Using symbolic language (including letters, numbers, graphics and various specific symbols) to describe the content of mathematics is symbolic thinking. The idea of symbols is to integrate all data examples into one, and express complex languages and characters with simple and clear letter formulas, which is convenient for memory and use. It is a process from concreteness to representation, and then to abstract symbolization, which abstracts things, phenomena and their relationships into mathematical symbols and formulas.

The mathematical language embodied by symbols is a worldwide language and a comprehensive reflection of a person's mathematical literacy.

In mathematics, all kinds of quantity relations, quantity changes, and the deduction and calculation between quantity and quantity are represented by lowercase letters, and a large amount of information is expressed in the condensed form of symbols, such as multiplication and division (A+B) × C = A× C+B× C; For another example, in the division teaching with remainder, a thinking question finally appeared: At the June 1st party, Xiaoming strung balloons to decorate the classroom in the order of 3 red balloons, 2 yellow balloons and 1 blue balloons. Do you know what color the 24th balloon is? To solve this problem, the letters A, B and C can represent red, yellow and blue balloons respectively, which can be converted into the following symbol forms according to the meaning of the question: aabbc aabbc aabbc aabbc .................................................................................................................... This is the concrete embodiment of symbolic thinking.

Change ideas

Transforming thinking is the most commonly used thinking method in mathematics. Its basic idea is to transform the solution of problem A into the solution of problem B, and then get the solution of problem A through the inverse of the solution of problem B ... generally refers to irreversible "transformation". Its basic forms are: turning difficulty into ease, turning life into maturity, turning complexity into simplicity, turning the whole into parts, turning music into straightness and so on. For example, when calculating the area of combined graphics, cut the combined graphics into simple graphics, and then calculate the sum or difference of the areas of each part, so that students can understand the essence of reduction.

Decompose thought

Decomposition thinking is a kind of thinking method that first decomposes the original problem into several easy-to-solve sub-problems, then decomposes several easy-to-solve areas, then decomposes several easy-to-advance problem-solving steps layer by layer, and then solves them one by one, so as to successfully solve the original problem. For example, the problem-solving strategy of "reverse thinking" in the teaching of "Problem-solving Strategy" in grade five embodies this idea.

change of heart

Transforming ideas is an important strategy to solve mathematical problems, and it is a way of thinking from one form to another. The transformation here is reversible bidirectional transformation. Reduction is a very useful strategy to solve mathematical problems. When the problem is transformed, both the known conditions and the conclusion of the problem can be transformed. Transforms can be equivalent or not. Solving mathematical problems with the idea of transformation is only the first step, the second step is to solve the problem of transformation, and the third step is to reverse the solution of the problem of transformation into the solution of the problem. If the equivalence relation is used for conversion, the solution can be obtained directly without inversion.

If the calculation is: 2.8 ÷113 ÷17 ÷ 0.7, the direct calculation is troublesome, and the multiplication and division of the fraction is more convenient than the decimal, then the original problem can be transformed into: 28/10× 3/4× 7.

Classification thought

The thinking method of classification is not unique to mathematics, but embodies the classification of mathematical objects and its classification standards. For example, the classification of natural numbers can be divided into odd and even numbers according to whether they can be divisible by 2; Divide prime numbers and composite numbers according to the number of factors. Another example is a triangle that can be divided by edges or angles. Different classification standards will have different classification results and produce new concepts. The correct and reasonable classification of mathematical objects depends on the correct and reasonable classification standards, and the classification of mathematical knowledge is helpful for students to sort out and construct their knowledge.

Inductive thinking

Mathematical induction is a mathematical proof method, which is usually used to determine whether an expression is valid in all natural numbers or another form is valid in an infinite sequence. There is a generalized formal view used in mathematical logic and computer science, pointing out that the expressions that can be found are equivalent expressions, which is the famous structural induction.

Analogical thinking

Mathematical analogy refers to the idea that according to the similarity between two kinds of mathematical objects, the known properties of one kind of mathematical object can be transferred to another kind of mathematical object, which can solve some seemingly complicated and difficult problems. Analogy not only makes mathematical knowledge easy to understand, but also makes the memory of formulas natural and concise, which can stimulate students' creativity. As the mathematician Paulia said, "We should discuss these processes of generalization, specialization and analogy, which are great sources of discovery."

