1. Mathematics education is the basic subject education in primary and secondary schools. Like other disciplines, its educational significance is not limited to mastering disciplines, but also reflected in its effective promotion of people's quality development, and people's quality is one of the most profound and effective parts of people's cultural literacy.

2. Reform direction of mathematics education in economically developed countries: The focus of school mathematics has shifted from dual tasks-teaching the least mathematics to most people and teaching advanced mathematics to a few people-to a single center, teaching the most important core of mathematics to all students. From the mode based on authority transfer to the student-centered practical activities characterized by inspiring learning. From the emphasis on preparing for the follow-up content to emphasizing what students need now and in the future. From the original emphasis on a piece of paper and a pen to the comprehensive use of calculators and computers.

3. Mathematics in primary and secondary schools contains factors that promote people's future development. This is people's mathematical quality, and its core is people's thinking quality.

4. Mathematics teachers' teaching experience has three levels: showing solutions, showing ideas and showing the process of finding ideas.

5. The significance of mathematics education lies in cultivating, enlightening and enriching people with the quality of the subject itself, and promoting the all-round development of people's quality.

6. Mathematics education is a kind of culture, which enables people to acquire mathematics literacy and better understand and appreciate the civilization of modern society; It is a methodology that makes people good at living and doing things and can improve work efficiency in modernization; It is a kind of spirit and attitude that makes people seek truth from facts, persevere and pursue unremittingly; It is "gymnastics of thinking", which makes people think sharply and express clearly.

7. Important features of mathematics-abstraction, rigor and systematicness.

8. The significance of mathematical thinking education lies in cultivating people's sense of numbers, mathematical concepts and mathematical thoughts. Mathematics education is to expand the mathematics space in people's minds.

9. Mathematics-related abilities-mathematization, axiomatization and formalization.

10, strive to mathematicize external phenomena, pay attention to the mathematical aspects of phenomena, pay attention to the relationship between space and quantity, and rely on functions everywhere.

1 1, mathematics, cultivate the will to learn, cultivate people's generalization ability, cultivate people's awareness of seeing problems, cultivate people's abstract consciousness, cultivate people's good thinking habits, form good thinking strategies, enhance people's reaction ability and improve people's thinking organs.

12, the purpose of mathematics education: (1), through the combination of "mathematical common sense" and "mathematical thinking ability", cultivate mathematical intelligence; (2) Cultivate people with mathematical literacy. "Mathematical literacy": knowing the value of mathematics, having confidence in one's own mathematical ability, having the ability to solve mathematical problems, learning mathematical communication and learning mathematical thinking methods. (3) Learning math skills through practice-suitable for learning facts and skills. By solving situations with certain characteristics, we can learn general methods to solve problems, and these characteristics are used to define a real problem-skills suitable for learning how to discover and explore, rediscover mathematics and learn how to learn.

13, the purpose of mathematics learning has changed from mastering "mathematical facts and skills" to mastering "general methods to solve problems", that is, "mathematical thinking", which is a major update of the concept of mathematics education.

14, four levels of understanding mathematics: (1), formal understanding. Logical thinking training should be the basic training in mathematics learning. (2) Understanding at the discovery level; (3) intuition-specific understanding; (4) intuitive understanding.

15, it is generally believed that mathematics is a science composed of strict logic, even if it is different from logic, it is roughly the same. But actually, mathematics has nothing to do with logic. Mathematics must follow logic, of course, but logic plays the same role in mathematics as grammar does in literature. Writing a grammatical article is completely different from writing a novel according to grammar. Similarly, carrying out correct logical reasoning and piling up logic to form mathematical theory are completely different problems. Mathematics and logic are essentially different.

16. Never put the logic car in front of the heuristic horse in mathematics.

17, only by knowing how the conclusion was drawn can we really understand the conclusion. Reproducing or experiencing the discovery process is a wonderful way for mathematicians to learn and study mathematics. The best way to learn is to do it-ask questions and solve problems. The best teaching method is to let students ask questions and solve problems, not just impart knowledge-encourage action.

18. Mathematics is very abstract. One aspect of understanding mathematics is to give it intuitive and concrete meaning.

It is wrong to overemphasize the formal structure of mathematics.

20. Abstraction is meaningful only on the basis of solid experience. In addition, after the introduction of abstract concepts, specific problems should be used to illustrate their usefulness.

2 1. The direction of learning modern algebra well is to emphasize several basic concepts, such as symmetry, continuity and linearity.

