On the paradigm significance and function of abstract thinking

Abstract thinking is logical thinking. The main characteristics of primary school students' thinking are concrete visualization, weak abstract thinking ability and slow development. I'll give you an example of abstract thinking, which I hope will help you.

An example of abstract thinking 1 Professor Chen Shengshen, a Chinese American, is a famous mathematician in the contemporary world. He was surprised by a lecture at Peking University:

? It is often said that the sum of the internal angles of a triangle is equal to 180 degrees. However, this is not right! ?

Everyone was shocked. What's going on here? The sum of the internal angles of a triangle is 180 degrees. Isn't this common sense in mathematics?

Then, the old professor gave an incisive answer to everyone's questions:? It is wrong to say that the sum of the angles in a triangle is 180 degrees, not that this fact is wrong, but that this way of looking at the problem is wrong. It should be said that the sum of the outer angles of a triangle is 360 degrees. ?

? Looking at the inside corner, we can only see:

The sum of the internal angles of the triangle is 180 degrees;

The sum of the internal angles of the quadrilateral is 360 degrees;

Is the sum of the internal angles of an N-polygon (n-2)? 180 degrees.

This has found a formula for calculating the sum of internal angles. The number of edges n appears in the formula. What if you look at the corner outside?

The sum of the external angles of the triangle is 360 degrees;

The sum of the external angles of the quadrilateral is 360 degrees;

The sum of the external angles of the Pentagon is 360 degrees;

The sum of the outer angles of any N-polygon is 360 degrees.

This sums up many situations with a very simple conclusion. A more general rule is found by replacing the formula related to n with a constant unrelated to n. ?

Understanding of abstract thinking;

After reading Chen Shengshen's story, we think of a passage by mathematician Bohr: Mathematicians tend to seek general solutions, and he likes to solve many special problems with several general formulas. ?

Example of abstract thinking 2 A farmer invited engineers, physicists and mathematicians to enclose the largest area with the least fences.

The engineer fenced a circle and declared that it was the best design.

The physicist said:? Remove the fence and form a long enough straight line. When it surrounds half the world, it has the largest area. ?

The mathematician gave them a big laugh. He surrounded himself with several fences and said, I am outside the fence now. ?

Understanding of abstract thinking;

The engineer's design is both practical and beautiful, isn't it? Optimal design? . Physicists have a strange imagination in their thinking, and fences can be decomposed and opened indefinitely. It seems that the closed area is already? The biggest? . Mathematicians surround themselves with few fences and say, I am outside the fence now. ? Engineers and physicists try to circle the largest area, while mathematicians circle the smallest area first. People say to take a step back and broaden their horizons, but mathematicians don't just take a step back, but do the opposite. ? The opposite? This is a quality of reverse thinking.

Reverse thinking is an integral part of creative thinking. Before we face it? Heavy mountains and heavy waters? At this time, reverse thinking often leads us to find? A bright future? This road. Mathematics teaching should make reverse thinking become students' conscious consciousness and practical behavior.

One day, the teacher wanted to see the IQ of the students, so he had the following dialogue.

The teacher asked: There were 10 birds in the tree, and 1 birds were shot. How many birds are left?

The student asked:? Are you sure that bird was really killed?

? Of course. ?

? Is it silent pistol?

? No?

? How loud was the shot?

? 80 ~ 100 decibel. ?

? So it hurts your ears?

? Yes ?

The teacher began to get impatient. Please, just tell me how much is left, okay?

? Ok, are the birds in the tree deaf?

? No?

? Is there anything about being caged?

? No?

? Are there any other trees around? Are there any other birds in the tree?

? No?

? Is it a bird in your belly?

? Not exactly. ?

? Are there any flowers in the bird swatter's eyes? Guaranteed to be 10?

? No flowers, just 10. ?

The teacher was sweating like a pig and the bell rang, but the students kept asking.

? Are you too stupid to be afraid of death?

? Fear of death. ?

? Would you kill two with one shot?

? No?

? Are all birds free to move?

? Absolutely. ?