For example, from learning additive commutative law A+B = B+A to learning multiplication and distribution law A× B = B× A.

For example, the area formula of a rectangle is length× width = a× b, and so on, and the area formula of a triangle can also be understood as length (bottom )× width (height) ÷ 2 = a× b (h) ÷ 2. Similarly, the formula of cylinder volume is bottom area × height, so the volume of cone can be understood as bottom area × height ÷3.

Hypothetical idea

Hypothesis is a commonly used thinking method of speculative mathematics. Some fill-in-the-blank questions, judgment questions and application questions can be solved with this idea. Some questions hide the quantitative relationship, so it is difficult to establish the relationship between quantities, or the quantitative relationship is abstract and impossible to start. A way of thinking can first make some assumptions about the known conditions or problems in the question, then calculate according to the known conditions in the question, and finally find the correct answer according to the contradiction in quantity. Hypothetical thinking is a meaningful imaginative thinking, which can make the problem to be solved more vivid and concrete after mastering it, thus enriching the thinking of solving problems.

Comparative thinking

People's understanding of everything is based on comparison, either distinguishing differences from similarities or seeking common ground from differences. Russian educator ushinski said: "Comparison is the basis of all understanding and all thinking." While learning mathematical knowledge, primary school students need to understand the essential meaning of new knowledge and master the connections and differences between knowledge by comparing mathematical materials.

In the application problem of teaching scores, teachers should be good at guiding students to compare the known and unknown quantities in the problem before and after the change, which can help students find the solution quickly.

Extreme thoughts

From quantitative change to qualitative change, the essence of limit method is to achieve qualitative change through the infinite process of quantitative change.

In the teaching of "area and perimeter of a circle", the ideas of "turning a circle into a square" and "turning a curve into a straight line" are applied. On the basis of observing finite division, students can imagine their limit states, so that they can not only master formulas, but also sprout the idea of infinite approximation from the contradiction transformation between curves and straight lines.

In the chapter "The World in Zhuangzi" in the Warring States Period, "A foot of pestle is inexhaustible." Full of extreme ideas. Liu Hui, an outstanding mathematician in ancient times, used the idea of limit to find the circumference of a circle. He first drew a circle inscribed with a regular polygon. When there are more sides of a polygon, the perimeter of the polygon is closer to the perimeter of a circle. Liu Hui concluded: "If you cut it carefully, you will lose less. If you cut and cut, you can't cut any more, then your round body has nothing to lose. " It is with this idea of limit that Liu Hui discovered π, that is, "emblem rate".

There are many places in the current primary school textbooks that pay attention to the infiltration of extreme thoughts: when teaching concepts such as "natural number", "odd number" and "even number", teachers can make students realize that natural numbers are infinite, and there are infinitely many odd numbers and even numbers, so that students can initially understand the idea of "infinity". In the circulating decimal part, 1 ÷ 3 = 0 in teaching. 333… is a cyclic decimal, and the number after the decimal point is endless. In the teaching of straight lines, rays and parallel lines, let students realize that the two ends of a straight line can extend indefinitely.

Deductive thinking:

Deduction is also a rational activity, but unlike intuition, they are not simple rational activities. We must first assume some truths (or definitions), and then draw some conclusions with these definitions. For example, after we know the definition and theorem of a triangle, we can deduce that the sum of the internal angles of a triangle is equal to the sum of two right angles. So the function of intuition is to provide the latest principles of science and philosophy. Deduction is to apply these principles to establish some theorems and propositions. Deduction does not need the direct proof of intuition, and its certainty is given to it by memory to some extent. It can draw a conclusion through a series of indirect arguments, just as we can know the last section of a long chain by holding the first section.

In other words, intuition is the basic principle of invention and deduction is the most basic conclusion. However, some philosophers think that deduction is flawed, because the same principle often leads to different conclusions, so there should be another way to correct it. The method of this correction is experience, which is called resorting to facts. In a word, intuition is to find the simplest, most unquestionable and undefended elements in human knowledge, that is, to find the simplest and most reliable ideas or principles. Then they are deductively reasoned and all reliable solutions are deduced.

