The Significance of Mathematical Intuition

Mathematical intuition is a form of psychological activity that directly reflects the structural relationship of mathematical objects, and it is the direct understanding or insight of mathematical objects by the human brain. When using knowledge chunks and intuition, we must deal with them properly, and connect the chunks stored in the brain that are similar to the current problem through different intuition. It creatively deals with the decomposition, transformation and integration of problems.

Mathematical intuition can be called number sense (many people think it belongs to image thinking), but it is not only mathematicians who can produce mathematical intuition. For those who have reached a certain level in learning mathematics, intuition can be generated and cultivated. The foundation of mathematical intuition lies in the chunk of mathematical knowledge and the growth of mathematical image intuition. Therefore, if a student can understand the conclusion directly and quickly when solving a new mathematical problem, then we should think that this is the performance of mathematical intuition.

Mathematics is the reflection of the objective world, it is the embodiment of people's intuition about the order of the life phenomenon world, and then the rational thinking process is formed in the form of mathematics. The original concept of mathematics is 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 "deductive reasoning elements", and a successful combination is like a passage from the beginning to the end. A basic operation and "deductive reasoning elements" are the chapters of this article. When a successful proof is in front of us, logic can help us determine 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 form a channel. 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 copied, I don't know what causes the consistency of the proof. ..... The order of these elements is more important than the elements themselves. Descartes believes that intuitive ability is indispensable in every step of mathematical reasoning. Just like we usually play basketball, we have to wait for the ball. In the fast movement, we have no time to make logical judgments. Action is only subconscious, and subconscious action is intuition generated by normal training.

In the process of education, because the teacher makes the process of proof too strict and procedural, students only see a rigid logical shell, and the aura of intuition is covered up. Success is often attributed to logic without thinking about their own intuition. Students' inner potential has not been stimulated, students' interest has not been mobilized, and they can't get the real pleasure of thinking. China Youth Daily reported that "about 30% junior high school students lose interest in mathematics learning after learning plane geometry reasoning", which should arouse the attention and reflection of mathematics educators.

Second, the main characteristics of mathematical intuitive thinking

Intuitive thinking has the following four 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 form of "jumping". 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 on the "essence" of things.

(2) empirical. The knowledge blocks and intuitive images used by intuition are the accumulation and sublimation of experience. Intuition constantly combines old experience to form new experience, thus constantly improving the level of intuition.

(3) rapidity. The process of intuitive problem solving is short, sensitive and direct.

(4) Possibility. The result of intuitive judgment is not necessarily correct. The result of intuitive judgment is not always correct, which is due to the fuzziness of lexical chunks themselves and their connections.

Thirdly, the cultivation of intuitive thinking in mathematics.

From the previous analysis, we can see that the focus of cultivating mathematical intuitive thinking is to attach importance to mathematical intuition. Professor Xu Lizhi pointed out: "Mathematical intuition can be cultivated. In fact, everyone's mathematical intuition is constantly improving. " In other words, mathematical intuition can be improved through training. Bruner, a famous American psychologist, pointed out: "The training of intuitive thinking and premonition is an important neglected feature of creative thinking in formal disciplines and daily life." And put forward "how is it possible to cultivate students' intuitive talent from the lower grades?" "Our students, especially poor students, have rich intuitive thinking potential, and the key lies in teachers' inspiration, induction and intentional cultivation. On the basis of clarifying the meaning of intuition, we can cultivate mathematical intuition from the following aspects:

1. Pay attention to the firm grasp and application of basic mathematical problems and methods, thus forming and enriching mathematical knowledge chunks.

Intuition does not depend on "opportunity". Although intuition is acquired by accident, it is by no means a fantasy for no reason, but is based on solid knowledge. Without a deep foundation, there will be no sparks of thought. Therefore, it is very important to master and apply the basic problems and methods of mathematics firmly. The so-called knowledge chunk is also called knowledge reaction block. They are composed of definitions, theorems, formulas and rules in mathematics, which are reflected in some basic problems, typical problems or methods. The solution of many other problems can often be reduced to one or several basic problems, become a typical question type, or be used in some way. Because these knowledge blocks do not necessarily appear in the form of theorems, properties, rules and so on. Instead, they are distributed in examples or problems, which are not easy to attract the special attention of teachers and students and are often submerged in the ocean of problems. How to screen them out and extract them is an important subject worthy of study in mathematics.

When solving a math problem, the subject often flashes an idea to describe the general idea of solving the problem after understanding the meaning of the problem and mastering the characteristics of the conditions or conclusions of the problem. This is something that top students often encounter. They have more knowledge chunks and intuitive images stored in their brains than ordinary students, so quick response mathematical intuition came into being.

