Mathematics is a science that studies the spatial form and quantitative relationship in the real world. It is divided into elementary mathematics and advanced mathematics. It is widely used in scientific development and modern life production, and is an indispensable basic tool for studying and studying modern science and technology.
Introduction of mathematical symbols
In a word, Mathematics is an infinite science.
2. The characteristics of mathematics
Rigidity
Mathematical language is also difficult for beginners. How to make these words have more precise meanings than everyday language also puzzles beginners. Words such as openness and domain have special meanings in mathematics. Mathematical terms also include proper nouns such as embryo and integrability. However, there are reasons for using these special symbols and proper terms: mathematics needs more accuracy than everyday language. Mathematicians call this requirement for linguistic and logical accuracy "rigor"
Rigidity is an important and basic part of mathematical proof. Mathematicians hope that their theorems can be inferred by systematic reasoning and axioms. This is for the following reasons. In order to avoid the wrong "theorem", there have been many examples in history. The expected rigor in mathematics varies with time: the Greeks expected careful arguments, but in Newton's time, the methods used were less rigorous. Newton's definition of solving problems was not dealt with again until the nineteenth century with careful analysis and formal proof. Today, Mathematicians are constantly arguing about the rigor of computer-aided proof. When a large number of measurements are difficult to verify, the proof can hardly be said to be effective and rigorous. Because of the differences of the times, a lot of knowledge has been erased, but mathematics will never be erased, and wisdom will spread forever.
3. The application of mathematics
Life cannot be separated from mathematics, and mathematics cannot be separated from life. Mathematical knowledge comes from life and is higher than life, and ultimately serves life. Indeed, learning mathematics is to be applied in real life. Mathematics is what people use to solve practical problems, in fact, mathematical problems arise in life. For example, there are countless problems such as adding, subtracting, multiplying and dividing when shopping on the street, and making floor plans when building houses. This knowledge is produced in life. In mathematics teaching, we should give students opportunities for practical activities, guide students to consciously use mathematical knowledge, analyze and solve practical problems in life with mathematical knowledge and methods, and make life problems mathematized, so that students can deeply appreciate the application value of mathematics.
Curriculum Standard emphasizes that starting from students' existing life experience, students should personally experience the process of abstracting practical problems into mathematical models and explaining and applying them. In fact, most of the teaching contents of primary school mathematics can be related to students' real life. Teachers should find out the "fit point" between the contents of each class and students' real life, and arouse students' interest in learning mathematics and their enthusiasm for participating in learning. In teaching, the teacher's responsibility is not only to induce students' desire to solve practical problems, but also to let students learn to choose the necessary conditions and information from many conditions and information to solve real-life problems and experience the success and happiness of applying mathematics to solve practical problems.
1. Solve the problems in life and apply what you have learned
The new curriculum standard points out that students should "realize that there is a lot of mathematical information in real life. Mathematics has a wide range of applications in the real world. When facing practical problems, we can actively try to use the knowledge and methods we have learned from the perspective of mathematics to seek strategies to solve problems ... ". We often encounter this situation, and students still don't understand a topic for a long time. If the teacher links this problem with real life, the students can solve it immediately. Therefore, as a teacher, we should think about how to make full use of students' existing life experience and guide students to apply mathematics knowledge to reality in order to realize the application value of mathematics in life.
Second, create life scenes to stimulate interest in learning
Application problems originate from life, and each application problem can always find its blueprint in life. Therefore, if we combine application problems with real life in application problem teaching, we can stimulate students' interest in learning.
Third, restore the essence of life and cultivate students' thinking
While paying attention to the life of mathematics, each of our teachers must fully realize that the essence of mathematics teaching is to develop students' thinking. Life-oriented does not mean simplification of mathematical knowledge. On the contrary, restoring mathematics to the essence of life is more conducive to the development of students' thinking.
