As an international competition, the International Mathematical Olympiad was put forward by international mathematical education experts, which exceeded the level of compulsory education in various countries and was much more difficult than the college entrance examination. According to experts, only 5% of children with extraordinary intelligence are suitable for learning the Olympic Mathematics, and it is rare to reach the top of the international mathematical Olympics all the way.
In recent years, all kinds of extracurricular mathematics classes and training courses in China, which are much higher than the classroom mathematics teaching content, have been named "Olympiad", which makes the cultivation of "Olympiad" gradually break away from the track of selecting Olympiad athletes and highlights the characteristics of universality. Although many well-known mathematicians and mathematics educators have called for vigilance against the deviation of "Olympic Mathematics", the subtle relationship between the results of "Olympic Mathematics" and secondary school enrollment makes the expansion trend of the connotation of "Olympic Mathematics" hard to stop. After-class mathematics training sponsored by various schools and organizations is all wrapped in the cloak of "Olympic Mathematics", which is synonymous with "Olympic Mathematics" because it is divorced from textbooks and emphasizes skills.
1, what do you study in Olympiad?
What exactly is "Olympiad"? What is the difference and connection between it and our usual math class? I think most parents and teachers are not very clear. They may think that only those so-called "questions" and "digressions" with novel and strange ideas and great difficulty are "Olympic numbers" Actually, it is not.
I think there is no doubt that Olympiad is still a mathematical subject. Of course, there are also some parts of the Olympic Mathematics related to the mathematics we usually study in class, which is the deepening and perfection of the classroom content; However, the olympiad is more of a seemingly unrelated content in the classroom. So what is this part and where does it come from?
The scope of mathematics is extremely extensive, and the most authoritative classification in the world probably divides mathematics into dozens of categories, with more than 100 subcategories. We started with the linear equation of the senior grade in primary school and graduated from high school. In 1978, the mathematical categories involved were plane geometry, trigonometric function, linear equation (group), analytic geometry, solid geometry, set theory, inequality, sequence and so on. As mathematics education, of course, we should focus on these contents, because they are the core methods and fields of mathematics, but these contents are not fully covered even in the category of elementary mathematics.
Ok, what's the Olympic number? In fact, it is the basic content of some branches of mathematics that we usually don't talk about in math class, such as graph theory, combinatorial mathematics, number theory, important mathematical ideas, such as structural thinking, specialized thinking, transformation thinking and so on. The selection of these contents is very scientific, because the basic methods and simple applications in these fields do not need special mathematical tools, and they are very interesting. These methods naturally help to cultivate students' interest in mathematics and expand their thinking and knowledge.
By the way, in fact, there are a lot of contents in the Olympic Mathematics, especially in the middle and lower grades, which are all derived from the methods and ideas of China's ancient mathematical monographs, such as "profit and loss problem", such as "the chicken and the rabbit are in the same cage", and the "China's surplus theorem" to be introduced into the Olympic Mathematics in high school or middle school. I think these methods seem simple, but they really condense the extraordinary wisdom of ancient mathematicians in China, which is quite different from the western mathematical equation thought and unique. I think this is also a part of China's outstanding cultural heritage, and it is naturally beneficial to learn it.
In the teaching practice of "Olympic Mathematics", we don't blindly pursue the difficulty and strangeness, but always operate with the purpose of "laying a solid foundation and using it flexibly", mainly to expand students' thinking and deepen their understanding of some seemingly insignificant common sense and small conclusions in mathematics, such as the multiplication and distribution law can be used to solve the area problem of any quadrilateral with vertical diagonal, and the summation of geometric series is essentially related to the method of cyclic fraction. But also involves a little idea of "structure" and so on. In the ordinary place, it is extraordinary, turning decay into magic, allowing students to deepen their understanding of mathematics in the exclamation of "Why didn't I think of it" and make progress unconsciously.
2. What kind of students are suitable for the Olympics?
In my opinion, the Olympic Mathematics is mainly aimed at students who have a solid knowledge of mathematics in class, have spare capacity for learning and have a certain interest in mathematics.
But at the same time, we should also see that there are differences among students who are suitable for studying Olympic Mathematics. Learning Olympic Mathematics should also be divided into levels and difficulties, and different contents and difficulties should be arranged according to different students, which should be adapted to different people, places and times. I think the choice of difficulty is best based on the fact that students can understand it in class and master it basically after class. On the other hand, I strongly disapprove of putting the cart before the horse. If you haven't learned the content in the usual math class well for the time being, you should still focus on the math content in the usual class, otherwise it will take time and effort to no avail.
