What is Mathematical Thought? How can I have both?

Mathematical thought refers to the results of the spatial form and quantitative relationship of the real world reflected in people's consciousness and produced through thinking activities. Mathematical IdeasMathematical Ideas.

Mathematical thoughts. It is the essential understanding of mathematical facts and theories after generalization.

Basic mathematical ideas are the foundational, summative and most extensive mathematical ideas that embody or should be embodied in basic mathematics, which contain the essence of traditional mathematical ideas and the basic characteristics of modern mathematical thoughts, and are developing historically. Through the cultivation of mathematical ideas, the ability of mathematics will be greatly improved. To master mathematical ideas is to master the essence of mathematics.

Mathematical ideas include arithmetic, reasoning, inverse and a series of basic knowledge. He has been honed and improved after blending negative reasoning and rigor, which allows mathematical ideas to be integrated into the human brain and then reused. A very important thing to do is to do the question, and then sort out the thinking methods in the question, such as using one knowledge point to deduce another, and then applying proof to solve the problem.

Secondly, learn more about the teacher's thinking method, this is a very abstract thing, don't always think about learning any mathematical ideas, learn the depths of nature. All hand hits, oh hope. Good luck with your studies.

The one downstairs can not ask Du Niang Just wanted to say that I admired the master and was discovered by me, there is really no one.

Mathematical ideas include: function ideas, number and form combination ideas, classification discussion ideas, equation ideas, whole ideas, reduction ideas, implicit conditional ideas, analogy ideas, modeling ideas, etc. Mathematical thought refers to the results of the spatial form and quantitative relationship of the real world reflected in people's consciousness and produced through thinking activities.

1. The idea of function equation: refers to the use of the concept and properties of functions to analyze and solve problems.

For example, in a series of equal differences and proportions, the formula for the sum of the first n terms can be regarded as a function of n.

2. The idea of combining numbers and shapes: the use of "combination of numbers and shapes" can make the problems to be studied difficult and simple.

For example, find the minimum value of the root number ((a-1) 2+(b-1) 2) + root number (a 2+(b-1) 2) + root number ((a-1) 2+b 2) + root number (a 2+b 2).

3. Classification and discussion ideas: When the problem may cause different results due to different situations of a certain quantity or graph, it is necessary to classify and discuss the various situations of this quantity.

For example: Solve inequality |a-1|> 4, it is necessary to classify and discuss the value of a.

4. Equation idea: When a problem may be related to a certain equation, the equation can be constructed and the properties of the equation can be studied to solve the problem.

For example, when proving Cauchy's inequality, you can convert Cauchy's inequality into a discriminant of a quadratic equation.

5. Overall thinking: Starting from the overall nature of the problem, highlight the analysis and transformation of the overall structure of the problem, and discover the overall structural characteristics of the problem.

For example, superposition and multiplication processing, global operations, complements in geometry, etc., are all holistic ideas.

6. The idea of naturalization: It is to transform unknown problems into known, familiar and simple problems through deductive induction.

For example: trigonometric functions, geometric transformations.

7. Implicit conditional thinking: There is no explicit expression or no explicit expression, but the condition is the truth.

For example, in an isosceles triangle, a line segment that passes the vertex is perpendicular to the base edge, then the line in which the line is located also bisects the base and the top corner.

8. Analogy: Compare two different mathematical objects and find that they are similar or similar in some aspects, and infer that they may also have similarities or similarities in other aspects.

9. Modeling idea: In order to describe an actual phenomenon in a more scientific and reproducible way, a language that is generally considered to be relatively strict is used to describe various phenomena.

Mathematical thought refers to the result of the reflection of the spatial form and quantitative relationship of the real world into human consciousness through thinking activities. It is the basic viewpoint of dealing with problems in mathematics, the summary of the essence of basic knowledge and basic methods of mathematics, and the guiding principle for the creative development of mathematics. Mathematical ideas have a higher level of abstract generalization than general mathematical concepts, the latter is more concrete and richer than the former, and the former is more essential and profound than the latter.