? If your answer is not a lie, the students say confidently. If the killed bird hangs on the tree and doesn't fall off, only1remains; If it falls, only 1 is left. ?

Understanding of abstract thinking;

After reading the above story, we also seem to feel dizzy. There are some birds in the tree. This is an interesting math problem. Mathematics needs fun, but I'm afraid this fun is a little childish and the answer is not thorough enough. ? Fun math? It is an effective material to stimulate students' mathematical imagination, interest and thinking. Once that interesting math problem? Sit tight? It has lost its vitality and vigor. The students in the story seem a little? Get possessed? Will this be related to rigid teaching?

If the open question is dismembered into closed questions, it will violate the original intention of opening up. Mathematics needs openness, the purpose of openness is divergent thinking, and the essence of openness is thinking. Mathematics teaching and learning need openness, including teaching organization and overall design. It can't be narrowly understood as a math problem, but a theme throughout the teaching process. The problem of openness is only the carrier and material, and it should be promoted to an idea.

Like what? How many birds are there in the tree? You may have different views on such a topic. Wise people see wisdom, interesting people see interest. Finally, let's look at the following two paragraphs:

? Even in mathematics, fantasy is needed. Even without fantasy, it is impossible to invent differentiation. ? (Lenin)

? Without bold speculation, there will be no great discovery. ? (Newton)

Example 4 of abstract thinking There are two sheep on the grass, but in the eyes of artists, biologists, physicists and mathematicians, they have different feelings and understandings. The following is their description.

Artist:? Blue sky, clear water, green grass and white sheep are beautiful and natural. ?

Biologist:? Men and women, endless. ?

Physicist:? The big sheep lay still and the little sheep walked. ?

Mathematician:? 1+ 1=2。 ?

Understanding of abstract thinking;

From the descriptions of two sheep by people of different occupations in the story, we feel that artists pay attention to natural beauty, biologists pay attention to life, physicists pay attention to movement and stillness, and mathematicians abstract the quantitative relationship from color, gender and state: 1+ 1=2, which is a highly abstract embodiment in mathematics.

In mathematics teaching and learning, what should students experience in mathematics learning? Appearance? In the process of abstraction, we should build a bridge between intuitive objects and abstract concepts in teaching, and guide students to grasp the most important and essential mathematical attributes of things.

Abstraction has a process that students experience, rather than directly telling students the abstract results. Mathematical abstraction itself is a process of continuous improvement, which is endless.

Example 5 of abstract thinking A nosy person asked such a question:? If there is a gas stove, faucet, kettle and matches in front of you, what should you do if you want to boil some water?

The questioner replied:? Put water in the kettle, light the gas, and then put the kettle on the gas stove. ?

The questioner affirmed the answer and then asked:? What should you do if other things remain the same, but there is enough water in the kettle?

At this time, the questioner confidently replied:? Light the gas and put the kettle on the gas stove. ?

But the questioner said:? Physicists usually do this, while mathematicians will empty the water in the pot, claiming to turn the latter problem into the former one. ?

Understanding of abstract thinking;

Mathematicians empty the pot of water? It seems unnecessary. Didn't the creator of the story ask us to go? Pour the water out of the pot? But to guide us to understand the unique way of thinking of mathematicians-transformation.

Learning mathematics is not the memory accumulation of problem solutions, but learning to turn unknown problems into known problems, complex problems into simple problems and abstract problems into concrete problems. The transformation thought of mathematics simplifies our thinking state and improves our thinking quality. Transformation is not a matter-of-fact, one-thing, one-policy, but a matter of digging the core and prototype of the most essential problem, and then turning a new problem into a problem that can already be solved.

The idea of conversion is the basic idea of mathematics and should run through our mathematics teaching.

