"Being good at mathematics" is an educational ideal and the other side we pursue, but we often run counter to it and drift away. "Good mathematics" is a kind of value transformation and a yardstick of our knowledge, but we often distinguish between true and false; "Good mathematics" is an objective existence, which exists in our educational life, but we are often in it and don't know what it looks like. So first of all, we should make clear what "good mathematics" is.
First, "good mathematics" is not only "mathematics", but also "human studies"
Our mathematics education is not only to let students master the necessary basic knowledge and skills, but more importantly, to let students feel more important basic ideas and basic life experiences. At the same time, students should learn to think by using mathematical thinking mode, understand the intrinsic value of mathematics, develop good study habits, have a preliminary sense of innovation and a scientific attitude of seeking truth from facts, and so on. In a word, our mathematics education is not only the training of knowledge, but also the accumulation of wisdom, and it is also the embodiment of life growth, life value and significance.
Humanism is an organic ideological system that takes human nature (human essence), meaning of life and code of conduct as the thinking object, takes the theory of human nature as the core, and includes the outlook on life (the theory of life value and code of conduct), the theory of rule by man (the theory of self-cultivation and the theory of governing politics by others) and the theory of human social ideal. It regards mathematics not only as "mathematics" but also as "admission", which is the dialectical unity of mathematical instrumentality and humanity. "Good Mathematics" is a mathematics with people as the core, and it is a real "human study".
Second, "good mathematics" not only teaches knowledge and methods, but also teaches ideas.
Teaching has three levels: teaching knowledge, teaching methods and teaching ideas.
The excellent quality of mathematical thinking method is that it supports the whole mathematical building, is everywhere, is everywhere, is widely used, and is easy to stay in people's long-term memory. Any subject should use mathematical thinking methods, but the ways and degrees of application are different.
Teachers' teaching and students' learning are a unity. "Good mathematics" should first ask four questions:
First, what is the content of teaching and learning (consider what should be taught and what can be taught respectively);
Second, why should we teach and learn these contents?
Third, what should teachers and students do;
Fourth, why do you want to do this? On this basis, teachers should read the text thoroughly and carefully, investigate and analyze the logical starting point of students' learning, so as to understand what knowledge and skills students should master in a class, and what methods and ideas should be understood and improved. Mastering the mathematical thinking method, understanding the quantitative change law of the objective world, and using it to understand and transform the world are the true meaning of mathematical science.
Third, "good mathematics" not only focuses on yesterday and today, but also points to tomorrow.
Mathematics always coexists with challenges and crises. With the rapid development of science and technology, the total amount of human knowledge is increasing, and the speed of knowledge updating is also accelerating. The emerging new technologies and disciplines are closely related to mathematics, especially because of the development of computer technology, the application scope of mathematics is more extensive. We should keep pace with the times and have a forward-looking vision. Being good at math means being moved. She will not stay in the past, nor will she stay in today.
Yesterday, it means basic points and repetition; Today means the starting point and departure; Tomorrow is hope and direction. Yesterday's "old boat ticket" is difficult to board tomorrow's "new passenger ship", and it is really terrible to have no future math learning activities. "Good Mathematics" will not make students become mechanical porters who copy and paste, but let students raise their sails, stimulate their desire to think and explore, and move towards an uncertain and creative future.
Fourth, "good mathematics" is not only memory and imitation, but also development and creation.
In a letter to Professor Zheng Yuxin, American scholar Sting sincerely pointed out: "An important difference between Chinese and American students is that China students are more adapted to the' algorithm' learning of specific solutions to specific problems, while American students are better at solving unconventional problems that are open, vague, have' realistic' significance and need more creativity."
Undoubtedly, our basic education has laid a solid foundation for students, and the important role of memory and imitation is irreplaceable. We can't turn a blind eye here. However, it is obviously short-sighted to be complacent only by lying on a "solid double base". Where is the competitiveness of talents? Of course, her core is innovation ability. "Good mathematics" will inevitably study students' thinking activities in depth and choose mathematics with development and creativity. We neither worship foreigners nor blindly exclude foreigners; We are neither arrogant nor impetuous. We should create a good mathematics education suitable for China's national conditions between "tradition" and "bringing".
5. "Good math" is not only "fun" but also "useful"
In August 2002, during the International Congress of Mathematicians held in Beijing, Mr. Chen Shengshen, a 9 1 year-old master of mathematics, wrote an inscription for the children, and wrote four big characters: "Mathematics is fun". Of course, there are different levels and levels of fun in mathematics, and the fun that math masters see is completely different from the fun that primary school students see.
Being good at math is not only fun, but also useful. -Mathematics curriculum reform makes those "complicated and difficult to understand". Delete the old "useless" content, so that students can learn "useful" mathematics. It can be said that "mathematics is very useful."
Whenever I see a small member of the school team who has played football for four or five hours in the scorching sun, I can't help asking, "Are you tired?" "Not tired!" -the little guys answered very simply. Obviously exhausted, but still excited, why? Because he likes it, because he is intoxicated, because he suffers, because he enjoys it more. He doesn't feel bitter, but he feels fun, and he doesn't feel tired when he is tired. Actually, it's that simple.
