What is mathematical thought? Help! !

The so-called mathematical thinking refers to the spatial form and quantitative relationship of the real world reflected in human consciousness, as well as the result of thinking activities. Mathematical thought is the essential understanding after summarizing mathematical facts and theories; The thought of basic mathematics is the basic, summative and most extensive mathematical thought embodied or should be embodied in basic mathematics. They contain the essence of traditional mathematical thought and the basic characteristics of modern mathematical thought, and are historically developed.

"Mathematical thought" has a higher level of generalization and abstraction than the general "mathematical concept", the latter is more concrete and richer than the former, while the former is more essential and profound than the latter. "Mathematical thought" is the spiritual essence and theoretical basis of its corresponding "mathematical method", and "mathematical method" is the technology and operating procedure for implementing relevant "mathematical thought". All kinds of mathematical methods used in middle school mathematics reflect certain mathematical ideas. Mathematical thought belongs to scientific thought, but scientific thought is not necessarily mathematical thought. Some mathematical thoughts (such as "one divides into two" and "transformation") and logical thoughts (such as the idea of complete induction) are "mathematized" because of their application in mathematics, which can also be called mathematical thoughts.

The basic mathematical ideas include: the idea of symbols and independent variables, the idea of set, the idea of correspondence, the idea of axiomatization and structure, the idea of combining numbers and shapes, the idea of transformation, the idea of functions and equations, the idea of entirety, the idea of limit, the idea of sampling statistics and so on. When we classify the research objects according to the spatial form and quantitative relationship, we also take the classification idea as the basic mathematical idea. Basic mathematics thought has two cornerstones-the idea of symbol and demonstration and the idea of set, and two pillars-the corresponding idea and the idea of axiomatic structure. Basic mathematical ideas and other mathematical ideas derived from them form a highly structured network.

Mathematics is permeated with basic mathematical ideas and is the soul of basic knowledge. If we can put them into our thinking activities of learning and applying mathematics, we can play a methodological role in developing our mathematical ability, which is very important for learning mathematics, developing our ability and developing our intelligence.

The so-called mathematical thinking refers to the spatial form and quantitative relationship of the real world reflected in human consciousness, as well as the result of thinking activities. Mathematical thought is the essential understanding after summarizing mathematical facts and theories; Basic mathematics thought is the most fundamental, generalized and extensive mathematical thought embodied or should be embodied in basic mathematics. They contain the essence of traditional mathematical thought and the basic characteristics of modern mathematical thought, and are historically developed.

"Mathematical thought" has a higher level of generalization and abstraction than the general "mathematical concept", the latter is more concrete and richer than the former, while the former is more essential and profound than the latter. "Mathematical thought" is the spiritual essence and theoretical basis of its corresponding "mathematical method", and "mathematical method" is the technology and operating procedure for implementing relevant "mathematical thought". All kinds of mathematical methods used in middle school mathematics reflect certain mathematical ideas. Mathematical thought belongs to scientific thought, but scientific thought is not necessarily mathematical thought. Some mathematical thoughts (such as "one divides into two" and "transformation") and logical thoughts (such as the idea of complete induction) are "mathematized" because of their application in mathematics, which can also be called mathematical thoughts.

Since the 20th century, due to the emergence of important thinking methods in the discipline of basic mathematics, especially the formation of mathematical axioms and the in-depth development of theoretical research of basic mathematics, people have gradually paid attention to the internal relations among various branches of mathematics, and began to pay attention to the generation and development of mathematical thinking methods themselves. Many famous mathematicians have engaged in the research of mathematical thinking and method theory, and have obtained rich research results, which provide a theoretical basis for us to study the teaching of mathematical thinking and method today and make it possible for the teaching of mathematical thinking and method to proceed smoothly.

Since 1950s, many famous mathematicians, especially those who have been engaged in education for a long time, have concentrated on the educational function of mathematics and achieved a series of theoretical research results. For example, Mathematics and Conjecture written by Paulia and Spirit, Thought and Method of Mathematics published by Mishan National Library are all research results.

