1. Abstract
The so-called abstraction is the process of separating some attributes and connections of things from other things ideologically. Abstraction helps us to put aside all kinds of secondary influences, extract the main and essential characteristics of things, and investigate them in a "pure" form, thus determining the development law of these things. Mathematics appears in a highly abstract form, first of all, the high abstraction of the basic objects it studies. Mathematical abstraction first occurred in the formation of some basic concepts, and Engels made an extremely incisive exposition on it: "The concepts of number and shape come from the real world, not from anywhere else. The ten fingers that people use to learn to count, that is, to do the first arithmetic operation, can be anything else, but they are not created by intellectual freedom. In order to count, we should not only have countable objects, but also have the ability to discard all other features except their quantity when examining objects, which is the result of long-term historical development based on experience. The concept of shape, like the concept of number, comes from the outside world completely, not from the mind. Objects with certain shapes must exist first, and these shapes can be compared before the concept of shape can be formed. Pure mathematics is based on the spatial form and quantitative relationship of the real world, that is, based on very realistic materials. This substance appears in an extremely abstract form, which can only cover up on the surface that it comes from the outside world. However, in order to study these forms and relationships from their pure forms, they must be completely separated from their own content and put aside the content as irrelevant things; In this way, we get lines without length, width and height, a and b and x and y, constants and variables; Finally, we get the free creation and imagination of intellectuality itself, that is, the concepts of imaginary number and [1] number. The concepts of geometric figures such as points, lines and surfaces belong to the most primitive mathematical concepts. On the basis of the original concepts, some more abstract concepts such as rational number, irrational number, complex number, function, differential, integral, n-dimensional space and even infinite dimensional space have been formed. From the perspective of mathematical research, the original materials of mathematical research can come from any field, and the focus is not on the content of materials but on the form of materials. Irrelevant things are on the edge, and the edges of the form can present similar patterns. For example, algebraic calculus can describe logical reasoning and even computer operations. The equations of fluid mechanics may also appear in the financial field. The great vitality of mathematics lies in the ability to transfer ideas from one field to other fields through the abstract process, and the research results of pure mathematics can often blossom and bear fruit in unexpected places. Some mathematicians abroad think that mathematics doesn't know what it means because the object of mathematical research is abstract, which is wrong.
The high abstraction of mathematical science determines that developing students' abstract thinking ability should be the goal of mathematical education. In the process of abstracting quantitative relations and spatial forms from concrete things and transforming practical problems into mathematical problems, students' abstract ability can be cultivated.
In the process of cultivating students' abstract thinking ability, we should not only pay attention to the teaching of abstracting mathematical concepts from real things, but also pay attention not to get involved in the discussion of a specific prototype. For example, the concept of straight line should be abstracted from the common and understandable practical background of students, such as tight lines, straight trunks, telephone poles, etc., which shows that the concept of straight line is a mathematical concept abstracted from many practical prototypes, but don't turn the teaching of this concept into a discussion on the specific background of straight line. Light is an important practical prototype of a straight line, but if the teaching of the concept of straight line falls into the exploration of the concept of light, it will lead to the entanglement of the concept of straight line. The concept of light involves many mathematical and physical problems, including the concepts of modern geometry and physics, including the long research history of the fifth postulate of Euclidean geometry, the appearance of non-Euclidean geometry, optics, electromagnetism, space-time, from the absolute concept of space-time in Newtonian mechanics to Einstein's special relativity and general relativity, etc. Trying to talk about the concept of straight line from the actual background of light, falling into the discussion of the essence of light, makes the concept teaching of straight line go astray. What needs to be clear is that light is not the only actual prototype of a straight line, and the actual prototype of a straight line is extremely rich.
When cultivating middle school students' abstract thinking ability, we should pay attention to controlling the abstract degree of middle school mathematics teaching content according to their psychological characteristics. Too much abstract content is not suitable for ordinary middle school students (such as some concepts of modern mathematics). In addition, the learning of abstract concepts should be based on the largest and most concrete concepts that abstract concepts can build. Otherwise, if the specific knowledge is not prepared enough, the abstract concept will become an empty thing with no actual content, and students do not know enough about the importance and necessity of learning this abstract concept.