I think that when our children think mathematics is "fun", mathematics will no longer be boring, mechanical and difficult to understand, its inner beauty will no longer be far away, and its application value will naturally be realized. "Good at math" should be like this.
Teaching thought of mathematics education
Mathematics teaching process is a complicated process, and its fundamental purpose is to enable students to master the necessary mathematical theoretical knowledge and develop their abilities. In the process of imparting knowledge and cultivating basic abilities, we should constantly strengthen ideological education and cultivate students into a generation of new people with ideals, morality, culture and discipline who are all-round developed morally, intellectually, physically and aesthetically.
In this regard, as a math teacher, it is incumbent on him to undertake the glorious responsibility of teaching and educating people. However, mathematics course has its own characteristics, and if it is taught vaguely without the characteristics of mathematics itself, it will greatly affect the teaching quality. Therefore, we must combine the characteristics of mathematics itself, dig deep into the ideological education content contained in mathematics content, and integrate ideological education into intellectual education. Practice has proved that ideological education through specific content is promising. Therefore, this paper puts forward some suggestions on ideological education in mathematics teaching for reference.
First, stimulate students to learn mathematics well in order to realize socialist modernization.
Mathematics is a science that studies the spatial form and quantitative relationship of the real world. The characteristics of all things or the relationship between things need to be described in different degrees through a certain quantitative relationship. As Hua said, "the size of the universe, the size of particles, the speed of rockets, the cleverness of chemical engineering, the change of the earth, the mystery of biology and the complexity of daily life are all inseparable from mathematics." Therefore, mathematics has become an essential scientific and cultural accomplishment for the general members of modern society, an important tool for participating in modernization and an important foundation for learning other science and technology well. With the development of science and technology, mathematical methods are more and more widely used in various disciplines. A science is truly developed only when it can use mathematics. The history of scientific development has proved the correctness of this assertion, so it is very important to learn mathematics well. Because of the wide application of mathematics, we can introduce new knowledge from the application of mathematics in production practice and daily life when introducing new classes, so that students can feel that mathematics is everywhere in their lives.
This can inspire students to apply mathematics to solve practical problems, thus cultivating their strong interest in learning mathematics. Teachers must guide students to realize the necessity and urgency of learning mathematics well, and at the same time cultivate students' strong interest, thus stimulating students' enthusiasm for learning mathematics well.
Second, cultivate students' patriotism and national self-esteem.
It is of great practical significance to strengthen the education of patriotism and national pride for the younger generation, and mathematics teaching should and can undertake its own tasks in this respect. China is one of the ancient civilizations in the world history, which once created splendid culture. During thousands of years of human civilization, China has been at the forefront of the world for most of the time. From the third century BC to the sixteenth century AD, our ancestors always occupied the leading position in the world in mathematical research.
In the past, there was glory in the field of mathematics. At present, China mathematicians or Chinese mathematicians are advanced in the world in a series of fields. In teaching, we should introduce the outstanding contributions of China mathematicians in combination with specific teaching contents, cultivate students' patriotic thoughts, and enable students to establish the necessary national self-esteem and self-confidence. For example, when we talk about the concept of limit, we first describe the idea of limit through the "tangent circle method" created by China ancient mathematician Liu Hui (Ren Wei in the Three Kingdoms period) in 263 AD. At that time, Liu Hui used the tangent circle method to calculate pi to 3. 14 15, which fully shows that the modern method of limit thought was initially formed and applied in the Three Kingdoms period of China.
Third, cultivate students' rigorous scientific attitude and tenacious perseverance.
Mathematics is characterized by rigor. In mathematics teaching, to give full play to this feature, students are required to describe the conclusion concisely and accurately, and the reasoning and demonstration of the conclusion should be gradual and consistent with the requirements of logical theory everywhere. Only in this way can we gradually cultivate students' scientific attitude of seeking truth from facts, persisting in truth, correcting mistakes and being meticulous.
Mathematics is inseparable from reasoning. Cultivating students' reasoning habits through mathematics teaching. To judge whether a proposition or conjecture is true or false in mathematics, we should rely on the definitions of concepts, axioms and theorems for rigorous reasoning and argumentation, rather than through practical tests. In teaching, we should firmly grasp this feature and purposefully cultivate students' reasoning consciousness, so as to achieve the purpose of cultivating students' scientific attitude. Mathematics is highly abstract. Abstraction does not mean that its concepts and research objects are divorced from the objective world and life practice. Through the mathematization of the formation process of mathematical concepts and conclusions, students are trained to grasp the essential characteristics of physical objects, abstract concepts, and gradually cultivate their ability to transform practical problems into mathematical problems. In the teaching process of the new curriculum, we should cultivate students' rigorous and accurate academic spirit through the introduction of concepts and the demonstration of theorems. The exploration of solving problems cultivates students' ability to think diligently and analyze problems comprehensively. When encountering problems, we should have perseverance and perseverance to seek solutions and cultivate students' tenacious perseverance in studying hard.
