, but also enrich our lives.

Third, strengthen writing training.

An important purpose of learning Chinese is writing. Improving writing ability should start from bit by bit. Extracurricular accumulation is the basis of writing. To learn to read articles carefully, you'd better recite wonderful chapters. You can't write a good article on an empty stomach. Besides, observing life, feeling life, keeping a diary and writing with emotion are all effective ways to help us write a good composition. The written composition should be revised repeatedly, or you can consult your teachers and classmates to keep improving.

I hope you can establish a correct learning purpose, master the basic learning methods, and stick to it for a long time from now on. With accumulated efforts, the results will naturally come.

Recently, people of insight from all walks of life (especially literary and academic circles) have put forward many sharp and pertinent criticisms on the current Chinese teaching. To sum up, it is nothing more than: the teacher is very tired and the students study very hard; What should not be learned is popular, and what should be learned has not really been learned; It consumes students' youth, affects their health, and even distorts the minds of the next generation to varying degrees. Through criticism and reflection, we have to pick up a teaching motto left by our predecessors again: teach people to fish, but don't teach people to fish.

At present, the so-called Chinese studies counseling books, workbooks, simulation papers, etc. , supplied in the book market, packed in students' schoolbags, piled on teachers' desks, advertised in the news media, and sold to schools by thousands of farmers who specialize in this line, mostly with the names of famous teachers, famous teachers and famous schools (some are false names, and some are indeed compiled by such prominent figures), but most of these flooded goods can be attributed to them. Really valuable Chinese reference books that can be classified as "fishing" are hard to find. In particular, we must face up to the fact that it is difficult for the common Chinese textbooks we are using to be included in the category of "fishing", at least the purpose of "teaching them to fish" has not been penetrated.

We might as well discuss what is "fishing" in Chinese learning and what it should consist of.

The author believes that in Chinese teaching, the "fishing" that teachers should teach students should be the most basic effort for students to learn Chinese well today, which is conducive to the development of all aspects in the future and will benefit them for life. Roughly as follows.

First, Chinese Pinyin.

Hanyu Pinyin is a simple and scientific tool to annotate and correct the pronunciation of Chinese characters, which has incomparable advantages compared with the ancient backcutting method.

Second, writing.

Writing should be standardized, clear and beautiful. Punctuation marks should be regarded as an integral part of the text and should also meet this requirement.

Third, read aloud and recite.

Reading language materials should be correct, clear and full of emotion. The object of reciting should be excellent classic texts, and should not be things that have no reciting value, such as the popular teacher's handouts and exercise answers.

Fourth, consult reference books.

Primary and middle school students should learn to skillfully use four Chinese reference books: Xinhua Dictionary (lower grade of primary school), Idiom Dictionary, Modern Chinese Dictionary (upper grade of primary school) and Dictionary of Ancient Chinese Common Words (junior high school). These dictionaries should be equipped and learned to use at different learning stages. In high school, we should teach more ways to consult professional large-scale reference books and documents (but it is impossible for everyone to have them, and students can go to the school library to consult them themselves). Once students use it, teachers must never interfere. Now teachers often check first and then copy them to students, which must be completely changed.

Fifth, take notes.

Chinese notes can be roughly divided into three categories: head notes, reading notes and diaries. Among them, taking notes at the beginning of a book is the simplest, most practical and most indispensable effort. Reading notes can be composed of a "reading outline" notebook and a card. The former is used to record the titles, sources and content summaries of valuable people in the books/articles read, while the latter is used to extract wonderful fragments, good sentences, aphorisms, etc. Diary records daily study and life, observing and thinking.

6. Collect, collate and accumulate data.

At present, this point is almost blank, most teachers don't talk about it, students don't know it at all, and they must start from scratch.

Seven, extracurricular reading.

Reading in class is just an example, and reading after class is the purpose and foundation. Reading should be a big reading, besides books and periodicals, it should also include watching/listening to film and television broadcasts.

Pay attention to life, participate in life, experience life and understand life.

