What are the main characteristics of mathematical thinking?

High-quality answers. Thinking is an indirect reflection of the human brain's generalization of the nature of things and the regular relationships between things. Thinking is the core component of cognition, and the level of development of thinking determines the structure and function of the entire knowledge system.

Therefore, it is of great significance to develop the thinking potential of high school students and improve the quality of thinking.

The quality of thinking mainly includes the flexibility, expansiveness, agility, profundity, originality and criticality of thinking. Flexibility of thinking is a good quality that is built on the breadth and profundity of the mind and provides a guarantee for mental agility, originality and criticality. In people's work and life, it is easy to do things according to the rules, but it is difficult to pioneer and innovate, and the difficulty lies in the lack of flexible thinking.

Therefore, the cultivation of thinking flexibility is particularly important.

Mathematical thinking is the thinking process in which the human brain and mathematical objects interact and understand mathematical laws according to general thinking rules. It is manifested in the fact that students start from the original cognitive structure, through a series of mathematical thinking activities such as observation, analogy, association, conjecture, etc., to show the problem and put forward the process in a three-dimensional way, generate a strong desire for knowledge in the process of reviewing the past and learning the new, participate in the formation of concepts and the development process of conclusions as much as possible, and master the methods of observation, experiment, induction, deduction, analogy, association, generalization and specialization.

The quality of thinking].

1) The profundity of thinking: refers to the depth of thinking, which is concentrated in whether it is good at thinking deeply, grasping the law and essence of things, and meeting the development and process of things.

2) The breadth of thinking: It refers to looking at the problem from the connection of all aspects of things based on rich knowledge and experience.

3) Agility of thinking: refers to the speed or rapidity of the thinking process, that is, the thinking quality of people to make decisions and solve problems quickly in a short period of time.

4) Flexibility of thinking: refers to the degree of improvisability to think about problems and solve problems.

5) Originality of thinking: whether it is good at analyzing and solving problems independently.

6) Critical thinking: good at criticizing the ideas and achievements of others and oneself.

7) Logical thinking: It refers to the clear thinking, clear organization, and strict compliance with logical laws when considering and solving problems. [Central Link].

Cultivate students' thinking qualities.

1) Strengthen the training of scientific thinking methods;

2) Use heuristic methods to emphasize the enthusiasm and initiative of students' thinking;

3) Strengthen verbal communication training;

4) play a positive role in the stereotype;

5) Cultivate students' thinking quality in solving practical problems.

Ten ways of thinking in mathematical thinking:

1. Control method.

According to the meaning of mathematical problems, the method of solving problems by relying on the understanding, memory, identification, reproduction, and transfer of mathematical knowledge is called the comparison method by comparing the meaning and essence of concepts, properties, laws, laws, formulas, nouns, and terms.

2. Formula method.

Methods of using laws, formulas, rules, and laws to solve problems. It embodies deductive thinking from the general to the particular.

3. Comparative law.

By comparing the similarities and differences between mathematical conditions and problems, the causes of similarities and differences are studied, so as to discover ways to solve problems, which is called comparative method.

4. Taxonomy.

The method of classifying things into different types based on their commonalities and differences is called taxonomy. Classification is based on comparison. Combine things into larger classes based on what they have in common, and divide larger classes into smaller classes based on differences.

5. Analytical method.

The method of thinking that decomposes the whole into parts, and the complex things into individual parts or elements, and studies and deduces these parts or elements is called the analytical method.

6. Comprehensive law.

The method of researching, deriving, and thinking about the various parts or aspects or elements of an object together and combining them into an organic whole is called the synthesis method.

7. Equation method.

Unknowns are represented by letters, and expressions containing letters (equations) are listed according to the equality relationship. Column equations are a process of abstract generalization, and solving equations is a process of deductive derivation.

The biggest feature of the equation method is that it treats the unknown number as the known number, participates in the column formula and operation, and overcomes the shortcomings of the arithmetic method that must avoid the number of knowledge to the column. It is conducive to the transformation from the known to the unknown, so as to improve the efficiency and accuracy of problem solving.

