Shi Ningzhong: the highlight of the revision of the new curriculum standard of compulsory education mathematics and the expression of core literacy

? On April 2 1 day, 2022, the new curriculum standards for compulsory education were officially promulgated and implemented in the fall semester of 2022. Based on core literacy, various disciplines put forward new requirements for education and teaching.

? Among them, the new curriculum standard of mathematics advocates expanding the connotation of mathematical core literacy with mathematical vision, thinking and language. What are the highlights of the new mathematics curriculum standard? How do front-line teachers understand and express core literacy from the perspective of mathematics? Professor Shi Ningzhong, head of the research group of compulsory education mathematics curriculum standards of the Ministry of Education, made a comprehensive interpretation of the above problems.

? 2 1 Century Great changes have taken place in the basic education in China. The basic education reform that began in this century is essentially a question of how to formulate and implement curriculum standards. It has been nearly 20 years since the revision of curriculum standards. The revised mathematics curriculum standard for compulsory education presents three highlights as a whole, and insists on implementing two fundamental tasks.

? Expand the curriculum objectives. In the past, the traditional teaching goal was double basics. Later, it was found that the double basics were not enough, so it was extended to the four basics, adding basic ideas and basic activity experience. In the past, it was emphasized to analyze and solve problems, but around 20 1 1, China has proposed to cultivate innovative talents, so it is not enough to have the ability to analyze and solve problems, thus increasing the ability to find and ask problems.

? Enrich the course content. There is no concept of geometry in 200 1 curriculum standard, which is expressed by space and graphics. Later, we changed it to figure and geometry, and added some basic facts, which made it possible to prove geometry. There has always been a problem in mathematics education-the proof of mathematics is only geometric proof without algebraic proof, and the curriculum standard is only the basic facts of algebra but lacks the basic facts of geometry. Everyone thinks that the proof that does not proceed from the basic facts is not proof, so the newly promulgated curriculum standard adds two basic facts on this basis, one is equivalence and the other is the basic nature of the equation.

? Emphasize core literacy. The 20 1 1 version of the curriculum standard classifies the three abilities of mathematics, namely, computing ability, reasoning ability and spatial imagination, and the 10 core words that the country has always said, that is, the eight core words in mathematics. It can be seen that in addition to maintaining these, the core vocabulary related to symbols, consciousness, number sense and abstraction has been added, which is a very important change. Traditional mathematics education does not emphasize abstraction, but abstraction is extremely important for mathematics, because the conclusion of mathematics is seen, not proved, so all disciplines are extremely important for cultivating intuition of this discipline.

? What does the basic idea of mathematics mean? The first is that learning mathematics is generally willing to give a thinking principle. The basic idea of mathematics is the idea on which the emergence and development of mathematics must depend. The thinking characteristics that students who have studied mathematics should have are abstract reasoning and model. Bring real-world things into mathematics through abstraction, because several research results have been obtained through abstraction. The main abstract objects are quantity and quantity, and the relationship between graphs forms the relationship between the research objects, which is represented by definition symbols.

? The development of mathematics itself is obtained through reasoning, and the conclusion of mathematics is also obtained through reasoning. Reasoning is mainly divided into two categories, one is to draw conclusions through induction and analogy, and the other is to demonstrate conclusions through deduction. I wrote an article demonstrating that inductive analogy and deductive reasoning are logical.

? I define logic as the transitive form of reasoning and prove that induction, analogy and deduction are transitive. Therefore, mathematics is rigorous because both the process of drawing conclusions and the process of verifying results are logical. The third is the mode. Mathematics builds a bridge to the real world through language, so mathematics returns to the real world through model layout. Many disciplines also use mathematical language to express the nature, relationship and law of a certain field.

? This revision of curriculum standards has two fundamental tasks, which are also the tasks that all disciplines must complete.

? The first is Lide Shu Ren. The document of the Ministry of Education in 2004 is to accomplish the fundamental task of cultivating people with the help of the implementation of core literacy, requiring senior high school curriculum standards to implement core literacy and guiding the implementation of curriculum standards from beginning to end. Compulsory education has also put forward this requirement, so one of the tasks of our curriculum standard revision this time is how to implement core literacy.

