I am a second-year girl, in a good junior high school, her grades are not bad, she has not fallen in the top fifteenth grade, twice first. Chinese is now above 135, Mathematics English is above 145, and the total score of eight subjects is first in the basic grade.

When I was in elementary school, I was like most of my classmates. Although I was excellent in mathematics, I was afraid and bored with “Olympics” and never showed any talent for mathematics. It wasn’t until the summer vacation of Xiaoshengchu that I came into contact with geometric proofs at home for seven times. From then on, I was obsessed with mathematics and became a class representative at school. In the first year of junior high, I was very passionate about geometry. The small dining table was chaotic. I sat on the stone steps diagonally opposite the school at noon in the name of arriving early to prove the theorems I learned before. The sum of the inner angle and 180, the sum of the two sides. Greater than the third side.. These seemingly naive properties have given me an initial understanding of mathematics research and established the dream of pure numbers. Thinking about it now, I really miss the mood at that time, calm, quiet, innocent, warm and beautiful.

The physics teacher in our class in the second grade is my most respected. He is not limited to taking exams, he has a deep understanding of mathematics and physics, he also deeply influenced and inspired me, and cultivated my initial academic thinking. He also appreciated my seriousness, hard work and passion, and explained obtuse-angle trigonometric functions to me. Later, I wrote some creative ideas as my first thesis. I also learned about the SCI paper division, the four major journals and PRL, and heard about arXiv, and I personally logged in to experience it. Last winter vacation, I first came into contact with number theory and proved a theorem a few nights. When I showed the process to the mathematics teacher expectantly, I was overturned by a few words. At that moment, I was really uncomfortable, but still resisted crying and nodded silently. That is the difficult and vast side of mathematics, presented to me for the first time.

This semester, I will have more in-depth exposure to mathematics, study high school mathematics in class, and understand complex numbers, derivatives, non-Euclidean geometry, and group theory outside class. I also started to study a mathematics monograph “Modern Euclidean Geometry”. Some friends may have heard of it. . Going farther and farther on the road of mathematics, I started to feel powerless. To be honest, in our school, as a girl, I don’t think there are a few boys who can completely crush me in mathematics. But what really left me at a loss was the sense of insignificance when faced with the huge peak of mathematics. I used to have unrealistic fantasies about mathematics because I only saw the tip of the iceberg.

The article I stumbled across a month ago, “Being an Ordinary Man in Peking University School”, also completely pushed my dream of mathematics into the cold valley, and also removed the climbing ladder.

I began to doubt myself. On the surface, everything was business as usual, and my grades did not decline, but my heart became sensitive, fragile, and confused. I even dreamed that I couldn’t solve the problem. I cried and woke up in the second half of the night. Until the last monthly exam, I only took 142 due to copying the wrong number. Fortunately, my Chinese was super-normal and saved me on the edge of the top ten. A classmate of our class took the 150 test, and then a math weekly test was not completed in the whole class. When the math teacher asked me, I casually replied that xx (the 150) did not finish. That day, I suddenly realized that this small monthly test failure made me subconsciously put myself in a position worse than others. I suddenly woke up and felt that my state was wrong.

I have always believed that mathematics is not for examinations. My dream has always been to become a mathematician like Galois. In my heart, he is a person who can really understand mathematics. He died in a duel at the age of 21, but he achieved a mathematics achievement that will be remembered forever: he founded group theory and proved that there is no general solution to equations of degree five or more.

But I also began to faintly, and more and more clearly know that I am not a genius, I have no extraordinary talent, and no keen sense of number. I suspect that I shouldn’t fall in love with mathematics, and it’s not suitable to take the road of research. But I was not reconciled to give up. After all, I vowed to devote my life to mathematics on the night of completing the trigonometric function thesis, without any regrets. And now, looking back on my naive belief two months ago, I feel like a world away.

I just want to know, without outstanding talent, can I grow into an ordinary mathematics researcher? Is it possible to realize the sincere expectation of physics teachers: I hope you can become an excellent researcher and let knowledge and ideas become our beliefs.

