"Standard" points out: "In mathematics curriculum, we should pay attention to developing students' sense of numbers, symbol consciousness, spatial concept, geometric intuition, data analysis concept, calculation ability, reasoning ability and model thinking. In order to meet the needs of the development of the times for personnel training, mathematics courses should pay special attention to cultivating students' awareness of application and innovation. " From this 10 core concept, it is not difficult to see that the core concept refers not to the specific content itself, but to the basic ideas and thinking methods reflected by the content itself, and also to the feelings, concepts, consciousness and abilities that students should have in mathematics learning. The core concept reflects the core of a kind of curriculum content, which is the goal of students' mathematics learning and the key to mathematics teaching. Compared with the "experimental draft", 10 has four new core concepts, namely, computing power, model thinking, geometric intuition and innovative consciousness; There are three names or meanings that change greatly. They are the concepts of number sense, symbol consciousness and data analysis. The remaining three not only retain the original name, but also basically retain the original connotation. Mathematics textbooks provide learning themes, basic clues and knowledge structure for students' mathematics learning activities, and are important resources for realizing mathematics curriculum objectives and implementing mathematics teaching. It is one of the important tasks of mathematics teachers to actively explore the development of teaching materials and better integrate the core concepts.
In the goal, we can see some specific explanations of these core concepts, which are equivalent to some elements of the goal. But at the same time, we can also find that they are closely related, so the core concepts have the function of connecting the preceding with the following. It is very important to connect the above goal with the following content, so it is also called the core concept.
(a) Why design core concepts?
In the revision process of curriculum standards, besides the above ideas, how to design curriculum standards has also been discussed. In the process of proposing a design, two things are very important. One is to hope that these things in the course form a whole, and how to grasp the course as a whole needs to be emphasized repeatedly. The whole mathematics course is composed of knowledge and skills, process methods, emotions, attitudes and values. This is an infiltration of the whole standard development process. The second thing is that in the process of development, I hope to highlight the mathematics content that needs to be highly valued, because it reflects the most important and essential thing in mathematics, not only as a goal, but also organically combined with the content. I remember when we were discussing, on the basis of compulsory education in the past, could we use some words to express these things? After discussion, we put forward ten core concepts.
(B) the understanding of the core concepts
1, number sense
"Standard" deleted some contents related to operation in the description of number sense in the original "Experimental Draft", making it become another core concept independently: computing power.
The standard defines number sense as a feeling, which includes both perception and understanding, both perceptual thinking and rational thinking.
The standard summarizes this logarithmic perception into three aspects: number and quantity, quantitative relationship, and estimation of operation results.
Numbers and quantities are actually establishing the relationship between abstract numbers and real numbers.
This includes the perception of * * * between quantities in the abstract process from quantity to number; It also includes whether it is reasonable to mention a number in the actual background, which can be linked with the number in the actual background.
Sense of number is an attitude and consciousness to solve and use numbers actively, consciously or automatically in geography, that is, we can observe reality from a mathematical point of view, study reality with mathematical thinking and solve practical problems with mathematical methods. It makes people associate numbers with the real situation, and the world they see has quantitative significance.
The sense of number is mainly manifested in: understanding the meaning of number; Numbers can be expressed in many ways; Be able to grasp the relative size relationship of numbers in specific situations; Able to express and exchange information with numbers; Can choose the appropriate algorithm to solve the problem; Can estimate the result of operation and explain the rationality of the result.
To cultivate and develop students' sense of numbers, we should pay attention to the following two aspects: 1. Guide students to get in touch with specific and interesting things around them; Focus on solving practical problems.
2. Symbolic consciousness
First of all, the standard renamed "symbol sense" as "symbol consciousness", which put more emphasis on students' psychological tendency of actively understanding and using symbols.
Symbol consciousness mainly refers to the ability to understand and use symbols to express numbers, quantitative relations and changing laws. This article emphasizes the role of symbolic expression. Knowing that symbols can be used for operation and reasoning, the conclusion is general. This paper emphasizes the general characteristics of "symbol".
Because all operations related to numbers are cases, and mathematics needs to study general problems, and it needs to be represented, operated and reasoned by symbols. Therefore, on the one hand, symbols can be operated and reasoned like numbers, and the conclusions obtained through symbol operation and reasoning are universal. Establishing symbol consciousness is helpful for students to understand that the use of symbols is an important form of mathematical expression and mathematical thinking.
The sense of symbol is people's understanding of the meaning and function of symbols, as well as their consciousness and habit of actively using symbols. The sense of symbol is mainly manifested in: it can abstract the quantitative relationship and changing law from specific situations and express it with symbols; Understand the quantitative relationship and changing law represented by symbols; Will be converted between symbols; Can choose appropriate programs and methods to solve the problem of symbol representation.
Cultivating students' sense of symbols can be carried out from two aspects at the same time: 1. Teaching students some mathematical symbols in time in combination with mathematical content; Encourage students to use their own unique symbols creatively.
3. The concept of space is the basic element to cultivate students' initial innovative spirit and practical ability.
Except that the last article in the experimental draft is independent as another core concept "geometric intuition", the interpretation of "spatial concept" in the standard basically maintains the original statement.
The concept of space is manifested in the understanding and grasp of the shape, size, position, change and relationship of objects in the real world. The concept of space is mainly manifested in: you can imagine the geometric figure from the shape of the object, imagine the shape of the object from the geometric figure, and transform the geometric body into its three views and expanded drawings. Can make three-dimensional models or draw graphics according to conditions; Can separate basic graphics from more complex graphics, and can analyze basic elements and their relationships. Can describe the movement and change of physical objects or geometric figures; Can describe the positional relationship between objects in an appropriate way; Can use graphics to describe problems vividly and use intuition to think.