? The second is to realize the integration of disciplines, which is the world trend. In the stage of compulsory education, we should not divide the subjects very carefully, but should organically combine science subjects, including natural science subjects, with mathematics. Mathematics needs to do five things:

(1) Division of learning cycle. It is difficult to divide the study period. The 20 1 1 version of the curriculum standard is divided into two learning periods, one is from grade one to grade three and the other is from grade four to grade six. In fact, it is more reasonable to set up study subjects in the first and second grades, study subjects in the third and fourth grades, and study subjects in the fifth and sixth grades.

? Compulsory education has a basic principle, that is, education should adapt to the laws of students' physical and mental development. Children in the first and second grades of primary school are not suitable for learning mathematics because they can't understand anything they say; According to the students in Grade Four and Grade Five, after careful investigation, it will be found that there is a great watershed in the thinking process of children in Grade Four and Grade Five, and it seems that children in Grade Five can understand some abstract things more or less. Therefore, it is more reasonable to divide the three study periods into two study periods, depending on the psychological quality of students.

(2) How to organically combine the four basics and four abilities with the core literacy advocated.

(3) How to adjust the structure and content of the course on the premise of strengthening geometric intuition and improving mathematical literacy.

(4) How to integrate interdisciplinary knowledge and traditional culture in integration and practice.

(5) The most important thing is how to reflect the consistency of mathematics. Maybe many teachers and students don't know what the current problem of consistency is. Let me explain.

? One is that the biggest problem in primary school mathematics is the lack of consistency in the understanding of logarithm, that is, the lack of a mathematical process. How to recognize integers? Number of passes; How to recognize fractions and decimals? Through decimeters and centimeters; Fraction emphasizes equal division, and no * * can connect these three numbers in series. Therefore, when studying scores, there is no unit that emphasizes scores. Such a problem will lie in the comparison size, such as 1/2: 1/3. What is the reason? The numerator is equal, so the denominator is big and the fraction is small. In fact, scores should not have such a truth, and should be compared with the same unit. Therefore, to compare the sizes of 1/2 and 1/3, they must have the same units, that is, 3/6 and 2/6, and 3/6 is greater than 2/6, so that 1/2 is greater than 65438+.

? Second, there is a lack of consistency in digital operations. For example, the division of fractions and decimals is the focus and difficulty of primary school mathematics, but each has its own reasons. Fractional division is based on the principle of inclusion and division. For example, if one contains three 1/3, then 1÷ 1/3=3. What about 4÷ 1/3? It becomes 4× 1/3 divided by this, and then 1÷ 1/3= 3, so 4÷ 1/3=4×3, which is actually very deep; Fractional division uses the method of constant quotient, that is, 0.4÷0.02, amplified by 100 times, and the correct result is obtained. Why is it right to keep the business the same? The examples given are all examples of real numbers and examples of integers. Why are integer examples good for decimals? This is a problem.

? The above two points make students think that mathematics can be divided into integer mathematics, fractional mathematics and decimal mathematics. There are many kinds of operations, such as integer operation, decimal operation and decimal operation. This not only confuses students' thinking, but also delays them for a long time. Friedenthal saw this problem long ago, and suggested that blindly relying on specific situations would make the problem of division more complicated. Because teachers and textbook writers lack understanding of how to progress from intuitive score to algorithmic score, the practical problem is even worse.

? Because the fundamental task of Lide Shu Ren has not changed, the core literacy should involve primary school, junior high school and senior high school. I think universities and graduate students are likely to participate in the future, and teachers and researchers may also be guided. To understand and express core literacy, I think we must have three basic characteristics for your reference.

? Keep the consistency of connotation. The core literacy of primary school, junior high school and senior high school is inseparable. Core literacy should be possessed by everyone who has studied mathematics from beginning to end, but it is the ultimate and can never be achieved.

? Each stage of the demonstration process. Every different learning stage should have different performances, which is related to physical and mental development, knowledge reserve and experience accumulation.

? The expression should be holistic. We should have the characteristics of both mathematics and mathematics education. More specifically, it should have the thinking characteristics of the subject, as well as the characteristics of psychology and cognition.

