1. acquire important mathematical knowledge (including mathematical facts and experience in mathematical activities), basic mathematical thinking methods and necessary application skills necessary to adapt to future social life and further development.
In the elaboration of this goal, the understanding of mathematical knowledge has changed-mathematical knowledge includes not only "objective knowledge", that is, mathematical facts (such as formulas and rules) that are not transferred by regions and learners, but also "subjective knowledge" that belongs to students themselves, that is, personal knowledge and mathematical activity experience with distinctive individual cognitive characteristics. Such as the understanding of the function of "number", the basic idea of decomposing graphics, and the customary method of solving a mathematical problem. They are only subordinate to a specific learner and reflect his understanding of the corresponding mathematical objects at a certain learning stage. They are empirical and not so strict and wrong. According to the standard, students' experience in mathematics activities reflects their true understanding of mathematics, which is formed in the process of students' own mathematics activities and develops with students' mathematics learning, so it should be an integral part of students' mathematics knowledge.
2. Initially learn to use mathematical thinking to observe and analyze the real society, solve problems in daily life and other disciplines, and enhance the awareness of applied mathematics.
This goal reflects that "Standards" position mathematics learning as a compulsory education stage to promote students' all-round development. In short, it is to train students to "understand their environment and society from a mathematical perspective" and learn "mathematical thinking", that is, to use mathematical knowledge and methods to analyze things and think about problems. Therefore, the "subject-oriented" mathematics curriculum structure with the basic goal of "imparting systematic mathematics knowledge" will give way to the "student-oriented" mathematics curriculum structure with the basic goal of "promoting students' development". In other words, the new mathematics curriculum will no longer focus on whether to provide students with systematic mathematics knowledge, but more on whether to provide students with mathematics with realistic background, including mathematics in life, mathematics that they are interested in and mathematics that is conducive to their learning and growth. The important result of students' mathematics learning is no longer just how many "standardized" mathematical problems they can solve, but whether they can "see" mathematics from the realistic background and apply mathematics to think and solve problems.
3. Experience the close relationship between mathematics and nature and human society, understand the value of mathematics, and enhance the understanding of mathematics and confidence in learning mathematics well.
This goal shows that a good mathematics course should make students realize that mathematics is a civilization of human society, which plays a great role in the development of human beings yesterday, today and tomorrow. The mathematics we study is not only in the classroom and examination room, but also around us. For example, what does "the probability of precipitation tomorrow is 75%" mean? Drop a drop of ink in the middle of a piece of paper, fold it in half along the middle of the paper, flatten it, and open it. What are the characteristics of the ink patterns on both sides of the crease? These things that we often encounter in our lives are related to mathematics.
Mathematics, as an educational content, should not be simply regarded as abstract symbolic operation, graphic decomposition and proof, but reflect various relationships, forms and laws existing in real situations. For example, function should not be regarded as a formal symbolic expression, and the study and research on it should not only discuss the characteristics and properties of abstract expressions, such as definition domain, expression form, range, monotonicity and symmetry. It should be regarded as a mathematical model to describe the changing relationship between variables in real situations. When discussing a specific function, we should also pay attention to its background, the mathematical law described, and the practical significance that this mathematical law may bring under specific circumstances. Learning mathematics well, in particular, is not the patent of a few people, but the right of every student. In the whole compulsory education curriculum structure, mathematics should not be used as a "sieve"-eliminating "unintelligent" students and leaving "intelligent" students. Mathematics course is designed for every student. Every student with normal physical and mental development can learn mathematics well, achieve the goal set by the standard, and enhance their confidence in learning mathematics well.
4. Have a preliminary spirit of innovation and practical ability, and can be fully developed in terms of emotional attitude and general ability.
This goal shows that starting from the realistic situation, learning mathematics and acquiring knowledge through a process full of exploration, thinking and cooperation will gain self-confidence, sense of responsibility, realistic attitude, scientific spirit, innovative consciousness and practical ability, which are far more important than entering a higher school. As we all know, the realization of quality education does not mean the need to set up "quality education courses", and quality education is not the patent of art, sports or social activities. In fact, under today's education system, the main channel for implementing quality education is subject education, and mathematics classroom is such a channel.