Mathematics has always been my belief, but the belief in “I can handle mathematics” has gradually dimmed. what do I do?

zhiwo

By zhiwo

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helpmekim
5 months ago

Such a sincere question deserves a sincere answer. Due to my limited proficiency, I cannot comment on mathematics research, but I can still express some opinions on mathematics learning and the problems mentioned by the subject. First of all, I am envious of the subject’s literary talent, and here are some responses to the subject’s article: Until the summer of Xiaoshengchu, I was exposed to geometric proofs at home for seven times. I was obsessed with mathematics and became a class representative in school. In the first year of junior high, I was very passionate about geometry. The small dining table was chaotic. I sat on the stone steps diagonally opposite the school at noon in the name of arriving early to prove the theorems I learned before. The sum of the inner angle and 180, the sum of the two sides. Greater than the third side.. These seemingly naive properties have given me an initial understanding of mathematics research and established the dream of pure numbers. Thinking about it now, I really miss the mood at that time, calm, quiet, innocent, warm and beautiful. Very good, I can see that you really like it. Regardless of the future outcome, these beautiful memories are also worthwhile. Later, I wrote some creative ideas as my first thesis. I also learned about the SCI paper division, the four major journals and PRL, and heard about arXiv, and I personally logged in to experience it. Last winter vacation, I first came into contact with number theory and proved a theorem a few nights. When I showed the process to the mathematics teacher expectantly, I was overturned by a few words. At that moment, I was really uncomfortable, but still resisted crying and nodded silently. That is the difficult and vast side of mathematics, presented to me for the first time. It is a good thing to have innovative ideas and want to do research. But unfortunately, junior high school students, high school students, and most college students in mathematics departments do not have the ability to do valuable mathematical work. Generally speaking, if you have selected the field you are interested in, have completed most of the graduate courses in this field, and read some classic papers in this field, you will be considered to have passed the threshold of research in this field. This is very different from biology, chemistry, computers, and physics. So my suggestion to the subject is to study as much as possible now, so that you can study ahead, but you can temporarily dispel the idea of ​​research. But what really left me at a loss was the sense of insignificance when faced with the huge peak of mathematics. I used to have unrealistic fantasies about mathematics because I only saw the tip of the iceberg. The mathematical system that was gradually established after the beginning of the 20th century was too grand, and each of us was too small in front of this system. It’s normal to have this feeling. In fact, after four years of undergraduate studies, it is possible to browse through the framework of this field. The article I stumbled across a month ago, “Being an Ordinary Man in Peking University School”, also completely pushed my dream of mathematics into the cold valley, and also removed the climbing ladder. I also read this article. In fact, this article is more about how to adjust your mentality as a very good but not talented mathematics student in a place where experts gather at Peking University. The school the subject will go to in the future may not be as fierce as the Peking University School of Mathematics. For example, many foreign universities are far easier than Peking University, but the mathematics department is also very good. However, this article is still valuable, that is, the main topic is to recognize the fact that oneself is likely to be an ordinary person, and then work hard with a calm mind and down-to-earth. I began to doubt myself. On the surface, everything was business as usual, and the grades did not drop. But my heart became sensitive, fragile, and confused. I even dreamed that I couldn’t solve the problem. I cried and woke up in the second half of the night so that I wouldn’t be awake…this article It just tells that there are a bunch of mathematics geniuses in a school in Beijing who do not know the subject. Their excellence does not deny the effort and value of the subject. If you are sad because of the fact that there are people better than yourself in the world (this is an inevitable thing that everyone needs to face), it is really unnecessary. I have always believed that mathematics is not for examinations. My dream has always been to become a mathematician like Galois. In my heart, he is a person who can really understand mathematics. Unfortunately, this dream is unlikely. One dream with a much higher success rate is to become a mathematician. Galois is a genius once in a century. Even among the best mathematicians in the world, few people dare