What are the eight ways of thinking in mathematics and the ten ways of thinking in mathematics?

The eight thinking methods in junior high school mathematics are as follows:1. Abstract thinking.

2. Logical thinking.

3. Combination of numbers and shapes.

4. Classification discussion.

5. Equation thinking.

6. Universal thinking.

7. Dig deep into your thinking.

8. Naturalized thinking.

Through the complete analysis and research of the textbooks, we can clarify and grasp the system and context of the textbooks, and control the overall situation of the textbooks. Then, the interface relationship between various concepts, knowledge points or knowledge units is established, and their special properties and internal general laws are summarized and revealed. Further determine the combination point between mathematical knowledge and its thinking methods, establish a set of rich teaching examples or models, and finally form an active knowledge and thought interconnection network.

There are eight thinking methods in mathematics: algebraic thinking, number and form combination, transformation thinking, corresponding thinking method, hypothetical thinking method, comparative thinking method, symbolic thinking method, and limit thinking method.

This is one of the basic mathematical ideas, the number x in elementary school, and a series of letters to represent numbers in junior high school, which are all algebraic ideas, and also the most basic root of algebra!

It is one of the most important and basic methods of thought in mathematics, and it is an effective idea for solving many mathematical problems. "When the number is lacking in form, it is less intuitive, and when there are countless shapes, it is difficult to get into the details" is a famous quote of Professor Hua Luogeng, a famous mathematician in China, which is a high summary of the role of the combination of numbers and shapes. In junior high and high school, there are many problems that involve the combination of numbers and shapes, such as solving problems by making geometric figures to mark data, with the help of function images, etc., which are all reflected in the number shapes.

Throughout junior high school mathematics, the idea of transformation (naturalization) runs through it. Transformation of thought is to solve an unknown (to be solved) problem into a solved or easy to solve problem, such as simplifying, making difficult easy, turning the unknown into known, turning high order into low order, etc., it is the most basic idea to solve the problem, it is one of the basic ideas and methods of mathematics.

Correspondence is a way of thinking about the connection between two set factors, and elementary school mathematics is generally a visual diagram of one-to-one correspondence, and uses this to conceive the idea of volt function. For example, the points on a straight line (number axis) correspond one-to-one with the number representing the specifics.

Hypothesis is a method of thinking that first makes certain assumptions about the known conditions or problems in the question, then calculates according to the known conditions in the question, adjusts them appropriately according to the number of contradictions, and finally finds the correct answer. Hypothetical thinking is a kind of meaningful imaginative thinking, which can make the problem to be solved more vivid and concrete after mastering, so as to enrich the idea of problem solving.

Comparative thinking is one of the common thinking methods in mathematics, and it is also a means to promote the development of students' thinking. In the teaching fraction problem, the teacher is good at guiding students to compare the situation before and after the change of the known and unknown quantities in the problem, which can help students find the way to solve the problem quickly.

The use of symbolic language (including letters, numbers, graphics, and a variety of specific symbols) to describe mathematical content is called symbolic thinking. For example, in mathematics, various quantitative relationships, changes in quantities, and derivation and calculus between quantities are all represented by small letters, and a large amount of information is expressed in the condensed form of symbols. Such as laws, formulas, etc.

Things change from quantity to quality, and the essence of the limit method is to achieve qualitative change through the infinite process of quantitative change. When talking about the "area and circumference of a circle", the idea of "turning a circle into a square" and "turning a curve into a straight" limit division, and imagining their limit state on the basis of observing the finite division, not only enables students to master the formula, but also germinates the idea of infinite approximation from the contradiction between curve and straight.

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: the oneness of things. Generality is found in particularity.

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The eight thinking methods of mathematics are algebraic ideasCombination of numbers and shapes, Transformation of Thought, Correspondence of Thought Method, Hypothetical Thought Method, Comparative Thought Method, Symbolic Thought Method, Limit Thought Method. Transformational thinking in solving math problems refers to changing the direction of the problem from one form to another from different angles when encountering obstacles in the process of solving problems, seeking the best way to make the problem simpler and clearer.

Mathematics is different from Chinese, English and other language disciplines, it requires greater thinking ability, as long as you master the same type of problem-solving thinking, no matter how the question type changes, we can quickly solve, mathematics comes from life and acts on life, the mathematical knowledge in the textbook can actually find the original form in real life, but you need to be transformed into mathematical language through abstraction, simplification, etc., therefore, when learning mathematics, we should understand the essence of the meaning in connection with the actual life.