The only effective way to cultivate attention from image to abstract thinking is to exert an effect on thinking, while intuition can only promote the development and depth of attention according to the degree to which it stimulates the thinking process. The intuitive image of the object itself may also attract students' attention for a long time, but the purpose of using intuition is by no means to catch students' attention for the whole class. The introduction of intuitive means in the classroom is to let children get rid of the image of a certain stage of teaching and transition to universal truth and regularity in thinking. It is a beneficial attempt to use intuitive means to break through the important and difficult points in teaching and quickly transition from image thinking to abstract thinking. In practice, we often encounter some unexpected situations, that is, intuitive teaching AIDS bind children's attention to a certain detail, which not only does not help but hinders them from thinking about the abstract truth that the teacher originally wanted to guide children to think. Once, I brought a movable model of water turbine to my children. The water driving the impeller forms a fine mist due to impact and splash, and the rainbow is reflected when the sun shines. I didn't notice the rainbow, but the children did. As a result, all their attention was attracted by this interesting natural phenomenon, but safety was accidental at that time, and the general conclusion I wanted to guide them to was useless. This class has poor grades. When using intuitive means, students' age characteristics and hobbies must be considered to avoid unnecessary obstacles in class. If a teacher wants his students to write a composition about animals, he will catch a bird and tie it in the classroom. As a result, the bird was scared to fly away, and the students were distracted by this interesting thing. After a class, I didn't finish my composition, so I just read a joke.

The use of intuitive means requires teachers to have higher scientific and educational literacy, understand children's psychology and know the process of mastering knowledge.

First of all, remember intuition-this is the general principle of mental work for young students. Don? Season? Ushinski once wrote that children are? Use shape, sound, color and feeling? Thinking. This age law requires younger children's thinking to develop in nature, so that he can see, hear, feel and think at the same time. Intuition is a kind of power to develop observation and thinking, which can bring a perceptual color to cognition. Because vision, hearing, feeling and thinking are carried out at the same time, something called emotional memory is formed in children's consciousness. Every representation and concept left in a child's memory is associated with not only thoughts, but also emotions and inner feelings. If there is no developed and rich emotional memory, there is no perfect intellectual development in childhood. I suggest to primary school teachers that you should go to the source of thinking and teach children to think in nature and labor. Let the words have a distinct emotional color when they enter the child's consciousness. Sue's suggestion 7: How important it is to make words come alive in children's consciousness and make words a tool for children to master knowledge with them. If you want to keep knowledge from becoming fixed and dead, please turn words into one of the most important creative tools. In nature, let children see more, listen more, touch more and smell more. Their thinking is active and rich. Children are developing their thinking ability, so they experience an unparalleled pleasure of thinking and enjoyment of understanding. They feel that they have become thinkers. Make vivid creation, use words to understand things and phenomena around the world, and combine this to understand the extremely delicate emotional color of words themselves. The principle of intuition should not only run through the classroom, but also run through other aspects of teaching, teaching process and the whole understanding. After school, we should create opportunities for students to learn, feel and create in nature. This is the source of their flexible knowledge.

Secondly, when using intuition, we must consider how to transition from concrete to abstract, and at which stage in the classroom intuition will no longer be needed. At that time, students should not pay attention to intuitive means. This is a very important principle of intellectual education: intuitive means are needed to promote positive thinking only at a certain stage. Thinking of the current open class, many teachers always pay attention to well-designed courseware, audio, video, animation and so on, while ignoring the teaching content, just seeking fresh excitement and attracting the attention of the audience. In fact, many of them are redundant and can be deleted completely, leaving students with enough time to think and discuss.

Third, we should gradually transition from the intuitive means of objects to the intuitive means of painting, and then to the intuitive means of providing things and phenomena that meet the description. As early as the first or second grade, children should be trained to get rid of the real thing step by step, but this does not mean that they can get rid of it completely. Experienced teachers use the principle of intuition in all academic years (from grade one to grade ten), but they embody this principle in more and more complicated working methods and ways year by year. Even in the tenth grade, experienced Chinese teachers will still lead students to the Woods, to the banks of the river and to the gardens where spring flowers are in full bloom-here, it can be said that the emotional color of words is handled more finely, which deepens and develops students' emotional memory. The knowledge accumulated at ordinary times will become a static commodity if it is not allowed to turn into turnover. When students go out of school, they actually make knowledge come alive and let knowledge enter a state of turnover. Only this kind of knowledge is useful. Students will create more new knowledge and form new ideas.

