Confucius said, "Knowing is not as good as being kind, and being kind is not as good as being happy." Indeed, children with mathematical genius are "good people" and "happy people" in mathematics.

Because they like math, they are very active in math class, scrambling to answer questions, scrambling to calculate and prove on the blackboard, and boldly standing on the podium to explain their math knowledge and problem-solving methods to their classmates. After class, the students gave each other some intelligence tests, played some math games and handed out some interesting math extracurricular books. Interest and hobby are the motivation of diligence, which drives them to calculate and prove those math problems that ordinary people think are boring.

Yan Huafei, a girl who won the silver medal in the 30th International Mathematical Olympiad in her sophomore year, said: "The charm of mathematics itself is attracting me. I plunged into the ocean of problems and I will try my best on all the problems I can get. " Another student talked about his views on mathematics and said, "Mathematics is a mysterious temple, a beautiful maze, and traveling there is very interesting."

Another student said: "Competition mathematics is simply an art and an outstanding performance of human creative thinking. In the ideal society of the future, participating in math competitions will definitely become a popular form of entertainment. "

Second, outstanding self-study ability.

This is due to their strong understanding of mathematics and teachers' conscious cultivation. A good teacher can not only teach mathematics, but also inspire students to learn mathematics by themselves.

In math class, I consciously cultivate their self-study ability, and I always want them to read new lessons first. These gifted children in mathematics can easily understand and learn the knowledge in books, and some of them can immediately explain, calculate and prove on the blackboard. Because of their strong self-study ability, they are eager to study ahead of time after class.

Some of them taught themselves high school mathematics in junior high school and some college mathematics courses in high school. There was a student named Qian who taught himself how to solve equations in junior high school, how to analyze geometry and trigonometry in senior high school, and how to teach himself college calculus in primary school. Although what he has learned is still incomplete and inaccurate, as a child of eleven or twelve years old, it is unimaginable to reach this level without strong self-learning ability in mathematics.

Third, a strong sense of independence

Children with mathematical genius generally do not believe in books, teaching reference books or teachers. Their math exercise books are wonderful, often with multiple solutions to one problem and multiple solutions to one problem, and sometimes they can come up with more beautiful and concise solutions than reference books.

When a student recalled his study at that time, he wrote: "Often we will find that the answers given in the' teaching reference' are not perfect, or there is another simple solution, so we will write our opinions and ideas in the exercise book. When teachers review, they usually pay more attention to our unique insights and mistakes, and sometimes they will take them out for discussion in the whole class. " When they have different opinions, they also dare to argue with the teacher. We think this is an extremely valuable personality trait. With this quality, it is possible to have scientists like Bruno and Galileo who dare to stick to the truth in the future.

Fourth, extraordinary memory.

Children with mathematical genius generally have an amazing memory for mathematical symbols and numbers. Many students I have taught can recite dozens or hundreds of digits after π decimal point. Many people have a strong memory of classical literary works and can recite hundreds of Chinese and foreign poems and essays.

Once, I asked Professor Wu from the Institute of Mathematics of Chinese Academy of Sciences to give my students a test. The test results show that Yan Huafei, a girl of 15 years old, has achieved incredible results. So the professor sent two papers in succession, and the professor was shocked by the evaluation results. He couldn't believe that the little girl would give such beautiful answers, because these questions involved mathematical knowledge far beyond the courses she had studied. Yan Huafei told the professor that the questions on the test paper had been read before! The professor exclaimed, "This little girl really never forgets anything!"

Fifth, extraordinary mental arithmetic ability.

The most prominent feature of mathematics gifted children in junior high school is their quick mental arithmetic. The teacher's question has just been written on the blackboard, and the students' answers have been shouted out.

Among them, the most prominent is the champion of the National Chinese Junior Mathematics Cup. In junior high school, he always worked hard to solve problems and wrote the answers in one step without using draft paper. If you ask him to write the process, he says no, and the answer can be worked out directly. Some questions are more complicated, and he can get the answer only by writing you a few steps. In order to train him to solve problems according to the standard, I asked him to write the process of solving problems on the blackboard in class. As a result, he can only write a few simple steps and the algorithm is unique. I didn't solve his problem until high school. Later, he won the second prize in the national high school mathematics league and the gold medal (first prize) in the all-Russian mathematics competition, but he failed to win the highest prize in the end, and he was often deducted for jumping too much in the process of solving problems. Jian Xu's situation reminds me of the research of Russian psychologist V·A· Curut, who also found that children with mathematical genius have the ability to "operate quickly from concrete examples and omit intermediate steps in the process of thinking, that is, to shift from' direct' sequence operation to opposite sequence operation".