Modern scholars advocate "the sound of wind and rain, the sound of reading, family affairs, state affairs and everything in the world are all concerned", which is not only the creed of life, but also an indispensable way to learn Chinese well. Chinese class can combine the teaching of literary works with the teaching of composition, guide students to actively pay attention to and participate in social life, and seriously experience and understand life. In this way, students' compositions will get a steady stream of "flowing water".

Nine, thinking, analyzing and questioning.

, film and television) income and living income. No matter where you come from, you should gradually develop the habit of independent thinking, dialectical analysis and bold questioning. Only in this way can we lay a solid foundation for future application and innovation.

X. language expression.

Language expression is divided into oral and written forms, both of which are equally important. For written expression, at present, only accuracy, coherence and appropriateness are generally emphasized. I think we should add simplicity and truth. Chekhov said: Simplicity is the sister of genius. And truth is the major premise. Nowadays, students' compositions are full of false things. What's the use of false expressions even if they are accurate, coherent and appropriate?

Looking at the current Chinese teaching, among the above ten items, except for the first item, students' pinyin ability and spoken Mandarin level are relatively good, the other nine items can be said to have not received enough attention and solid training, let alone formed habits, while others are completely blank (not only middle school students, but also some university graduates of Chinese Department, leaving some gaps in this respect). Our Chinese teaching, if we can make practical efforts from the above ten aspects (strict requirements, scientific training and making it a habit), it is not difficult to completely change the "strange phenomenon" of "time-consuming, heavy burden and low efficiency" and "China learns his own language, but he can't pass the exam after more than ten years".

Senior high school students should strengthen their awareness of autonomous learning.

Some students do well in junior high school, but they slip away after high school. The more they learn, the harder it becomes. The main reason is that the self-study consciousness is weak, and it is not clear what is the difference between the learning methods of high school and junior high school.

Chinese learning methods in senior high schools are different, mainly in three aspects; First, junior high school Chinese texts are short, while senior high school texts are mostly long; Junior high school teachers generally use the "intensive lecture" method to explain in detail, while senior high school Chinese classes should generally use the "guided reading" method to teach, focusing on inspiration, induction and inspiration. This requires high school students to constantly improve their self-study ability, "self-study, self-composition" (Ye Shengtao). Second, most junior high schools give priority to imparting knowledge, while senior high schools require not only learning knowledge, but also emphasizing the application of knowledge, training in application, and improving self-study ability and application ability. Third, senior high schools should greatly improve their thinking ability, require students to think more about problems, independently complete reading and writing tasks, and improve their ability to analyze and solve problems. All these require senior high school students to strengthen their self-study awareness and improve their self-study ability.

Sociologists point out that yesterday's illiterates were uneducated people, today's illiterates can't use computers, and tomorrow's illiterates can't teach themselves. Colleges and universities also have a soft spot for people with strong self-learning ability, but they despise high school students who are highly dependent (actually inert) and even deliberately come out when they put out college entrance examination questions. No matter from which angle, it is very important to cultivate and improve their self-study ability.

Generally speaking, self-learning a text can be divided into three steps. The first step is to learn "tips" (including "preview tips" and "self-study tips"), take into account "thinking and practice" and unit knowledge essays, clarify the key points and difficulties, and establish self-study goals. Step 2: Read the text and notes, and master the text content and writing skills as a whole (including the profound meaning of the topic). Step 3: independently complete the exercises behind the text and the supplementary "one lesson and one practice" related exercises to consolidate and expand the results of self-study. In self-study, we should pay attention to the principle of combining intensive reading with skimming. Read the central paragraph, wonderful paragraph and characteristic paragraph of the text intensively, and skim the rest. For example, in the article "Climbing Mount Tai in the Rain", the author's vivid description of Mount Tai pine trees is very wonderful and distinctive, which needs intensive reading. The author uses personification and other rhetorical devices to describe the various postures of pine trees, and also implies the author's life experience and thinking: stubbornly facing the wind, frost, rain and snow. Repeatedly pondering this passage should be plagiarism and imitation. Mao Zedong highly praised the famous educator Xu Teli's emphasis that "without pen and ink, there is no reading". This is especially true when reading texts. Self-study texts must be diligent in pen and ink, and adopt corresponding self-study skills, such as writing outlines, excerpts, cards, lists, charts, comments, circles, comparisons, notes, and feelings after reading. In addition, you can also use the "self-test method" (using the "test questions" in this paper to test yourself) to consolidate the results of self-study and strive for the greatest self-study effect.