8. Parametric method.

The method of expressing the relevant quantity with letters or numbers that only participate in the column and operation without solving it, and listing the equation according to the meaning is called the parametric method. Parameters are also called auxiliary unknowns, also known as intermediate variables. The parametric method is the product of the extension and expansion of the equation method.

9. Exclusion method.

The elimination of the result of opposition is called the method of elimination.

The logical principle of the method of elimination is that everything has its opposite, and among the many outcomes that are right and wrong, all the wrong results are excluded, and all that remains is only the correct result. This method is also called the elimination method, the screening method or the counter-evidence method.

This is an indispensable method of formal thinking.

10. Special case law.

For problems involving general conclusions, the method of solving the problem by taking special values, drawing special diagrams, or setting special positions is called the special case method.

The logical principle of the special case method is that the generality of things exists in the particularity.

The characteristics of mathematical thinking of children of different ages are as follows:

The development of children's thinking around the age of 3 is generally characterized by actions to dominate thinking.

When children use their brains to think, they will use various parts of their bodies to arrive at the answer, and sometimes you ask your child how many dishes there are on the table, and at this stage children often have to use their fingers to count one by one to answer you.

6-year-old children's thinking characteristics are more specific images.

At this stage, the child cannot understand the abstract numbers, but can understand the specific objects, sometimes there are four toys and two chairs in front of the child, the child can not fully answer how many objects you have, can only tell you that there are four toys, two chairs, and only know to stretch out four fingers first, and then two fingers.

After the age of life, children develop from concrete figurative thinking to abstract thinking.

In this process, the child will gradually learn to generalize and summarize, and will have a deeper understanding of some quantities, and can do some simple addition and subtraction.

Therefore, many parents are anxious to let their children think like adults when they are doing mathematical thinking enlightenment, which will delay their children's understanding of mathematics, so they should have different plans and methods to do a good job in children's mathematical enlightenment, and use them according to their children's learning progress and efficiency.

Here's how to plan the math thinking initiation of children aged 3-8:

1. Use "correspondence" to recognize quantity.

For young children often do not understand the true meaning of numbers and quantities, so if we want children to learn mathematical thinking, we must let children understand what numbers and quantities are first, this process is to concretize numbers, and in this link children can better understand the concept of numbers and quantities.

We can use toys or fruits at home, respectively, take four fruits, and four bowls, you can tell the child a known condition, so that the child can learn to correspond, a fruit and a bowl, so that the child will understand the relationship between each other, visually, the number of two items is equal, so the child can understand the specific number of items well through simple correspondence.

2. Use tangram skillfully.

Not only can numbers help children improve their mathematical thinking, but graphics can also help children better understand mathematics. The different shapes allow the child to construct and disassemble in their minds.

In the plane figures, triangles, squares, and hexagons, this kind of basic graphics can give children a preliminary understanding of the relationship between lines and surfaces.

Parents can use tangram puzzles to piece together different shapes, tell children about the relationship and function of these figures, and tangram is more like an entertainment project for children, so children will not feel bored when facing these tools, but will be more willing to operate them.

Mathematics is a basic subject throughout the learning process, when parents are worried about their children's mathematics learning, have you ever thought that maybe you should first understand the characteristics of children's mathematical cognition in each period?

4—5 years

Children begin to be interested in numbers, numbers, numbers. For example, when dividing food, children will be particularly concerned about whether they have more or less portions. During this period, it is necessary to grasp the sensitive period of children's numbers, and do guidance training in a purposeful manner, so as to lay the foundation for their future arithmetic ability and numerical co-ordination ability.

5—6 years

Children begin to develop an interest in mathematical logic, especially the sequence of numbers, and the relationship between the idea and the idea. For example, children in this period especially like to test their parents' arithmetic, or when describing the quantity, they especially like to say: "The sky is so big, the earth is so big" This period needs rich props and professional guidance, so that children can fully experience the fun of "addition and subtraction", and stimulate children's motivation to explore the relationship between quantities, rather than constantly taking arithmetic questions to test children, once children think that "mathematics = arithmetic" may become the biggest stumbling block for children to learn mathematics in the future.