I once saw a report that a professor asked a group of foreign students, "How many times did the minute hand coincide with the hour hand between 12 o'clock and 1 o'clock?" Those students all took off their watches from their wrists and began to set their hands; When the professor tells the same question to the students in China, the students will apply the mathematical formula to calculate. The commentary said that it can be seen that China students' mathematical knowledge is transferred from books to their brains, so they can't apply it flexibly. They rarely think of learning, applying and mastering mathematical knowledge in real life.
Fourth, realize the needs of life and promote the development of the subject
From the perspective of educational psychology, there are five different levels of needs in life, and the highest need is the need of self-realization and the need of decision-making. Once we link the application problem teaching with life in teaching, the potential needs of students will be even stronger.
five. The importance of mathematics
is proved by the famous saying:
Everything is counted-Pythagoras
In the world of mathematics, what matters is not what we know, but how we know it.-Pythagoras
The beauty of mathematical symbols
Numbers rule the universe.-Pythagoras
There is no king in geometry.- I don't think about those problems that are only used to practice my thoughts. I do this to study another kind of geometry, that is, geometry that aims to explain natural phenomena. —— Rene Descartes 1596-165) imaginary number is a wonderful human god's sustenance. It seems to be an amphibian between existence and non-existence.-Gottfried Wilhelm von Leibniz (1646-1716) Things that don't work will not exist.-Leibniz The whole thing boils down to pure geometry, which is a goal of physics and mechanics.-Leibniz Although we are not allowed to see through the secrets of nature, we can know the real reason of the phenomenon. However, it may still happen that certain fictitious assumptions can explain many phenomena. —— Leon Hard Euler (177-1783) Because the structure of the universe is the most perfect and the creation of the most wise God, if there is no certain maximum or minimum law in the universe, Then nothing will happen at all. —— Some beautiful theorems in Euler's mathematics have such characteristics: they are easy to be induced from facts, but the proof is extremely deep. Mathematics is the king of science. —— Gauss Mathematics is the head of natural science, while number theory is the queen of mathematics. —— Gauss This is the advantage of a well-structured language. Its simplified notation is often the source of abstruse theories. —— Pierre Simon Laplace 1749-1827) Read Euler, read Euler, he is our teacher. —— Laplace Only mathematics flourishes in a country. Only in this way can she show her powerful national strength. —— Laplace's research method of knowing a giant is not less useful for the progress of science than the discovery itself. Scientific research methods are often extremely interesting parts. —— Laplace It would be a serious mistake to think that it is necessary only in geometric proofs or sensory evidence. —— Augustin Louis Cauchy (1789-1857) Give me the sixth coefficient, and the elephant will wag its tail. —— Cauchy People must be convinced that if he is adding many new terms to science and let readers continue to study the wonderful and difficult things in front of them, science has made great progress. —— Cauchy Geometry sometimes seems to take the lead in analysis, but in fact, geometry takes the lead in analysis, just like a servant walking in front of his master. It opens the way for the host. —— James Joseph Sylvester (1814-1897) Maybe I can claim the title of Adam in mathematics without undue demands. Because I believe that there are more mathematical rational creations named by me (which has become popular and universal) than all other mathematicians in the same period put together. —— Sylvester A mathematician who has no talent as a poet will never become a complete mathematician. —— Karl Weierstrass 1815-1897) As long as a branch of science can ask a lot of questions, it is full of vitality, while the lack of questions indicates the termination or decline of independent development.-Hilbert Music can stimulate or soothe feelings, painting can make people pleasing to the eye, poetry can touch people's hearts, philosophy can make people gain wisdom, and science can improve material life. But mathematics can give all of the above.-Klein No subject can explain the harmony of nature more clearly than mathematics. ---Carus,Paul Problems are the heart of mathematics-P.R. halmos Where there are numbers, there is beauty! -Plocque Lars Logic is invincible, because logic must be used to oppose it.-Boutroux Mathematics subsystem is as vast as nature itself-Fourier Logic can wait, because it is eternal-Havisham A science, only when it successfully uses mathematics, Only in this way can we reach a truly perfect level.-Marx Mathematics is an infinite science.-Herman Weil History makes people wise and poetry makes them witty. Mathematics makes people subtle.-Bacon The scientific level of a country can be measured by the mathematics it consumes.-Rao No subject can explain the harmony of nature more clearly than mathematics.