How to tell if someone has mathematical talent?

For those who think that they or a friend or relative is talented in mathematics, 80% to 90% of them refer to mathematics in middle and high school. Later, they did not enter the field of mathematics. They have never seen real mountains, so they are not in awe. The people who really study mathematics that I have met and got to know basically feel that they are very ordinary. Although they also know that they are not ordinary, they feel that they are really ordinary in front of mathematics, which they are best at. We are all ordinary people, so we dislike emphasizing talent. We always feel that hard work is more important. By the standards of studying mathematics, the sample size of real geniuses is too small, and it is not your turn to summarize any unique characteristics. The answer to this question about what is good in math and the habit of using the stroke input method is really ridiculous. There was a time in my life when I thought I had a talent for math, and looking back I think I was stupid as hell. Not only me, but also few of my classmates from the mathematics department who study with me think that they are gifted. There are indeed a few, but everyone treats them as a joke because they can't even pass the exam. One of them thinks that Goldbach It is very simple to guess that people have stopped doing mathematics for a long time and went to an unknown university to study for a PhD in engineering. On the contrary, one of my junior students who is studying for a PhD in mathematics at Oxford does not feel that he has any talent. They also think I have talent, which makes me embarrassed. In fact, I just studied for a while longer than him.

Of course, this kind of "self-thinking" is unreliable, and often can only be seen as self-confidence or arrogance. Then it only depends on actual performance, whether you can win an award, and what level of work you have done is the most direct. It's a pity that apart from colleagues, it's hard to comment on others. After all, it’s unfair to just look at impact factors. A bit crude, but fairly fair, it depends on how many highly cited articles you have published in high-level magazines in this direction. What a pity, it wouldn’t have been possible to see this without n years. At least it is difficult for undergraduates to be included in this system. Judging from my life experience, those who have struggled with talent issues for a long time will quickly give up on mathematics. Because they don’t actually like mathematics, they just like the glory and honor brought by being “good at mathematics”. When they were young, they were often good at math, the kind who could do well without working very hard, so they felt that they were talented and that math was a shortcut to simply enjoy honors. Then, they were slapped in the face by reality and discovered that high-level mathematics was not simple.

So they need an excuse to give up, and talent is naturally the best excuse. They are very afraid of the humiliation caused by failure after hard work, and giving up is the best way to avoid failure. After giving up, you can look back and laugh at those who are still persisting. However, loving something is not like this. To love is not to like the aura it gives you but to like the thing itself. You have to find mathematics fun. If you really love, even if you can't do a job that satisfies you in your life, at least you will do something you like. This itself is a privilege that few people enjoy. My undergraduate algebra teacher was always in high spirits during class. Even though he was over 70 years old, he lectured for two hours without getting tired. I admire him very much, not for his physical strength, but at the age of 70, he still shines like a child when he talks about basic algebra knowledge. This kind of innocent love is what I admire. For those who just want to enjoy the halo, I advise you to play a game of krypton gold, make some money, and then block people and kill people in it. Isn't it easier to kill Buddhas if they block Buddhas? Why go to join in the fun on the destined thorny road of mathematics?

Elementary talent: There is not much pressure in mathematics learning (excluding competitions) at the middle school level, and you have your own learning experience that you can share with other students; you are unanimously considered by your classmates, teachers, and parents to be a good student in mathematics, but not among the best. There is still a bit of a gap between students and students; if you perform normally in relatively easy provinces, 140 in mathematics in the college entrance examination will not be a big problem (assuming a full score of 150). Intermediate talent: When I was asked to share my mathematics learning experience in middle school, I felt like there was nothing to say, because many students have never experienced the frustrations and difficulties they experienced in middle school mathematics learning. . It feels like I don’t have to put in any special effort, I just understand it naturally and get used to it. There is no need to do special knowledge summarization, write wrong question sets, etc., because you can remember it. You can win some substantial awards in middle school competitions, such as first place in the provincial league; you can smoothly transition from middle school mathematics to undergraduate mathematics. Some advanced undergraduate mathematics courses require more time to understand and "familiarize", such as functionals. , topology, etc., but it is not an insurmountable obstacle, and it is entirely possible to get an A grade.

I self-assess this level. Undergraduate students who have majored in mathematics are basically at this level. It is also almost the mainstream level of outstanding answerers on Zhihu’s mathematics section.

Advanced talent: Can make a smooth transition from mathematics learning to mathematics research, have certain research capabilities, have published pure mathematics papers that are not water-worthy, or have the potential to produce water-free results ( Because it is still difficult to publish papers in many mathematics disciplines). The talent for mathematics is also accompanied by enthusiasm and diligence, and he is a qualified mathematics worker. Great talent: 16-year-old accessible reading EGA; a contender for various mathematics research awards and a rising star in academia. The first time I read the story of Gauss, it was said that when he was in fourth grade, his math teacher gave him a question: "1+2+3+4+...+98+99+100". When I first saw this question, I After a little thought, I came up with the idea that I could use the method "1+99+2+98+3+97"..." to calculate it. If I continued reading, it turned out that Gauss had the same idea as me.

At that time I was in the third grade, and I was 9 years old. Later I learned that Gauss became such an awesome mathematician, and I always felt that I was born three hundred years too late. (People's Education Press version). I solved this problem without looking at the textbook, even though I didn't know how to set up two unknown numbers at that time. When I was learning cross multiplication in seventh grade, I often needed to add a four-digit number. Splitting two two-digit numbers and multiplying them, I can almost see how to split it. In fact, this method is okay, but the teacher has never taught it, or the teacher thinks that most people can’t learn it even if he teaches it. Even the teacher himself can’t do it. So I think that even if my mathematical talent is not great, it is at least better than most people.

However, many years later, I became a Chinese teacher. Occasionally I meet my former teachers on the road and talk about my current situation. When I talk about my current career, they almost all look confused and can't believe it. In fact, it is very easy to judge whether a child has math talent. If you encounter a question that you don't know, you can learn it once, and you can draw inferences from one example, which is good. Some children can understand it from the beginning, and this is very good. , he can do the questions that others cannot do or require high-end knowledge to do with limited knowledge. This is very awesome. To add, I have never taken any mathematics or Olympiad courses. In the cram school, I have never read any books about mathematics except textbooks. The sentence "I feel like I was born three hundred years too late" is just a joke. I know my level and there is no comparison with Gauss.