"Symbolic thinking" is the basic idea of mathematics. Mathematics, as a subject language, is a tool to describe the world, and symbols can make the research object of mathematics more concrete and vivid, and can simply express the essential characteristics and laws of things. The use of symbols determines the progress of mathematics to a great extent, which has the ability to cultivate people's highly abstract thinking. For example, the content of "one-dimensional equation" in primary school mathematics books suggests that students use letters to represent numbers, which is an abstraction in essence. Its purpose is to explore and reveal the mathematical law more deeply, express it more accurately and concisely, and affirm its correctness in a wider range. The commutative law of addition is a+b=b+a, the circular area is represented by S=πr2, and so on. In addition, the equation solution is used to solve application problems, and the solution itself also contains symbolic ideas, which are mainly reflected in the following aspects: (1) algebraic assumption, using letters to replace unknowns and participating in operations on an equal footing with known numbers; (2) Algebraic translation, that is, translating the known conditions expressed in natural language into equations expressed in symbolic language. (3) Solving algebraic equations. Taking letters as known numbers, four operations are carried out to achieve the purpose of solving.
It can be seen that mathematical symbols are the pillars that run through mathematics. Mathematical symbols are condensed with unique simplicity, abstraction and generality, so they are relatively difficult to master and use. As a mathematics teacher, it is of great significance to deeply understand the idea of mathematical symbols and study the teaching of mathematical symbols to promote mathematics teaching and improve its quality.
The second is the infiltration of "return to thought".
"Transforming thought", also known as "transforming thought", is one of the most critical mathematical thoughts in primary school mathematics. It often transforms a practical problem into a mathematical problem and a more complicated problem into a simpler one by means of observation, reasoning and analogy until it is solved or easy to solve. Its basic forms include turning life into maturity, turning difficult into easy, simplifying complexity, breaking the whole into parts, turning the unknown into known, turning the general into special, and turning abstract into concrete. Infiltrating this idea into students is conducive to improving their logical thinking ability.
For example, in the area calculation of teaching plane graphics, the assimilation and adaptation among the area calculation formulas of rectangle, square, parallelogram, triangle, trapezoid and circle are realized based on the theory of reduction transformation, thus the cognitive structure of students for area calculation is constructed and improved. Fractional division is classified as division with integer divisor by "quotient invariance"; Addition and subtraction of fractions with different denominators are divided into addition and subtraction of fractions with the same denominator; The comparison size of scores with different denominators is classified into the comparison size of scores with the same denominator through "general scores" and so on. The research on this knowledge is permeated with the idea of reduction.
Third, the infiltration of the idea of "combination of numbers and shapes".
"Combination of numbers and shapes" is the idea of solving mathematical problems through the mutual transformation of numbers and shapes according to the corresponding relationship between numbers and shapes. The idea of "combination of numbers and shapes" can make some abstract mathematical problems intuitive and vivid, change abstract thinking into image thinking, and help to grasp the essence of mathematical problems. In primary school teaching, it is mainly manifested in the transformation of abstract quantitative relations into appropriate geometric figures, from the characteristics of intuitive figures to the discovery of the relationship between quantities, so as to achieve the purpose of turning abstract into concrete, turning hidden into obvious, and solving problems simply and quickly.
It can promote the coordinated development of students' thinking in images and abstract thinking, communicate the relationship between mathematical knowledge with the help of simple charts, symbols and words, and highlight the most essential characteristics from complex quantitative relations. For example, we often use the method of drawing line segments to solve application problems, which is a method of replacing quantitative relations with graphics. We can also study the perimeter, area and volume of geometric figures by algebraic method, which embodies the idea of "combination of numbers and shapes"
Fourth, the infiltration of "extreme thoughts"
Extreme thinking is an important mathematical thinking method. With the help of limit thought flexibly, we can simplify some mathematical problems, avoid some complicated operations, and explore the direction of solving problems or transformation ways. In the process of deriving the formula for calculating the area of a circle and the formula for calculating the volume of a cylinder, the ideas of "turning a circle into a square" and "turning it straight into a square" are adopted. On the basis of "observing finite segmentation" and "imagining infinite subdivision", according to the changing trend of graphic segmentation and assembly, imagine their final state. In this way, students not only mastered the formula for calculating the area of a circle and the volume of a cylinder, but also naturally sprouted the "limit thought" of infinite approximation in the contradiction transformation between "curve" and "straight line".
In addition, there are many places in the current primary school textbooks that pay attention to the infiltration of extreme ideas. When teaching the concepts of "natural number", "odd number" and "even number", teachers can make students realize that natural numbers are infinite, and there are infinite odd numbers and even numbers, so that students can understand the idea of infinity initially; This part of the cyclic decimal, 1 ÷ 3 = 0.33… is a cyclic decimal, the number after the decimal point is endless, and the limit of 0.99 … is equal to1; In the teaching of straight lines, rays and parallel lines, let students realize that the two ends of a straight line can extend indefinitely.
Fifth, the infiltration of "collective thinking".
quadrilateral
"Collective thinking" is an early way of thinking of human beings. It puts a group of related objects together as the scope of discussion, and then lists the abstract thinking objects in an orderly way to a certain extent, which makes people clear at a glance. For example, after teaching parallelogram, rectangle and square, let the students make it clear that rectangle is a special parallelogram and square is a special rectangle, and it is more vivid to express it with correct pictures. In order to deepen students' understanding of this assembly drawing, let's give another example: our whole school students are like this largest circle, our grade students are part of the whole school, our class students are part of the whole grade, and the first group students are a small part of the whole class, that is, the smallest circle inside. Let students really understand the meaning of set diagram and learn to apply it. Set's mathematical thinking method permeates all stages of primary school 1 ~ 6 grade. Mathematical ideas such as subsets and intersections permeate the divisibility of a number. Set theory can make mathematics and logic more unified, which is beneficial to the research of mathematical theory and application. Using set theory to solve problems can prevent repetition and omission in the process of classification and make abstract mathematical problems concrete.