For example, the proof of mathematical theorems is a kind of deductive reasoning.

Model thinking

It refers to a specific object in the real world, starting from its specific life prototype, making full use of the so-called processes of observation, experiment, operation, comparison, analysis, synthesis and generalization, and getting simplification and hypothesis. It is a way of thinking to turn practical problems in life into mathematical problem models.

It is the highest realm of mathematics and the goal of students' high mathematical literacy to cultivate students to understand and deal with the surrounding things or mathematical problems with mathematical vision.

Mathematical model method is not only a classic method to deal with pure mathematical problems, but also a general mathematical method to deal with various practical problems in natural science, social science, engineering technology and social production. In order to solve some practical problems by mathematical methods, practical problems are usually abstracted into mathematical models first. The so-called mathematical model refers to the mathematical equation that describes the characteristics, relationships and laws of the real prototype as a whole. According to a broad interpretation, all mathematical concepts, mathematical theoretical systems, various mathematical formulas, various mathematical equations and an algorithm system consisting of a series of formulas are called models. But in a narrow sense, only those mathematical relationship structures that reflect specific problems or specific things systems are called mathematical models. For example, according to the quantitative relationship in specific problems, a mathematical model is established and equations are listed for solving.

Corresponding thought:

Correspondence means that an item in one system is equivalent to an item in another system in nature, function and position. Corresponding thinking can be understood as a way of thinking about the connection between two set elements. Infiltrating corresponding ideas in primary school mathematics teaching is helpful to improve students' ability to analyze and solve problems.

The concept of "correspondence" has a long history. For example, we correspond a pencil, a book and a house to an abstract number "1", and two eyes, a pair of earrings and a pair of twins to an abstract number "2". With the deepening of learning, we also extend "correspondence" to a form, a relationship, and so on.

Another example is the one-to-one correspondence between points on the number axis and real numbers, and the correspondence between functions and their images. In addition, in the lesson "More and Less", each teacup corresponds to a teacup cover, which intuitively shows that "compared with teacups, there are not many teacups, and there are not many teacups and teacups covers". Let students come into contact with the idea of one-to-one correspondence and feel that there is one-to-one correspondence between the two groups of elements, and their numbers are "the same". The idea of "correspondence" will play an increasingly important role in future research.

Collective thought:

Combining some different things (whether concrete or abstract) into a whole is called a set, in which each thing is called an element of the set. In layman's terms, that is to say, the whole is a collection of all objects that can be determined as a whole.

The characteristics of collective thinking:

(1) Determinism: Given a set, determine whether any object is an element of this set. That is to say, given whether an element in this set is or not, it cannot be ambiguous.

(2) Reciprocity: the elements in the set must be different, that is, the elements in the set are not repeated.

(3) Disorder: There is no fixed order for the elements in the set.

According to the different genera of the elements contained in the set, the set can be divided into the following categories:

(1) A set without any elements is called an empty set.

(2) A set with finite elements is called a finite set.

(3) A set containing infinite elements is called an infinite set.

The expression of set: enumeration; Block diagram method; Description method

For example, a number divisible by 2 is a set.

Combination of numbers and shapes:

According to the internal relationship between the conditions and conclusions of mathematical problems, it not only analyzes its algebraic significance but also reveals its geometric significance, so that the quantitative relationship and spatial form of the problem can be skillfully and harmoniously combined, and the mathematical problems can be solved through the mutual transformation of numbers and shapes. Its essence is to combine abstract mathematical language with intuitive images, and the key is the mutual transformation between algebraic problems and graphics, which can make algebraic problems geometric and algebraic. The idea of combining numbers with shapes includes two aspects, namely, "helping numbers with shapes" and "helping shapes with numbers". Its application can be roughly divided into two situations: one is to clarify the relationship between numbers with the help of the vividness and intuition of shapes. For example, the basic nature of P60 score in the second volume of fourth-grade mathematics is to clarify the relationship between molecules and denominators in the score with the help of the vividness and intuition of graphics; Or clarify some properties of the shape with the help of the accuracy and rigor of numbers.