to say that they have the talent and spirituality of Galois. As mentioned earlier, most of us are destined to be ordinary people, so it is normal not to be the next super genius like Galois, and there is no shame. In addition, how to define a “person who really understands mathematics”? There are also many unknown mathematicians in the world who have worked hard in their fields for a lifetime and made contributions. In my opinion, these people are also people who really understand mathematics. I wonder if the subject agrees with my idea? But I also began to faintly, and more and more clearly know that I am not a genius, I have no extraordinary talent, and no keen sense of number. I suspect that I shouldn’t fall in love with mathematics, and it’s not suitable to take the road of research. But I was not reconciled to give up. After all, I vowed to devote my life to mathematics on the night of completing the trigonometric function thesis, without any regrets. And now, looking back on my naive belief two months ago, I feel like a world away. Not a genius, no talent is a fact that 99.9999…% of people in the world need to accept, because by definition, only a handful of people are geniuses. It is not shameful to admit that you are not a genius and to reconcile with yourself. On the contrary, this is a very courageous thing. But for this reason, you should not feel that you should not fall in love with mathematics and not suitable for the road of research. Love this kind of thing is beyond all rational interpretable categories. So if the subject feels that he is in love with mathematics, it can only be said that it is your inner decision, then just follow your heart and let the flow go. If you give up mathematics because of this little setback, it is really not suitable for research, because the difficulties you encounter on the research road are much greater than this, and you may not be able to bear it. I just want to know, without outstanding talent, can I grow into an ordinary mathematics researcher? Is it possible to realize the sincere expectation of physics teachers: I hope you can become an excellent researcher and let knowledge and ideas become our beliefs. It is possible that, in fact, the group of “ordinary mathematics researchers” is much larger than the subject imagined. But becoming a mathematician is more difficult than you think. And this difficulty comes not only from the difficulty of the subject and the limitation of one’s own ability. The competition in the academic circle is too cruel, and the front is too long. You may question your choices because of many unexpected things: lifestyle, money, family, career… If you find yourself hesitating in your heart, or your thoughts have changed in the middle of the process, hope that the main topic will be honest with yourself. . Mathematics has always been my belief, but the belief in “I can handle mathematics” has gradually dimmed. what do I do? The belief that “I can handle mathematics” is not true, and it should not be the belief of the subject. No mathematics learner has the guts to say “I can do mathematics”. No matter how smart people are, no matter how talented they are, with the deepening of mathematics learning, they will inevitably encounter a frustrating moment. At this point, your intuition is no longer accurate, your brain starts to tire, and your logical chain begins to break. You will find that you have no way to easily control an abstract mathematical concept, just like riding an untamed wild horse. difficult. In many cases, you need to spend hours, or even days, to fully understand the content of one or two pages of a math textbook. No one can escape this node, although some people may encounter it later than others. But this is inevitable: mathematics is not easy for anyone. Then with your repeated derivation, your brain is constantly running at a high speed unconsciously, considering the logic between theorems, you will crush your original incorrect intuition, and your understanding of these objects and logic will deepen, suddenly At a certain moment, you will suddenly realize that all the pieces start to combine and become a beautiful work of art, and you finally understand this theorem. After that, you will build up new intuitions, and then continue to turn to the next page of the math book and repeat the same process… So my advice to the subject is not to think so much, because no one can learn from yourself Early performance predicts performance after ten years, and there is no way to predict whether one can achieve the goal within time (proof: not a Nash equilibrium). So the decision to embark on the path of mathematics research is actually an incomplete information game, which can only be said to be a pony crossing the river. But given that the subject likes mathematics so much, I think no matter how talented you are, you will unconsciously embark on the road of mathematics, so you have not many choices because you have already made a decision in your heart (laughs). In this case, follow your heart, but ask for no regrets. Finally, I hope the subject will give up the fantasy and prepare to fight.