In the 1980s, mathematical methodology, as a new discipline, which studies the laws of mathematical development, mathematical thinking methods and the laws of mathematical discovery, invention and innovation, has been widely concerned in China's mathematics field, especially in the mathematics education field. During this period, Xu Lizhi's Lecture Notes on Mathematical Methodology and Zheng Yuxin's Introduction to Mathematical Methodology are of great significance. These works are groundbreaking and groundbreaking. These works have directly promoted the research on mathematical thinking methods and teaching in China's mathematics education circles.

In the 1990s, with the deepening of education reform, many domestic experts and scholars became more and more interested in the research of mathematical thinking methods and teaching, and published many new books, such as Introduction to Mathematical Methodology by Mr. Wang and First Draft of Mathematical Methodology co-authored by Mr. Wang and Mr. Guo. Many valuable papers are published in many newspapers and magazines. Especially after the Mathematics Teaching Outline for Nine-year Compulsory Education 1992 formulated by the State Education Commission in August made it clear that mathematics thinking method is an integral part of mathematics knowledge, people paid more attention to the teaching of mathematics thinking method, and the teaching research of mathematics thinking method was deepened and broadened, which solved many practical teaching problems, greatly promoted the process of mathematics education reform in China and became a unique and far-reaching research topic. So, what exactly is a mathematical thinking method?

The word "method" comes from Greek, which literally means to follow the road. Its semantic interpretation refers to the interpretation of some adjustment principles that must be followed in order to achieve a certain goal. The encyclopedia of the Soviet Union said: "Method refers to the method, theory or doctrine of research or understanding, that is, the sum of means or operations to solve specific problems by grasping reality in practice or theory." The Encyclopedia of Philosophy of McClellan Company in the United States interprets the method as "the steps that must be taken to achieve a given result according to a given procedure." The definition of "law" in China Ci Yuan is "law, illusion or illusion". From the perspective of scientific research, methods are the means and tools people use to study and solve problems. These means and tools are closely related to people's knowledge, experience and theoretical level, and are the principles guiding people's actions. At the beginning of China's ancient art work "Thirty-six Plans", he wrote: "Six, six, thirty-six plans, there are techniques in the number, and there are numbers in the operation." It shows that the ancients have long realized the close relationship between mathematics and strategies and methods. We think that mathematical method is a general strategy to put forward, analyze, deal with and solve mathematical problems.

In modern Chinese, "thought" is interpreted as the result of objective existence reflected in people's consciousness through thinking activities. In Ci Hai, "thought" is called rational knowledge. Encyclopedia of China holds that "thought" is the result of rational knowledge relative to perceptual knowledge. The encyclopedia of the Soviet Union pointed out: "Thought is the principle to explain objective phenomena." Mao Zedong said in the article "Where does man's correct thinking come from": "If more materials of perceptual knowledge are accumulated, it will make a leap and become rational knowledge, which is thought." Taken together, thinking is the advanced stage of cognition, and it is an essential and advanced abstract summary of things. We believe that mathematical thought is a rational understanding in mathematics, the essence of mathematical knowledge, and a highly abstract and generalized content in mathematics, which is contained in the process of analyzing, handling and solving mathematical problems by using mathematical methods.

Mathematical thought is an essential understanding of mathematical facts, concepts and theories, and a high generalization of mathematical knowledge. Mathematical method is the concrete reflection and embodiment of mathematical thought in mathematical cognitive activities, and it is the means and tool to solve mathematical problems and realize mathematical thought. Broadly speaking, mathematical thoughts and methods are part of mathematical knowledge.

(A) the structure of mathematical thinking

The range of mathematical ideas is very wide, and the basic mathematical ideas commonly used in middle schools are:

The idea of (1) transformation. Mathematics is full of contradictions, complex and simple, difficult and easy, general and special, unknown and known. Through transformation, we can simplify the complex, make the difficult easy, turn the general into the special, turn the unknown into the known, and solve the contradiction. The process of solving mathematical problems is actually the process from conditions to conclusions. Starting from the conditions, we first draw a transitional conclusion, and then gradually transform it into the final conclusion. Therefore, reduction is the most basic idea in mathematics. Specifically, there are transformations such as addition, subtraction, multiplication and division, multiplication and root, exponent and logarithm, high-order to low-order, multivariate to unary, three-dimensional to two-dimensional and so on.