Example: Known, Verified:

After analyzing and observing the formula structure of the topic conditions and conclusions, two thoughts will flash: (1) When a, b and c are arbitrary values, the equation usually does not hold, so there is a simpler relationship between a, b and c than the given conditions; (2) As a special case, it is obvious that when two of the three numbers are opposite, the conditions and conclusions are both valid, which means that the conditional formula contains the factor (a+b) or (b+c) or (c+a), and because of the rotational symmetry, it must contain (a+b)(b+c) (c+a), thus forming a mathematical intuition. This intuition comes from the past operating experience-knowledge chunk, and also from the direct perceptual transformation of schema representation given by the problem.

2. Emphasize the combination of numbers and shapes, and develop geometric thinking and quasi-geometric thinking.

Mathematical image intuition is one of the sources of mathematical intuitive thinking, and mathematical image intuition is a form of geometric intuition or spatial concept. For geometric problems, we should cultivate the intuitive feeling ability of geometric transformation and deformation. For non-geometric problems, we should examine and analyze them from the perspective of geometry, and then we can gradually transition to geometric-like thinking.

Example 2: if a < b < c, find the minimum value of the function y = | x-a |+| x-b |+| x-c |.

Analysis: The distance formula between two points on the number axis is AB = | XA-XB |, and the approximate positions of numbers A, B and C on the number axis are shown in the figure.

a

b

c

Find the minimum value of y = | x-a | +| x-b |+| x-c |. That is to say, find the point X on the number axis to minimize the sum of its distances to A, B and C. Obviously, when X is between A and C, | X-A |+| X-C | is the smallest. therefore

When x=b, the value of y = | x-a | +| x-b |+| x-c | is the smallest.

3. Pay attention to overall analysis and advocate block thinking.

When solving mathematical problems, we should teach students to analyze the whole problem from a macro perspective, grasp the framework structure and essential relationship of the problem, and determine the starting point and thinking of solving the problem from the perspective of thinking strategies. On the basis of overall analysis, let students think big, so as to change and reduce problems, analyze and identify the knowledge integration blocks that constitute problems, and cultivate their thinking jumping ability. In practice, we should pay attention to the exploration of methods, the search of ideas and the identification of types, and cultivate the epiphany ability to simplify the logical reasoning process and make intuitive judgments quickly.

Example 3: I is the heart of △ABC, and the extension lines of AI, BI and CI intersect the circumscribed circle of △ABC in D, E and F respectively. Proof: AD+BE+CF > AB+BC+CA.

D

E

F

B

A

C

I

Analysis: Observe the graph carefully and seek the applicable knowledge chunks. There are two intuitions: (1) Di = DB = DC comes from inner nature; (2) The proper combination of trigonometric inequalities should BE used to form characteristic inequalities, which enlightens that AD can be divided into two stages (be and CF are similar), that is, DB+DC > BC can lead to DI> BC and AI+IB > AB. After four other similar inequalities are obtained, they can be added in the same direction to draw a conclusion.

4. Encourage bold guesses and develop mathematical thinking habits that are good at guessing.

Mathematical conjecture is a thinking process of conceiving mathematical propositions before mathematical proof. "Mathematical facts are first guessed and then confirmed." Guess is a reasonable reasoning and a supplement to the logical reasoning used in argument. For mathematical problems with no conclusion, the formation of conjecture is conducive to the correct induction of problem-solving ideas; For the problems that have been concluded, conjecture is also an important means to seek thinking strategies for solving problems. Mathematical conjecture has certain rules, and it should be based on the experience of mathematical knowledge. But cultivating the thinking habit of daring to guess and being good at exploring is the basic quality of forming mathematical intuition, developing mathematical thinking and obtaining mathematical discovery. Therefore, in mathematics teaching, we should not only emphasize the rigor of thinking and the correctness of results, but also ignore the exploration and discovery of thinking, that is, we should attach importance to the rationality and necessity of mathematical intuitive guessing.

Example 4: As shown in the figure, in a square ABCD, BC = 2cm, and there are two points, E and F, starting from point B and point A respectively. BA moves to point A at a speed of 1cm/s along point E, and point F moves to point C at a speed of 2cm/s along dotted line A-D-C. Set the time for point E to leave point B as t (seconds) (66. If there are any changes, please explain the reasons; If there is no change, please give proof and find the value of AP∶PC.

Guess: the position of point P remains unchanged. Analysis: Because the time from point E to point B is t (seconds), AE = (2- 1t) cm. Since the time from point F to point A is t (seconds), the speed is 2cm/s, and CF = (4-2t) cm. Then:

E

F

D

A

B

C

P

Because AE‖FC, the factor AP∶PC=AE∶CF= 1∶2, the position of point P remains unchanged.

The cultivation of mathematical intuitive thinking ability is a long-term process. To be a good teacher, we must cultivate students' intuitive thinking into every corner of mathematics education, so that students can have agile thinking, flexible problem-solving ideas and strong comprehensive utilization ability of previous knowledge structures. This is not only conducive to the intellectual development of students, but also conducive to the cultivation of students' logical thinking.