6—8 years

Children already have a certain level of abstract thinking, and are able to complete simple reasoning and spatial imagination. For example, children in this period like to ask about some abstract and difficult points. Most children have a desire for some molds and models, and children at this stage need free imagination and space to explore.

Although there is a score evaluation for each subject in primary school, don't let your child feel that learning mathematics is just to get a good grade, imagination and desire to explore can stimulate children's enthusiasm for independent learning, and only then can they really make achievements in mathematics in a longer learning career.

8—12 years

Children's abstract thinking is basically formed, and children in this period begin to become independent and assertive. In terms of mathematics learning, they already have their own ideas and methods for solving problems, for example, during this period, children do homework and look at the homework or task, and especially like to say "Oh! This is simple", or when you see a practical problem, before you finish reading the question, you will immediately know "what to set as x first", and guide children in this period to make them have cognitive conflicts.

The teacher's teaching method must be innovative, only in this way can the child follow the teacher's train of thought, and have the intellectual curiosity to explore advanced methods while having opinions and ideas. The quick formula is not intuitively placed in front of you, and you are forced to memorize it in order to cope with the exam, and you will be out of the sky in a few days. The advanced method is that the child practices reasoning by himself, and the spirit of knowing what it is and why it is true can benefit the child in the long run.

After understanding the characteristics of children's mathematical cognitive development, the next step is to make the daily mathematical thinking training more in line with the growth law of children's thinking, emphasizing learning motivation and interest, so that parents and children no longer talk about "number" color change.

3 4-year-old children belong to the stage of intuitive action thinking, this stage, the child is trained in reverse thinking, mainly by creating a relaxed, interesting and pleasant game environment for the child, so that he can germinate his interest in thinking, and do it himself, so that the child is often in a state of active activity.

4 5 years old is a critical stage in the development of children's thinking activities, and at this stage, children's thinking has entered the concrete image stage. Reverse thinking training for 4 to 5-year-old children is mainly to continuously enrich the child's knowledge, develop his language, and help the child learn to think about problems from both positive and negative aspects, and make judgments.

At the age of 5 or 6, the child's abstract logical thinking develops relatively rapidly, which lays the intellectual foundation for him to enter school. At this stage, children are already able to use thinking forms such as concepts, judgments, and reasoning. Reverse thinking training for 5 and 6-year-old children is mainly to help children look at the inherent and conventional views from the opposite perspective, learn the correct way of thinking, and develop their reverse thinking through various creative activities.

Mathematics is a discipline that studies the spatial form and quantitative relationship of the real world, and the characteristics of this discipline that guide life make us think from two aspects: on the one hand, children grow up gradually with his continuous understanding of the world around them, then, just imagine, if children do not have the concept of number and shape, they will not even know how many people in the family, how many hands they have, how many balls they play, and what shape the building blocks are; If there is no concept of measurement, there will be no distinction between the size, thickness, height, etc. of objects; If there is no concept of spatial orientation, it is not clear to distinguish up and down, left and right; Without a little concept of time, you can't distinguish between yesterday, today and tomorrow. Obviously, mathematics education is the need for children to understand objective things, and children not only need to understand the external characteristics, uses and interrelationships of things, but also often encounter problems with numbers and shapes.

On the other hand, because of the uniqueness of mathematics, the precision, abstraction, and logic can help children to understand various things in life and their relationships in general, so that children can gain a way of thinking and learn to use mathematical methods to solve practical problems. Promote children's mathematics and intelligence to get a better development for further learning to lay a good foundation, for example, three or four years old children can not write Arabic numerals, do not understand the meaning of the "+" representative, can not give the exact meaning of triangles, circles, can not accurately measure the weight of objects, length, but people are also increasingly aware that in daily life children have been able to count the number of objects in a shorter number of sequences, can use the physical object or the representation of the physical object to calculate a simple addition and subtraction, correct identification of geometric figures, Compare and measure objects with natural objects, and it is possible to find out the pattern of object placement. These facts make people realize that mathematical cognitive ability does not begin with the individual's understanding of abstract symbol systems, and that the ability to quantitative, shape, and spatial aspects based on concrete physical representations is the initial manifestation of mathematical cognitive ability.