heloword
5 months ago

If you have the ability to make money, you can do it. To be honest, even if you are not a genius, you can’t get a great work certificate. The simple and unpretentious little theorem is something that most people can do. The only thing that is worse in mathematics is that the cost of failure is relatively high, and there is no corresponding industry to transfer, so if there is no retreat, it is not recommended to all in pure numbers. But on the contrary, if you can accept the pure number PhD, you may not be able to stay in the academic circle or you can only go to an ordinary school to be a lecturer with a meager salary, then you can take a gamble.

helpyme
5 months ago

The students from Peking University Academy came to give a strong answer. First, it is rare to see children who love mathematics who are particularly good in middle school Chinese. The main reason is that I have always felt that aesthetic thinking and mathematics are mutually exclusive. It seems that there is a problem with my understanding. The subject’s language description ability is still very strong. Second, junior high school grades really cannot explain the problem. You don’t have to worry about the difficulties encountered at this stage. People with high talents are all those things that have received serious mathematical training, including those geniuses. Without the guidance of a teacher, there is no need to be too advanced to look at future knowledge. For example, there is no pressure for excellent junior high school students to understand high school textbooks, but problem-solving training in the form of exam-taking is actually impossible What’s missing, excellent students can do less than ordinary people, but they have to do it. You need to internalize common problem transformation, algebraic deformation, geometric structure and tool proficiency. In fact, problem-solving training will last until the end of the second year of college. Prior to this, follow the normal rhythm, and the students who study hard can be mastered. There is no situation where they can’t learn because they are not geniuses. If they don’t learn, even the imo gold medal will be left behind. Third, today’s mathematics, including other subjects with high mathematics requirements, are highly detailed. In each specific direction, there may be only a few people in the world who are doing it. If you want to make something, it often does not need to reach our mouth and ears. According to legend, the level of talent of those people who are said to be smarter is enough. Therefore, excluding those top geniuses (the Peking University School of Mathematical Sciences may not have such a degree in a year), relatively ordinary people want to do mathematics, in fact, a normal mentality, a love of mathematics (no fanaticism), and hard work, It is enough for people to do something in scientific research. Fourth, Peking University School, Versailles has a strong atmosphere, including that article. The fact is that for people who have passed the basic talents, becoming a small celebrity requires high talent and hard work (such as Chusay n gold medals, but it will definitely not reach the people you see in the book), and the top 20 may require Higher talents and hard work, the first 70 in the exam (this level is basically whether to do the academic watershed, the total number of people is 200) really only requires hard work and concentration. The gap between the experts and the discomfort of advanced mathematics mentioned in the article is really just a pain, which can be wiped out in a semester, and even for those who really want to go to the library every day, this pain does not exist. Most of the colleges and universities fail to work hard (just playing, or enjoying university life, academic work in associations, etc.). For example, in the mathematical analysis of the freshman year, we divide the degree of effort into the following categories. Did not finish reading the book or did not do the homework, did not do the homework seriously after reading the book, did the homework after reading the book, finished the after-class questions after reading the book, finished the after-class questions after reading the book and finished Xie Huimin, a Peking University The basic IQ scores in other schools are generally 70, 85, 90+, 90+, 90+, and in Peking University, they are generally 50, 50, 70, 85, 90+. I have never seen low scores even after hard work. of. Fifth, recommendations. There are two ways. The first is not to learn to compete. Then look ahead based on interest. The high school entrance examination must be good, high school subjects are balanced, moderately advanced learning, and the college entrance examination is good. The second is to learn competitions, find reliable teachers, do not be greedy fast, lay a good foundation, learn high school content in advance, plus after the holidays, most of the high school mathematics before going to high school, and at the same time test a competitive school, in the process If you feel strenuous, you can give up at any time.

sina156
5 months ago

Actually, I feel that learning mathematics is just playing. For the sake of fun, it is not important to have talent, or even to understand whether to learn or not. After all, people have to find something interesting for themselves to do while alive. There is a saying that is very good: the competition itself is meaningless. I came to the world to follow in the footsteps of the Creator, not to be smarter than others. Too perceptually, taking mathematics as a belief will not have any good results. In a word, it is good to be happy in learning mathematics.

yahoo898
5 months ago

Mathematical Olympiad is mainly technical practice. Skills are important, but there are skills at every level at each level. Although it cannot be said that there is no similarity between elementary school mathematics and middle school mathematics, there is a huge difference, and there is a big difference with university skills. Those who are not good at using small files may not be able to use CNC machine tools, and those who are good at using small files may not be able to use CNC machine tools well. Classical plane geometry has no place in modern mathematics research, only as a kind of logical exercise or archeology. In general, you can try if you like. If you don’t want to, give up and be free. There is no need to care too much about the score, the key is whether the problem will be done or whether there are difficulties beyond the control of your ability. Why does classical plane geometry have no place in modern times? Because the Descartes era began to have a coordinate system. In theory, the problems of straight lines and circles can be solved by first-order and second-order equations. Although the amount of calculation may be relatively large, it is almost mechanical. Elementary number theory is different. Advanced theories cannot fully mechanize elementary number theory problems, but it is still possible to get in touch with functions of complex variables earlier.