Many students have long been accustomed to the baby-like foraging method that teachers chew and feed, and their self-learning ability in high school is still very poor. Rome was not built in a day. Thawing three feet was not built in a day. To overcome the impetuous mentality of thinking at once, few people make a fortune, and many people make a fortune: self-study requires long-term unremitting efforts to achieve great gains.

High school mathematics learning methods

First of all, we should have a good interest in learning.

More than 2,000 years ago, Confucius said, "Knowing is not as good as being kind, and being kind is not as good as being happy." It means that it is better to love something than to do it, to know it, to understand it, and to enjoy it than to like it. "Good" and "happy" mean willing to learn and enjoying learning, which is interest. Interest is the best teacher. Only when you are interested can you have hobbies. If you like it, you have to practice and enjoy it. With interest, we can form the initiative and enthusiasm of learning. In mathematics learning, we turn this spontaneous perceptual pleasure into a conscious and rational "understanding" process, which will naturally become the determination to learn mathematics well and the success of mathematics learning. So how can we establish a good interest in learning mathematics?

1. preview before class, and have doubts and curiosity about what you have learned.

2. Cooperate with the teacher in class to satisfy the excitement of the senses. In class, we should focus on solving the problems in preview, regard the teacher's questions, pauses, teaching AIDS and model demonstrations as appreciating music, answer the teacher's questions in time in class, cultivate the synchronization of thinking and teachers, improve the spirit, and turn the teacher's evaluation of your questions into a driving force to spur learning.

3. Think about problems, pay attention to induction, and tap your learning potential.

4. Pay attention to the teacher's mathematical thinking when explaining in class, and ask why you think so. How did this method come about?

5. Let the concept return to nature. All disciplines are summarized from practical problems, and mathematical concepts also return to real life, such as the concept of angle, the generation of rectangular coordinate system and the generation of polar coordinate system are all abstracted from real life. Only by returning to reality can the understanding of concepts be practical and reliable and accurate in the application of concept judgment and reasoning.

Second, form a good habit of learning mathematics.

Establishing a good habit of learning mathematics will make you feel orderly and relaxed in your study. The good habits of high school mathematics should be: asking more questions, thinking hard, doing easily, summarizing again and paying attention to application. In the process of learning mathematics, students should translate the knowledge taught by teachers into their own unique language and keep it in their minds forever. Good habits of learning mathematics include self-study before class, paying attention to class, reviewing in time, working independently, solving problems, systematically summarizing and studying after class. Let your math study get used to all aspects of math classroom learning.

Third, timely understand and master the commonly used mathematical ideas and methods.

To learn high school mathematics well, we need to master it from the height of mathematical thinking methods. Mathematics thoughts that should be mastered in middle school mathematics learning include: set and correspondence thoughts, classified discussion thoughts, combination of numbers and shapes, movement thoughts, transformation thoughts and transformation thoughts. With mathematical ideas, we should master specific methods, such as method of substitution, undetermined coefficient method, mathematical induction, analysis, synthesis and induction. 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.

1, turn your attention to ideological learning.

People's learning process is to understand and solve unknown knowledge with mastered knowledge. In the process of mathematics learning, old knowledge is used to lead out and solve new problems, and new knowledge is used to solve new knowledge when mastered. Junior high school knowledge is the foundation. If you can answer new knowledge with old knowledge, you will have the idea of transformation. It can be seen that learning is constantly transforming, inheriting, developing and updating old knowledge.

2. Learn the mathematical thinking method of mathematics textbooks.

Mathematical thinking methods in learning mathematics textbooks. Mathematics textbooks melt mathematics thoughts into mathematics knowledge system by means of suggestion and revelation. Therefore, it is very necessary to summarize and summarize the mathematical thoughts in time. Summarizing mathematical thought can be divided into two steps: one is to reveal the content law of mathematical thought, that is, to extract the attributes or relationships of mathematical objects; The second is to clarify the relationship between mathematical ideas, methods and knowledge, and refine the framework to solve the whole problem. The implementation of these two steps can be carried out in classroom listening and extracurricular self-study.