Step 2 be strict
The so-called mathematical rigor means that any mathematical conclusion must be drawn by logical reasoning in strict accordance with the correct reasoning rules and according to the correct conclusions (axioms, theorems, laws, rules, formulas, etc.). ) has been proved and confirmed in mathematics, which requires that the conclusions obtained cannot be subjective and one-sided. The rigor of mathematics is closely related to the abstraction of mathematics. Because of the high abstraction of mathematics, whether its conclusion is correct can not be proved by experiments, but by strict reasoning like physics, chemistry and other disciplines. Once the conclusion is proved by reasoning, the conclusion is correct.
The general feature of mathematical science is strict logic, but there are many typical examples in the history of mathematical development. For example, the deepening of the understanding of the concept of infinity, the Pythagorean school's discovery of irrational numbers, Newton and Leibniz's calculus and its rigor, the construction of functions that are continuous everywhere but not derivative everywhere, and the construction of set theory paradox all well illustrate this rigorous style and spirit of mathematics.
Strict reasoning in mathematics makes every mathematical conclusion unshakable. The rigor of mathematics is the requirement and guarantee of mathematics as a science, and the rigorous reasoning method in mathematics is widely needed and widely used. Learning mathematics is not only about learning mathematical conclusions, but also emphasizes that students should understand mathematical conclusions, know how mathematical conclusions are proved, and learn mathematical science methods, including strict reasoning methods and other thinking methods. If we don't talk about the proof process of some important conclusions in mathematics teaching, the teaching value will be greatly reduced. Students often have difficulty in understanding some important and basic mathematical conclusions, and can't get the guidance of teachers in time, thus losing interest and confidence in mathematics learning. According to some surveys of mathematics teaching in new senior high schools, the derivation of some formulas and the explanation of some contents in the new textbooks are too simple to meet the learning requirements of students. In particular, some relational judgment theorems in typical solid geometry only give conclusions without proof, and the method of verifying experimental conclusions by experimental science is not in line with the spirit and method of mathematical science. Teachers have many opinions, which is the problem faced by mathematics teaching practice a few days ago. An important goal of mathematics teaching is to teach students the process and method of thinking, so that students can fully understand the authenticity and scientificity of mathematical conclusions and develop strict logical thinking ability.
Of course, we should implement the principle of teaching students in accordance with their aptitude and make adjustments according to students and teaching practice. Mathematics textbooks (including teachers' teaching books) can provide different rigorous teaching schemes for selection and reference. For example, the proportion theorem of parallel lines and line segments in plane geometry can adopt three different teaching schemes according to the actual teaching situation. The first one is widely used in junior high school mathematics textbooks (such as the second volume of Geometry of Nine-year Compulsory Education and Three-year Junior High School compiled by the Middle School Mathematics Room of People's Education Publishing House), that is, reasoning from special situations and generalizing conclusions to general situations without proof; The second method is to prove the theorem by using the area method (for example, the proving method of the second volume of geometry, a mathematics experimental textbook for junior high schools in compulsory education, compiled by the middle school mathematics room of Education Press); Thirdly, it is proved that the ratio is rational and irrational, which is a more rigorous teaching scheme for students' thinking ability (such as the teaching requirements of some junior high school mathematics textbooks in the former Soviet Union). To be sure, the long-term differences in teaching requirements at different levels naturally lead to great differences in students' mathematical ability. From the perspective of cultivating talents, it is of course necessary to design different teaching programs for different students, so as to be conducive to the all-round development of students.
In addition, the rigor of logic in mathematical science is not absolute, and the rigor is gradually strengthened in the history of mathematical development. For example, Euclid's Elements of Geometry was once regarded as a model of logical rigor, but later generations also found that it was not rigorous, and the process of proof often depended on graphic intuition. On the issue of cultivating students' logical thinking ability in middle school mathematics teaching, we should pay attention to the problem of strictness and moderation. In this regard, Chinese middle school mathematics textbook workers and teachers have done a lot of research on the teaching of elementary mathematics content. Many processing methods reflect the understanding level of middle school students and are very valuable. For example, many operations in algebra teaching in middle schools are not as strict as mathematical theory in scientific sense, and the processing methods are basically reasonable.
In addition, the pursuit of logical rigor in mathematics teaching needs the guarantee of teaching time, and middle school students have limited learning time. At present, after the implementation of the new mathematics curriculum in senior high schools, the actual teaching in various places reflects the contradiction between more teaching contents and tight class hours, and appropriately reduces some teaching contents that are more abstract, difficult or comprehensive for middle school students, so as to make the teaching time more abundant and facilitate students to digest and absorb knowledge. In the current high school mathematics new curriculum experiment, how to make the amount of teaching content more reasonable, how to make some high school students learn new mathematics elective courses (especially some topics of elective series 4) effectively implemented, so as to implement the diversity and selectivity of the new high school curriculum is also an important issue worthy of in-depth discussion.