It can be seen that compared with the previous mathematics curriculum objectives, the curriculum objectives set by the standards are richer in connotation, more reasonable in structure and more closely related to national rejuvenation and development.
Second, understand the target areas of each course and their relationships.
In order to further understand the overall goal, we need to understand the connotation of each specific goal and the relationship between them.
The overall goal of mathematics curriculum is refined into four aspects: knowledge and skills, mathematical thinking, problem solving, emotion and attitude, which is the concrete embodiment of the three-dimensional goal of "knowledge and skills, process and method, emotional attitude and values" in the Outline in mathematics curriculum. The standard elaborates the connotation of each target area and their relationship.
The goal of mathematics course is not only to enable students to acquire necessary mathematical knowledge and skills, but also to develop inspiring thinking, problem solving, emotion and attitude. This result stems from the new mathematics curriculum idea possessed by the Standard-the fundamental purpose of setting up mathematics curriculum is not only to let students master the basic knowledge, skills and methods of mathematics, but also to make students willing to be close to mathematics, understand mathematics, use mathematics, learn to "know their living environment and society from a mathematical perspective", learn to "do mathematics" and "think mathematics", and develop their rational spirit, innovative consciousness and practical ability. Therefore, the standard clearly lists "mathematical thinking, problem solving, emotion and attitude" as the curriculum target area, and gives them a more specific explanation. This is a feature of the Standard-in the past, these goals were only regarded as a "by-product" in the process of students' learning mathematical knowledge and skills, that is, the main task of students' mathematics learning is to master mathematical knowledge and skills, and the cultivation of their abilities, especially the development of their emotions and attitudes, can only be carried out "by the way" in the process of knowledge learning. Once there is a conflict between "knowledge learning" and "the development of their emotions and attitudes", the latter will naturally abdicate. The "Standard" clearly regards the four goals as the overall goals of the mathematics curriculum in the compulsory education stage, which effectively restricts the occurrence of "abdication" and ensures the balanced and sustainable development of students.
1. About knowledge and skills
The standard still thinks that basic knowledge and skills are the focus of students' mathematics learning, but what needs to be reflected is, in today's society, what basic knowledge and skills should students spend time and energy to master firmly? In the past, it was thought that the formalization, standardized expression and application of concepts and theorems (rules), fast and accurate skills in complex numerical calculation and algebraic operation, and various types and routines of problem-solving skills were such knowledge and skills. The standard holds that with the progress of society, especially the rapid development of science and technology and mathematics, the understanding of basic knowledge and skills should keep pace with the times, and some "basic knowledge" and "basic skills" that were valued many years ago are no longer the focus of students' mathematics learning today or in the future. For example, some complicated calculation skills and proof skills far beyond the students' understanding level and ability are artificially fabricated and only related to the exam? Quot question type "and so on. On the contrary, some previously neglected knowledge, skills or mathematical thinking methods should become "basic knowledge" and "basic skills" that students must master. For example, the ability to choose an appropriate algorithm in combination with the actual background, the ability to use a calculator to process data, the ability to read data, the ability to process data and make inferences based on the obtained results, and the awareness of grasping and applying the changing law between variables in the process of change are all basic mathematical literacy that a citizen should have, and are the basic knowledge and skills that must be mastered.
It is worth noting that the process goal of knowledge and skills appeared for the first time, and experienced the process of abstracting some practical problems into numerical and algebraic problems, exploring the shape, size, positional relationship and transformation of objects and graphics, asking questions, collecting and processing data, making decisions and forecasting.
Our previous teaching practice of "Knowledge and Skills" has roughly gone through two stages:
The first stage: as long as the result, not the process. That is to shorten the formation process of knowledge, and make students familiar with relevant knowledge and skills quickly through a lot of imitation, memory and practice. For example, for the study of solving equations, by understanding various methods of solving equations and solving a large number of various types of equations, you can be familiar with the procedures of solving equations and finally be proficient in solving "various" equations.