Mathematical methods refer to the operational rules or patterns contained in the means, pathways, and ways of behaving that people adopt in order to achieve a certain end. Mathematical ideas and methods are both unified and distinct. For example.

In junior high school algebra, the system of multivariate equations is solved using the "elimination method"; To solve the higher-order equation, the "descending method" is used; Solve biquadratic equations. The "substitution method" is usedThe "elimination", "descent" and "substitution" here are all specific mathematical methods, but they are not mathematical ideas, and these three methods together embody the mathematical idea of "transformation", that is, the idea of transforming complex problems into simple problems.

Specific mathematical methods cannot be crowned with the word "thought". For example, "matching method" cannot be called a mathematical idea. Its essence is identity deformation, which embodies the mathematical idea of "transformation".

However, every mathematical approach. all embody certain mathematical ideas; Each kind of mathematical thought is expressed on different occasions through certain means, and the means here are the mathematical method. That is, mathematical thought is rational cognition.

It is the spiritual essence and theoretical basis of the relevant mathematical method. The mathematical method is directed towards practice. It is instrumental, and it is a technical means to implement the relevant ideas.

Therefore. Mathematical ideas and methods are often seen as a holistic concept – mathematical methods of thought and methods. In general, there are three levels of mathematical thinking methods:

Low-level mathematical thinking methods (such as elimination method, exchange method, generation method, etc.), high-level mathematical thinking methods (such as analysis, synthesis, induction, deduction, generalization, abstraction, analogy, etc.), high-level mathematical thinking methods (such as transformation, classification, combination of numbers and shapes, etc.).Lower-level mathematical ideas and methods can be abstractly generalized to higher-level mathematical ideas and methods, and there is no clear boundary between the levels.

Mathematical thought is the result of the reflection of the spatial form and quantitative relationship of the real world into people's consciousness and through thinking activities. So what are some common mathematical ideas?

1. Symbolic thinking: In mathematics teaching, the relationship between various quantities, the changes of quantities, and the derivation and calculation between quantities are all represented in the form of symbols (including letters, numbers, graphs and charts, and various specific symbols), that is, a set of formal mathematical language is running.

2. Classification idea: On the basis of comparison, according to the similarities and differences in the properties of things, objects of the same nature are classified into one category, and objects of different properties are classified into different categories - this is classification, also known as division. The idea of classification in mathematics embodies the classification of mathematical objects and their classification criteria.

3. The idea of function: The concept of function profoundly reflects the dependence between the movement and change of the objective world and the quantity of actual things.

4. The idea of naturalization: "Naturalization" is transformation and reduction. When solving mathematical problems, people often reduce the problem that needs to be solved to another problem that is relatively easy to solve or has a solution program through some means of transformation, in order to obtain the solution to the problem.

Mathematics in elementary school.

It is the most basic and commonly used way of thinking to solve problems.

5. Inductive thinking: When studying general problems, we should first study a few simple, individual, and special situations, and then summarize the general laws and properties from them, which is a way of thinking from special to general.

It is called inductive thought.

6. Optimization thinking: "selecting the best among the many, selecting the best and using the best" is not only a natural law, but also a good way of thinking. Algorithm diversification is an important embodiment of the diversification of problem-solving strategies.

Calculating the perimeter of a rectangle is a problem with multiple solutions, seeking common ground while reserving differences, and choosing the best method among the right methods, figuring out the right and the good, and choosing the good.

7. Combination of numbers and shapes.

Idea: Mathematics is the science that studies the relationship between spatial forms and quantities in the real world. The idea of combining numbers and shapes is the idea of combining the quantitative relationship and the spatial form of the problem to investigate.

That's it for some of the most common mathematical ideas.

The "Compulsory Education Mathematics Curriculum Standards" develops the "double base" in mathematics teaching into "four bases", that is, in addition to "basic mathematics knowledge" and "basic mathematics skills", "basic mathematical ideas" and "basic mathematics activity experience" are added. So, what are the basic ideas of mathematics?