22. Geometrical intuition is still the most effective way to understand mathematics. Geometric intuition means that abstract things can be described and thought in your mind like painting.

23. The combination of mathematics teaching and people's quality development is the most important purpose of mathematics education.

24. Geometry is a mathematical coincidence, a "symbol of intuitive space to help memory" and a "graphic formula".

What mathematics really wants to do is to solve specific problems. The best way to understand a theory is to find a specific problem, and then study a sample example of this theory, a typical example that can explain everything.

26. For a mathematical theory, give typical examples, counterexamples, special cases (that is, special cases) and so on. , merlin city specialized in understanding this method of mathematical theory.

27. Logic is used to prove and intuition is used to invent.

28. In the process of understanding mathematics, it is called intuition to understand the wholeness, orderliness and harmony implied in the reasoning chain, to achieve the overall grasp of the reasoning chain, and even to be able to foresee the proof.

29. Memory is very important in mathematics, but it is not necessary to remember mathematical facts.

30. Mathematical intuition means not strict; Means visible; It means that the proof lacks rationality and credibility; Means incomplete; It means relying on physical models or some major examples; It means generality or comprehensiveness as opposed to detail or analysis.

3 1, understanding is more important than proof.

32. Mathematical thinking education requires students to learn through their own thinking.

33. Shortcomings of current education: Some adopt the tactics of injecting and asking questions, and regard learning mathematics only as perception and cognition, weakening or canceling its central link-thinking. Some mathematical thinking activities are only regarded as formal logical thinking, ignoring the dialectical and developing thinking activities that look at the problem as a whole.

34. If the question provides students with a suitable thinking situation, it will greatly arouse their enthusiasm for thinking.

35. There is a vast, middle and gray area between understanding and not understanding.

36. Students' practical thinking process from ignorance to knowledge is much more responsible than we thought. Students' thinking process is not completed at one time, but full of dialectical nature such as movement, change and relativity.

Teachers often hope that students' understanding will be fixed in the mode of "correct", "reasonable", "strict" and "concise" from the beginning, ignoring the dialectical psychological process that they don't know, know little and know much.

38. Learn "static" with "dynamic" in mathematics education, so that static theorems, formulas and laws have dynamic life and can be active in students' thinking.

39. There are three stages in the history of mathematics: first, in the first stage of arithmetic and geometry, the concreteness of the object is abandoned; Second, in the second stage leading to arithmetic symbols, specific numbers and specific quantities are omitted; Third, and finally, the third stage of modern mathematics, which not only omits the characters of objects, but also omits the dependence between them.

40. Holistic thinking refers to the thinking tendency of paying attention to the overall grasp of the object-geometric thinking.

Factorial thinking refers to the thinking tendency that focuses on decomposing a problem into a series of sub-problems and then solving them step by step-algebraic thinking.

4 1. Holistic thinking mode is often neglected in practical teaching. On the one hand, people do not realize that holistic thinking is indispensable in people's mathematical thinking; On the other hand, it is often difficult for adults to recall the process of the emergence and development of their own thinking in those years, so they think that children's learning is based on discrete thinking, which is manifested in the fact that the textbooks and workbooks written by adults for children are all small-step western thinking methods.

42. In the higher level of thinking in images, we have made some concessions to form and logic, such as the accuracy of language, the adoption of symbols, the basis of reasoning and so on. In other words, it replaces the fuzziness of quantity and the fuzziness of reasoning form with the vividness and vividness of quality.

43. The cultivation of mathematical thinking in images is an important part of mathematics teaching reform.

44. In practical thinking, when abstract thinking cannot be continued by algorithms, we must find the direction of abstraction with the help of images and find new opportunities for abstract thinking (problem solving). The result of abstract thinking can also be expressed in the form of images, so there is a so-called "simple explanation" expression. Simple explanation is a process from image to abstraction and from abstraction to image.

In order to make students full of creative spirit, we must pay attention to the cultivation of thinking from seeking common ground to seeking differences.

46. We often overemphasize students' deductive thinking, but neglect to guide students to make reasonable reasoning.

47. Reasonable reasoning includes inductive reasoning and analogical reasoning.

48. Reasonable reasoning is a kind of possibility reasoning, which draws a conclusion of possibility based on people's experience, knowledge, intuition and feelings.

49. Practice shows that among a large number of graduates, the common sense and instrumental function of this subject are far from being brought into play. The reason is not the uselessness of knowledge, but the lack of mathematical concepts to guide knowledge. To unify knowledge, formal training and social significance of knowledge, it is necessary to carry out mathematical concept education.

