First, the scientific value of mathematics
The scientific value of mathematics refers to the role and significance of mathematics in the emergence and development of natural science. Since the 1920s of 19, the research objects and methods of mathematics are more and more different from those of (natural) science, and mathematics is separated from science and has become its own "portal". However, this separation is not the separation between mathematics and science, but shows that the application of mathematics is more extensive, including not only (natural) science, but also humanities and social sciences such as political science, history, economics, linguistics, military science, and artistic sciences such as music, painting and sculpture, and also involves many fields of technology, economic construction and even society. Especially in today's era, with the rapid development of science and technology, the trend of scientific mathematicization is becoming more and more obvious, and modern science is developing in the direction of widely applying mathematics.
The value of mathematics to science is manifested in many aspects of the emergence and development of physics, chemistry, biology, astronomy and other disciplines. If you look at the elements of mathematics, it is embodied in the following four aspects.
(A) the application of mathematical knowledge
In the emergence and development of science, the application of mathematical knowledge is the most direct and extensive. This can be seen from the development of astronomy. When Copernicus put forward Heliocentrism, there was not much observational evidence. Even to some extent, some results are not as accurate as the original geocentric theory. It was he who established a new astronomical theory based on mathematical theory and applied mathematical methods. Kepler further applied mathematics to astronomy. He used a lot of observation data from Tycho and Bula, and through a lot of calculation and mathematical analysis, he abandoned the idea that planets had circular orbits since ancient Greece, thus establishing a new theory of planetary operation. By Galileo and Descartes, mathematics became a universal scientific method. 19th century, the achievements in mathematical application are more prominent: Gauss proposed the calculation method of planetary orbits (1809), Poisson established the differential equation for calculating electric potential (18 1 1) and the state equation of ideal gas (1820/). In addition, the combination of science and mathematics has produced some interdisciplinary and marginal disciplines, such as mathematical physical equations (methods), biomathematics, mathematical ecology and so on.
(B) the application of mathematical (symbolic) language
Mathematics is the main term of science. The connection between mathematical language and science was emphasized in ancient Greek natural philosophy. "Greek philosophy has discovered a new language-the language of numbers. This discovery marks the birth of our modern scientific concept. " In modern times, mathematics "is regarded as a new and powerful symbol system, which is superior to the speech symbol system for all scientific purposes" [1]. Galileo, known as the "father of modern natural science", also believes that the universe in front of us is like a big book written in mathematical language. If you don't master the symbolic language of mathematics, it is like wandering in a dark maze, and you can't know anything clearly. For example, the expression of the basic laws of contemporary physics-Newton's law of motion, Newton's law of universal gravitation, electromagnetic field principle, the first and second laws of thermodynamics, statistical mechanics principle, special relativity principle, general relativity principle, quantum mechanics law, relativistic electron wave principle, gauge field theory, etc. It is unthinkable without a mathematical language.
(C) the application of mathematical thinking methods
Mathematical calculation, mathematical proof, mathematical model and other methods play a vital role in the emergence of science. For example, calculation is one of the most important methods in various sciences (technologies). 1846, Levi predicted Neptune by calculation, which is a story in the history of science. In modern science, due to the wide application of mathematical thinking methods, a large number of marginal and interdisciplinary disciplines related to calculation have emerged, such as computational mechanics, computational fluid mechanics, computational structural mechanics, computational physics, computational chemistry, computational biology, computational embryology, computational geology, computational seismology, numerical meteorology and so on.
