Our country pedagogical colleges and universities mathematics professional shouldering the task of cultivating primary and secondary mathematics teachers. In math teaching in primary and secondary schools to reveal the thinking process of mathematical knowledge acquisition, students to show mathematical knowledge of the occurrence and development process, to guide students to realize the real process of mathematical thinking, reveal contains the mathematical thinking and methods, stimulate the students' desire for knowledge and reconstruction of university mathematics education philosophy, it for cultivation of talents of good mathematical thinking and innovative spirit has great significance and value.
1, reconstruction of the concept of Mathematics Education
Modern higher education both to impart knowledge as the center and to create knowledge as the focus, Custers professor at the University of California, Berkeley, this made a very vivid metaphor: if the knowledge and information is the current new world economy, then the university is the current generator. Throughout the University's history, the 13th century, the first batch of University in Europe established. However, the medieval university due to disseminated theological ideology at that time, the mission is learning knowledge and the cultivation of personality. To the 17th century, to the University of Halle, Germany on behalf of the true meaning of the university there, breaking the monopoly of the medieval universities of higher education, since then, the university has become the center of profound knowledge of institutions and scientific, scholarly, and "academic freedom" and "academic autonomy" become the World University of basic principles.
In the undergraduate teaching of Chinese mathematics, mainly oriented to impart knowledge, to mining in the textbooks of history of mathematics and mathematics method of thinking and how to reveal the acquisition of mathematical knowledge thinking seriously enough, but in 1998 the Ministry of Education promulgated the "ordinary higher school undergraduate professional directory and professional presentation", the included in the history of mathematics undergraduates in mathematics and Applied Mathematics (teacher training) one of the main courses, and in "business training requirements" clearly put forward math graduates should understand the overview of the development of modern mathematics and in social development ". Ministry of education will also be included in the history of mathematics in senior high school mathematics elective course, in the "general high school mathematics curriculum standard (Experiment)"
It also made clear that "mathematical inquiry, mathematical modeling, mathematical culture should run through the whole high school mathematics curriculum", "mathematics curriculum should reflect the history, application and development trend of mathematics." Therefore, academic is the logical starting point of the University, for knowledge transfer, critique and exploration is the eternal theme, a profound mathematics educational and teaching reform in Colleges and universities in our country are widely spread, reform of the center is training what kind of talent, how to cultivate such talents. Offer solutions, however, because mathematics is a very old subject teaching education situation is not ideal, in most people's eyes, mathematics is only a pile of formula and symbol, obscure abstruse and boring and difficult to learn, there is no practical use, so the mathematics is at a distance of attitude to mathematics spirit and thought also may not be able to comprehend and understand, most students will be in addition to the book of mathematical problems, few would have thought in the textbook contains the history of mathematics and mathematics thinking method, and the high colleges and universities mathematics departments of the leadership, teachers and students in general are not attaching too much importance to the log school history curriculum and teaching, but the mathematics basic course and specialized course is very important. Therefore, in mathematics teaching to cultivate mathematical talents with innovative spirit, in addition to strengthen the study of basic courses and professional courses must also be pay attention to cultivation of students with good mathematics thinking quality, mining materials contained in the mathematics history and mathematics thinking method, reveal the mathematical knowledge of the occurrence, development and settlement process, put forward a question often than solving a problem is more important, stimulates the student to study mathematics of curiosity and of scientific spirit of exploration and methods, students cultivate the spirit of innovation.
2, mathematics history and mathematics thought method of mining
At present, in the mathematics classroom teaching competition in primary and secondary schools, primary and secondary school math teacher together as much as possible to the teaching contents and the history of mathematics thought and method, in the classroom teaching process, reflect the occurrence of mathematical knowledge, develop and solve the whole thinking process, and the majority of university teachers is to impart knowledge, very few to analysis the textbooks contain the history of mathematics knowledge and mathematics method of thinking, not to think of mathematics knowledge from primary schools to universities and to the process of occurrence and development of mathematics in the forefront of the University, to study the lack of overall understanding and grasp, knowing but not the why, most students do not think how much contact the university mathematics curriculum in primary and secondary schools and the number of teaching, not much to guide future elementary school mathematics teaching, lack of learning motivation, learning a lost one The door, so that good mathematics thinking has not been formed, but the history of mathematics and the way of thinking in the textbooks can be seen everywhere, the key is to dig it, the need to look at the history of mathematics books, to master the knowledge of mathematical history is a prerequisite for a qualified primary school mathematics teachers, stimulate students' interest in learning, knowledge reduction and the way of thinking process and other aspects of course impossible, the history of mathematics knowledge plays an irreplaceable role in mathematics, especially it plays a guiding role in high school students, not only the knowledge and mathematics knowledge linked elementary school mathematics, understanding mathematics knowledge and development of the original. But also to the students after graduation engaged in the teaching of primary and middle school education to provide ideas and methods of instruction, and materials contained in the history of mathematics and the way of thinking too much, the following two cases To explain.