leexin
5 months ago

Answer the questions first. I personally think that talent is not necessary, but the process of becoming a talented person will be easier. It’s like running. People with good talents can run very fast, and those with good talents may easily reach the same distance. But learning math depends on the result, that is, how far you ran, not how fast you ran. Although talent is not important, there are certain thresholds for learning mathematics. In other words, you can’t run too slowly, otherwise the goal will always be indefinitely, and it will only waste time. Although talent is not that important to mathematics, I still have to pour cold water. Mathematics is a broad and profound subject. Even a college student who has graduated from a bachelor’s degree and has been exposed to systematic training may still need one or two years of study to get in touch with the frontiers of some fields, let alone a junior high school student. In short, reaching a level where you can start research may not be as easy as you think. I am also studying hard myself, hoping to touch the cutting edge one day. I personally feel that the sooner I recognize and accept the limitations of my abilities, the better I can face mathematics. Therefore, “Dedicating my life to mathematics, no regrets.” I think this idea is very dangerous. Because of this, you will naturally associate your self-esteem with your so-called mathematical talent. Over time, one’s own interest will be wiped out under the successive blows. Compared with talent, I feel that learning mathematics requires more endurance, the kind of perseverance that can stick to problems without thinking for dozens of hours. The sooner you reconcile with yourself and realize your limitations, the more you may be able to sit still and go further along this road. But I think these are too early for you. I think it’s hard for a junior high school student to use my spare time to understand some mathematics concepts, read some popular science articles, and maintain my interest in mathematics. Don’t be discouraged because of your limited abilities now. Because this is inherently unrealistic for middle school students, don’t doubt yourself. I think what you have to do now is to study hard and lay a solid foundation while maintaining an interest in mathematics. As for the realization of your own mathematical ideals, you should leave it to the future that has a solid foundation, mature academics, and the more frustrated you are, the more courage you will realize it yourself. I hope I can help you.

greatword
5 months ago

The article is very touching! Tell me about my feelings for your reference. When I was in middle school, I won first place in the district, but the city’s Olympiad didn’t even make it into the quarterfinals. Mathematical Olympiad is not the same thing as mathematics, it is only a small part of mathematics. But I still entered the mathematics department at university. After I went in, I found that there were too many smart heads, and I was shocked compared to many classmates. However, college mathematics values ​​concepts most, so hard work is the first element of learning mathematics. It is not easy to persist for four years or even longer. It takes the support of ambition and perseverance to persist. In the four years of university, about one-third of the students who directly enter the university after graduation, girls have better grades. Basically, those who can go directly to school are hard-working. It is not easy to rely on a smart brain. Hard work is the foundation. Some students who did not get good grades at the time of admission also went directly to higher education, so hard work is the most important thing. There is basically no plane geometry in universities. Although analytic geometry is a course, one semester is over. The focus of university learning is on concepts and systems, which is actually quite boring. The things that outsiders are more interested in, such as operations research, graph theory, mathematical statistics, econometrics, etc., are relatively superficial things. The department of mathematics learns the axiom system of these things, while the departments of computer science, economics, and mathematical statistics will Learn these things as the core. After entering society, only less than 20% of people continue to study or teach mathematics, and everyone else is doing other things. It is relatively easy for people who have studied mathematics to learn other subjects. Those parts of cybernetics and probability are the places for fun, and only the places that need to be memorized are the places where the effort is spent. Therefore, there were quite a few people who were experts in computers and economics in our session.

loveyou
5 months ago

To be honest, it is too early to take mathematics as one’s faith in the second grade. It’s okay to like mathematics, it’s okay to like the Olympiad, it’s okay to have time to see advanced mathematics, and it’s okay to even register for a mathematics major after the college entrance examination. You will have four years of undergraduate time to judge whether you really like mathematics. . During the undergraduate study period, if you really like mathematics and have a clear understanding of the future employment environment, then you can read a straight Ph.D. You may be able to achieve something; if you find that you don’t like mathematics that much, change to operations research/statistics/finance, the same is true There is no problem. Let me take “a classmate I know” as an example. This classmate has always had good grades in physics in middle and high school. He also likes physics very much. He feels that he can make a big difference in physics research in the future and passed the self-recruitment of engineering physics. I was still very happy; then, about a certain period of time in the sophomore year, this student was basically sure that he could not pursue a direction that was more related to physics. One of the interests gradually lost, and the talents of the two were indeed limited. So, at the age of the second grade, learning is still the main thing. Engaging in the Olympiad and looking at high numbers can only be used as an adjustment. And your writing is really fluent and natural, presumably the language is also very good. Rather than thinking about the question of “Is your belief bleak”, it is better to do two more questions-a better high school/university platform can give you more choices and a broader future.