The content of the eight ways of thinking in mathematics

Reverse thinking. It is also called different thinking.

It is a way of thinking in reverse, which is commonplace and seems to be a foregone conclusion.

Dare to think the opposite, let the thinking develop in the direction of the opposite, explore deeply from the opposite side of the problem, establish new ideas, and create a new image.

Logical thinking. It is the thinking process in which people observe, compare, analyze, synthesize, abstract, generalize, judge, and reason things with the help of concepts, judgments, reasoning and other forms of thinking in the process of cognition.

Extensive and innovative thinking is used when asking questions.

It refers to the thinking process of solving problems in a novel and original way, through which the boundaries of conventional thinking can be broken through, and the problem can be thought about from an unconventional or even unconventional perspective and come up with a different solution, which can be divided into four types: differentiation, exploration, optimization and negation.

The mathematical thinking methods include: the idea of function, the idea of classification discussion, the idea of reverse thinking, the idea of combining numbers and shapes, functions and equations, reduction and transformation, the idea of wholeness, the idea of transformation, the idea of implicit conditions, and the idea of limit.

1.Functional ideas.

Functional thinking is a thinking strategy for solving "mathematical" problems. Since people have used functions, after a long period of research and exploration, the scientific community has generally had an awareness, that is, the idea of functions, and when using this thinking strategy to solve problems, scientists have found that they all have a common property, that is, the connection between quantification and variables.

2.Categorize the idea of discussion.

The idea of classification discussion is an important method of thinking, and its basic idea is to decompose a relatively complex mathematical problem into several basic problems, realize the ideological strategy of the original problem through the solution of the basic problem, classify and integrate the problem, and the classification standard is equivalent to adding a known condition, realizing the effective addition, decomposing the comprehensive problem into small problems, optimizing the solution ideas, and reducing the difficulty of solving the problem.

3.The idea of thinking backwards.

Reverse thinking, also known as divergent thinking, is a way of thinking about things or opinions that seem to be a foregone conclusion, daring to "think the opposite", so that thinking in the opposite direction of development, from the opposite side of the problem to explore deeply, establish new ideas, create a new image.

4.Numbers and shapes combine ideas.

Numbers and shapes are the two oldest and most basic objects of study in mathematics, and they can be transformed into each other under certain conditions. The object of secondary school mathematics research can be divided into two parts: number and shape, and number and form are related, and this connection is called the combination of number and shape, or the combination of form and number.

Eight Ways to Think in Mathematics:

1. Transformational thinking in solving math problems refers to changing the direction of the problem from different angles to transform the problem from one form to another and seeking the best way to make the problem simpler and clearer when encountering obstacles in the process of solving problems.

2. Reverse thinking is also called different thinking. It is a way of thinking that, in turn, has become the final conclusion about a common thing or point of view. Dare to "do the opposite", let the thinking develop in the opposite direction, explore deeply from the opposite side of the problem, establish new ideas, and create a new image.

3. Logical thinking is the thinking process in which people observe, compare, analyze, synthesize, abstract, generalize, judge and reason things with the help of concepts, judgments, reasoning and other forms of thinking in the process of understanding. Logical thinking, which is widely used in solving logical reasoning problems.

4. Innovative thinking refers to the thinking process of using innovative and novel methods to solve problems. Through this kind of thinking, we can push the boundaries of traditional thinking, think about problems with unconventional or even unconventional methods and perspectives, and propose different solutions. It can be divided into four types:

Difference, Exploration, Optimization, and Negation.

5. Analogical thinking refers to the way of thinking that compares unfamiliar and unfamiliar problems with familiar problems or other things according to some similar properties of things, so as to find out the commonality of knowledge and find its essence.

6. Correspondence thinking is a thinking method that establishes a direct connection between quantitative relationships (including quantity difference, quantity multiple, and quantity rate). The more common ones are general correspondence (such as the correspondence between two or more quantities and the difference multiple) and quantity rate correspondence.

7. Image thinking mainly refers to the formation of people's choice of the expression of things in the process of understanding the world. It refers to the way of thinking that solves problems with intuitive visual representation. Imagination is an advanced form and basic method of figurative thinking.

8. Systems thinking is also called holistic thinking. Systems thinking refers to the systematic understanding of the knowledge points involved in specific topics when solving problems, that is, first analyzing and judging which knowledge points belong to which knowledge points, and then recalling the types of such problems and corresponding solutions.