The transition to intuitive painting is a long process. This is not to say that teachers simply bring pictures of kittens to class, not live kittens. The visual means of painting, even if it vividly expresses the shape, color and other characteristics of physical visual means, is always a generalization. Therefore, the teacher's task is to gradually transition to more and more complex generalization when using the intuitive means of painting. It is particularly important to teach children to understand the corresponding paintings-sketches, schematic diagrams, etc. These means play a great role in developing abstract thinking. Combined with this, I want to put forward some hopes on how to use the blackboard.

There is a blackboard in the classroom, which can not only write on it, but also draw sketches, schematic diagrams and detailed drawings on it when the teacher is talking, talking and talking. When I teach history, plants, animals, physics, geography and mathematics, almost all classes use blackboards and colored chalk (history is about 80%, plants, animals and geography are 90%, physics and mathematics 100%). In my opinion, without this, it is impossible to imagine a process of developing abstract thinking. In my opinion, the intuition of painting is not only a means to concretize appearances and concepts, but also a means to get rid of the world of appearances and enter the world of abstract thinking. Combining the subject content, designing high-quality blackboard writing is also a problem that should be paid attention to when preparing lessons. We should consider the key points and difficulties to effectively guide students from the appearance to the abstract world.

Painting intuition is also a means to educate students' self-intelligence. In the second and third grades, my students always divided the arithmetic exercise books in the middle? Two halves? The left half is used to solve exercises, and the right half is used to draw application problems in an intuitive and graphic way. Before solving the exercises, students should? Draw application questions? . Teach students to put application problems? Painting? Its purpose is to ensure the transition from concrete thinking to abstract thinking. Children draw some objects (apples, baskets, trees, birds) at first, and then turn to the schematic diagram, which is represented by small squares and circles. I am particularly concerned about, who are the students who find it difficult to learn? Painting? Application problem. If this teaching method is not adopted, these students may not learn to solve application problems and think about their conditions. If a child learns. Painting? I can safely say that he can learn to solve application problems. There are still some students who can't learn to use pictures to express the conditions of application problems for several months. This means that they not only can't think abstractly, but also can't? Have image, sound, color and feeling? To think. This requires teaching them to think in images first, and then gradually guiding them to think abstractly. With pictures? Painting? It is really ingenious to apply questions to show the process of students' thinking formation. No wonder when I was at school, I had a little knowledge of math application problems. In fact, this is fundamental. Painting? It's not good for these problems, because I can't think visually with images, sounds, colors and feelings.

If some students in your primary school class find mathematics difficult to learn, please try to teach them first? Painting? Application problems. Children should be guided from vivid images to their symbolic descriptions, and then from descriptions to their understanding of the relationships and interrelationships between things.

Fourth, we should guide students to gradually transition from painting intuition to writing intuition. Words-what is this image made of? Have image, sound, color and feeling? Thinking is a step towards conceptual thinking. Experienced primary school teachers not only use words to create images of things that can't be seen directly (such as ice groups in the Arctic, volcanic eruptions, etc.). ), but also use words to create images of things that can be seen in nature and human labor around us. The images of these words are of great significance to the formation, content and speech enrichment of emotional memory in psychology. From image to abstraction, and then express abstract things with image words, this is creation.

It is also necessary to talk about the work of students with learning difficulties. Experience shows that the intellectual development of such students depends largely on the transition from image thinking to conceptual thinking, how long it takes and what steps to take. Individual students with learning difficulties are in a hopeless situation, and teachers don't know what to do with them and how to stimulate their thinking. This situation is mainly because these students have not experienced it for a long time? Thinking in images? However, the teacher is urging them to turn to abstract thinking quickly, and the students are unprepared for abstract thinking. Students with learning difficulties often can't relate the examples they give to the rules that they have spent a lot of time memorizing. This situation is one of the consequences of the disconnection between image thinking and conceptual thinking, and it is the result of the teacher's hurry. I often make this kind of teacher's mistake, so I should pay attention to it in the future. Before students write a composition, let them say it first, and then start writing, so that students will not complain that there is nothing to write in the composition.