2.1, from the Pythagorean theorem to Diophantine equation
Talked about indefinite equation, people first thought is the Pythagorean number, that equation (1) integer solutions. And Newton made "a cow eats grass" and Han soldiers and belong to indeterminate equation, and the oldest adventitious equation is the Pythagorean number the, equation (1) is also known for Beeta Pythagorean equation, (3,4,5) is known as the Pythagorean number. Generally, if (... Is the Pythagorean number, (so)... ) is a Pythagorean number, more than 3000 years ago, people in the study of the Pythagorean number, but also master the knowledge on the Pythagorean number. Used in Princeton University, Columbia Museum of No. 322 babylonian mathematics mud board in 1945 by new green Bauer (Otto Neugebauer) and sacks (Sachs) explained, people surprised to found the Babylonians lists 15 groups of Pythagorean number.
In the 3rd century AD Diophantine (diophante, 246~330) is found for all Pythagorean number method, the method is: design, two positive integers, is a perfect square, then... Is a group of Pythagorean number. And and Diophantine are with the era of the Wei and Jin period in China mathematicians Liuhui by geometrical method found for the Pythagorean formula: design, with positive parity, and then... This result is set out in the notes of the nine chapter of the 263 chapter of the "chapter" by Liu Hui.
Equation (1) to solve the problem, people should consider is, even more general problems, in 1621 the French mathematician Felma made in Diophantine academic monograph "blank" arithmetic in this book to write his a reading notes: "a more than two times the power is divided into two same power, this is not possible, for this, I am sure have found a wonderful proofs, but here the gap is too small, can't write." that is, for all positive integers, Diophantine equation has no nonzero solutions, and this is the famous Felma theorem. Early studies are from algebra to carry out, although some progress, but this problem can not be solved fundamentally, later it was discovered that the relationship between the Felma theorem, into: (a) curve without any rational point.
As a result, Fermat's theorem and geometric contact on the line, the British Mathematical Mo Dyer (L.J.Mordell) in 1922 proposed a conjecture: rational algebraic curve of genus has at most finitely many understanding. German mathematician Va Eling J (Faltings) in 1983 to prove that the modal conjecture: at that time, is the algebraic curve of genus the Va Eling J won the highest prize in Mathematics in 1986 Fields Award (Fields), on the surface, the results of Va Eling J from Fermat's last theorem the end seems to be only one step away, but the arrival of the road is another Japanese Valley in 1955 proposed a conjecture: "elliptic curves are modular curve" now, just to prove that the Taniyama conjecture, then Fermat's theorem to obtain the certificate.1993 in June, Professor Wiles of Princeton University in New Jersey in the United States (A.Wiles) at the University of Cambridge Newton Mathematical Science Research Institute held "Iwasawa theory, mode and adic said" the academic conference, was invited to give a "elliptic curves, modular forms and Gawa said" series of reports, valley conjecture is proved in the end of the report, to prove Fermat's last theorem, all mathematicians feel excited and shocked than a sleepy. From the mathematics community for more than 300 years to solve the problem of the world, however that there are loopholes in need of repair, repair of 1994 Wiles finally completed his thesis, America's "annals of mathematics" published in 1995 by Wiles on Fermat's last theorem 2 papers to Fermat's last theorem was finally dust settles, and in 1994 Wiles just over 40 old, missed the opportunity to obtain fields prize in 1996, became the youngest winner of the Wolf prize.
What is worth mentioning is that Fermat's last theorem is a will the hen that laid the golden eggs, prove Fermat's assertion is not very important, important is in up to more than 300 years long exploring produced in the process of thought, method and theory, such as Fermat's infinite descent method, Euler's second type theory and factor decomposition, the kullmer of Cyclotomic number factor decomposition and so on, because this 300 years for Fermat's last theorem and struggling to find mathematicians wisdom produces the many new branch of mathematics, has enriched the treasure house of mathematics, found wide application fields of mathematics. The Pearl in the crown of even and odd and even in Chinese people's memory and never forget the famous mathematical questions of Goldbach's conjecture, 1743, Goldbach to mathematician Euler's letter put forward two questions: "whether there is any small 6 can be expressed as two odd primes? Is not any not less than 9 can be expressed as three odd primes?" Euler in letter wrote: "no more than 6 are two odd prime and, although I can't prove it, but I had no doubt that this is a perfectly correct theorem." this is known for mathematical Goldbach's conjecture.
Chinese mathematician Wang, pan Chengdong therefore made outstanding contributions, especially in 1965, received the "1 + 2" the Chen Yingrun 1933-1996. His achievements so far no one beyond, but formal papers are published in the 1973 Science in China, more than 40 years past the, this last way away is still not over, the future removal of mathematical crown pearl who will be anyone?? in the process of exploring this may create some new mathematics thoughts, methods and theory, creative wonders, to further enrich mathematical treasure, to the cause of human science to make greater contribution.