② The idea of functions and equations. Function describes the dependence between quantity and quantity in nature. The idea of function is to abstract the characteristics of quantitative relationship from practical problems and establish functional relationship, so as to study the changing law of variables.

The idea of an equation is to set some unknowns when solving a problem, then find out the equivalent relationship between known numbers and unknowns according to the conditions of the problem, list the equations, and finally solve the problem by solving the values of the unknowns of the equations.

③ The idea of logical division. Also known as the idea of classified discussion, its essence is to determine the classification standard according to the requirements of the problem, classify the research objects, then solve each category separately, and finally draw a comprehensive conclusion.

④ The idea of combining numbers with shapes. The combination of numbers and shapes is to combine quantitative relations with spatial graphics, abstract thinking with image thinking, and to transform quantitative relations into graphic properties and solve algebraic problems with geometric methods, or to transform graphic properties into quantitative relations and solve geometric problems with algebraic methods.

(2) the structure of basic mathematical methods

There are usually two basic mathematical methods:

① Mathematical thinking method. This is a higher-level method in mathematical methods and a thinking method in mathematics, including analysis, synthesis, abstraction, generalization, observation, experiment, association and analogy, conjecture, induction, deduction, generalization and specialization.

② Mathematical problem solving methods. This is a general mathematical problem-solving method, which is universal compared with special problem-solving skills.

Laws include collocation method, method of substitution method, elimination method, substitution method, undetermined coefficient method, parameter method and so on.

Earlier, I talked about the important role of teaching that attaches importance to the occurrence, formation and development of mathematical knowledge in effectively forming students' cognitive structure. At the same time, we also know that problem is the heart of mathematics, method is the behavior of mathematics, and thought is the soul of mathematics. Whether it is the establishment of mathematical concepts, the discovery of mathematical laws, the solution of mathematical problems, or even the construction of the whole "mathematical building", the core problem lies in the cultivation and establishment of mathematical thinking methods. Therefore, in teaching, I not only attach importance to the process of knowledge formation, but also attach great importance to discovering the important thinking methods contained in the process of the occurrence, formation and development of mathematical knowledge. The reason why "mathematical science" is separated from natural science and becomes one of the top ten departments of modern science is not because of mathematical knowledge itself, but because of the important role of mathematical thought and consciousness. The most useful thing in a person's life is not only mathematical knowledge, but also mathematical thoughts and consciousness. Therefore, we should lose no time in infiltrating thinking methods into primary school mathematics teaching.

(A) the penetration of the concept of "unit"

In mathematics, the calculation of number and quantity benefits from the idea of unit.

1. Pay attention to the idea that "1" is the unit of natural numbers.

It can be said that there is no natural number without "1", and there is no complete mathematical system. So from the first grade, I paid great attention to the infiltration of the idea of "unit" into students.

(1) Before I knew the numbers within 10, I attached great importance to the teaching of "1" and "duo". The teacher showed a basket of apples and said there were "many" apples in the basket. And let the students put the apples in the basket into each small plate one by one, then each small plate is "1" apples. Then put an apple from each plate in the basket, and there will be a lot of apples in the basket. In the above demonstration process, let students experience the relationship between "duo" and "1": "duo" consists of a "1"; "Duo" can be divided into "1". "Duo" stands for "1".

(2) In the understanding stage of numbers within 10, pay attention to the relationship between each number and "1", and emphasize that several "1" can synthesize this number. For example, when teaching "7", I don't display "6" first, then add "1" to explain to students that this is "7"; Instead, show seven objects at a time and compare them directly with one object, so that students can understand that "7" is seven "1"; Secondly, it reveals the relationship between "7" and previously known numbers, especially the closest number "6" in front of it.