A related problem is that mathematics teaching should deal with the relationship between process and result. The basic and important goal of learning mathematics is to solve various problems. Too much emphasis on logic and proof in mathematics teaching will lead to a wide range of knowledge, so that we don't know enough about many far-reaching and widely used mathematical methods. This shows that on the one hand, mathematics education should pay attention to the cultivation of logical thinking ability, but also to the cultivation of scientific spirit and the understanding of mathematical thinking methods. The rigor and rigor of mathematical conclusions need a rigorous attitude. However, in some specific teaching stages, as long as it does not lead to the reduction of logical thinking ability and affect students' understanding of the conclusion, the proof of some similar mathematical theorems can be omitted and should not affect the cultivation of mathematical ability.
Other scientific work often turns to experiments to prove its own conclusions, while mathematics relies on reasoning and calculation to draw conclusions. Calculation is an important way of mathematics research, and middle school mathematics teaching must cultivate students' quantitative concept and operational ability. Nowadays, computing tools are more advanced, and large computing systems can be used, which can greatly enhance computing power. The new mathematics curriculum in senior high school has increased the content of algorithm, enriched the content of probability statistics and data processing, and the high school technology curriculum has added the module of "algorithm and programming", which embodies the new requirements of the computer and information age for cultivating computing ability. Judging from the actual teaching situation of middle school mathematics at present, due to the limitation of technical conditions, the teaching implementation of algorithm content is not enough. To solve practical difficulties in teaching, such as the real operation of algorithms on computers and the implementation of teaching, this involves the problem of computer language. However, the direct introduction of computer programming language into middle school mathematics curriculum seems to make the content of middle school mathematics teaching too technical and professional, which is a problem worthy of study.
3. Widely used
In daily life, work, productive labor and scientific research, problems in quantitative relations and spatial forms are common, and the application of mathematics is universal. Mathematics is a discipline with a long history and has experienced unprecedented prosperity since the Second World War. While major breakthroughs have been made in the research of various branches, new connections between various branches of mathematics and between mathematics and other disciplines are constantly emerging, which has changed the face of mathematical science more significantly. The most far-reaching is the revolutionary change in the role of mathematics in social life, especially in the technical field. With the development of computer, mathematics has penetrated into all walks of life and materialized into all kinds of advanced equipment. From satellites to nuclear power plants, from weather forecasting to household appliances, the characteristics of new technologies, such as high precision, high speed, high automation, high safety, high quality and high efficiency, are all realized through mathematical models and methods and with the help of computer calculation and control. Computer software technology occupies a large proportion in high technology, and in the final analysis, software technology is actually the all-digital development process of mathematical technology, digital TV system and advanced civil aircraft. A large number of examples show that mathematics has shown the nature of the world's first productive force. She is not only a "behind-the-scenes hero" supporting other sciences, but also directly active in the front line of technological revolution. Mathematics is also very important for contemporary science, and various disciplines are becoming more and more quantitative, so it is more and more necessary to express their quantitative and qualitative laws with mathematics. The appearance and progress of computer itself strongly depend on the progress of mathematical science. Almost all important disciplines, such as adding the word "mathematics" or "calculation" before the name, are the names of an existing international academic magazine, which shows that a large number of interdisciplinary fields are constantly emerging, and various disciplines are making full use of mathematical methods and achievements to accelerate the development of their own disciplines. On the extensive application of mathematics, Arthur Jaffe, a professor of mathematical physics at Harvard University, made an incisive exposition in his famous long paper "Clarifying the Order of the Universe-The Role of Mathematics" (this article is the appendix of the report "Further Prospering American Mathematics" by the National Research Council of the United States), and he fully affirmed the important role of mathematics in modern society; "In the past quarter century, mathematics and mathematical technology have penetrated into science and technology and production and become an inseparable part of it. In today's technologically advanced society, the task of eliminating mathematical illiteracy has replaced the past literacy task and become an important goal of today's education. People can compare the contribution of mathematics to our society to the role of air and food in life. In fact, it can be said that we all live in the age of mathematics-our culture has been mathematized. Around us, computers with magical power can best reflect the existence of mathematics ... If you want to write the practical value of mathematical research to our society and explain how some specific mathematical ideas affect the world, you can write several books. " He pointed out: "(1) brilliant mathematics, no matter how abstract, will eventually be applied in the white world;" (2) It is impossible to accurately predict where a mathematical field will be useful. " [2] Many mathematicians are often surprised by the application of their ideas. For example, G·H· Hardy, a British mathematician, studied mathematics purely to pursue the beauty of mathematics, not because it had any practical use. He confidently claimed that number theory would not have any practical use, but forty years later, the nature of prime numbers became the basis for compiling new passwords, and abstract number theory was closely related to national security. Computer scientists report that every point of mathematics is helpful in practical application in one way or another, while physicists are surprised by the unusual effectiveness of mathematics in natural science.