The second stage: begin to pay attention to learning knowledge in the process of knowledge formation (application process). At this time, the orientation of "process" is mainly to serve the learning of knowledge, that is, grasping "process" is conducive to understanding and mastering the corresponding knowledge. For example, to learn to solve equations, we should start with understanding the meaning of equation solutions, explore the ideas and methods of obtaining solutions, and finally form the basic strategies of solving equations. This is undoubtedly correct. The question is, how can this process be realized? For example, can the "exploration process of solving equations" mentioned above be realized through the direct teaching of teachers? It saves time and effort, but mathematics learning has changed from "listening to the results" to "listening to the process", and the "process" here has lost the meaning of inquiry.
The standard gives a deeper meaning to "process" and defines its orientation: the process itself is a curriculum goal, that is, students should first "experience the process" in mathematics learning activities. The process must be associated with some specific knowledge, skills or methods, but the process is not only for these results. If so, isn't it more labor-saving for teachers to "talk" about the process? The experience process will bring students the experience of exploration, the attempt of innovation, the opportunity of practice and the ability of discovery, which are more important than those specific results.
The goal of "knowledge and skills" has different requirements for students in different classes. For example, for the students in the third class, the focus of knowledge learning of "Number and Algebra" is to understand the origin of related concepts, understand the reasoning of corresponding operations, be able to operate skillfully, and be able to engage in activities to explore quantitative relations and changing laws. And can master related mathematical models (algebraic expressions, equations, functions, etc. ): The key point of learning the knowledge of "Space and Graphics" is to learn how to study and express the related properties and basic relations of geometry (graphics) by different methods (operation, transformation, drawing, demonstration, etc.). ), and master the method of expressing the position relationship of objects in the plane rectangular coordinate system; The key point of the knowledge learning of Statistics and Probability is to go through the data processing process completely, collect, sort out and analyze the data one by one, infer according to the analysis results, and learn how to calculate the probability of some events.
2. About "Mathematical Thinking"
The connotation of this goal does not simply point to the pure mathematical activity itself, but should directly point to the development of students' general thinking level related to mathematics. In fact, mathematics education in compulsory education stage is a kind of civic education, which brings students more than just solving more mathematics problems. Students will encounter different challenges in the future-some people need to learn or study more mathematics, and it is very important for them to "think about mathematics"; Others (the vast majority of educated students) basically don't need to solve pure math problems after employment (except taking math exams). For them, "thinking about mathematics" is a need, but it is more likely to be able to "think about mathematics", that is, when facing various problem situations (especially non-mathematical problems), they can think about problems from a mathematical point of view, find out mathematical phenomena and use mathematics. For all future citizens, abstract thinking and image thinking ability, statistical concepts, rational reasoning consciousness and deductive reasoning are indispensable. They should be an important goal for students to learn mathematics.
As the two aspects of this goal-thinking about mathematics and thinking about mathematics, their meanings are very different from "knowledge and skills": on the one hand, it is realized in the process of learning mathematical knowledge and solving mathematical problems (we don't need and can't set up a special course of "mathematical thinking"), on the other hand, it is not realized by whether we know a certain concept or theorem or whether we will use a certain formula. Moreover, the realization of this goal can not only be carried out by studying "pure" mathematical phenomena, but should be gradually completed in the process of studying various phenomena and problems (mathematical and non-mathematical). Specifically, the significance of these goals and their realization should pay attention to the following issues.
(1) experienced the process of describing the real world with mathematical symbols and graphics, established a preliminary feeling of numbers and symbols, and developed abstract thinking.
The significance of this goal mainly lies in being able to describe the real world in mathematical language and discover the universal laws hidden behind specific things. Compared with students in different classes, the focus of this goal is different. For example, in the third class, students should be able to describe specific problems with various mathematical relationships (equations, inequalities, functions, etc.) in addition to completing the previous tasks at a more complex level. ) and establish an appropriate mathematical model.
(2) Enrich the understanding of real space and graphics, establish a preliminary concept of space, and develop thinking in images.
The main significance of this goal is to let students establish a preliminary concept of space and think with the help of graphics, which is also the primary goal for students to learn Space and Graphics. It is also worth noting that compared with students in different classes, the focus of this goal is different. For example, for the students in the third class, it is more important to be able to construct geometric space by various methods (including operation, graphic transformation, pattern design, etc.). ) and try to use graphics for reasoning activities.
(3) Experience the process of describing information and inferring with data, and develop statistical concepts.