Fifth, intuitive means should make students focus on the most important and essential things.

Once again, using the principle of intuition requires high skills, and it is necessary to understand students' thinking and mood.

Key points:

1. Rational use of intuitive means requires teachers to have high scientific literacy and pedagogy literacy, understand children's psychology and know the process of mastering knowledge.

2. Who is the child? Use shape, sound, color and feeling? Thinking. Children's thinking should be developed in essence, so that they can see, hear, feel and think at the same time.

3. Intuitive means is a means to promote positive thinking. Lead students to nature and let their accumulated knowledge enter the state of volcanic eruption.

4. Attach importance to the important role of classroom blackboard writing, so that students can experience it in blackboard writing and break through the teaching difficulties.

5. Do students with learning difficulties need to be strengthened? Thinking in images? Training.

The significance and function of abstract thinking method Abstract thinking is an advanced form of thinking, also known as abstract logical thinking or logical thinking. Abstract thinking method is a method of thinking by using concepts and language symbols. Its main feature is the coordinated application of basic methods such as analysis, synthesis, abstraction and generalization, so as to reveal the essence and regular relationship of things. From concrete to abstract, from perceptual to rational understanding, we should use abstract thinking methods.

The basic unit of abstract logical thinking is concept, through which people judge and reason. Concept, judgment and reasoning are the basic forms of abstract thinking. Abstract logical thinking is a unique thinking form of human beings, and abstract thinking method is the basic method of human thinking. In study, life and work, people use abstract thinking to judge and solve various problems.

Abstract thinking can be divided into empirical thinking and theoretical thinking. People's thinking based on daily life experience or daily concepts is called empirical thinking. Children often use empirical thinking, such as? Birds are flying animals? ,? Is fruit an edible plant? Equality belongs to empirical thinking. Due to the limitation of life experience, experience is prone to one-sidedness and wrong conclusions. Theoretical thinking is thinking based on scientific concepts and theories. This kind of thinking activity can often grasp the key characteristics and essence of things. Middle school students should strive to master scientific concepts and cultivate and develop theoretical thinking.

Abstract logical thinking can also be divided into formal logical thinking and dialectical logical thinking.

The so-called formal logic thinking is thinking based on concepts and theoretical knowledge and following the laws of formal logic. The form of this kind of thinking is concept, judgment and reasoning. In learning, formal thinking plays a very important role. Formulas, theorems, rules and laws in any discipline must be mastered through formal thinking, and their application and solving homework tasks are also inseparable from formal thinking. Therefore, in a sense, the process of mastering knowledge is the process of mastering concepts, judgments and reasoning by using formal thinking.

The so-called dialectical logic thinking means thinking according to the laws of dialectical logic with the help of concepts and theoretical knowledge. Thinking is the reflection of objective reality. The objective reality has its relatively stable side, and it also has its constantly moving and developing side. Formal thinking is a reflection of objective things that are relatively stable and have little change in development; Dialectical thinking is a reflection of the continuous development and change of things. Therefore, the form of dialectical thinking, that is, the process of concept, judgment and reasoning, is also dialectical For example, Newton's three laws belong to formal thinking; Einstein's theory of relativity belongs to the category of dialectical thinking. Dialectical thinking gets rid of intuition and concreteness.

In learning, we should abide by the laws of logical thinking, but we should not be limited to formal thinking, but also develop dialectical thinking, because objective things are interrelated and constantly developing and changing. Only by dialectical thinking can we obtain new theories and discover new disciplines. Many interdisciplinary and marginal subjects are summarized through dialectical thinking. Therefore, some advanced subjects can't learn well without dialectical thinking, and some low-level subjects can't do without dialectical thinking, such as solving multiple problems and understanding the variation of concepts. A person's dialectical thinking (also called seeking difference thinking abroad) is relatively developed, so his intelligence is relatively high and his creativity is relatively strong, and his study will be much more effective. If we continue to develop and insist on dialectical thinking, then this person may achieve greater success.