Secondly, mathematics education should focus on cultivating students' awareness and ability of applying mathematics, which has become the knowledge of mathematics education in China. On the other hand, we should note that mathematics is widely used. In the limited time of primary and secondary schools, we must grasp the degree of introducing mathematics application. Mathematics is widely used, and any mathematical concept, theorem, formula and rule are widely used. Too many and too detailed mathematical application problems will inevitably affect the teaching of basic mathematical theory, and weakening the study of basic theory will lead to the weakening of mathematical application. In middle school mathematics teaching, it is important for students to understand the application of mathematics in some fields, understand the value of mathematics learning and attach importance to mathematics learning. In addition, the application of mathematics is not limited to the practical application of specific knowledge, but also includes the application of some mathematical concepts and ideas in practical work. Primary and secondary schools are the time to lay the foundation. The so-called laying a foundation is mainly to lay a good foundation for basic mathematics knowledge and skills. In order to let students have a broader vision of mathematics, we should not decide the teaching content directly based on whether there is a choice in practice, nor should we ask students to consider the most applied problems without learning much mathematics knowledge. The practice of junior high school mathematics teaching reflects that some traditional teaching contents have been deleted, which has a bad influence on students' mathematics learning; The experimental return visit to the new high school mathematics textbook also reflects that some practical problems in the high school mathematics textbook are too heavy, and the background of many practical problems and exercises is too complicated. It takes a lot of time to help students understand the actual background in teaching, which dilutes the study of main mathematical knowledge. In fact, the practical problems faced by students after taking part in the work will be very different, and their working and living backgrounds will be different, and their interests in the actual background and problems will be different. In addition, practical problems often involve many factors, which are often complicated for primary and secondary school students, especially those in compulsory education. The application of mathematics in a special field will inevitably involve a lot of professional knowledge in this field, which will become a great difficulty for students. In addition, although school education is to prepare students for work and production in the future, it is not necessary for students to spend too much time thinking about some practical problems that they will encounter in adulthood. Some practical problems are better left to adults to consider. On 200 1, the Middle School Mathematics Room of People's Education Publishing House invited Professor Tian Gang from Peking University Institute of Mathematical Sciences and others to talk about related issues of mathematics education. When talking about their views on mathematical science and its teaching, they pointed out that mathematics is mainly about calculation and reasoning, and the most important things that can be learned from mathematics are logical thinking and abstract methods, which are universally useful things; Mathematics education should strengthen the cultivation of logical thinking ability. As far as application is concerned, the current information technology needs strong logical thinking ability, especially programming, which is short and long, and is less likely to make mistakes. How to solve problems in a short time without making mistakes requires logical thinking; The teaching reform of calculus in the United States has advanced graphic calculators, which can be viewed and approached intuitively; Technology can help you master mathematics intuitively, but it is really important and useful to deduce formulas logically. Mathematics education should teach some basic things.
Thirdly, mathematics is widely used, but not all students will engage in jobs that require profound mathematical knowledge. From the perspective of direct application of mathematics, not every student needs to learn advanced mathematical theories. Ordinary people often apply the most basic mathematical knowledge, and the important purpose of learning mathematics is to improve their thinking ability through learning. Therefore, in primary and secondary schools, on the one hand, mathematics teaching should be oriented to all students so that everyone can get a good mathematics education; On the other hand, according to students' reality and their hobbies, according to each student's academic and intellectual development expertise, different students should get different development in different aspects. Of course, students who plan to develop in the field of science and technology must lay a good foundation in mathematics. It has been noticed that a considerable number of students who have laid a good foundation in middle school mathematics, including some winners of international and domestic middle school mathematics competitions, did not continue to regard mathematics as their main development direction in the subsequent learning stage, but chose other fields. Students who choose science and engineering majors often still study many courses in mathematical science at the university level, which also shows the wide application of mathematics and its important value for students' development.