(3) When teaching the understanding of numbers within 100 and 10000, we still emphasize that "1" is the unit of natural numbers, and pay attention to the differences from counting units such as "ten", "hundred", "thousand" and "ten thousand".

2. Attention should be paid to the introduction of "measuring unit" in the teaching of quantity measurement.

The primary problem of quantitative measurement teaching is to introduce measurement units reasonably. Historically, the introduction of any unit of measurement has a long historical process. As a textbook, it is impossible and unnecessary to make great efforts to explain this process. However, as a teacher, according to the actual situation of teaching, it is beneficial to cultivate students' creative thinking quality and the spirit of exploring and pursuing truth by properly showing their simple processes and thinking methods. For example, in the teaching of "area and area unit", when students can't directly compare the sizes of two graphs, they introduce "small squares" and expand them one by one on the two graphs compared. This not only compares the sizes of the two graphs, but also "quantifies" the areas of the two graphs. Turn the problem of shape into a problem of number. In this process, students personally experience the role of "small squares". Then through the teaching process that the size of the "small square" must be unified, let the students deeply realize that there must be a standard for the quantification of any quantity, and the standard should be unified. Nature has infiltrated the idea of "unit".

For another example, in the teaching of "hours, minutes and seconds", I designed the following process at the beginning of introducing a new lesson: (1) The teacher makes two "ah" sounds (the two times are obviously different) and asks the students which "ah" takes longer? Then, the teacher raised his left hand and right hand respectively (the length of the left hand and right hand grip is obviously different). Ask students how long it takes to raise their hands left and right. The purpose of designing this teaching process is to let students experience that although time is invisible and intangible, we can feel that time does exist with our eyes and ears. (2) The teacher made the sound of "Ah" twice and raised his left and right hands, but the time was almost the same, so it was difficult for students to judge the time of "Ah" twice and the time of raising their hands. So that students feel that feeling alone can't solve the problem. (3) The teacher raised his left hand and right hand again, and counted the time of raising his left hand and right hand. When you raise your left hand, you count five times, and when you raise your right hand, you count six times at the same speed, so that students will soon know that it takes longer to raise their right hand. Here, although the difference between the left hand and the right hand is still small, it is easy to judge, because students know that "number" is a "unit". Let students feel the necessity of introducing objective "standards". Naturally, there should be a "unit" to calculate the length of time, thus infiltrating the idea of "unit" in a timely manner.

(B) the infiltration of thinking mode transformation

The thought of conversion is one of the important thinking methods in primary school mathematics. The so-called "transformation" can be understood as "transformation" and "reduction". I think that as a primary school math teacher, paying attention to and correctly applying "transformation thought" in teaching can help students grasp the development process of things and have a deeper understanding of the internal structure, vertical and horizontal relations and quantitative characteristics of things. Here are a few examples.

1. Four arithmetic "rules of clever use".

There are many problems in the four operations. Although the correct results can be calculated step by step according to the conventional operation order, the calculation is very complicated because of the complex data. If we can use identity transformation to make the structure of the topic suitable for a "model" and use the laws and properties we have learned to solve it, it will be done overnight.

For example, calculate 1.25×96×25.

It is very convenient to decompose 96 into 8×4×3, and then calculate it by multiplicative commutative law and associative law.

1.25×96×25= 1.25×8×4×3×25

=( 1.25×8)(25×4)×3

= 10× 100×3

=3000

It is convenient to convert the second factor 18 into (17+ 1) and solve it by multiplication and division.

2. Area calculation "conversion diagram".

To solve the area of some combined geometric figures, using the idea of transformation, the original figures are "deformed" by means of rotation, translation, folding and shearing, which can make the problem difficult and solve it naturally.

For example, the picture on the left below. The area of a large regular triangle is 28 square centimeters. Find the area of a small regular triangle.

It is difficult to see the area relationship of equilateral triangles in the figure. If the small equilateral triangle is "rotated" to the right, four congruent small equilateral triangles will appear, and the answer will be easily obtained. The area of a small regular triangle is:

28÷4=7 (square centimeter).