strongman
5 months ago

The subject can private me, I am now a junior in ustc mathematics. I know the answers above are too messy, and I don’t have so much time to write them down seriously. But I would be happy to share some of my learning experience with you, and even write a version of “Ordinary People at the University of Science and Technology Mathematics” by the University of Science and Technology of China. (Beijing University School of Numbers is indeed crushing the existence of other colleges and universities. It is incomparable. I can only look up hahahaha.) In fact, I am also very confused now. I very much hope that there will be someone who will guide me. Similarly, I am also happy to provide you with some help, such as the expansion of some mathematical knowledge and answering questions.

stockin
5 months ago

Children, looking at the problem description, your school does not even have a competition class? Really, in this case, your starting point is much lower than others. In this case, the “superiority” will not bring you much benefit. Look, now you are experiencing a little setback. The “love” of mathematics is about to cool down. To put it bluntly, there are too many dramas. Let’s make a joke, let’s be more serious. Many people always think that the world is a story when they are young. As long as they dramatically “make up their minds” and “dedicate their life to xx” and think about those historical “starring actors” in their hearts, they can be the protagonist in the same way-at least It is a character in one’s own world-to perform a life of high spirits, but reality is often not the case. The drama (or sense of mission) that you imagined when you were young can become a motivation to act, and can support you to overcome difficulties and move on to a larger stage; but also, when you exceed this limit, the BGM that inspires you will also take you Entering the current trough, entangled in the scene of the loser can not extricate themselves. After growing up, some people will gradually discover that the world is not always full of exciting stories, and your success or failure will not have a beautiful transition. Many wonderful plots that you once thought about are calm and tasteless. life. Plenty of emotions will not affect the operation of reality, and moving forward silently will not reduce the gains due. The same is true for learning. You have to know that learning is objective. Whether you “love” or not only determines whether you learn or not. After all, when you are studying, your mind should not pretend to be “love” but unremitting thinking. Learning is just learning. Many people read the book while thinking about how difficult it is. It’s weird to be able to read it clearly. It is more meaningful to reduce distracting thoughts, concentrate on the study itself, and think about where you haven’t understood and why you didn’t understand, than thinking about how difficult it is for nothing. In other words, under the debuff of these messy thoughts in your head, your learning is likely to be inefficient. You might as well let go of these meaningless thoughts and try to really learn, at least to make you really recognize your own learning ability. . In fact, according to your current level, it is really not enough to explain anything. I haven’t finished learning anything in high school, let alone competitions. I don’t even know what math is, so how can I “like” and “talent”. Mathematical talent is the manifestation of the foundation. The best way to judge is university knowledge. There are not so many tricks in competitions, but simple mathematical thinking. Abstraction and topology are good tests. Basically, they are abstract concepts. If you don’t have abstraction problems, you don’t even need to learn number theory. Topology can even be seen without any mathematical foundation. In this way, you can see what the entry line of mathematics is like. After that, it is worth considering whether you like it or not, and the talent is not talent. It is really necessary to get in touch with modern knowledge as early as possible. Finally, if you really want to dedicate your life to mathematics, why would you retreat because of these things? The biggest feature of knowledge is that everyone can learn it, and talent only determines the speed. If you really like mathematics itself, you won’t care about these irrelevant things. If learning mathematics is only based on the illusory “love” and superiority than others, let it go as soon as possible, and talk about it later, if you know the possibility If you can continue to move forward unconsciously after the difficulties, then you will be able to make your own choices. But don’t be too confused. In many cases, everything is a natural result. You may not be able to come out now, but in a few years you may understand it naturally. Even the ability to learn may be the same. What you don’t understand now will be very simple in the future.
You are only in the second grade, precious time is like running water to you, it is the most hopeful time, you should cherish it.

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