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.
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.