First, the way of thinking in primary school mathematics The so-called mathematical method is the way to solve mathematical problems. That is, the methods, ways and means to solve specific mathematical problems. It is a concrete act of learning mathematical knowledge and using mathematical knowledge to solve practical problems. The so-called mathematical thought is the essential understanding of mathematical knowledge, methods and laws, and some viewpoints extracted from some specific mathematical understanding processes, which are more abstract, more general and more essential than mathematical methods. Therefore, mathematical thought is the soul of mathematics and the theoretical basis of mathematical methods. Because primary school mathematics is the most basic mathematical knowledge, its content is relatively simple, and it is difficult to completely separate the thinking methods it contains. More reflected in the connection, the essence is often the same. Therefore, it is easier for everyone to accept and understand the mathematical ideas and methods in primary school mathematics as a whole concept. Starting from the first volume, primary school mathematics textbooks present mathematical knowledge and skills in stages, and at the same time contain vertical mathematical thinking methods, mainly including: symbolic thinking method, corresponding thinking method, set thinking method, transformation thinking method, combination of numbers and shapes thinking method, model thinking method, extreme thinking method, system structure thinking method, statistical thinking method, mathematical beauty thinking method and so on.
Second, the role of mathematical thinking methods in primary schools The core of mathematical quality is mathematical thinking. To improve students' mathematical literacy, we should pay attention to the teaching of mathematical thinking methods contained in textbooks, which has the following functions.
1. is helpful to cultivate and develop students' cognitive ability.
As we all know, all mathematical concepts, formulas, laws and rules can be regarded as mathematical models. In mathematics teaching, starting from the realistic prototype, we use the methods of experiment, operation and observation, and express the thinking process in mathematical language through basic thinking methods such as comparison, analysis, synthesis and abstract generalization, so that students can obtain accurate mathematical models and develop their cognitive ability. For example, the teaching of "9 plus several" leads to such a mathematical model: when students master the method of "adding ten", they can move towards "8 plus several" and "7 plus several" ... to develop students' cognitive ability in learning mathematics.
2. It helps to construct and improve students' cognitive structure.
Piaget thinks: "All mathematics can be considered according to the structure." To form a knowledge structure, it is convenient for students to form a cognitive structure. Therefore, we should combine mathematics teaching with knowledge structure with mathematical scientific order. When designing the teaching process, the knowledge structure is gradually transformed into the cognitive structure in students' minds. Mathematical thinking method is the theoretical basis of constructing cognitive structure. For example, in the teaching of the area formula of plane figure, based on the theory of reduced thinking and reduced thinking, the assimilation and adaptation among the area calculation formulas of rectangle, square, parallelogram, triangle, trapezoid and circle are realized, thus the cognitive structure of students is constructed and improved.
3. Help to guide students to master learning methods.
Students have formed personality differences due to various factors, so they should teach students in accordance with their aptitude. If we pay attention to inspiring students from mathematical thinking methods, students will not only learn, but also learn new knowledge and have a further rational understanding. For example, when teaching fractional division, students often just turn the divisor into an integer, but fail to correctly handle the decimal point position of the dividend. For this kind of students, we should use the idea of "constant transformation" to guide them to apply the "quotient invariance" they have mastered to fractional division, so as to solve the problem and grasp the essence of the law of fractional division. It can be seen that students can't solve problems without the guidance of mathematical thinking methods.
4. Help students inspire dialectical materialism.
Mathematical thinking method is the embodiment of dialectical materialism in mathematics. For example, teaching the circumference and area of a circle with the extreme idea of "turning joy into straightness" is not only beneficial for students to master knowledge, but also inspires the dialectical materialism of "finiteness and infinity" and "quantitative change to qualitative change" in essence.
5. It is helpful to cultivate and develop students' aesthetic taste.
Mathematician Klein once described the beauty of mathematics as follows: "Music can stimulate or soothe feelings, painting can make people pleasing to the eye, poetry can impress people, philosophy can make people gain wisdom, technology can improve material life, but mathematics can provide all of the above." The main characteristics of mathematical beauty are order, conciseness, symmetry and unity. The method of synthesis and analysis in mathematical thinking embodies the order; Symbol thought fully embodies the simplicity and clarity of mathematical expression; The combination of numbers and shapes, the knowledge structure fully embodies the beauty of unity; The golden section fully embodies the singular beauty of mathematics, and the essence of mathematical thinking method embodies the beauty of mathematics. In teaching, students are influenced by the beauty of mathematics while consciously learning mathematics.
