Professor Fujitsu Hiroshi, a Japanese mathematician, pointed out in the report of the Ninth International Conference on Mathematics Education that there were four mathematical peaks in human history: the first time was the deductive mathematics period in ancient Greece, which represented the birth of mathematics as a scientific form and the first major victory of human "rational thinking"; The second is Newton-Leibniz's calculus period, which was born to meet the needs of the industrial revolution and achieved great success in the fields of mechanics, optics and engineering technology. The third is the axiomatic period of formalism represented by Hilbert; The fourth is the new mathematics period marked by computer technology, and we are now in this period. The three major crises in the history of mathematics are the measurement of incommensurability in ancient Greece, the demonstration based on calculus in 17 and 18 centuries, and the paradox of set theory in the early 20th century, which are closely related to the first three peaks. This connection is by no means accidental, but the necessity of mathematics as a science that pursues perfection. From this connection, students can find that mathematics pursues clarity, accuracy and strictness, and it is not allowed to be messy or vague. At this time, students can easily understand the three basic characteristics of mathematics-abstraction, rigor and extensive application.
At the same time, introducing the necessary knowledge of the history of mathematics can make students have a deeper understanding of the background of the problems they have learned in their usual study and realize that mathematics is by no means isolated. 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. In the new mathematics period in which we live, mathematics (not only natural science) has gradually entered the field of social science and played an unexpected role. It can be said that all high technologies are supported by some kind of mathematical technology, which has become an important feature of the knowledge economy era. These understandings are necessary and indispensable for a high school student who has studied mathematics for more than ten years.
Second, studying the history of mathematics is conducive to cultivating students' correct mathematical thinking mode.
Nowadays, mathematics textbooks are generally scrutinized repeatedly, and the language is very concise. In order to keep the knowledge systematic, the teaching content is arranged in the order of definition, theorem, proof, inference and example, lacking natural thinking mode and introducing the connotation of mathematical knowledge and the creation process of corresponding knowledge. Although it is beneficial for students to accept knowledge, it is easy for students to have the wrong view that mathematical knowledge is defined first, then properties and theorems are summarized, and then used to solve problems. Therefore, there are such contradictions in the process of teaching and learning: on the one hand, educators systematize knowledge in order to let students master mathematics knowledge faster and better; On the other hand, systematic knowledge can't let students know that knowledge is mostly matured step by step by asking questions, guessing, demonstrating, testing and perfecting. Affect the formation of students' correct mathematical thinking mode.
The study of the history of mathematics is conducive to alleviating this contradiction. By explaining some related mathematical history, students can have a clear understanding of the generation process of mathematical knowledge while learning systematic mathematical knowledge, thus cultivating students' correct mathematical thinking mode. There are many such examples, such as the generation of calculus: the traditional deduction system of Euclidean geometry can not produce calculus, but it was created by Newton and Leibniz in order to meet the needs of the first industrial revolution, inspired by the ancient Greek ideas of "exhaustive method" and "finding the arcuate area of parabola". At the beginning of its emergence, the definition of "infinitesimal" was vague and not as strict as what we see now. With the continuous supplement and improvement of mathematicians,
The study of the history of mathematics can guide students to form the habit of exploration and research, to discover and understand what has really been created in the process of a problem from its emergence to its solution, and what ideas and methods represent the substantial progress of the content compared with the previous content. Understanding this creative process can help students experience a vivid and real mathematical thinking process, and help students to form a deeper understanding of some mathematical problems and understand the realistic source and application of mathematical knowledge, instead of simply accepting the knowledge taught by teachers, so that they can gradually form a correct mathematical thinking mode in this process of continuous learning, exploration and research.
Thirdly, studying the history of mathematics is conducive to cultivating students' interest in mathematics and stimulating their motivation to learn mathematics.
Motivation is a force that inspires and pushes people to act. From a psychological point of view, motivation can be divided into two parts. People's curiosity, thirst for knowledge and hobbies constitute the internal motivation conducive to creation; Social responsibility is the external motive force of creation. Interest is the best motivation. Japanese middle school students get the first place in the total score of international IEA survey, and at the same time, they find that the proportion of Japanese students who don't like mathematics is also the first, which shows that their good grades are obtained under the pressure of society, parents and schools. What's the situation in China? There is no comprehensive report, but a survey of high school students studying mathematics in four middle schools in Xinxiang City, Henan Province shows that the proportion of students who "I don't like mathematics, but I have to learn it well for the college entrance examination" is as high as 62.2 1%, while only 23. 12% are interested in mathematics. It can be seen that the current middle school students' learning motivation is not clear and their interest in mathematics is not enough, which greatly affects the effect of learning mathematics. But this is not because mathematics itself is boring, but because it is ignored by our teaching. Combining the history of mathematics properly in mathematics education is conducive to cultivating students' interest in mathematics and overcoming the negative tendency of motivation factors.
There are many contents in the history of mathematics that can cultivate students' interest in learning, mainly in these aspects: First, small games related to mathematics, such as skillfully holding matchsticks, Rubik's Cube, and businessmen crossing the river, are highly operable and can achieve good results as classroom activities or after-school research. Second, some famous mathematical problems in history, such as the Seven Bridges problem and Goldbach's conjecture, often have vivid cultural backgrounds and are easy to arouse students' interest. There are also some famous mathematicians' lives and anecdotes, such as the stories that some young mathematicians have become useful. From Abel mentioned in the standard to Galois, Abel is 22 years old, which proves that there is no formula for finding the roots of algebraic equations with more than five degrees. Galois was only 65,438+08 years old when he founded Group Theory. Pascal, a French mathematician, became one of the founders of projective geometry at the age of 16, and invented the original calculator at the age of 19; /kloc-at the age of 0/9, the German mathematician Gauss solved the problem of drawing regular polygons; At the age of 20, he proved the basic theorem of algebra; At the age of 24, he published Arithmetic Research, which influenced the development of number theory in the whole19th century and is still very important. There are also many examples that many mathematicians from poor and humble backgrounds have finally made outstanding achievements in mathematical research through their own efforts. For example,/kloc-Steiner, a great geometer in the 9th century, was born in a peasant family and grew up as a farmer. He didn't learn to write until 14 years old, and 18 years old officially began to study. Later, he made a living as a private school teacher. After hard work, he finally got an important job in mathematics at the age of 30 and became famous in one fell swoop. If these students are interested and knowledgeable in teaching, eliminate students' fear of mathematics and increase the attraction of mathematics, mathematics learning may no longer be compulsory.