? Core literacy is now defined as "the ultimate goal of mathematics education, which is related to human behavior (thinking and doing things)"; It is gradually formed and developed by students in their own teaching activities; It is the accumulation of experience, the expansion of process objectives, and the inheritance and development of the four foundations. Furthermore, core literacy can be expressed as "observing the real world with mathematical eyes, thinking about the real world with mathematical thinking, and expressing the real world with mathematical language". This expression has both the characteristics of mathematics itself and the cognitive and psychological characteristics that education should describe.

? Now the core literacy contains more things than the mathematical ideas I once said. Let's take a concrete look.

? What is the vision of mathematics? Why should we observe? Because mathematics provides an observation method for people to know and explore the real world, with this observation method, students can intuitively understand the knowledge and background of mathematics, discover mathematics in daily life and other disciplines, study things in other fields by mathematical methods, and show the relationship between these research objects and him. With this vision, he can find meaningful mathematical problems and trigger mathematical inquiry. Emotion, attitude and values can develop curiosity, imagination and innovative consciousness. In this sense, it is superior to the abstraction of mathematics.

? What is mathematical thinking? Mathematics provides people with a way of thinking to understand and explain the real world. Through learning, students can understand the basic concepts of mathematics, the process of the occurrence and development of laws, the relationship between basic concepts of mathematics, the relationship between mathematics and the real world, some conclusions of mathematics, study the laws of the real world by mathematical methods, the process of mathematical rediscovery, the cultivation of critical thinking, and the cultivation of scientific attitude and rational spirit of seeking truth from facts.

? What is the language of mathematics? How is mathematics expressed? Mathematics provides people with a way to describe and communicate with the real world, which is superior to models. When a student has mastered the language of mathematics, he has initially mastered the way of communication between mathematics and the real world, can consciously express the essence, relationship and law in real life and other disciplines with mathematics, and feel the significance and value of data. Only in this process can students understand the simplicity and beauty of mathematical language, including expressing concepts, including equations and inequalities, developing the expression and communication of mathematical language, and increasing interdisciplinary application awareness and practice.

? What are the mathematical characteristics of core literacy? Although mathematical vision is a way for people to observe the world, it is essentially an abstraction of mathematics, and its mathematical characteristics are general because of the abstraction of mathematics. The essence of mathematical thinking is logical reasoning, that is, transitive reasoning, which cultivates logical thinking ability and transitive thinking. For example, from small to large, it is inductive analogy; Since childhood is deduction, this reasoning is logical, so mathematics is rigorous. If the subject of modern society wants to go to science, it is necessary to use mathematical language and establish mathematical model as much as possible, which makes mathematics form a new feature, that is, the universality of mathematical application. This new curriculum standard organically combines core literacy, mathematical thinking and mathematical characteristics.

? What are the phased characteristics of core literacy? The low school stage is based on senses, more concrete and more conscious, which is based on sensibility; High school is based on concepts, which are more general and more focused on concepts. Concept is based on concept, with the help of thinking and ability cognition. Therefore, we keep 10 core words in version 201and 6 expressions in version 20 17. The concrete manifestations of core literacy are abstraction, imagination, logic, reasoning and calculation, which correspond to the mathematical vision, thinking and language of senior high school. Junior high school is more about concrete abstract ability, spatial concept and geometric intuition; Elementary school hardly talks about abstraction, but about symbol consciousness and number sense. This time, a word-sense of quantity was added.

? However, teachers must know that these words are concrete expressions of abstraction. Only in junior high school can we talk more about abstraction. As long as primary school has the meaning of reasoning, junior high school should emphasize the ability of reasoning, and senior high school should clearly state the form of reasoning. One of the abilities required by primary schools is operation, and it is enough for primary schools to have model awareness and data awareness. Junior high school should form ideas and concepts, and senior high school requires ability.