(D) the application of mathematical thinking mode
Symbolization, mathematicization, abstraction, axiomatization, structuring, logical analysis, reasoning calculation, data inference and optimization play a very important role in the construction and development of scientific theories. For example, Newton's mathematical principles of natural philosophy, Lagrange's analytical mechanics and Clausius' theory of thermomechanical motion are all written in an axiomatic way. Another example is the development of biology. At first, it "had to start with a simple classification of facts like other natural sciences …", and then it gradually "progressed to a new stage of deductive formulaic theory". 〔2〕
Second, the scientific literacy value of mathematics education
The scientific literacy value of mathematics education refers to the significance and role of mathematics education in forming people's scientific literacy (such as scientific consciousness, scientific thought, scientific method, scientific spirit, scientific attitude and scientific quality). Mathematics education has this value because mathematics still retains many characteristics of science, such as "everyone has faith in understandable rules;" The interaction between imagination and strict logic; Honest and open mind; The extreme importance of peer review; The value of being the first to make a major discovery; Internationally, with the development of high-power electronic computers, we will use computer technology to open up new research fields. " [3] Specifically, it has the following characteristics.
(A) the scientific characteristics of mathematics
As early as in ancient Greece, mathematics and science were the same; Modern mathematics and science are learning to find general laws and relationships. The world-recognizable view of science, the essential view of truth, goodness and beauty of science, the external confirmation and internal perfection of scientific theory evaluation, the development and uncertainty of scientific knowledge, observation, experiment, verification and evidence in scientific exploration, the function of scientific explanation and prediction, and many other scientific characteristics.
(B) the scientific way of thinking in mathematics
Whether it is empirical method, rational method, perfect method, or analogical reasoning, rational reasoning, intuition and inspiration in scientific discovery, it is completely the same and consistent with the methods and modes of mathematical discovery. Poincare, a famous French scientist and philosopher, elaborated on the role of "mathematical beauty" and "mathematical intuition" in mathematical discovery and learning, and pointed out that "the aesthetic feeling of mathematics, the harmony of numbers and shapes, and the beauty of geometry are all known to all real mathematicians ... those who lack this aesthetic feeling will never become real creators"; [4] "Without intuition, young people can't begin to understand mathematics; They can't learn to love it, and what they see from it is empty rhetorical argument; In particular, without intuition, they will never have the ability to apply mathematics ... If intuition is useful for students, it is even more indispensable for creative scientists "[5].
(C) the scientific spirit in mathematics
What exactly does the scientific spirit include? There are different opinions so far. The scientific spirit embodied in mathematics includes the spirit of seeking truth, being realistic and objective, the spirit of reasonable doubt and critical innovation, the spirit of democracy, equality and cooperation, the spirit of constant exploration, perseverance and perseverance, and so on.
Scientific application of mathematics
The emergence and development of mathematics, like other sciences, comes from problems. The problems here can generally be divided into two categories: practical problems and theoretical problems. The nature studied by science is undoubtedly the source of practical problems. For example, ancient astronomy in China, as one of the earliest and oldest astronomy in the world, is closely related to mathematics in calendar compilation and astronomical observation. In fact, mathematicians at that time, that is, astronomers, obtained many mathematical achievements in the process of compiling calendars, such as fractional operation, pythagorean measurement, residue theorem, interpolation, higher-order equations and so on. Moreover, scientific theoretical problems are also the source of problems in mathematical research. A famous example is that the theoretical problems of Einstein's theory of relativity contributed to the emergence of Riemannian geometry.
Thirdly, the scientific value of mathematics in mathematics education.
There should be no doubt about the "scientific value of mathematics" in mathematics education, but now it has become a complex subject. With the continuous development of people's understanding of the essence and value of mathematics, people are rethinking how to understand the relationship between science and humanity in mathematics education, how to treat the natural orientation and value orientation of mathematics content in primary and secondary schools, and what kind of mathematics should be taught in primary and secondary schools.