3. "Olympiad" does not mean "learning in advance"
I saw an article on the Internet called "The Olympic Mathematics in Primary Schools is Too Hot", written by Mr. Zhou Jiguang, a special math teacher in Shanghai. In Mr. Zhou's view, Olympic Mathematics seems to be synonymous with "learning in advance". In this article, he said: Recently, the author randomly bought a book "Sprint Gold Medal-Selected and Answered the Latest Excellent Test Questions of the National Primary School Mathematics Olympics" in the Olympic Mathematics "Book Sea" of the Book City, which almost covered the national primary school mathematics competition questions in 2000-2002. I found 38 questions about geometry and did them all. I found that 30 questions need knowledge above grade two, such as Pythagorean theorem, radical operation, proportional line segment, equal product transformation and so on. There are still seven problems that can only be solved with the relevant knowledge of Grade One. Primary school mathematics knowledge can only solve one problem. The algebra problem in the book has a similar situation. Imagine, isn't it difficult for ordinary primary school students to chew on these topics? Such inappropriate advanced training is not only not conducive to the development of students' thinking, but also makes most students afraid, far away from and even hate mathematics. Heavy psychological pressure will hinder the healthy development of students' body and mind, which is deeply worried by many teachers and parents.
I don't agree with what Teacher Zhou said above. First of all, the triangle with the same base and height (or the same base and height) is equal in area, which is the content of the fourth grade of primary school. The so-called "equal product transformation" is actually such a content in the primary school Olympics. Take one more step at most. The ratio of the area of a triangle with the same height is equal to the ratio of the base. As for rotation transformation and reflection transformation, there is no such thing. Proportion is also the content of primary schools. Of course, the content of Shanghai primary school may be less than that of other places, because it has preparatory classes for junior high schools, which is actually equivalent to the sixth grade of ordinary primary schools. The national primary school mathematics competition cannot reduce the outline content because of the special situation in Shanghai. If Mr. Zhou thinks this part of the content is also junior high school, then the problem is really unclear; Secondly, the proportion of line segments is naturally the content of primary schools, as long as it does not involve similar triangles or the proportion theorem of parallel lines. As far as my teaching practice is concerned, the geometric problems in the national primary school mathematics competition can basically be solved by simple transformation of triangle area, and at most, a simple linear equation or letters are added to represent numbers, which is also the content of the fifth grade of primary school. As for Pythagorean theorem, it generally only involves three strands, four chords and five, and it is not necessary to really calculate the square. Even if the calculation is a good number, any radical operation will not appear at all. The author once wrote an article "Analysis of Primary School Geometry" by selecting several competition questions, which introduced some difficult problems in the middle cup. As long as the knowledge of primary school is used, it is only much more flexible.
How about "learning in advance"? I don't feel good either. It's not necessary. So, is there any mathematical knowledge that Olympiad has learned in advance? Yes However, the proportion is very small. In the first part of this article, I explained most of the contents of the Olympiad. It does not overlap with the content of orthodox mathematics class. Do you usually teach the principle of pigeon hole in math class? Can you talk about the problem of the seven bridges at Gettysburg? Can you talk about "chickens and rabbits in the same cage" and "profit and loss problem" in ancient China? Don't talk. At the same time, in our teaching practice, we have always avoided talking about the content of junior high school; What absolute value, real number, algebraic expression (of course, the most basic square difference, complete square sixth grade is still to be taught next semester), strict geometric argument and so on will not be discussed. Some proof problems involved in the sixth grade are also some dyeing problems, such as pigeonhole principle. And the proof of geometric algebra in middle school was not involved in advance.
Let's talk about equations first. As far as the students I contacted are concerned, most students didn't really understand the thinking mode of equations when they learned letters to represent numbers and equations once in primary school. Through the study of olympiad, their cognition has been improved, and they have cultivated good equation thinking, and they also understand that equations and solutions are two completely separate mathematical thinking activities. Of course, there are a little more equations for primary school Olympic math requirements than primary school textbooks. The sixth grade requires flexible use of linear equations in the last semester and simple binary linear equations in the next semester, but it will never involve the solution of quadratic equations and radical operation.
Therefore, the Olympiad is not "learning in advance", let alone "acrobatics in mathematics" as some people say. It is mathematics outside the classroom, and mathematics in the classroom is the relationship between the trunk and branches, which is not only the promotion and deepening of the classroom, but also a mathematical garden that broadens the horizon. The so-called "learning in advance" brings all kinds of burdens and adverse effects to students, which does not apply to "Olympic Mathematics", at least not to most of its contents.