In order to develop students' concept of space, the following corresponding measures can be taken: 1. Increase the knowledge of translation, rotation and symmetry, relative position of objects, cognitive direction and road map, measurement of irregular figures, etc. 4. Weaken simple quadrature calculation, reduce the amount of calculation, control the number of calculations, and allow students to use calculation tools appropriately; 3. Change the traditional teaching methods.
4. Geometrically intuitive
Geometric intuition is a new core concept in the standard, which mainly refers to "describing and analyzing problems with graphics". With the help of geometric intuition, complex mathematical problems can be made concise and vivid, which is helpful to explore the solution ideas and predict the results. Geometric intuition can help students understand mathematics intuitively and play an important role in the whole process of mathematics learning. "
5. The concept of data analysis
The standard renamed "statistical concept" as "data analysis concept" and pointed out that the core of statistics is data analysis.
Furthermore, the concept of data analysis highlights the unique thinking method of statistics and probability: understanding the information contained in data; Select the appropriate method according to the background of the problem; Experience randomness through data analysis.
6, computing power
"Standard" points out: "Computing ability mainly refers to the ability to correctly perform operations according to laws and algorithms. Cultivating the ability of operation helps students understand the arithmetic of operation and seek reasonable and concise methods to solve problems. "
As mentioned earlier, computing power is a new core concept in the standard. The basic characteristics of computing power are correct, well-founded, reasonable and concise. Correctness is the basic requirement of operation; Evidence is the premise of correct operation; Rationality is the condition of operation; Conciseness is the quality description of operation. Operation is different from calculation. It is necessary to correctly understand relevant knowledge, identify and distinguish operating conditions, reasonably select operating methods, and effectively design operating steps, so that operators can obtain operating results economically and reasonably and finally as concisely as possible. It is the combination of "calculation" and "thinking", and the integration of operation and speculation. The cultivation of computing ability is a long-term task. Starting from the characteristics of the first and second mathematics courses, we need to go through a process of repeated training and circular rise from simple to complex, from concrete to abstract, from simple to comprehensive.
7, the cultivation of innovative consciousness
The cultivation of innovative consciousness is the basic task of modern mathematics education, which should be reflected in the process of mathematics teaching and learning. Students' finding and asking questions themselves is the basis of innovation; Independent thinking and learning to think are the core of innovation; It is an important method of innovation to get conjectures and laws through induction and verification. The cultivation of innovative consciousness should start from the compulsory education stage and run through the whole process of mathematics education.
To cultivate innovative consciousness in existing mathematics teaching, we should change the way of teaching and learning. Make the teaching of some mathematics content change from teachers' teaching to students' exploration. Encourage students to guess and verify; Experiments and discoveries; Questioning and exploring; Cooperation and communication. Teachers often discover and construct new knowledge under their guidance and organization, and their sense of innovation is properly cultivated. In the process of cultivating innovative consciousness, we should also pay attention to the evaluation of students. Its main purpose is to fully understand the process and results of students' mathematics learning, motivate students to learn and improve teachers' teaching.
8. Cultivate students' awareness of application.
Application consciousness is to comprehensively use existing knowledge and experience, and solve challenging and comprehensive problems closely related to life experience through independent exploration and cooperation and exchange. There are two meanings: on the one hand, consciously use mathematical concepts, principles and methods to explain phenomena in the real world and solve problems in the real world; On the other hand, it is recognized that there are a lot of problems related to quantity and graphics in real life, which are abstracted as mathematical problems and solved by mathematical methods. The application consciousness is mainly manifested in: recognizing that there is a lot of mathematical information in real life and that mathematics has a wide range of applications in the real world; In the face of practical problems, we can actively try to use the knowledge and methods we have learned from the perspective of mathematics to find strategies to solve problems; When faced with new mathematical knowledge, we can actively look for its actual background and explore its application value.
To cultivate students' application consciousness, we should pay attention to the following points: 1. Guide students to choose good topics; 4. Define the activity objectives; 3. Emphasize the requirements of autonomy and communication; 4. Summary and evaluation.
9. Model idea
The standard first explains the value of model thinking, that is, it establishes the connection between mathematics and the outside world. The process of establishing and solving the model includes: abstracting mathematical problems from real life or specific situations, and establishing equations, inequalities and functions. Use mathematical symbols to express the relationship and changing law of mathematical quantities in mathematical problems, find the results and discuss the significance of the results.
There are two typical modes in primary schools: "distance = speed × time" and "total price = unit price × quantity". With these models, we can establish equations to explain "stories" in the real world and help us solve problems.
10, focusing on the development of students' reasoning ability
Reasoning is a basic way of thinking in mathematics, and it is also a way of thinking that people often use in their study and life. Generally including rational reasoning and deductive reasoning. Reasonable reasoning is based on the existing knowledge and experience, in a certain situation and process, to deduce the conclusion of possibility. Inductive reasoning, analogical reasoning and statistical reasoning are the main forms of perceptual reasoning. Deductive reasoning is based on existing facts and certain rules, and is proved and calculated according to the laws of logical reasoning. Reasoning ability is mainly manifested in: being able to obtain mathematical guesses through observation, experiment, induction and analogy. , and further proof, proof or counterexample; Be able to express your thinking process clearly and methodically, and be reasonable and well-founded; In the process of communicating with others, I can discuss and ask questions logically in mathematical language. To cultivate pupils' reasoning ability, we should do the following two things: First, cultivate students' reasoning ability in daily mathematics teaching. Secondly, the cultivation of reasoning ability is implemented in the four content areas of the standard.