Basic ideas refer to the ideas on which mathematics is born and developed; The thinking ability that you have after learning mathematics (the difference between thinking that has studied mathematics and those who have not).

There are three basic ideas in mathematics: one is the idea of mathematical abstraction, one is the idea of mathematical reasoning, and the other is the idea of mathematical modeling.

The top 10 mathematical thinking methods in primary school mathematics are as follows:

1. Corresponding thinking methods.

Correspondence is a way of thinking about the connection between two elements of a set. Primary school mathematics is generally a one-to-one correspondence of intuitive diagrams, and this is the idea of a pregnant volt function.

In the teaching of primary school mathematics, the main number deficit should use dotted lines, solid lines, arrows, counters and other graphics to connect elements with elements, objects with objects, numbers with equations, quantities with quantities, and infiltrate corresponding ideas.

For example, in the first grade textbook, the rabbit and the deer, the monkey and the bear, and the rabbit and the bird are corresponded one by one, and the comparison is carried out, which penetrates the correspondence between things to the students and provides students with ideas and methods for solving problems.

2. Methods of Transforming Thoughts:

This is an important strategy for solving math problems. It is a method of thinking that transforms one form into another. And its size itself is constant. Through transformation, it can turn the difficult into easy, the new into the old, the complex into simple, the whole into zero, and the curved into straight.

3. Symbolic thinking methods.

The Symbolic Thought Method uses symbolic language (including letters, numbers, graphics, and a variety of specific symbols) to describe mathematical content, which is the Orange Town Symbolic Thought.

4. Classification of thinking methods.

The method of thinking of classification is not unique to mathematics, and the method of thinking of classification in mathematics embodies the classification of mathematical objects and the criteria for their classification.

5. Comparative thinking methods.

Comparative thinking is one of the common thinking methods in mathematics, and it is also a means to promote the development of gross thinking.

6. Analogical thinking methods.

The idea of analogy refers to the idea that it is possible to transfer the properties of one type of mathematical object to another based on the similarity of two types of mathematical objects.

7. Substitution of thinking methods.

He is an important principle of equation solving, which allows one condition to be replaced by another.

8. Hypothetical thinking methods.

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.

9. Reversible thinking method.

It is the basic idea in logical thinking, when it is difficult to solve the problem in forward thinking, you can seek a way to solve the problem from conditional or problem thinking, and sometimes you can use the line diagram to extrapolate.

10. Naturalized thinking method.

Naturalization is a common method of thinking for solving mathematical problems. Reduction refers to the transformation of unsolved or unsolved problems into a class of solved or relatively easy to solve problems through the process of transformation, in order to solve them.

There are some basic ideas and principles in mathematics that play an important role in the field of mathematics as a whole. Here are some common math basics:

1.Abstraction: Mathematics extracts the key concepts in practical problems through abstraction to form an abstract mathematical structure, so that the problem can be studied and solved more clearly.

2.Reasoning and Proof: Mathematics is based on logical reasoning and proof, and establishes the correctness of mathematical conclusions through a rigorous reasoning process. Proof is a core activity in mathematics that ensures the accuracy and credibility of mathematics.

3.Induction and deduction: Induction is often used in mathematics to derive general laws from individual situations, and deductive methods are also used to deduce specific conclusions from general principles.

4.Pattern Recognition and Problem Modeling: Mathematicians often discover the essence of mathematical problems by observing patterns and patterns in problems, and transform them into models that can be mathematically analyzed.

5.The interrelationship between intuition and abstraction: Intuitive images and geometric intuitions in mathematics often interact with abstract notation and algebraic structures, and promote the development of mathematics through mutual transformation.

6.Continuity and discreteness: Concepts of continuity and discreteness are involved in mathematics, such as continuous functions and discrete data, and these two aspects of research complement each other and constitute two important branches of mathematics.

7.Balance between reason and intuition: Mathematics is both a rigorous discipline and requires the intuition and creativity of mathematicians. In the study of mathematics, reason and intuition interact and balance each other.

These basic ideas run through all branches and fields of mathematics, and they are intertwined with each other and together form the basis for the development and application of mathematics.