So is it necessary for only those who study mathematics to understand the history of mathematics? Or is it only beneficial for those who study and study mathematics to understand the history of mathematics? ?
As a culture, mathematical science is not only an important part of the whole human culture, but also an important force to promote human culture from beginning to end. It is closely related to many other disciplines, even the foundation and growing point of many disciplines, and plays a great role in the development of human civilization. From the history of mathematics, mathematics and astronomy have always been closely related, and the discovery process of Neptune is a good example; It is also inseparable from physics. Newton, Descartes and others are all famous mathematicians and physicists. For everyone who wants to know the whole history of human civilization, the history of mathematics is a must-read chapter. A.Whitehead, a famous philosopher, used a metaphor to illustrate that mathematics is one of the elements in the history of human thought when criticizing the past thinkers for neglecting the position of mathematics. He said: "If someone says that writing a history of ideas without in-depth study of the mathematical concepts of each era is equivalent to removing the role of Hamlet from the play Hamlet, this statement may be a bit excessive, and I don't want to go that far. But doing so is absolutely equivalent to removing the role of O 'filia. Ophelia is very important to the whole plot [2]? . "He is only in terms of the history of thought. In fact, we can say that it is impossible to fully understand the whole history of human civilization without understanding the history of mathematics. ?
Studying the history of mathematics plays an inestimable role in the development of mathematics itself. As we all know, Academician Wu Wenjun, who won the first National Science and Technology Award in 2000, is an advocate of mathematical mechanization research. He has made basic achievements in the research of demonstrative class and embedded class, and has been widely used in many problems. The method of proving geometric theorems by computer is fundamentally different from the commonly used methods based on mathematical logic, showing unparalleled advantages, changing the face of automatic reasoning research in the world, being called the pioneering work in the field of automatic reasoning, and thus winning the Gerbrand Award for Outstanding Achievement in Automatic Reasoning. When analyzing the results, Professor Wu Wenjun pointed out, "We followed the enlightenment of China's ancient mechanized mathematics-algebraic geometry, and transformed the non-mechanized geometric theorem proof into polynomial equation, thus realizing the mechanized proof of geometric theorem." There are countless examples of studying mathematical thought seriously and using it to guide mathematical research and making great achievements. Even for the teaching of advanced mathematics, the role of mathematical history cannot be underestimated. ?
If the whole mathematics is compared to a big tree, then elementary mathematics is the root, all branches of mathematics are branches, and the main part of the trunk is calculus. This shows the importance of calculus and its relationship with other disciplines. Therefore, "Calculus" has always been an important compulsory course for science and engineering in colleges and universities. Generally, it is a two-semester teaching plan. It contains the basic knowledge of differential calculus, integral calculus, spatial analytic geometry, infinite series and ordinary differential equations. Mathematics teaching in China has always paid attention to the training of formal deduction of mathematical thinking, but neglected to cultivate students' understanding of mathematics as a scientific ideological system, cultural connotation and aesthetic value. And because of the influence of traditional teaching hours and content arrangement, the teaching of advanced mathematics often has the contradiction of less class hours and more content. Therefore, in order to complete the teaching task and achieve the effect of "being able to test", teachers often only pay attention to the teaching of mathematics knowledge in class, while ignoring the ideological and interesting nature of mathematics. Courant, a famous contemporary mathematician, once pointed out: "Calculus, or mathematical analysis, is one of the great achievements of human thinking. Its position between natural science and humanities makes it a particularly effective tool for higher education. Unfortunately, the teaching method of calculus is sometimes mechanical, which cannot reflect that this subject is the crystallization of a shocking intellectual struggle. " ?
As teachers of advanced mathematics, we have had such experience. Although we carefully prepare lessons and explain them comprehensively, we find that the teaching effect is not ideal, and some abstract concepts are difficult to understand, which is generally reflected as incomprehensible. In the long run, some students even lose confidence in learning advanced mathematics and lose interest in learning. After several generations of continuous research on the teaching methods of advanced mathematics, the role of mathematical history in advanced mathematics teaching has been recognized by everyone. Those who think that it is unnecessary and a waste of time to talk about the history of mathematics in teaching, and that "precious time" should be spent more on exercise training have become a thing of the past. In teachers' teaching, introducing the theme of mathematical history related to the theme will have a very positive significance for students' learning, which can not only arouse their enthusiasm for learning, but also help students to concretize abstract concepts. Because the evolution of important concepts of mathematics has practical significance in the application of science and technology and the breakthrough of ideas, it is necessary to teach the history of mathematics in an enlightening way. ?
Based on the above understanding, in recent years, a lot of research results have been made in this field. Domestic and international exchange activities are also increasingly frequent. The history of mathematics has been set as an elective course in some schools. This paper systematically introduces the origin and development of mathematics. This has played a very good auxiliary role in the teaching of advanced mathematics. However, due to the shortage of talents in this field, some schools cannot offer this elective course. Moreover, as a separate elective course, it should systematically reflect the origin and development of mathematics, but it can't keep up with the content taught by advanced mathematics in time. Therefore, this requires our teachers who teach advanced mathematics to combine the history of mathematics in their usual teaching. ?