Fourthly, studying the history of mathematics provides a stage for moral education.
Under the requirements of "Standards", moral education no longer focuses on politics, Chinese and history as before. With the addition of the history of mathematics, mathematics education has a stronger moral function. Let's discuss it from the following aspects.
First of all, studying the history of mathematics can educate students in patriotism. Most of the current middle school textbooks talk about foreign mathematical achievements, and seldom talk about China's contribution to the history of mathematics. In fact, China's mathematics has a glorious tradition, including a number of outstanding mathematicians such as Liu Hui, Zu Chongzhi, Zuxuan, Yang Hui, Qin, Zhu Shijie, and China's remainder theorem, Zuxuan axiom, "Secant" and other mathematical achievements with world influence, many of which were studied many years earlier than foreign countries. The third topic in the selected lecture on the history of mathematics in the Standard is "the treasure of ancient mathematics in China", which refers to the representative achievements of ancient mathematics in China, such as Nine Chapters of Arithmetic and Sun Tzu's Theorem.
However, at this stage, patriotism education can't just stop lamenting the glory of ancient mathematics in China. Since the Ming Dynasty, China's mathematics has gradually fallen behind the West. At the beginning of the 20th century, mathematicians in China began the difficult course of learning and catching up with western advanced mathematics. The "Special Lecture on the History of Mathematics" in the standard 1 1-The Development of Modern Mathematics in China also mentioned that it is necessary to introduce "the glorious course of mathematicians in modern China trying to catch up with the advanced level of mathematics in the world". Under the requirements of the new era, in addition to enhancing students' national pride, we should also cultivate students' "international consciousness" and let students realize that patriotism is not reflected in "drawing inferences from others". In scientific discovery, all mankind should learn from each other, learn from each other and improve together. We should respect foreign mathematical achievements, study with an open mind and "make foreign things serve China".
Secondly, studying the history of mathematics can guide students to learn the excellent qualities of mathematicians. The progress and development of any science is not smooth sailing. The discovery of irrational numbers, the creation of non-Euclidean geometry, the discovery of calculus and so on all illustrate this point. Mathematicians either stick to the truth and are not afraid of authority, or persevere and fight for it, and many even devote their lives to it. When the enemy invaded the city and endangered his life, Archimedes was still immersed in mathematical research, in order to "I can't leave an unproven theorem to future generations." Euler 3 1 year-old was blind in his right eye and had poor vision in his later years. However, he continued his research with strong perseverance, and many papers were long, so that his papers were still published in the Journal of Chinese Academy of Sciences 10 years after his death. For those students who give up when they encounter a little complicated calculation and a little complicated proof in their study, it will play an important role for them to correctly look at the difficulties encountered in the learning process and establish their confidence in learning mathematics by introducing stories about how some great mathematicians persist in pursuing when they encounter setbacks.
Finally, studying the history of mathematics can improve students' aesthetic quality. Mathematics is beautiful, and countless mathematicians are impressed by its beauty. Being able to appreciate beautiful things is a basic quality of human beings, and learning the history of mathematics can guide students to understand the beauty of mathematics. Many famous mathematical theorems and principles shine with the brilliance of aesthetics. For example, Pythagorean Theorem (Pythagorean Theorem) is a very concise and profound theorem that everyone is very familiar with in elementary mathematics, and its application range is very wide. For more than two thousand years, it has aroused countless people's interest in mathematics. The famous Italian painter leonardo da vinci, Indian King Bascara, and the 20th President of the United States have all given proof. 1940, American mathematician loomis collected 370 proofs of Pythagorean propositional art in the second edition, which fully demonstrated the infinite charm of this theorem. The golden section is also beautiful and charming. As early as the 6th century BC, Pythagoras School studied it. In modern times, people are surprised to find that it has a very close internal relationship with the famous Fibonacci sequence. At the same time, when we sigh and appreciate the symmetrical beauty of geometric figures, the simple beauty of ruler drawing, the unified beauty of volume triangle formula and the singular beauty of non-Euclidean geometry, we can form a good mathematical emotional experience and improve our mathematical literacy and aesthetic quality, which is a new breakthrough in moral education.
refer to
1 The people of China * * * and the Ministry of Education of People's Republic of China (PRC) have formulated the Mathematics Curriculum Standard for Ordinary Senior High Schools (Experiment). People's Education Press, 2003.
2 Introduction to Mathematics Education edited by Li Jun Higher Education Press, 2003.
3 Introduction to the History of Mathematics, Li Wenlin Higher Education Press, 2002.
4 Mathematics and Culture Higher Education Press 1999, edited by Zhang Chuting, Department of Higher Education, Ministry of Education.
5 Li Huaxuan's Investigation on Mathematics Learning of Senior High School Students Journal of Xinxiang Institute of Education, 04, 2003
This paper is an exchange paper of the 2004 annual meeting of the National Research Association of Mathematics Education in Normal Universities.