Under the traditional examination system in China, "elite education" and "genius education" have a long history, which seems to have formed "the tradition of China" and become more and more intense since 1990s (obviously, basic education should not be "elite education" or "genius education"). This educational thought and social trend of thought have had a profound and significant impact on mathematics curriculum and mathematics teaching, and caused many people to criticize the past mathematics education. Some people think that this kind of mathematics education is "the education of cultivating mathematicians" and "the education of mathematical geniuses". Some people think that it only pays attention to the scientific value orientation of mathematics and ignores the humanistic value orientation; Wait a minute. These criticisms are reasonable to some extent. Obviously, mathematics education should not be "the education of cultivating mathematicians" or "the education of mathematical geniuses". However, we should also carefully analyze and think about the following questions: In what sense did the past mathematics education mean "education to cultivate mathematicians" or "education to mathematical geniuses"? Are the ideas and reforms such as "popularizing mathematics" and "solving problems" advocated by the United States necessarily fair and reasonable value orientation of mathematics curriculum (or must conform to China's national conditions)? How to grasp the "scientificity" and "humanity" of mathematics in mathematics curriculum —— The relationship between "mathematics aspect" and "education aspect" of mathematics education? These problems need further analysis and thinking.
Mathematics education is not a simple addition of "mathematics" and "education", it includes at least two aspects, that is, "mathematics" is both the "purpose" and the "means" of education. As a means, students learn mathematics (mainly knowledge, theory and corresponding mathematical activities, such as mathematical problem solving and mathematical proof). ) improve their thinking ability and ability to analyze and solve problems, form a good personality quality and psychological structure, and enhance national self-esteem and pride; As an aim, students should learn, understand and master mathematics, that is, through mathematics education, students can acquire basic mathematical knowledge, basic mathematical skills and important mathematical thinking methods, and form a correct mathematical view and a certain mathematical consciousness. According to the philosophical principle of "unity of purpose and means", it is very important to master mathematical knowledge; Ignoring knowledge is actually "the product of metaphysical thinking mode to a great extent, which separates the relationship between knowledge and method, knowledge and ability" [6]. "It is believed that at any time, a solid knowledge base, a broad knowledge horizon, a reasonable knowledge structure and a good knowledge literacy are all the goals that education should pursue, and in the era of knowledge surge, it is no exception or even more important. Development through knowledge is an irrefutable educational truth. " [7] This shows that the "mathematical aspects" and "educational aspects" of mathematics education are unified, and a certain balance must be maintained between them. It is unreasonable and unfair to ignore any aspect.
1989, the United States promulgated the School Mathematics Curriculum and Evaluation Standard, and later promulgated the Professional Standard for Mathematics Teachers (19 1) and the School Mathematics Evaluation Standard (1995), which implemented the mathematics curriculum reform. As far as the overall result is concerned, it is counterproductive. According to the third "International Research on Mathematics and Science Education" survey, the performance of American students is far from people's expectations. Among them, the test scores of eighth and twelfth grade students are far lower than those of other countries, and the fourth grade students only reach the average level. In this regard, Professor Fran Kursio of new york University pointed out that there are seven reasons, namely: ignoring basic calculation; It is enough to have a general answer to the question; There is only one way to teach mathematics; Standards-compliant textbooks support reform; There is no effective research to support reform; Concrete experience can automatically lead to abstraction; The application of modern technology in mathematics is equal to teaching reform. Scholars at home and abroad have also rationally analyzed the limitations and consequences of "popular mathematics", "open problems" and "over-emphasizing application". They think that making mathematics more and more "simplified", "practical" and "life-oriented" will eventually make students learn things other than mathematics, and will not really arouse their enthusiasm for learning mathematics, but will make them feel that mathematics is meaningless and useless. 〔9〕
At present, mathematics education in China (including other subjects) not only increases the burden on students, but also becomes a "sieve" for screening students. This is due to the comprehensive effect of many factors such as society, economy and traditional culture in our country, and we must not blame all the responsibilities on the mathematics curriculum without analysis.
To sum up, in any case, mathematics is still mathematics (mathematics is culture, it should be "mathematical science" at first, and its core is also "mathematical science"), and its "mathematical scientific value"-basic knowledge, basic skills and mathematical activities that embody the essence of mathematics (such as mathematical reasoning, mathematical proof, mathematical thinking and mathematical rationality) can not be ignored in mathematics education. Only in this way can we truly realize the "humanistic value" of mathematics education.