How to combine the teaching of mathematical knowledge with the introduction of mathematical history in the case of heavy teaching tasks and tight classroom teaching time? How can the introduction of the history of mathematics not only ensure the completion of teaching tasks within limited classroom time, but also improve everyone's interest in learning and convey mathematical ideas? ?
Throughout the long river of historical development, the birth of important thoughts cannot be separated from important people. So is the development of mathematics. H.Weyl, a famous German mathematician, said: "If we don't know the concepts, methods and achievements established and developed by our predecessors, we can't understand the goals of mathematics in the past 50 years, and we can't understand its achievements." Therefore, in the study of mathematical history, it is very important to study mathematical graphics. ?
In the textbooks of advanced mathematics, we will come into contact with some theorems and concepts of fundamental importance. Such as Newton-Leibniz theorem, Lagrange mean value theorem, Fourier trigonometric series and so on. Learning these theorems and concepts is not only important for learning advanced mathematics knowledge, but also necessary for improving mathematics quality. They are the essence of calculus and the necessary content of higher mathematics teaching. Most of these theorems and concepts are named after important mathematical figures. They are also the founders and pioneers of calculus. This reminds teachers that properly introducing the life and achievements of pioneers in the classroom teaching process can not only complete our teaching tasks in a limited time, but also improve everyone's interest in learning and convey the role of mathematical thinking. It plays an important role in our classroom teaching. ?
Newton [3]? (1642~ 1727) is a British mathematician, physicist and astronomer. He comes from a peasant family. 166 1 was admitted to Trinity College, Cambridge University. 1665, the plague was prevalent in London, and Cambridge University was temporarily closed. Newton returned to his hometown and lived in the countryside for two years, thinking about various problems all day and exploring the mysteries of nature. The three great inventions of his life, calculus, gravity and spectral analysis, all sprouted here. Later, when Newton recalled this eventful autumn, he said with deep feelings, "My success should be attributed to the exploration of energy." "Without bold speculation, there can be no great discovery." Newton's calculus theory is mainly embodied in three works: using infinite equation analysis, flow and infinite series, and finding the area of curved polygon. In the book Analysis by Using Infinite Equation, he gave a general method to find the instantaneous rate of change, and clarified that finding the rate of change and finding the area are two mutually inverse problems, thus revealing the relationship between differential and integral, that is, the so-called basic theorem of calculus that is still used today. In Flow Number and Infinite Series, Newton made a more extensive and in-depth exposition of his calculus theory. For example, he changed the static view in the past and thought that variables were generated by the continuous movement of points, lines and surfaces. However, Newton tried to eliminate the confusion caused by infinitesimal in the classic literature about integrable curves, Finding the Area of Curved Edge. There is a way of thinking based on limit in calculus. Newton also said: "If I see farther than Descartes and others, it is only because I stand on the shoulders of giants." ?
Leibniz [3]? (1646~ 1746) is a German mathematician, naturalist philosopher and natural scientist. His first paper on differential calculus, "A new method for finding minimax and tangents, which is also applicable to fractions and irrational numbers, and the calculation of the wonderful types of this new method", is the earliest published document on differential calculus in history. He is also the greatest symbolist in history. He once said, "If you want to invent, you must choose the right symbol. To do this, you have to use a few simple symbols to express or describe the inner essence of things more faithfully, thus minimizing people's thinking labor. " For example,? dx、dy、∫、log？ Wait, he started it all. His superior symbols brought great convenience to the development of analytical science in the future. ?
These are just two of the most dazzling pearls we picked in the sea of mathematical figures in Wang Yang, and they are just a brief introduction to their lives and achievements. The introduction of these contents does not occupy much "precious" time in class, but through these, we can see vividly that the development of mathematics is tortuous, and the emergence of an important concept cannot be separated from practical problems. Only by thinking about practical problems can we find the essence of the problems and abstract mathematical ideas. There are also concepts such as "infinitesimal" and "flow number" that the author often uses in solving practical problems, which makes us realize the importance of mastering basic concepts correctly and skillfully for understanding mathematical ideas. For the mathematical symbols that we usually regard as boring, it is the most direct and concise tool to express mathematical thinking. And from the words and deeds of the pioneers, we can feel the scientists' academic attitude and persistent pursuit of knowledge, which can often inspire everyone to study hard and forge ahead. ?
Finally, we believe that as advanced mathematics teachers, our purpose is not only to impart mathematical knowledge to everyone, but more importantly, to let everyone master mathematical ideas and improve their mathematical literacy in the process of learning mathematical knowledge. Combining the history of mathematics with the teaching of mathematical knowledge can achieve the above goals well. After years of teaching practice, the introduction of mathematical graphics into higher mathematics teaching can play a very good auxiliary role in higher mathematics teaching. We believe that if the teachers of advanced mathematics are familiar with their life, achievements, attitudes, methods and anecdotes, it will be beneficial to the teaching of advanced mathematics and will certainly make it more vivid, interesting and philosophical. For many students who are studying advanced mathematics, once they know the academic achievements and moral demeanor of these predecessors, they will be encouraged, thus improving their interest in learning and achieving greater results.