The Mathematics Curriculum Standard for Compulsory Education (20 1 1 Edition) puts forward ten core concepts, which some researchers agree are the ten core qualities that mathematics needs to develop. Teacher Cao Peiying of Shanghai Jing 'an Institute of Education put forward a model as shown in figure 1, which basically conforms to the mathematical reality. Of course, mathematics learning is based on problems or tasks, the development of learning content is based on problem situations, and the goal of learning is to solve problems. In the process of solving problems, mathematical abstraction, mathematical reasoning and communication, mathematical model thinking, students' self-monitoring in solving problems, etc. In other words, problem solving is the carrier of carrying out scientific spirit, learning to learn and practicing innovation. Therefore, the author prefers to regard abstract ability, reasoning ability and problem-solving ability as the core literacy of mathematics. It should be noted that the problems here are not only problems in real life, but also problems in the development of mathematics itself; Problem solving here also means not only analyzing and solving problems, but also discovering and asking questions. The goal of mathematics teaching is to form certain concepts of computing ability, spatial imagination ability and data analysis through the study of specific knowledge, and to form certain abstract ability, reasoning ability and other superior and hidden abilities on the basis of these three abilities, and then comprehensively use these abilities to solve problems.
Therefore, in the process of learning specific knowledge, we must pay attention to the problem as the carrier and pay attention to the development of students' abstract ability, reasoning ability and application ability. Below, the author takes the study of "number" as an example to illustrate.
First, in the study of "number", the whole process runs through solving problems.
Appropriate problem situations can stimulate students' interest in learning and make them feel the significance of learning new knowledge; Through problem solving, students can not only acquire new knowledge smoothly, but also improve their mathematical thinking and learning ability in the process of problem solving. Therefore, it is necessary to solve problems throughout the study of numbers.
"Number" and its operation are based on actual needs. Natural numbers are generated based on the needs of counting in real life; Decimal is the product of conversion between different units in various measurement activities, and it is also a natural extension of the result of division of natural numbers. Fraction is used to represent the number of non-integers as needed, and can also be used to describe the result of integer division, ratio and so on. The operation of numbers is the product of practical needs, and the problems of comparison, merger and distribution of numbers are all generated in real situations, so it is natural to study the operation of addition, subtraction, multiplication and division of numbers. Therefore, in the study of "number" and its operation, students should be allowed to discover, put forward, analyze and solve problems spontaneously from the situation based on real problems, and naturally acquire new knowledge. For example, for the "addition and subtraction of two decimal places", the textbook of Jiangsu Education Edition presents the situation as shown in Figure 2, and classroom teaching can be roughly carried out through the following questions:
(1) What information did you get? Based on this information, what one-step calculation questions can you ask?
(2) Can you classify these formulas according to decimal places?
(3) Which of these formulas is easier to calculate? What have you studied? Can you calculate it in detail?
(4) What formulas should we study next? Tell your reasons and communicate with your peers.
(5) Looking back, what questions were raised today? What problems have been solved? What is the following question? What have you gained from studying in the whole class?
Starting from the situation, we will go through a complete problem-solving process of finding and asking questions, then sort out the problems properly, solve simple problems first, think more complicated problems with the help of the experience of solving simple problems, and finally sort out the experience of solving problems. This learning experience will benefit students for life.
Second, feel the abstraction in the cognitive learning of "number"
Abstraction is to abandon the non-essential attributes of things and grasp the essential attributes of things. Mathematical abstraction is to extract the essential attribute of quantitative relationship or spatial form from the research object. Therefore, mathematics is a highly abstract subject. Because of this, mathematics has become a good carrier to cultivate students' abstract ability, and abstraction has become the core literacy of mathematics. The process of extracting mathematical concepts from real questions and abstracting mathematical problems is a good opportunity to develop students' abstract ability. Let's take "understanding of natural numbers" as an example to illustrate.