In teaching, it is found that the intelligence of most poor students is not bad, which shows that the factors that determine poor students are often non-intelligence factors. The so-called non-intelligence factors refer to factors that affect students' learning enthusiasm, such as motivation, interest, emotion, personality, will, habits and so on. Therefore, in teaching, while inspiring students' thinking, developing students' intelligence and cultivating students' ability, we must also integrate the cultivation of non-intelligence factors into teaching, and take the cultivation of students' non-intelligence factors as one of the goals of subject teaching. Let's talk about some methods of cultivating students' non-intelligence factors in combination with the content of teaching materials in teaching.
First, educate students with typical examples and cultivate students' good learning will.
Will is an important aspect of non-intellectual factors, and students' good will quality can strengthen and promote their intellectual development. In teaching, students are purposefully educated with vivid examples of words and deeds, and their tenacious learning will is cultivated. For example, it tells the stories of famous mathematicians in China, such as Hua and Chen Jingrun, and their success stories. It is pointed out that they can climb the peak of mathematics because they have strong perseverance, educate students to learn the valuable qualities of scientists, and cultivate their perseverance to overcome difficulties and study hard and tenaciously. In teaching, students are provided with opportunities for independent activities to overcome difficulties, and teachers actively inspire and induce them to explore ways and means to overcome difficulties independently through students' own efforts. At the same time, pay attention to cultivating students' self-control ability. Junior high school students are unstable in thinking, easy to divert interest and easy to be distracted in class. In class, constantly warn them by changing their eyes, expressions, gestures and voices or making necessary pauses, so that they can feel that they have been under the teacher's attention, thus consciously controlling their attention. Some students are highly dependent and don't like to use their brains and copy their homework. They should be educated to realize that doing homework is a true record of the learning process and a consolidation of what they have learned. Although it is a long-term and arduous task to finish homework independently, it is good for learning, so that they can be good at controlling their bad behavior and "force" themselves to finish their homework independently on the basis of careful review and develop good self-control.
Second, stimulate students' interest in learning and arouse their enthusiasm for learning.
Interest is the best teacher. A strong interest in learning can make the brain in the most active state and enhance people's observation, attention, memory and thinking ability. Introduce some ancient and modern mathematical history or interesting mathematical knowledge combined with the content of the textbook to stimulate students' initiative and thirst for knowledge. Pay attention to the interest, exploration and application of fabricating teaching content, such as the story of Diao Fandu, an ancient Greek mathematician, using equations to solve application problems. When talking about the form of distance, we make our own parallelogram teaching aid, and use the instability of parallelogram to turn a parallelogram into a parallelogram with a right angle. Through demonstration and observation, the following questions are put forward for students to argue and explore: what kind of figure does a parallelogram become when its four sides remain unchanged? What happened to the parallelogram? (Angle), what hasn't changed? (edge). What is the definition of rectangle? What kind of quadrilateral is it? What special attributes do you have besides graphic attributes? A series of questions inspire students to think and explore actively. When talking about the positional relationship between two circles, students can see the changing process of the two circles' external separation-external cutting-intersection-internal cutting by using visual teaching AIDS and sports. Thus, five positional relationships between two circles are summarized, which enhances the intuition.
Third, use both hands and brains to cultivate students' hands-on operation ability.
Nowadays, junior high school students have excellent family conditions, parents arrange substitute classes, and their hands-on ability is poor, which brings obstacles to mathematics learning. In teaching, students are allowed to operate and make teaching AIDS, and intuitive thinking is raised to abstract thinking in the process of completing the operation. For example, when talking about the internal angle and theorem of a triangle, let each student prepare a triangle made of cardboard first, and let the students cut off the two angles of this triangle in class, and then put them together with the third angle to form a right angle. In this way, we can quickly find a way to prove the theorem. For example, when learning the judgment method of triangle congruence, guide students to start drawing experiments and cut two triangles with two included angles, two sides sandwiched in the middle and three corresponding sides respectively. By comparison, inspire students to sum up their own judgment theorems. By allowing students to participate in more practical activities, make teaching AIDS, hold objects with their own eyes and touch them with their own hands, they can deeply remember their characteristics, which not only enlivens the classroom atmosphere, but also broadens students' thinking.