The standard clearly points out that the consciousness and methods of statistics should be necessary for every future citizen, which is exactly what this goal is concerned about. Moreover, the elaboration of the goal also clearly shows that the realization of the goal is achieved by students in a series of activities. Specifically, for the students in the third period, they need to be able to collect and process some useful information in real situations as needed, and make reasonable inferences according to the results of information processing. At this time, students need to go through a relatively complete process of statistical activities: formulating indicators for collecting data, collecting and expressing data, mathematically processing the data, and making statistical inference according to the processing results.
(4) By observing, experimenting, guessing anger, proving and other mathematical activities, we have developed a reasonable reasoning knife and a preliminary deductive knife, which can explain our views in an orderly and clear way.
As a rational citizen with systematic education, an important sign is that he can make reasonable judgments and choices through reasoning and express his views clearly in the process of communicating with others. As far as the development of deductive reasoning ability is concerned, it is gradually carried out with the development of students' logical thinking level, so the realization of the goal has obvious stages. For example, for the students in the third period, we should try to test the credibility of a guess in different ways, form a reasonable guess through different types of reasoning activities, and be able to express our deductive reasoning process in a more standardized form.
3. About solving problems
Our students "solve problems" almost every day and solve many problems. However, the "problem solving" concerned by the standard is not the same as these problem solving activities.
First of all, from the content point of view, the "problems" mentioned in the standard are not limited to pure mathematical problems, especially those that can only be solved through non-thinking activities, such as "identifying questions, recalling solutions, and imitating examples". The problems mentioned here can be both pure mathematical problems and various problems presented in the form of non-mathematical problems. But no matter what kind of problem, its core needs students to solve through activities with rich thinking elements such as "observation, thinking, guessing, communication and reasoning".
Secondly, in terms of specific connotation, the requirements of the standard are various, including the initial ability to ask and understand problems from a mathematical point of view, and the ability to comprehensively use the knowledge and skills learned to solve problems.
(1) Initially learn to ask and understand questions from the perspective of mathematics.
It first requires students to "ask questions from a mathematical point of view" when facing different phenomena (including mathematical and non-mathematical). In other words, they initially have a mathematical vision, and can identify mathematical phenomena or daily, non-mathematical phenomena and problems that exist in mathematical problems or mathematical relationships, and put them forward. Then, they can apply knowledge and skills to solve problems. In fact, in the past, students thought about the solution to a problem, that is, asking questions is the responsibility of textbooks or teachers, and the task of solving problems is students. At this point, the standard is a precedent. Therefore, our textbooks should provide students with opportunities to observe, think and guess, and our teaching should ask students more "What did you find?" Such a question. For example, for the students in the third period, they can find mathematical relations or problems from mathematical phenomena, other subjects or problems in life, and secondly, they can comprehensively use relevant mathematical knowledge and methods to solve some problems, which is the primary connotation of the goal.
(2) Form some basic problem-solving strategies, experience the diversity of problem-solving strategies, and develop practical skills and innovative spirit.
For the development of students, the value of problem-solving activities is not only to get concrete conclusions, or the main value is not here. Its significance is more to make students realize that there are different strategies in solving problems. Everyone should have their own understanding of the problem and form their own basic strategies to solve the problem on this basis. In this sense of encouraging individuality, it is possible to cultivate innovative spirit. In order to achieve this goal, the teaching materials in each period should provide students with opportunities for thinking and communication, and all teaching activities should also allow students to express their understanding of the problem and adopt appropriate problem-solving strategies. Specifically, different classes have different requirements for students. For example, in the third class, students can try to evaluate the differences between different methods, and understand that the formation of different methods mainly comes from different perspectives on the problem.
In addition, it is also an important goal to cultivate practical ability and innovative spirit. Personal innovation is based on independent thinking, and one of the basic elements of innovative spirit is the non-imitatability and uniqueness of thinking activities; Practical ability is not "heard" or "seen", but gradually formed in the process of independent activities. If students have enough time and space to think, a relaxed atmosphere to freely express their ideas to solve problems, and opportunities to communicate with their peers in the process of mathematics learning … then they are engaged in an "enlightenment" activity, which is helpful to cultivate their innovative spirit; On the contrary, if students' mathematics learning process is full of mechanical learning activities such as "imitation, memory, recognition and practice", then they are engaged in a kind of "closed door" activity, which will gradually dilute the innovative consciousness contained in every student's nature. Therefore, it is worth advocating that students seek their own understanding of knowledge and methods. In the process of solving problems, all students can have a successful experience, and they all face challenges to varying degrees. There is no ready-made formula or question type to solve problems, so students should be given enough time and space to think and opportunities to communicate with their peers. However, the teaching strategy of "question type+sea of questions" must be strongly controlled.