In fact, in primary school textbooks, except for the rectangular area calculation formula, the area calculation formulas of other plane graphics are obtained by transforming the original graphics. In teaching, we should seize the opportunity and use these graphic transformations to infiltrate our thoughts.

3. Understand the quantity of "from here to there".

Some topics, according to the analysis and combination of the usual known quantities, often find it difficult, and even suffer from "insufficient conditions." But as long as we break the mindset and analyze the quantitative relationship from a new angle, we will find the correct way to solve the problem.

For example, the picture below shows a right-angled trapezoidal wall. Try to draw the shadow with 2 kilograms of paint. According to this calculation, how much paint does it take to paint this wall?

If the relationship among area, unit quantity and total quantity is solved by convention, the wall area must be calculated first. Compared with the known conditions, you will be at a loss. If we change the method, first calculate the proportion of the shadow area to the whole wall area, and then calculate the total amount of the whole wall according to the known amount of the shadow part, it will be easy to solve the problem.

Shaded area: the whole trapezoidal area.

4. Mathematical language "interchangeable expressions".

There are three kinds of mathematical languages: general language, graphic language and symbolic language. For example, "the volume of a cone" is expressed as v = 1/3sh in symbolic language and as "the volume of a cone is equal to one third of the volume of a cylinder with the same height as its bottom surface" in popular language. The textbook is also equipped with graphic language. Because the three forms of mathematical language have their own characteristics, the graphic language is intuitive, the symbolic language is concise and accurate, and the common language is easy to understand. In primary school, because students' thinking is still in the transition stage from image thinking to abstract thinking, graphic language and common language are the main textbooks, but symbolic language has also appeared in many places. Therefore, in mathematics teaching, strengthening the transformation of various mathematical languages can deepen the understanding and memory of mathematical concepts and propositions, and help students to examine questions and explore solutions.

(C) the infiltration of symbolic thinking

Mathematical symbols play a very important role in mathematics. Russell, a famous British philosopher and mathematician, also said, what is mathematics? Mathematics is symbol plus logic. Faced with a common mathematical formula: S=πr2, anyone with primary education knows what this means, no matter where he comes from. The symbolic language of mathematics can be used everywhere regardless of country or race. World communication needs mathematical symbolic language.

In a simple inequality: 3+□ < 8, □ can mean many numbers (0, 1, 2, 3, 4) for primary school students, and countless numbers (0 ≤□ < 5) for senior students. Then, students can see: □ I deeply understand that symbols can express a lot of information in their condensed form. At the same time, symbolic thinking can greatly simplify the operation or reasoning process, speed up thinking and improve the efficiency of unit time.

The essence of symbolic thinking has two aspects: one is to have the consciousness of expressing practical problems with mathematical symbols as much as possible; The second is to fully grasp the rich connotation and practical significance of each mathematical symbol. Therefore, whether it is an element symbol, an operation symbol, a relation symbol, a combination symbol, etc. I noticed the above two points. For example, when explaining the number symbol "5", on the one hand, the number of objects that emphasize "so many fingers" in one hand can be represented by the symbol "5". At the same time, let primary school students look at "5" and say its connotation. For example, name five people, five pens and five cars. For the mathematical formulas and algorithms in primary school textbooks, I not only try to let students express them with symbols, but also ask them to say the meaning of each formula and algorithm completely.

It is not easy for primary school students to abstract things and phenomena existing in objective reality and their relationships into mathematical symbols and formulas. This is because symbolization has a process from concrete-representation-abstraction-symbolization. Therefore, it is necessary to gradually cultivate the abstract generalization ability of primary school students. For example, in the teaching of practical problems, I often train students to condense and refine quantitative relations from complex plots and relational narratives. This is not only conducive to solving problems, but also conducive to cultivating and improving corresponding abilities.

At the primary school stage, there are not many existing digital symbol languages in textbooks, so there should be no excessive requirements for primary school students to master how many symbol languages. But in daily teaching, we math teachers should have such a strong consciousness: pay attention to the infiltration of symbolic thinking; Attach importance to the cultivation of primary school students' abstract generalization ability.