? Implement core literacy and change the expression of curriculum standards. Mathematics originates from the abstraction of the real world, and its main research objects are quantity and its relationship, as well as the relationship between figures; On the basis of abstract structure, mathematical symbolic operation, formal reasoning, model construction and other methods to form mathematical conclusions are obtained. I mainly hope to help people understand and express the real world and express the essential relationships and laws. Therefore, mathematics is not only a tool for calculation and reasoning, but also a language for expression and communication. Language carries ideas and culture. If mathematics is also a language, then mathematics can also carry ideas and culture, so mathematics is an important part of human civilization.

? This way of understanding mathematics began in Galileo and Newton's time. Galileo once said: "The universe is described in mathematical language. If we don't understand mathematical language, then we can only grope in the dark "; Einstein also commented: "Because Galileo saw this, especially because he tirelessly taught this to the scientific community, he became the father of modern physics and, in fact, the father of modern science." So the development of modern science is because the laws of the real world are expressed by mathematical principles.

? 1. Emphasize abstract structure

? We emphasize abstraction and emphasize abstract structure on the basis of abstraction. Abstract structure is a very basic thing in the development of modern mathematics. We should not only know what the research object is, but also pay attention to its nature. This concept was first put forward by Aristotle. He said: "Mathematicians study things in an abstract way, removing perceptual things ... lines, angles or other quantities, not as existence but as relations", that is, the existence of this thing is not important in itself, but the relationship between them.

? In the same way, Hilbert described it vividly: "Euclid's definitions of points, lines and surfaces are not important in mathematics, they become the center of discussion only because axioms tell the relationship between them." In other words, whether it is a point, a line, a surface, a table, a chair or a beer bottle, the final conclusion is the same. " There is something wrong with the concept-only teaching method. While teaching concepts, we should teach their properties, relationships and laws, or one of them, so concepts need to spiral up.

? For example, set is the basic language of modern mathematics. On this basis, the relationship between numbers is constructed by talking about order. The essence of number is the relationship between size, number is the abstraction of quantity, and the essence of quantity is more and less, so the essence of number is abstracted as big and small, so the real number space is the basic space of constructor.

? Measurement is very important in universities. Defining distance space and defining measure, such as probability theory, are mainly based on two strategies, one is technical measure, the other is Lebesgue degree, combined with operation.

? Operation is also very important for basic education. After understanding the research object, we must understand the operation of the research object. Especially the number domain is directly related to the research object and operation. Ensure the operation, mainly to ensure the closure of the reverse operation. In this case, subtraction can expand natural numbers into integers, and division can expand integers into rational numbers.

? Based on this idea, we merged the six themes of number and algebra into two themes, and the field has not changed, but the theme has become bigger. For example, the four themes of graphics and geometry become two themes, and the understanding and measurement of graphics are put together, and the understanding and operation of numbers are put together.

? 2. Promote the integrity of teaching and research.

? In this curriculum standard, we advocate preparing lessons as a whole, including preparing lessons for the whole grade, preparing lessons for the whole time and preparing lessons for the whole school teachers. Corresponding to the integrity, consistency and stage of core literacy, it should be reflected in the integrity, consistency and stage of daily teaching.

? Sacredness in daily teaching refers to the overall grasp of the knowledge system and the corresponding core literacy; Consistency means that the concept should be consistent from the initial proposal to the final practical application; The stage is that we need to know how the mathematics knowledge we study is advanced and how the core literacy is advanced.

? 3. Increase interdisciplinary content

? In addition to dividing the study period into three study periods, the content of synthesis and practice has also increased, mainly interdisciplinary content, emphasizing traditional culture. For example, in the story of elephants and pi in Cao Chong, it is very important to talk about mathematics. The traditional culture in mathematics is different from that in other disciplines. For example, Cao Chong can speak elephant language, but elephants in Cao Chong should not only know the unit of weight, but also talk about the truth that the equivalent is equal to the equivalent and the total is equal to the sum of the components. The mathematical truth contained in traditional culture is worthy of teachers' serious discussion.

? 4. Emphasize algebraic reasoning and geometric intuition.

? This curriculum standard emphasizes algebraic reasoning and geometric intuition. Algebraic reasoning is to draw conclusions through induction and analogy. Can this idea be sorted out a little in primary school? I hope the textbook editor will seriously think about it. For example, we talked about two digits multiplied by one digit, two digits multiplied by two digits, or three digits multiplied by one digit, so whether the calculation method of three digits multiplied by two digits can be drawn by students themselves, so that students can draw inferences about the algorithm.