Fourthly, cultivate students' dialectical materialism.
Engels once pointed out that "dialectics in the real world can be embodied in mathematical concepts and formulas, and students can meet the expression of these dialectical laws anywhere". This shows that we should not introduce dialectics into mathematics as an exotic product, but should look for dialectical factors from the contents and methods of mathematics. Such as finite and infinite; Continuity and discontinuity; Straight lines and curves; Approximation and accuracy; Differential and integral; Convergence and divergence, etc. These contents are full of dialectical factors. In mathematics, we must make full use of the dialectical factors of mathematics itself, cultivate students' dialectical materialism and develop students' dialectical thinking ability.
1. Practice first.
The emergence of mathematics is due to the needs of practice, and the needs of production practice and technological development directly or indirectly stimulate the development of mathematics. The reality and origin of mathematics and the history of its development due to the needs of production practice should be expounded in combination with teaching materials. As we all know, mathematical concepts and formulas are the reflection of objective reality, and they all have their actual models. Therefore, when talking about new knowledge, we should list the things that students are familiar with to introduce concepts and formulas, or let students operate by hand to enrich their perceptual knowledge, and then solve practical problems with what they have learned. This will greatly arouse students' learning enthusiasm and make them understand the view of practice first and the relationship between mathematics and practice theoretically.
2. The view of unity of opposites
Chairman Mao pointed out: "All contradictory things are interrelated, not only unified under certain conditions, but also transformed into each other under certain conditions". The idea of unity of opposites can be seen everywhere in mathematics, such as: positive and negative integers and positive and negative fractions are unified in rational numbers; Rational numbers and irrational numbers are unified in real numbers; Real numbers and imaginary numbers are unified in complex numbers. In mathematics, the opposition and transformation between the two sides of contradiction are frequent, and the whole process of mathematics development is a process of constant unity of opposites. We should always grasp the transformation of opposites in teaching. There are various types of transformation, such as operational transformation; Transformation of numbers and shapes; The transformation of opposing concepts (constants and variables, known and unknown). The key to solving mathematical problems by this transformation method is to analyze the contradictions in the problem, find out the relationship between different conditions in the problem, and then seek the transformation method to achieve the purpose of solving the problem.
3. The viewpoint of movement change
Dialectical materialism holds that movement and change are absolute, and static and unchangeable are relative, but human beings gradually understand these movements and changes in countless relative stillness. This is like human beings knowing absolute truth from countless relative truths, such as curves passing through straight lines, variables passing through constants, precision passing through approximation, abstraction passing through concrete understanding and so on. In mathematics teaching, we should consciously consider, analyze and understand things with changing viewpoints, and then reveal the essential attributes of things. For example, when discussing variable-speed movement, how can we understand variable-speed movement in essence? In calculus, it is to study the instantaneous velocity of motion at a certain point (that is, at a certain moment) and describe the motion state of that point with instantaneous velocity. The definition process of instantaneous speed is the process of knowing the variable speed movement. For another example, if the curve is regarded as the trajectory of a point, if the coordinate system is established and the moving point coordinates are introduced, the curve and the equation can be linked, so that analytic geometry can be formed through the development of algebra and geometry.
4. The viewpoint of mutual change of quality
Everything has a certain quality and quantity, which is the unity of quality and quantity. Quality and quantity are interdependent and mutually restrictive. When the equivalent increases or decreases to a certain extent, the substance will undergo qualitative change. It is not only possible but also necessary to help us understand the changes of things through their changes. For example, the sum of finite infinitesimals or infinitesimals. When we change the word "finite" to infinity, there will be a qualitative change. In fact, the sum of infinite infinitesimals is not necessarily infinite.
5. Negative view of negation
Negation is the decisive link in the development of things. Without negation, there will be no qualitative change, and there will be no death of old things and the emergence of new things. At the same time, negation is also "sublation". The so-called "sublation" includes the meaning of abandoning, retaining and carrying forward; It is overcoming and retaining, criticizing and inheriting; On the basis of overcoming the negative factors of old things, some positive factors conducive to the development of new things are retained. The expansion of Tu Tu's concept runs through the whole elementary mathematics content. The introduction of the concept of new numbers is always carried out on the premise of denying the concept of old numbers, and at the same time, the old and new numbers are unified at the corresponding stage, so that the concept of numbers is continuously enriched, thus solving new problems. For another example, the popularization of the concept of angle is very similar to the development of the concept of number. The negation of negation is two transformations of contradictions and opposites within things. That is, yes-no-no again. That is, the development process of things can be divided into the first stage (affirmative stage); The second stage (negative stage); The third stage (again negative stage). After things develop to the third stage, some features of the first stage may reappear in the third stage, and new things that were not in the first stage may appear in the third stage, that is to say, the negation of negation is not a simple repetition of things, but a spiral rise.