The understanding of "number" begins with comparison, and on the basis of comparison, the concepts of more and less, equality and inequality are produced. Based on the commonness of "equality", an abstract natural number is formed, and the core idea of knowing more and less, equality and inequality is correspondence. Because preschool children have rich experience in recognizing numbers, textbooks generally present a big situation directly, requiring students to see the numbers of various objects separately, which has actually skipped the abstract link, but teachers had better be able to let students moderately feel the abstract process contained in it through some activities. For example, in a graphic background, students find that there are as many animals as there are. At this time, they can ask "How do you know they are the same", and students may mostly compare them in quantity, such as "all three". Then, students can be guided to explain from other angles. As shown in Figure 3, students can be guided to feel the corresponding relationship between giraffes and sika deer from the graphics, and then continue to guide students to find as many animals as giraffes from the background graphics, and connect giraffes with as many animals as them one by one with lines, so that the essence of feeling equality is to be able to correspond one by one. Finally, you can drag three pictures of other objects from the background graphic and superimpose them on the picture of sika deer, so that students can think about whether they are as many as giraffes. In this process, it is irrelevant for students to realize other characteristics of specific objects. Here, what we care about is whether they correspond to each other one by one, and what we care about is their number. On this basis, we draw a "3" to represent this number. In short, in the primary school stage, we should pay attention to guiding students to experience the process of gradually abstracting mathematical concepts or problems from concrete, intuitive and realistic backgrounds, so that students can form abstract preliminary experience and develop their initial abstract ability. However, it should be noted that primary school students are young and weak in abstract ability, so we should grasp the degree of abstraction in teaching, not to mention the abstract word "abstract".
Third, pay attention to reasoning ability in the operation study of "number"
Deducing another unknown judgment from one or several known judgments is called reasoning. Reasoning includes not only strict deductive reasoning, but also reasonable reasoning (such as analogical reasoning, inductive reasoning, statistical reasoning, etc. Deductive reasoning is mostly used to sort out mathematical knowledge, while perceptual reasoning is helpful to mathematical discovery. They often work together and cannot be neglected. Paulia, an American mathematics educator, pointed out in his masterpiece Mathematics and Guess that a student who seriously wants to take mathematics as his lifelong career must learn to reason, which is a special symbol of his major and his subject. However, in order to achieve real success, he must also learn reasonable reasoning: or this is the kind of reasoning on which his creative work depends. Ordinary students or students who are interested in mathematics should also experience reasoning. Although he may not have the opportunity to apply it directly, he should get a standard through which he can compare all kinds of so-called evidence in modern life. Many people think that geometry is a good carrier to develop students' reasoning ability. In fact, the study of number is also a good carrier to develop students' reasoning ability, especially in the study of operation, which can guide students to participate in the construction of operation rules and develop their reasoning ability in the process of understanding operation rules.
When teaching "addition of a decimal", teachers usually present a situation first to guide students to draw corresponding formulas from the situation. For example, the following questions are presented: How much did it cost to buy 1 bag Miaocuijiao 4.8 yuan and 1 bottle Screaming 2.8 yuan? It is not difficult for students to list the formula of 4.8+2.8. This is a new problem, but students have certain life experience, which becomes an important basis for them to solve the problem. According to life experience, students all know that it costs about 7 yuan, and this guessing process already includes reasoning, such as "Miao Cuijiao is close to 5 yuan, plus screaming 2.8 yuan, it must cost 7 yuan more". Of course, we need accurate values. Therefore, students can explain 7 yuan from six angles with the help of life experience. These explanations may be varied: 4.8 yuan +2.8 yuan, 4 yuan and 2 yuan together are 6 yuan, and two octagons together are 16, which is the six angles of 1 yuan; 4.8 yuan and 2.8 yuan both converted angles into angles of 48 and 28, 48 plus 28 into angles of 76, and converted elements into 7.6 yuan ... These explanations are good reasoning processes in themselves. On the basis of these explanations, students can be further guided to sum up their own experience, explore the vertical operation of one-digit decimal addition, and explain the reasons for decimal point alignment. Obviously, the process of arithmetic is a very important reasoning process.
To sum up, in primary school mathematics teaching, we must take problems as the carrier, let students experience the whole process of finding, asking, analyzing and solving problems, better show students' thinking process in communication and reflection activities, and thus better cultivate students' abstract ability, reasoning ability, application consciousness and application ability.