? In order to make such an induction, we should pay attention to the fact that in the past, it was impossible to write vertical textbooks without writing horizontally. Horizontal is arithmetic, vertical is algorithm. Without arithmetic, there is no algorithm. For example, multiplication and vertical calculation need to decompose numbers and use the distribution law. It can be concluded that the law of arithmetic determines arithmetic, and arithmetic determines algorithm. This idea is very important.

? Geometry should be drawn with a ruler and ruler, and the existence of abstract objects should be understood. What is the essence of geometric abstraction? I think the essence of geometric abstraction is to express three-dimensional objects with two-dimensional graphics, that is, the essence of geometry is the relationship between two dimensions and three dimensions. Therefore, primary schools should draw rulers and rulers to let students feel the existence of abstract things, such as the existence of circles; For example, line segments with the same length can be measured with compasses in addition to the scale; Or to give students a line segment that can be made into an equilateral triangle, it is very important to know what the perimeter is.

? 5. About Mathematicization

? First of all, about mathematicization, the curriculum standard is described as follows: we should initially understand the conceptual consistency and operational consistency of numbers. That is, how to achieve consistency is how to achieve mathematicization. This time, the concept of counting unit was introduced in particular. Counting unit is a special unit of measurement, which is the unit of measurement of number and order, and is regarded as the starting point of mathematical consistency. The understanding of numbers, whether fractions, decimals or integers, is the expression of counting units. For example, 4/3 of 1/3 and 1/3 are counting units, which solves the problem of false scores, otherwise the false scores will never be clear. The same is true for operations, and the comparison size should be carried out under the same counting unit, so in the addition and subtraction operation of fractions, the general fractions always get the same counting unit.

? 6. Reorganization of course content

? The new curriculum standard moves the content of the equation to junior high school. There are two main reasons. The first is that the essence of the equation is not involved. In the past, primary school mathematics regarded the equation as only one letter indicating the unknown, but in fact, indicating the unknown is not the essence of the problem. In this equation, letters only represent coefficients, not unknowns, so it is very important to express properties, relationships and laws with letters. In the past, the notation of letters only taught half a class, and the People's Education Edition taught less than one class at most. Now I have 6-8 classes. In terms of how to represent numbers by letters, I feel that letters are abstract.

? The second is that there is no need for perceptual equations. In the past, it was very inappropriate to understand simple equations and study them with equations like 5-x=2. There is a basic principle now, which may be followed by teachers when compiling textbooks and even giving lectures in the future, that is, the introduction of all new concepts and methods should make students aware of its necessity. It is not that I teach you to learn, but that I teach you this thing is very useful, so the teacher should guide the students to have interest in learning. So we can only introduce the equation when it is very difficult to solve the four operations, and it is very convenient to introduce the equation.

? The traditional definition of an equation containing unknowns is called an equation, which is untenable because the mathematical definition must be sufficient and necessary. For example, 2-x=x is an unknown equation, but it is not an equation, it is only the result of calculation and transmission. The equation must tell two or more stories, and the two stories are equal in quantity, so the equal sign in the equation has two functions, one is to indicate transitivity, and the other is to indicate equal in quantity.

? Considering the demand of big data, the curriculum standard moves the percentage to the statistics and probability part. Because percentage is becoming more and more important in the processing of big data, such as how the deterministic percentage of juice industry transitions to randomness, such as shooting percentage. The introduction of percentage can be used to make decisions on random phenomena, and decisions on random phenomena appear more in real life than those on certainty. Therefore, in primary school, children have a better understanding of how to make decisions about random phenomena.

? For example, the percentage is used to set the skipping standard for the fourth grade children, so that the children can write it down after jumping, then queue up from small to large, and the number of people passing is the top 25% or 50%, and then the skipping standard is determined according to the percentage, which is also the basis for the state to formulate the blue sky plan. Therefore, percentage is introduced into statistics and probability to better meet the needs of big data.