Classroom learning is the main battlefield of mathematics learning. In class, teachers explain and decompose mathematical ideas in textbooks, train mathematical skills, and enable high school students to acquire rich mathematical knowledge. Scientific research activities organized by teachers can make mathematical concepts, theorems and principles in textbooks be understood and excavated to the greatest extent. For example, in the teaching of the concept of reciprocal in junior high school, teachers often have the following understanding in classroom teaching:

① Find the reciprocal of 3 and -5 by definition, and the reciprocal is _ _ _ _ _.

② Understanding from the angle of number axis: Which two points indicate that numbers are opposite? (a point symmetrical about the origin)

③ In terms of absolute value, the two numbers of absolute value _ _ _ _ _ are opposite.

④ Are the two numbers that add up to zero opposite?

These different angles of teaching will broaden students' thinking and improve their thinking quality. I hope that students can take the classroom as the main battlefield for learning.

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. The commonly used mathematical thinking strategies in senior high school mathematics include: controlling complexity with simplicity, combining numbers with shapes, advancing forward and backward with each other, turning life into familiarity, turning difficulties into difficulties, turning retreat into progress, turning static into dynamic, and separating and combining.

Fourth, gradually form a "self-centered" learning model.

Mathematics is not taught by teachers, but acquired through active thinking activities under the guidance of teachers. To learn mathematics, we must actively participate in the learning process, develop a scientific attitude of seeking truth from facts, and have the innovative spirit of independent thinking and bold exploration; Correctly treat difficulties and setbacks in learning, persevere in failure, be neither arrogant nor impetuous in victory, and develop good psychological qualities of initiative, perseverance and resistance to setbacks; In the process of learning, we should follow the cognitive law, be good at using our brains, actively find problems, pay attention to the internal relationship between old and new knowledge, not be satisfied with the ready-made ideas and conclusions, and often think about the problem from many aspects and angles and explore the essence of the problem. When learning mathematics, we must pay attention to "living". You can't just read books without doing problems, and you can't just bury your head in doing problems without summing up the accumulation. We should be able to learn from textbooks and find the best learning method according to our own characteristics.

Five, according to their own learning situation, take some concrete measures.

Take math notes, especially the different aspects of concept understanding and mathematical laws, as well as the extracurricular knowledge that teachers expand in class. Write down the most valuable thinking methods or examples in this chapter, as well as your unsolved problems, so as to make up for them in the future.

Establish a mathematical error correction book. Write down error-prone knowledge or reasoning in case it happens again. Strive to find wrong mistakes, analyze them, correct them and prevent them. Understanding: being able to deeply understand the right things from the opposite side; Guo Shuo can get to the root of the error, so as to prescribe the right medicine; Answer questions completely and reason strictly.

Recite some mathematical rules and small conclusions, so that your usual operation skills can reach the level of automation or semi-automation proficiency.

Often organize the knowledge structure into plate structure and implement "full container", such as tabulation, so that the knowledge structure can be seen at a glance; Often classify exercises, from a case to a class, from a class to multiple classes, from multiple classes to unity; Several kinds of problems boil down to the same knowledge method.

Read math extracurricular books and newspapers, participate in math extracurricular activities and lectures, do more extracurricular math problems, increase self-study and expand knowledge.

Review in time, strengthen the understanding and memory of the basic concept knowledge system, carry out appropriate repeated consolidation, and eliminate learning without forgetting.

Learn to summarize and classify from multiple angles and levels. Such as: ① classification from mathematical thoughts, ② classification from problem-solving methods, ③ classification from knowledge application, etc. , so that the knowledge learned is systematic, organized, thematic and networked.

Often do some "reflection" after doing the problem, think about the basic knowledge used in this problem, what is the mathematical thinking method, why do you think so, whether there are other ideas and solutions, and whether the analytical methods and solutions of this problem have been used in solving other problems.

Whether it is homework or exams, we should put accuracy first and general methods first, rather than blindly pursuing speed or skills. This is an important problem to learn mathematics well.