2.2, from the radical solution of the equation to the permutation group
Junior high school students will root formula for solving a yuan quadratic equation, however, back in about 1700 BC, the Egyptians will solve a yuan a equation, most late 6th century BC, the Babylonians will solve a yuan quadratic equation, but because when people on the negative, no physical number and complex, the lack of awareness until the 16th century, problem solving a quadratic equation was solved. In sixteenth Century, people have some special solution three times and solve the problem of general equation is three equations a world-class problem.1535 in February 22nd, the Church of Duomo in Italy held a high-profile Mathematics Tournament, is a professor of mathematics at Venice tower Ertaliya, one side is Italy Poirot and mathematics President Ferlo of the Russian students Philippines rules of the game, each of the 30 road three order equation problem, give the answer, who finished first wins the result only two hours, tal Talia answers out of the 30 party to solve three equations problem, and the Philippines Russian there is no problem to do, shocking, this means that tal Talia learned three equations to solve. "Tal Talia" is the meaning of "stutter", tall tower in Asia the young year period is the value meaning law war, on 19 February 1512, French army looted Brescia, tal Talia hide in the church also not spared, with his tongue and the head was chopped, fall the stutter, tal Talia is his nickname, few people know his real name Darrow (Fontana, he put method for solving a cubic equation as a secret weapon, not to tell anybody. But in many ask for advice, a man named Pierre Cardin is a famous doctor, is also a famous mathematician, tal Talia classics do not live of his rhetoric, and he swore to lifelong conservative secret under the, finally in a first statement of obscure poetry in the solution of a cubic equation method to tell Cardin, ten years later, Cardan fails to comply with his promise, published in a book he wrote "Dafa" the solution of a cubic equation, the mathematical community to solving a cubic equation roots formula called Cardan formula.
In the 16th century have been solved a yuan three times and yuan a four equation to solve the problem, then many mathematicians are eager to solve the equation more than five times and five times, we are convinced that a quintic equation a radical solution. Time past 200 years, one of the five equations of the radical solution is still not found, in eighteenth Century, the French mathematician Lagrange found three times, four times the equation's success is the use of the auxiliary equation of a low frequency (resolvent), and its replacement times all the root of equation on the 1 - 1 yuan to construct a mapping, however five equations resolvent has not been found, Lagrange concluded that the solution of equation of five degree or beyond human intelligence, or is the root of the expression is different from everything that was known at that time. The first proof of algebraic equations five times and five times more no radical solution is Norway's young mathematician Abel, he is the conversion of the idea, try to prove that the solution does not exist, but the problem has not ended, he said five times and five times more than the mean equation coefficients are all written in general The equation can be solved by radicals is for some specific equations, but some of it seems very simple Fang Chengru cannot use radical solution, so there is a question of a specific equation of what is the necessary and sufficient conditions for radical solution? This problem by the French mathematician Galo young (E.Galois, 1811-1832) draw a satisfactory conclusion, he is no one was able to understand the permutation group to solve, and the concept of permutation group at the time including the top mathematicians feel be rather baffling, unfortunately Galo died in a fight for love, but he wrote in a duel on the eve of his proof please, friend to give Gauss or Yalobi (C.G.J.Jacobi, 1804-1851), after 14 years, Liu Neville (J.Liouville, 1809-1882) published Galo's paper was first published in 1846, the manuscript is not long "Guan Yufang In the condition of the radical solution from the record ", but very deep thought of very simple, finally solved the equations out around the difficulties with the crux of radical, and this difficulty is great mathematicians fruitless struggle over the Galois theory, it is gradually recognized by people, in 1870, Jordan in the" replacement and algebraic methods on "a book a comprehensive introduction to the Galois theory, and Galois, ultra era thought has finally been recognized, in the process of solving the equation of Galois radical solution, Galois equation in the achievement is his theory first introduced the general concept of new important, these concepts later in the whole mathematics plays an important role in a brand-new subject in this extension of the mathematical generation.
Therefore, mathematical thought is extremely important for a mathematics worker, a good mathematical thinking will have a profound influence on the development of mathematics, as long as we try to tap the materials contained in the history of mathematics thought and method, give full play to the role of teachers organizers, guides and participants in teaching activities, the history of mathematics and the content of the textbook knowledge and re combination of materials for two times the development, re organization, cultivate the spirit of exploration, pioneering consciousness and innovation ability, make mathematics more vivid and interesting, which will be conducive to help students understand the background and development process of knowledge, so that students in the process of learning knowledge fully formed feel the hidden history of mathematics thinking method and the wisdom of promoting students' autonomous learning, exploration of cultivating thinking spirit and mathematics and science quality.
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