Another gifted child named Liu told me about one of his phenomena. He said: "I don't know why my brain is always in an advanced state, that is, what I think in my brain is always ahead of my handwriting." But I feel comfortable in thinking activities that don't need handwriting, such as solving difficult problems.

Sixth, strong will quality.

Children with mathematical genius generally have superhuman perseverance.

I have done such a survey and asked them to talk about how long they can "hold back" themselves without asking others in order to understand a difficult problem or prove a theorem. The students talked about their own situation one after another, some said that they would persist for several hours, some said that they would persist for one day, and some said that they would persist for one or two months. A classmate said that he had done a problem independently for half a year. This shows that they are persistent in their studies.

Some students also show superhuman perseverance in physical exercise. I have a student named Cha Sang Yuan. In high school, he won the first prize in the national mathematics competition, the first prize in the national physics competition, and many awards in the national and Beijing computer competitions. When he first entered the gifted children's class, he saw that there were many strong players in the class, so he studied hard and collapsed. Later, under my advice and the specific guidance of the PE teacher, he began to practice long-distance running. He runs four times around the 400-meter track every morning, and then runs two times at top speed, which keeps him going for several years, not only strengthening his physique, but also exercising his will.

VII. Creativity

I found that gifted children in mathematics all showed some creativity in the learning process.

For example, a girl named Yan entered the Chinese Department of Beijing Normal University after being interviewed by two professors. She once said: "I have tried to apply the methods I learned in Chinese class, such as writing paragraphs, finding centers and outlines, to the reading of science books, and I have also received good results." Under the infiltration of complementary arts and sciences, my knowledge structure has been effectively adjusted and balanced. "This is obviously the creative behavior caused by the positive transfer of learning.

Another student, Wang Kuanhong, entered the Department of Biochemistry in Peking University without examination after graduating from high school, and was admitted as a graduate student by Harvard University on a full scholarship with excellent TOEFL scores. He wrote: "There is a vague indirect but comparable connection between two seemingly different disciplines. Only by being good at association and analogy can the learning methods and characteristics of one subject be applied to another course. For example, I think there is such a connection between chemistry and English. Just as the laws of chemistry exist in the ever-changing nature, you can find out the laws of language from the complicated language phenomena (not necessarily limited to grammar, sometimes just a habit, and a law is like an empirical formula in chemistry). Using this law, we can understand and clarify various phenomena and create new content. Just as English emphasizes the sense of language, chemistry also has its own unique sense of chemistry. Through more practice and practice, we can cultivate this feeling. Many details and characteristics can be exchanged with each other, which is easy to remember, and we can also find the key points of many problems directly. This generalized associative analogy method can organically interweave all disciplines into a whole, which is similar to other disciplines and develops together. "

How valuable this creative experience is, I seem to have a hunch that they will make creative contributions to mankind in the future.

Eight, ambitious and negative.

Students in math gifted children's classes generally have a high level of ideals and ambitions.

This first stems from their strong self-confidence. Every student has a self-confidence that "each has his own strengths, and I can do what you can through hard work"; If they fall behind in the exam results, they can treat them correctly and will not be discouraged. One student wrote: "It can't be said that there is no pressure in the exam, but most students have a kind of self-confidence. This poor performance only shows that they have not learned enough in this field. As long as they study hard, they will definitely improve. " In fact, everyone's grades and rankings in the class are constantly changing, and many students used to take the lead.

Secondly, the high-level ideal ambition of gifted children in mathematics lies in the requirements and encouragement of teachers. Yan Huafei remembered the teacher's words: "You should have such an ambition to be the first in the country! First of all, you must strive to stand in the top of the Austrian school. " Yan Huafei recalled: "What a great spirit! I think from teachers, we can not only learn knowledge, but more importantly, we can learn the momentum that pervades Changhong ... Teachers often tell us:' If people want to do great things, they must sacrifice some small aspects'. Only those who have lofty goals and broad minds can do great things. "

Above, I summarized the characteristics of gifted children in mathematics. Below, according to my teaching practice for many years, I would like to talk about some understandings of the research on the characteristics of extraordinary children:

First, the early characteristics of gifted children in mathematics are very distinct, and in the middle school stage, these characteristics are more prominent and show extraordinary. Mathematics is boring or even difficult for quite a few people, but it is the pursuit of fun and self-confidence for children with mathematical talent.

Secondly, the characteristics of gifted children in mathematics need to be identified and guided in time, and the environment and conditions suitable for their development need to be created. In the past eight years, it has been found that children with mathematical genius have high requirements for teachers and education and teaching environment.

Thirdly, gifted children in mathematics have strong migration and development ability. They are not only excellent in mathematics, but also can be transferred to many aspects. The previous training was often too narrow, ignoring the cultivation of their talents in many aspects and the possibility of their development in many aspects. Mathematics is sometimes their purpose, but it can also be used as a tool.