(3) Learn to cooperate with others and communicate the process and results of thinking with others.
Communicating with people is a basic skill that every citizen must master in the future. We cannot unilaterally think that asking others is a kind of "laziness" in thinking. Specifically, it is to encourage students to communicate with others on the basis of independent thinking-to communicate their own understanding of the problem, ideas and methods to solve the problem, and the results obtained. Only in this way can we develop the ability of "thinking and communication" in the process of problem-solving activities. At this point, students in different classes have different requirements. For example, the third class can try to benefit from communication with others and learn to respect others' opinions on the basis of the first two classes.
(4) initially form the consciousness of evaluation and reflection.
We believe that it is impossible for people to make essential progress without reflection. For students, the reflection mentioned here is a relatively preliminary requirement, and its purpose is only to let students understand the significance of reflection, experience the activities of reflection, and initially appreciate the benefits brought by reflection. These goals should be developed in the process of students solving problems. Therefore, we should pay attention to this goal consciously in the actual teaching process. For example, the third issue can focus on the reflection and collation of experience. To this end, we can ask more questions in the teaching process: under what conditions can this (successful) method be effective? In other cases, how to modify this method to make it still effective? What is the main reason why this problem has not been solved?
4. About emotions and attitudes
This goal is related to the understanding of quality education in mathematics classroom. According to the standard, mathematics classroom is a quality education classroom. Many basic qualities of qualified citizens, such as curiosity about natural and social phenomena, thirst for knowledge, attitude of seeking truth from facts, rational spirit, ability of independent thinking and cooperative communication, self-confidence and willpower to overcome difficulties, innovative spirit and practical ability, can be cultivated through mathematics teaching activities.
(1) can actively participate in mathematics learning activities, and have curiosity and thirst for knowledge about mathematics.
Children's curiosity and thirst for knowledge about natural and social phenomena is an important quality, which can make a person continue to learn and develop, and may also make a person enter the hall of science; On the other hand, it will make a person not strive for progress and accomplish nothing for life. Although mathematics education in compulsory education stage does not take training mathematicians as its own responsibility, nor does it expect all students to love mathematics and contribute a lot of time and energy to learning mathematics, it should make students have a more comprehensive and objective understanding of mathematics, be willing to get close to mathematics, understand mathematics and talk about mathematics, and maintain a certain curiosity about mathematical phenomena. All these are actually a way to cultivate students' curiosity about natural and social phenomena. Similarly, the realization of this goal is also hierarchical. For example, in the third period, students' understanding and application of mathematics are cultivated by enumerating examples of solving real-life problems with mathematics and some wonderful mathematical problems.
(2) gain successful experience in mathematics learning activities, exercise the will to overcome difficulties and build self-confidence.
In the previous mathematics teaching practice, we emphasized that "failure is the mother of success" and the hardship of mathematics learning, and thought that only by creating difficulties and obstacles for students in the process of mathematics learning can we cultivate their self-confidence and willpower to overcome difficulties. Theory and practice show that this is a one-sided understanding of students in compulsory education. The feedback formed by many students in this learning process is that mathematics learning is "failure, failure, failure again, until complete failure" for me. Therefore, I have lost confidence in mathematics learning and even other courses, not to mention the willpower to overcome the difficulties encountered in the learning process. The standard emphasizes that in cultivating students' self-confidence and willpower to overcome difficulties, two points should be paid attention to: (1) Provide students with challenging questions and give them the opportunity to experience activities to overcome difficulties; (2) Let them gain a successful experience in the process of engaging in these activities, or solve related problems, or find effective ways to solve problems, or solve some problems, or have a further understanding of problems ... Therefore, textbooks (or teachers' teaching) should provide a "ladder" type of question string as far as possible when introducing new mathematics knowledge and designing the situation of applying what they have learned to solve problems, so that every student can, for example, the third period. Even if the problem can't be completely solved, as long as we get an effective solution or have a further understanding of the problem, it will help students to establish self-confidence in learning mathematics well.