Thirdly, combining with the content of teaching materials, consciously infiltrating mathematical thoughts and knowledge is the "carrier" of mathematical thinking methods. Mathematics teaching in primary schools should infiltrate mathematics thoughts into students according to their thinking characteristics and knowledge teaching, that is, in the process of imparting knowledge, some basic mathematics thoughts should be organically infiltrated into students, so that students can form mathematics thoughts while acquiring knowledge.
1. Combine the contents of the textbook and consciously infiltrate the corresponding ideas.
Correspondence is a way to think about the connection between two set elements. There are a lot of corresponding ideas in primary school mathematics textbooks. There are mainly single-value correspondence, one-to-one correspondence and inverse correspondence. In teaching, combining the relevant contents of teaching materials, creating scenarios and consciously infiltrating corresponding ideas will help to cultivate students' flexibility and creativity in thinking, understand mathematical concepts, master mathematical skills, prevent students from thinking in a fixed way and improve their dialectical thinking ability. For example, it is necessary to find out the corresponding quantitative relationship in the application problem of teaching scores, such as the simple application problem of teaching "Mom bought 10 apples and 8 pears". How many apples are there than pears? "For the first-year students who have just come into contact with the application problem, in order to let the students fully understand the meaning of" who is more than who ",the teacher posted a physical picture: through the intuitive comparison of graphic images, one apple corresponds to a pear, and the students found that there are two apples that do not correspond to pears, thus enlightening the students to understand the meaning of more apples than pears, and then entering determinant calculation. In this way, students can clearly find out the quantitative relationship, find out the law of solving problems, and let students unconsciously establish corresponding ideas.
2. Combined with the content of teaching materials, consciously infiltrate collective thinking.
Set theory is an important theory and problem-solving tool in mathematics. There are a lot of stereotypes in primary school mathematics textbooks. Therefore, in the process of implementing quality education, we should not only impart knowledge to students, but also consciously infiltrate the established ideas contained in textbooks, which is conducive to cultivating students' abstract generalization ability and improving their ability to analyze and solve problems. The textbook adopts intuitive means and uses the idea of infiltration and collection of graphics and objects. For example:
Through charts, students can clearly and intuitively understand and master mathematical concepts, not only their attribute relations, but also the set thought (proper subset and Bing). For example, when talking about common divisors, make a slide that can be drawn:
Students can clearly and intuitively know that the common divisors of 12 and 15 are 1 and 3, and the greatest common divisor is 3, thus giving birth to the idea of intersection. For another example, when recognizing numbers in teaching, it is common to connect the same number of lines (as shown below). These problems are essentially to let students further establish the idea of set and correspondence through practice.
3. Combined with the content of teaching materials, consciously penetrate into the mind.
Reduction method 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, if there is a largest circle in a square with an area of 15 square centimeter, find the area of this circle. Because 15 is an incomplete square number, if you want to solve it directly, you need to use the root sign, which seems impossible to solve in primary school. However, we can turn the original problem into: knowing the side length of a square is 1 cm, find the area of the largest circle in this square. In this way, we can easily solve the problem: the area of the largest circle in a square with a side length of 1/22× 3. 14 = 157/200 (square centimeter), that is, the circle accounts for 157/200 of the square area. So the circular area in the original problem is157/200×15 =11.775 (square centimeter). For example, when calculating the area of a combined graph, first cut the combined graph into simple graphs, and then calculate the sum or difference of the areas of each part. Can make students understand the essence of reduction.
4. Combine the contents of the teaching materials, and consciously infiltrate and transform ideas.
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. 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.
For another example, the number of people absent from a class in the morning is 0/7 of 65438+ attendance, and the number of people absent from class is 0/6 of 65438+ attendance in the afternoon because 1 person asks for sick leave. How many people are there in this class? It is difficult to solve this problem because the number of people who took part in it changed yesterday afternoon. If the number of people absent from class in the morning is converted into 1/7+ 1= 1/8 of the class number, the number of people absent from class in the afternoon is 1/6+ 1/7 of the class number, and the essential relationship can be found soon: 65438+.
