The cultivation of students' thinking ability is one of the three abilities in mathematics teaching. In the usual teaching, we should pay attention to the cultivation of logical thinking ability besides observation, intuition and imagination. In particular, the cultivation of intuitive thinking ability has been neglected for a long time, and students tend to misunderstand the essence of mathematics in the learning process and think that mathematics is boring; At the same time, they also lack the necessary confidence to succeed in mathematics learning, thus losing their interest in mathematics learning. Cultivating intuitive thinking ability is the need of social development and talents to adapt to the new era.
First, the interpretation of mathematical intuitive thinking
Mathematical intuition is a conscious human brain's direct understanding and insight into mathematical objects (structures and their relationships).
Intuition Intuition is the feeling or perception obtained directly through various sensory organs with real things as the object. For example, the definition of the concept and properties of an isosceles triangle, such as a triangle with two equal base angles and two equal angles, is not a strict proof, but an intuitive perception. The research object of intuition is abstract mathematical structure and its relationship. Poincare said: intuition does not have to be based on understanding. Feeling will soon become weak. For example, we still can't imagine polygons, but we can think of polygons intuitively. Polygons including polygons are a special case. This shows that intuition is a deep psychological activity, and thinking has no specific intuitive image and logical order of operation. As Divadone said, what distinguishes these creative scientists is that they have a vivid idea and a deep understanding of the research object. When these ideas are combined with understanding, they are called "intuition" ... because the objects they apply to are usually invisible in our sensory world.
Thinking mode can be divided into logical thinking and intuitive thinking. For a long time, people deliberately separated the two. Actually, this is a misunderstanding. Logical thinking and intuitive thinking are never separated. There is a view that logic is more important than deduction and intuition is more important than analysis. From the point of view of emphasis, this statement is not unreasonable, but emphasis does not mean completeness. Will there be intuition in mathematical logic? Is Mathematical Intuition Logical? For example, there are many inexplicable things in daily life, and people can't judge and guess various events without intuition. It can even be said that intuition works all the time. Mathematics is also a reflection of the objective world. It is the embodiment of people's intuition about life phenomena and the running order of the world, and then the rational process of formatting thinking in the form of mathematics. The original concepts of mathematics are all based on intuition. Mathematics is developed in solving problems to a certain extent, and solving problems cannot be separated from intuition. We take the proof of mathematical problems as an example to investigate the role of intuition in the proof process.
A mathematical proof can be decomposed into many basic operations or many deductive reasoning elements. A successful mathematical proof is a successful combination of these basic operations or deductive reasoning elements, just like a passage from the beginning to the end. Every basic operation and deductive reasoning element is a section of this passage. When a successful proof is in front of us, logic can help us to be sure that we can reach our destination smoothly along this road, but logic can't tell us why the choice of these paths and such a combination can. In fact, soon after we set out, we will encounter a fork in the road, that is, we will encounter the problem of correctly choosing the road sections that make up the passage. Poincare believes that even if a successful mathematical proof can be reproduced, I don't know what leads to the consistency of the proof ... the order of these elements is more important than the elements themselves. Descartes believes that intuition is indispensable in every step of mathematical reasoning. Just like we usually play basketball, it depends on the sense of the ball. In the fast movement, we have no time to make logical judgments. Action is only subconscious, and subconscious action is only an intuition generated in normal training.
In the process of education, teachers make the proof process too strict and procedural. Students just see a rigid logical shell, and the aura of intuition is covered up. They often attribute their success to logic, but don't think about their intuition. Students' intrinsic potential has not been stimulated, their interest in learning has not been mobilized, and they can't get the real pleasure of thinking. China Youth Daily reported that about 30% of junior high school students lost interest in mathematics learning after learning plane geometry reasoning, which should arouse the attention and reflection of mathematics educators.
Second, the main characteristics of students' intuitive thinking
Intuitive thinking has the characteristics of freedom, flexibility, spontaneity, contingency and unreliability. Judging from the necessity of cultivating intuitive thinking, the author thinks that intuitive thinking has the following three main characteristics:
(1) Simple
Intuitive thinking is a sharp and rapid assumption, guess or judgment made by investigating the thinking object as a whole and mobilizing all your knowledge and experience through rich imagination. It omits the intermediate link of step-by-step analysis and reasoning, and adopts the jumping form. It is a flash of thought, a sublimation of long-term accumulation, an inspiration and epiphany of thinkers, and a highly simplified thinking process, but it clearly touches the essence of things.
(2) Creativity
Modern society needs innovative talents. Because of long-term reference to foreign experience, China textbooks pay too much attention to the cultivation of logical thinking, and most of the trained talents are accustomed to step by step, stick to the rules, and lack creativity and pioneering spirit. Intuitive thinking is based on the overall grasp of the research object, and it is not devoted to the scrutiny of details. It is a big thinking. It is precisely because of the unconsciousness of thinking that its imagination is rich and divergent, which makes people's cognitive structure expand infinitely outward, thus having the originality of abnormal law.