(3) Understand the close relationship between mathematics and human life and its role in the development of human history. Experiencing mathematics activities is full of exploration and creation, feeling the rigor of mathematics and the certainty of mathematical conclusions.
In the history of human development, there are many examples that show the great impetus of mathematics. Knowing this will help students have a more comprehensive understanding of the value of mathematics, and sometimes it will stimulate students' desire to learn mathematics. Therefore, teaching materials and teachers should introduce relevant mathematical historical facts to students in a timely manner, such as the deeds of famous mathematicians, classic cases, mathematical masterpieces and so on. The specific content design should consider the age characteristics and knowledge background of students, and choose the forms of introduction of mathematical characters, introduction of mathematical stories, introduction of mathematical applications, and solution of mathematical problems. For example, in the third period, we should introduce the important role of mathematics in the process of human development and the field of contemporary science and technology, so that students can understand the necessity of proof in mathematical activities, learn to prove it, and rationally understand the correctness of relevant mathematical conclusions.
(4) Form the attitude of seeking truth from facts and the habit of questioning and thinking independently.
Basic thinking ability, scientific attitude and rational spirit are the most basic and important qualities for the survival and development of future citizens. Mathematics education undoubtedly bears an important responsibility for the development of these qualities of students. However, this does not mean that we should set aside specific class hours to teach them, or mention them from time to time: this is thinking ability, this is scientific attitude, this is rational spirit ... In fact, as long as we have such an idea in our minds, we can create many opportunities in mathematics teaching to promote the realization of this goal. For example, when students learn a new mathematical knowledge, they are encouraged to adopt an exploratory method and gain an understanding of the new knowledge through their own efforts or cooperation with their peers rather than "telling"; When students are faced with difficulties, guide them to find solutions to problems and sum up the experience gained in the process of solving problems, rather than directly giving solutions to problems; What do students get from themselves or their peers? Quot Mathematical Conjecture requires and helps them to find evidence for their conjectures and correct their conjectures according to the actual situation, instead of directly affirming or denying their conjectures; When students have doubts about the opinions and methods of others (including textbooks and teachers), they are encouraged to find evidence for their doubts and engage in research activities aimed at denying or correcting other people's conclusions. Even if students' doubts are denied, we must first fully affirm their awareness of respecting facts and daring to challenge "authority". For the students in the third period, our main task is to make them dare and be good at expressing their views, understanding the significance of others' views and communicating with others.
5. The Standard clearly explains the relationship between the four aspects of curriculum objectives.
(1) "The above four goals are a closely linked organic whole, which plays a very important role in human development." In other words, mathematics teaching activities in the classroom, as the main way to achieve the curriculum objectives, should take these four aspects of the curriculum objectives as our "teaching objectives" at the same time, instead of focusing only on one or several of them, or taking one of them (such as emotion and attitude) as a "by-product" in the process of achieving other objectives.
(2) "They are realized in colorful mathematical activities. Among them, the development of mathematical thinking, problem solving, emotion and attitude can not be separated from the learning of knowledge and skills. At the same time, the learning of knowledge and skills must be based on the premise of being conducive to the realization of other goals. " There are two meanings here: ① the goal of "mathematical thinking, problem solving and emotional attitude" is achieved through the study of mathematical knowledge, and it is unnecessary and impossible to offer special courses for it; (2) What kind of knowledge and skills to learn should first consider whether it is conducive to the realization of the other three goals. For example, purely from the perspective of knowledge and skills, it seems that students "can skillfully do complex algebraic operations better than not" and "can prove difficult geometric propositions better than not". However, when we consider the development of students as a whole, the answer may not be so simple. First, is this knowledge necessary for all students in the future? Secondly, the acquisition of these skills requires a lot of practice. Do they help students to have positive feelings about mathematics learning? Can you deepen students' understanding of relevant knowledge? Can students be promoted to apply mathematics in their own lives and other disciplines? Is there anything more important to learn?
Seek adoption.