Fourth, strengthen the guidance of learning the law and cultivate good study habits.
Students acquire knowledge and ability in the process of learning behavior. A certain learning behavior, repeated many times, will form certain learning habits, and forming good habits will benefit people for life. And bad habits will seriously affect students' mathematics learning and hinder the overall improvement of students' mathematics quality. Therefore, as long as students want to learn, optics is not enough, and they must also "learn". Pay attention to learning methods, improve learning efficiency, and change passivity into initiative. In teaching, we should pay attention to strengthening the guidance of mathematics learning methods, mainly adopting the following methods:
1, preparation method guidance.
Preview is a process that students explore, do, think and read the text by themselves, which can cultivate students' reading and self-study ability. A preview outline should be arranged before class, so that students can read through the text first, then carefully read and understand the general content, define some marks of "marking" and "remembering", and mark the key sentences, keywords, concepts, formulas and theorems in the textbook, so that they can develop the habit of reading, marking and calculating while reading.
2. Guidance on listening methods.
Students are required to do "one expert and three moves" in class, that is, to concentrate on listening to the teacher's analysis of key and difficult points, listening to examples, thinking, analyzing and skills. Be diligent in thinking, actively raise your hand to speak, and dare to express your opinions. Do a good job in class exercises, listen carefully to the teacher's comments and after-class summary, and actively participate in teaching activities with your brain, hands and mouth.
3. Summarize the guidance of review methods.
In unit summary or term summary review, guide students to comprehensively summarize the contents of each knowledge point and chapter they have learned, pay attention to the old and new connection of knowledge, the front and back connection of knowledge, and the horizontal connection of knowledge, and write a concise summary to make knowledge systematic, organized and thematic. Solve some exercises of various types and grades selectively, so that students can master the laws and methods of solving various problems and consolidate what they have learned.
4. Cultivate students' quasi-accelerated computing ability.
Mathematics is a subject with high requirements for operation, and operation ability is one of the basic abilities that students should have. Students use mental arithmetic more in primary school, but in junior high school mathematics stage, the difficulty and steps of the corresponding operation have increased, and there are often errors caused by mental arithmetic without writing, such as one-step error and one-step error. Some students often "look" at the problem and are too lazy to start work. In the long run, the speed of doing problems will slow down, leading to a decline in ability. In teaching, we should first pay attention to the training and guidance of operation, educate students who make mistakes in exercises or exams, make them feel "heartbroken" when they lose them, let students insist on writing, draft when calculating, and organically combine mental arithmetic with written arithmetic, which can greatly improve the accuracy of operation and reduce mistakes. Secondly, we should strengthen the limited training of operation, such as five-minute evaluation, improve the speed of operation and cultivate students' good operating habits.
In short, while doing a good job in teaching reform, we should also pay attention to the cultivation of students' non-intellectual factors, arouse students' enthusiasm for learning mathematics, enable students to master scientific learning methods, develop their ability to acquire mathematical knowledge independently, and comprehensively improve their mathematical ability and quality.
Although the time for face-to-face teaching is less and the requirements for autonomous learning are higher, you can improve your learning ability through this new learning form.
In the information society, people's living and working environment is changing faster and faster, and they need to face the emerging new knowledge and technology. One-off school education is increasingly unable to meet the lifelong social needs of individuals. Only by continuous learning can we keep up with the rhythm of life and work.
Modern distance education provides equal learning opportunities for all scholars, so that receiving higher education is no longer a right enjoyed by a few people, but a basic condition for individual survival; The universality of educational resources, educational objects and educational time and space provides the possibility for mass lifelong learning.
● Receiving education is not only about learning knowledge, but more importantly, learning to learn and cultivating good study habits for further study; Master the necessary learning skills. Learning to use modern information technology for autonomous learning is very beneficial to continuously acquire knowledge and improve education level in the future.