Ian Stuttgart said: Intuition is what real mathematicians live on, and many important discoveries are based on intuition. The five postulates of Euclid geometry are all based on intuition, thus establishing the glorious building of Euclid geometry; Hamilton started the spark of constructing four elements on the road of walking; Archimedes found a way to distinguish the true crown from the false crown in the bathroom; Kekule found that the cyclic structure of benzene is a successful example of intuitive thinking.
(3) Confidence
Students are interested in mathematics for two reasons. One is the teacher's personality charm, and the other is the charm of mathematics itself. There is no denying the important role of emotion, but the author's point of view is that interest comes more from mathematics itself. Success can cultivate a person's self-confidence, and intuitive discovery is accompanied by strong self-confidence. Compared with other material rewards and emotional incentives, this self-confidence is more stable and lasting. When a problem is proved by intuition rather than logic, then the shock brought by success is enormous, and there will be a strong motivation for learning and research in the heart, so that you have more confidence in your ability.
Gauss can solve the problems in primary schools. 1+2+...+99+ 100 =? This is based on his extraordinary grasp of logarithmic sensitivity, which has an indelible impact on his life's success. Nowadays, middle school students have little intuitive awareness and are dubious about limited intuition. They can't control the problem as a whole, so they can't form self-confidence.
Third, how to cultivate students' intuitive thinking
A person's mathematical thinking and judgment ability mainly depend on the level of intuitive thinking ability. Professor Xu Lizhi pointed out that mathematical intuition can be cultivated. In fact, everyone's mathematical intuition is constantly improving. Mathematical intuition can be improved through training.
(! A solid foundation is the source of intuition.
Intuition does not depend on opportunity. Although the acquisition of intuition is accidental, it is by no means a fantasy for no reason, but is based on solid knowledge. Without a deep foundation, there will be no slightest thinking. Attiya said: Once you really feel that you understand something and have gained enough experience in dealing with that problem through a large number of examples and contacts with other eastern countries, you will have an intuition about the situation in the development process and what conclusions should be correct. Adama once said humorously: Can a monkey be printed into the entire American Constitution to meet the opportunity?
(2) Infiltrating the philosophy and aesthetics of mathematics.
Intuition is based on the overall grasp of the research object, while philosophical views are conducive to grasping the essence of things. These philosophical viewpoints include the unity of opposites, movement changes, mutual transformation, symmetry and so on, which are common in mathematics. For example, (a+b)2= a2+2ab-b2, even if you haven't studied the complete square formula, you can judge the truth of the conclusion from a symmetrical point of view.
Aesthetic feeling and aesthetic feeling are the essence of mathematical intuition. Improving aesthetic ability is conducive to cultivating intuitive awareness of the harmonious relationship and order among mathematical things. The stronger the aesthetic ability, the stronger the mathematical intuition ability. 193 1 year, Dirac boldly put forward the antimatter hypothesis from the perspective of mathematical symmetry. He thinks that the antielectrons in a vacuum are positrons. He also questioned Maxwell equation. He once said that if a physical equation looks unattractive mathematically, then the correctness of the equation is questionable.
(3) Pay attention to problem-solving teaching.
Choosing appropriate topic types in teaching is conducive to cultivating and testing students' intuitive thinking.
For example, multiple-choice questions only need to be selected from four branches, which omits the problem-solving process and allows reasonable guessing, which is conducive to the development of intuitive thinking. Implementing open question teaching is also an effective way to cultivate intuitive thinking. The conditions or conclusions of open-ended questions are not clear enough, so we can seek the reasons from the results and guess from the reasons. Because of the divergence of answers, it is conducive to the cultivation of intuitive thinking ability.
(4) Setting the artistic conception and motivation induction of intuitive thinking.
This requires teachers to change their teaching concepts and return the initiative to students. Fully affirm students' bold ideas, encourage their reasonable components in time, care for and support students' spontaneous intuitive thinking, and don't dampen their enthusiasm and understanding of intuitive thinking. Teachers should make use of the situation in time, eliminate doubts in students' hearts and make students happy with their intuition.
Follow your feelings is what teachers often say. In fact, this sentence already contains the bud of intuitive thinking, but it has not risen to a thinking concept. Teachers should clearly put forward intuitive thinking, formulate corresponding activity strategies and analyze the characteristics of problems as a whole in classroom teaching; Attach importance to the teaching of mathematical thinking methods, such as method of substitution, the combination of numbers and shapes, induction, guessing ideas and reduction to absurdity. It is of great benefit to infiltrate intuitive concepts and the development of thinking ability.
Four. Concluding remarks
Intuitive thinking is as important as logical thinking. Deviation from either side will restrict the development of one's thinking ability. Edith Stewart once said that the whole power of mathematics lies in the ingenious combination of intuition and strictness, controlled spirit and inspirational logic. Controlled spirit and aesthetic logic are the charm of mathematics and the direction of mathematics educators' efforts.