50. Due to the influence of examinations, the traditional subject teaching will gradually shift to the end of the teaching process. The so-called "periphery" refers to those topics with non-basic skills and technology as the backbone. Therefore, it has little effect on forming a person's mathematical concept and stimulating people's most positive thinking.

5 1. Once creative thinking is taught, it loses the meaning of creation.

52. Thinking depends mainly on enlightenment, not on teaching. The clearer you teach, the less you need to think. Even if the teaching is an example, it only increases the knowledge storage, which may not make people ask for it in the new situation.

53. The working face for teachers to enlighten thinking: (1). Stimulate learning interest, stimulate learning motivation and create an atmosphere of successful education; (2) Creating problem situations and enhancing the internal driving force to solve problems; (3) Transforming new problems.

54. One of the criteria to measure the quality of mathematics teaching is whether teaching can effectively expand people's realistic mathematics space. Mathematical space is not only formed by some acquired knowledge, but more importantly, with the help of the growth point, open face and mathematical thinking process of the learned knowledge, a mathematical-related ability is obtained, so that the mathematical space has a certain degree of openness, including: mathematization-people observe the real world by mathematical methods, analyze and study various mathematical phenomena, and organize the real world process. We study mathematics, and the most important thing is to learn mathematicization. Similarly, we learn axiomatic knowledge instead of "axiomatization" and "formalization" instead of formal system.

The formulation of "cultivating mathematical intelligence" points out that the composition of mathematical intelligence and the way to cultivate it are the combination of "mathematical common sense" and "mathematical thinking ability".

56. After mathematics teaching, students' knowledge of mathematics will be forgotten more and more. However, if the teaching method is correct and students' understanding of what they have learned in the process of mathematics teaching reaches the level they should reach, then they will almost extract the most basic, essential, most important and usually the simplest part from all the contents they have learned, and remember it forever, to the extent that they want to forget it. This small part is "common sense of mathematics". Therefore, students' mathematical knowledge should go through a process of "less-more-less".

57. Exam-oriented education is often impossible for students to reach the level of understanding they should reach, so students will soon forget the mathematics they have learned after completing the exam-oriented task.

58. For a long time, due to the influence of exam-oriented education, mathematics education only pays attention to learning ready-made knowledge conclusions, skills and techniques, while ignoring the cultivation and training of the basic spirit of the subject and the basic attitude and methods of mathematics. One aspect that is particularly neglected is the education of mathematical concepts. Mathematical concept refers to people's views and consciousness on the origin and ontology of a mathematical object or process, including why, how and people's views on this mathematical knowledge.

Yuan Mei, a poet in A Qing, pointed out in Poems with the Garden: "Learning is like a crossbow, like an arrow, you know how to lead it and release energy to catch it". Talent-intelligence, learning-knowledge, knowledge-insight, knowledge. Knowledge is the basis of solving problems, intelligence is the tool to transform knowledge into solving problems, and insight leads the application direction, methods and ways of knowledge and ability. Without the latter, knowledge and ability will not find a place to use.

60. The main direction of thinking education in mathematics teaching is: 1. How to cultivate students' creative thinking; Second, how to combine imparting knowledge with cultivating thinking ability.

6 1, for students, as long as the knowledge to be learned is taken as the result to be created, they can unify learning knowledge and acquiring creative ability.

62. We should consciously strengthen the following kinds of education: First, the education of reasoning consciousness. Let students know that any laws and formulas have certain basis and reason. Second, describe the education of harmonious consciousness in the objective world. Third, the education of the principle of form invariance.

63. The mistake of mathematics education is that it is easy to turn the inquiry part into the reappearance part, which makes it lose the meaning of thinking education.

64. Stimulating learning interest and motivation is a problem that teachers must pay attention to from beginning to end in mathematics education. Instruct students in teaching: 1. Love mathematics and respect the intelligent activity process of mathematics. Mathematics, as a gift from nature and the creation of human wisdom, has duality. On the one hand, nature and human society always maintain and present a law, a harmony and an unchangeable conservation in their movements; On the other hand, human beings have created a beautiful material world by using the laws described by mathematics. 2. Create an atmosphere of successful education, so that students can get the joy brought by thinking results.

65. Create problem situations and enhance the internal driving force to solve problems. The difficulty of creating problem situations requires students to work hard to achieve it. The deep-seated purpose of creating problem situations is to stimulate students' potential.

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