5. Combining with the content of teaching materials, consciously infiltrate the idea of combining numbers with shapes.
Number and shape are the two main objects of mathematical research, which are both different and related. On the one hand, abstract mathematical concepts and complex quantitative relations are visualized, visualized and simplified through graphics; On the other hand, complex geometric shapes can be expressed by simple quantitative relations. In practical problem teaching, the combination of numbers and shapes can transform the quantitative relationship given in the problem into graphs, and graphs can reveal the quantitative relationship intuitively, which is conducive to activating students' thinking, broadening students' problem-solving ideas, improving their problem-solving ability and promoting intellectual development. For example, a batch of goods has been shipped 100 tons, and the remaining110 tons are all missing. How many tons are there in this shipment? Draw a line drawing:
The correspondence between quantities in this question is very clear: 1- all goods? many
1-110-(100-1) tons
It is convenient to list the formula (100-1) ÷ (1-10).
The combination of numbers and shapes can promote the flexibility and creativity of students' thinking, obtain more optimized solutions, and even stimulate students' inspiration, generate epiphany and directly obtain results. Such as1/2+1/4+1/8+116 =? This problem is not difficult, but with the help of drawing:
The solution is simple:1/2+1/4+1116 =1-kloc-0//kloc-6 =
6. Combining with the content of teaching materials, consciously infiltrate the idea of mathematical model.
The so-called mathematical model refers to a mathematical structure obtained by using mathematical tools to simplify and assume a specific object in the real world for a specific purpose, which provides the optimal decision or control for the object. In fact, primary school mathematics teaching can be regarded as the teaching of mathematical model. The life experience of primary school students is limited, and many practical problems cannot be directly related to their own experiences. Therefore, we can't rely on life experience to turn practical problems into mathematical problems to answer. In the teaching of application problems, students can be guided to build practical models according to the plots of application problems, help students to establish appearances, understand the quantitative relationship between application problems, and grasp the essence of problems, so as to turn practical problems into mathematical problems and achieve the purpose of solving practical problems. For example, a pedestrian is 100 meters long and 6 meters wide, and the ground is paved with square bricks with a side length of 40 cm. How many pieces do you need? Although such problems are often encountered in daily life, students cannot solve them in the right way. At this time, I guide students to reasonably imagine the actual scene of the sidewalk and build the following sidewalk model:
With the help of imagery, students turn the actual problem into a mathematical problem of "How many 0. 16 square meters are there in 600 square meters", and accurately capture such a problem-solving method: (100×6)÷(0.4×0.4)=3750 (block).
7. Combining with the content of teaching materials, consciously penetrate the extreme thoughts.
Things change from quantitative to qualitative, and there is a "joint point" in the process of this change. For example, when we talk about "knowledge of circle area", we will make a circular teaching aid with the limit as the "joint point" and divide it into many sectors with different shares. For example, if the circle is divided into eight parts, the assembled figure is similar to a parallelogram, and the shape of the side is wavy; Divide the circle into 16 parts, the figure is closer to a parallelogram, and the shape of the side is straight; Continue to divide the circle into 32 parts on average, and the edge of the spelled figure becomes straighter and straighter, and the figure gets closer and closer to the parallelogram; By comparing the spliced graphics, students can intuitively see that the more the number of equally divided sectors, the closer the spliced graphics will be to the parallelogram. If we continue to divide them into 64 equal parts, such as 128, the spliced graphics are no different from rectangles. In this way, in the process of observation and comparison, students not only understand the dialectical thought that the area of the spliced rectangle is equal to the area of the original circle, but also the initial contact quantity changes qualitatively, from finite to infinite, which cultivates students' spatial concept and develops their thinking ability. Then guide the students to analyze and compare the relationship between the length and width of a rectangle and the perimeter and radius of the original circle, and then get S=πr2.
The thinking method of system structure embodies the systematicness, orderliness and integrity of mathematical knowledge. Symbolic thinking method is the carrier of mathematical information, and it is also the carrier of quantitative analysis and systematic analysis. With the development of modernization, data processing methods are more and more deeply involved in all fields of social life. Therefore, in teaching, structural thinking, symbolic thinking and statistical thinking should be consciously infiltrated in combination with the content of teaching materials.
Mathematical thought always takes concrete mathematical knowledge as the carrier. Therefore, in the teaching process of specific mathematical knowledge, according to students' cognitive rules and age characteristics, combined with the contents of teaching materials, mathematical ideas can be organically infiltrated, which can be single or comprehensive, so as to deepen students' understanding of basic knowledge, broaden their knowledge, master mathematical methods and skills, inspire students to explore new knowledge, and understand the objective world with dialectical thoughts, laying a solid foundation for effectively implementing quality education and cultivating talents for cross-century construction.