-Carlos Mathematics is the judge and master of laws and theories.-Benjamin VI. Mathematics and culture The cultural value of mathematics. Many important debates in the field of philosophy in history often involve the understanding of some fundamental problems in mathematics. Thinking about these problems will help us to correctly understand mathematics and the relevant debates in philosophy. (A) mathematics-rooted in the external performance of practical mathematics, more or less related to human intellectual activities. Therefore, in the relationship between mathematics and practice, it has always been argued that mathematics is "the free creation of human spirit", denying that mathematics comes from practice. In fact, all the development of mathematics comes down to practical needs to varying degrees. From the Oracle Bone Inscriptions of the Yin Dynasty in China, we can see that our ancestors had already used the decimal counting method at that time. In order to meet the needs of agriculture, they matched "ten branches" and "twelve branches" into sixty jiazi to record the years, months and days. Thousands of years of history showed that this calendar calculation method was effective. Similarly, due to the calculation of business and debt, the ancient Babylonians had multiplication tables and reciprocal tables, and accumulated a lot of materials belonging to the category of elementary algebra. In Egypt, due to the need to re-measure the land after the Nile flood, a lot of geometric knowledge of calculating area has been accumulated. Later, with the development of social production, especially the astronomical survey to meet the needs of agricultural cultivation and navigation, elementary mathematics gradually formed, including most of the mathematical knowledge we learned in middle schools today. Later, the industrial revolution caused by the invention of steam engines and other machinery required a more detailed study of motion, especially variable speed motion, and a large number of mechanical problems appeared, which prompted calculus to emerge after a long period of brewing. With the rapid development of modern science and technology since the 2th century, mathematics has entered an unprecedented period of prosperity. During this period, many new branches of mathematics appeared: computational mathematics, information theory, cybernetics, fractal geometry and so on. In a word, the need of practice is the most fundamental driving force for the development of mathematics. The abstraction of mathematics is often misunderstood by people. Some people think that the axioms, postulates and theorems of mathematics are only the products of mathematicians' thinking. Mathematicians work on a piece of paper and a pen, which has nothing to do with reality. In fact, even as far as Euclid's geometry, the earliest axiomatic system, is concerned, the geometric intuition of practical things and the phenomena developed by people in practice, although not in line with the various axiomatic systems of mathematicians, still contain the core of mathematical theory. When mathematicians take the establishment of the axiomatic system of geometry as their goal, his mind must also be linked to geometric drawing and intuitive phenomena. A person, even a gifted mathematician, can get scientific results in the study of mathematics. Besides receiving strict mathematical thinking training, he will be guided by practice consciously or unconsciously in the process of mathematical theory research, such as raising questions, choosing methods, prompting conclusions and so on. It can be said that without practice, mathematics will become water without a source and a tree without a foundation. In fact, even in Euclidean geometry, the earliest axiomatic system, the geometric intuition of practical things and the phenomena discovered in practice, although not in line with the program of mathematician's axiomatic system, still contain the core of mathematical theory. When a mathematician takes the establishment of the axiomatic system of geometry as his goal, his mind must also be connected with geometric drawing and intuitive phenomena. A person, even a gifted mathematician, can get scientific results in the study of mathematics. Apart from his strict mathematical thinking training, he will be guided by practice consciously or unconsciously in the process of mathematical theory research in many aspects such as raising questions, choosing methods and prompting conclusions. It can be said that without practice, mathematics will become water without a source and a tree without a foundation. However, because of the characteristics of mathematical rational thinking, it will not be satisfied with only studying the realistic quantitative relations and spatial forms, but also tries to explore all possible quantitative relations and spatial forms. In ancient Greece, mathematicians went beyond the method of measuring line segments within the realistic limited scale precision and realized the existence of incommensurable measuring line segments, that is, the existence of irrational numbers. This is actually one of the most difficult concepts in mathematics-continuity and infinity. Until two thousand years later, the same problem led to the in-depth study of limit theory, which greatly promoted the mathematical