In primary school teaching, it is mainly manifested in transforming the abstract quantitative relationship into appropriate geometric figures, and discovering the relationship between quantity and quantity from the intuitive characteristics of drawing, so as to achieve the purpose of turning the difficult into the easy, simplifying the complex into the simple, turning the hidden into the obvious, and solving the problem simply and conveniently. Usually, the quantitative relationship is transformed into a line chart, which is a basic and natural means, such as the relationship between the number axis and the corresponding point of grade one.

For some problems, if the line segment diagram can't clearly express its quantitative relationship, we can analyze, reform, design and construct a geometric figure that can clearly express its quantitative relationship. For example, try P72 in the sixth grade math book. The calculation is:1/2+1/4+1/8+1/6, which can be solved by a square graph.

In mathematics teaching, the idea of combining numbers with shapes to help numbers has the advantage of presenting problems intuitively, which is conducive to deepening students' memory and understanding of knowledge; When solving mathematical problems, the combination of numbers and shapes is helpful for students to analyze the relationship between numbers in problems, enrich appearances, trigger associations, enlighten thinking, broaden their thinking, and quickly find solutions to problems, thus improving their ability to analyze and solve problems. Grasping the ideological teaching of the combination of numbers and shapes can not only improve students' ability to transform numbers and shapes, but also improve students' ability to transfer their thinking.

Statistical thought

The significance of adding statistics and probability courses in primary school mathematics lies in forming the ability to explain data reasonably, improving the ability to understand the objective world scientifically and developing the ability to solve practical problems in realistic situations. The composition of the preliminary knowledge of statistics and probability mainly includes the following basic contents: 1. Understanding the value of data in describing, analyzing, predicting and solving some phenomena and problems in daily life; Second, learn some simple basic abilities of data collection, sorting, analysis, processing and utilization; Third, I will interpret and make some simple statistical charts; Fourth, recognize some random phenomena and predict the possibility of these random phenomena with appropriate methods.

Systematic thinking

Systematic thinking is an organic whole with specific functions, which is composed of several elements (or components) that think of correlation and function. The method of systematic thinking is to ask people to study the object from the relationship between system elements, the relationship and interaction between system and elements, and from the system to the external environment, so as to get the best scheme for studying and solving problems.

A system is a unity with overall function and comprehensive behavior, which is composed of a number of interrelated, interdependent, mutually restrictive and interactive things and processes. Elements are the basic units of the system, and the elements in the system are an organic whole that is interrelated and influenced each other. If one element changes, so will other elements.

For example, the "shopping problem" in the teaching of applied problems. Unit price, quantity and total price constitute a system. Quantity unchanged, unit price increased, total price increased; The unit price remains unchanged, the quantity increases, and the total price increases; The unit price remains unchanged, the total price increases and the quantity increases. "Unit price, quantity and total price" have the following relations:

Unit price × quantity = total price; Total price/unit price = quantity; Total price ÷ quantity = unit price

Through contact, we can grasp several concepts as a whole, from concrete to abstract, and then from abstract to concrete, find their laws, and better understand and master concepts and their relationships. These elements are not isolated and scattered, but interrelated and influenced each other. In the teaching process, students should be guided to understand concepts, find connections and discover laws. Only in this way can they better master what they have learned, achieve mastery through a comprehensive study and get twice the result with half the effort.

Third, some explanations.

Professor Zhou Chunli, director of China Mathematical Science Methodology Research and Exchange Center, said in his book:

Traditionally, people often use mathematical thoughts to refer to some mathematical achievements with great significance, rich content and complete system.

What's the difference between mathematical thought and mathematical method? Generally speaking, mathematical thought is people's essential understanding of mathematical content, a further abstraction and generalization of mathematical knowledge and methods, and belongs to the category of rational understanding of mathematical laws, while mathematical methods are the means to solve mathematical problems, with the meaning of "rules of conduct" and certain operability. When the same mathematical achievement is used to solve other problems, it is called a method; When it comes to its value and significance in the mathematical system, it is called thought.

It is difficult to strictly distinguish between mathematical thought and mathematical method. Therefore, it is more convenient for people to call it mathematical thoughts and methods indiscriminately.