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When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also rigorously (know how to prove or derive). However, I really find many results (even elementary results like euclidean algorithm or things like $ax+by=\gcd(a,b)$) hard to intuitively understand. I usually will be thinking about these theorems most of the spare time, whenever I am not doing anything requires my thought...

Even if I do comprehend, it takes a long time until I fully comprehend the theorem. And sometimes I may even forgot them(maybe I didn't actually fully comprehend) I feel like I am spending tons of time more than others on number theory. I mean, many of people online claim that introductory number theory is easy. Is that I am not smart enough to do mathematics and make contribution to the world of math later on since great mathematicians must have great intuition, or is that other people are not fully comprehending the theorems and it really does take a lot of time to just figure out one theorem completely?

I am very confused. I like math, but I really want to know if I am capable of doing it and make contributions. And I wish to make the best choice for myself. I appreciate any good comments or advices!

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closed as off-topic by Andrés E. Caicedo, beep-boop, user642796 Oct 10 '14 at 3:24

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – Andrés E. Caicedo, beep-boop, user642796
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ YMMV. Just keep working on it and you will get faster at learning new concepts. $\endgroup$ – graydad Oct 8 '14 at 0:11
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    $\begingroup$ In posting this question here where people are expected to be polite, you won't find anyone say that you're not smart enough. Think about that. $\endgroup$ – nigel Oct 8 '14 at 3:51
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    $\begingroup$ @nigelvr I'd add to that: If someone on the internet tells you you're not smart enough, what then? You will take their word for it and give up? $\endgroup$ – Superbest Oct 8 '14 at 5:11
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    $\begingroup$ Brain is like muscle and practice makes it stronger and faster. I got this the hard way while studying engineering physics - in the beginning it was awfully hard and I thought I will never finish, but after hundreds of problems solved, books and articles read and etc. I graduated with excellent results for my own surprise. For example, after solving more than 1000 surface and volume integrals and analyzed ton of functions I got it right - not only that I can solve problems only in my mind but I have developed intuition and understanding about the theory and how to apply it in practice. $\endgroup$ – Ognyan Dimitrov Oct 9 '14 at 8:47
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    $\begingroup$ I found myself to learn things slow, but I seemed to remember things when I did learn them, better than my colleagues or at the least, I performed better on exams and in the courses in general, gradewise. I knew that I learned slow. I have some really bad anxiety and I am on a kind of continuous fight or flight panic from the moment I would wake during the semester, and it made it difficult to concentrate. So, although this doesn't seem very popular, I studied during the breaks when I was less stressed. I did very well but ended up leaving grad school because anxiety was too great to deal with $\endgroup$ – Frudrururu Oct 9 '14 at 22:49

11 Answers 11

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When you say that you're experiencing some difficulties understanding intuitively some elementary things, there are a couple of possibilities:

1 - By "understanding intuitively", you actually mean the point in which you have devoted enough time and effort to certain topic that everything becomes clear and straightforward. That is what understanding something really means. Most people don't reach this stage as they stop their learning process when something "makes sense".

2 - If you regard "intuition" as an immediate understanding in the same way we know that 1+1=2, let me tell you that most mathematical concepts are not amenable to that kind of intuition. As timur said, many concepts in mathematics don't have parallel in the real world. Therefore, you can not expect the euclidean algorithm and 1+1=2 to produce the same "result" in your brain. As Von Neumann said:

Young man, in mathematics you don't understand things. You just get used to them.

That's what happens. You get used to dealing with very elaborate concepts and they become second nature to you.

Finally, if you forget something you spent a great deal of time studying it but after refreshing your knowledge, you are able to regain that understanding quickly and somewhat effortlessly, it means you actually understood it very well the first time. That's how the human brain works when it comes to nonessential things.

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    $\begingroup$ Till now I wasted a lot of time trying to prove that point 2 is wrong. I have succeeded a lot of times. $\endgroup$ – N.S.JOHN Mar 7 '16 at 6:01
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Let me give you a personal story. As a young kid, I was always very strong in math but was pretty hampered by one of the worst educational environments in the USA. I ended up entering a magnet school for junior high and had to take a math placement exam to determine which of three math classes I would join: regular math, pre-algebra and algebra. I didn't do quite well enough to fully justify being placed in algebra but did a bit too well to justify holding me back in pre-algebra. So as a seventh grader, I was placed in algebra. I struggled with it immensely. I had a private tutor and studied tons but to no avail. Ended up getting a 36 average or so and dropped down to pre-algebra in which I got a 105 average. Eighth grade came around and I had to take algebra; again, I didn't do that great but was better than before. I ended up with a low 70.

I did very well in geometry in high school but again not so great in algebra 2. Pre-calculus was hit-and-miss: some topics I did very well in, some not so well in. It was not until calculus that I really began to understand math at an intuitive and deep level.

Since taking calculus, I've excelled in mathematics. I ended up with nearly a 4.0 GPA in my math courses in undergrad (one A-) and I am currently in graduate school doing pure math after all of the struggle I went through. I'm pursuing very difficult and unique research and am very fluent in various aspects of mathematics. Just because you are struggling now does not mean you are incapable. Plenty of good mathematicians had trouble with math at some point for one reason or another. Don't throw in the towel so soon if you really like the material!

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    $\begingroup$ +1 I love this answer! @Kun Being good at math doesn't mean you'll find all of it easy. I've had similar experiences, where I found math super easy at first, until I hit abstract algebra and my brain couldn't handle it. I found it weird having to struggle in a math class, but eventually I pulled through and so will you! $\endgroup$ – Bobo Oct 8 '14 at 18:45
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    $\begingroup$ I can understand the point, yet the details are mostly incomprehensible for non-US(?) readers. "a 36 average" of what, what is the maximum? 200? Again "4.0 GPA", GPA? and is 4.0 good or bad? $\endgroup$ – Rolazaro Azeveires Oct 8 '14 at 19:51
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    $\begingroup$ @RolazaroAzeveires Missing the point. $\endgroup$ – layman Dec 2 '14 at 1:08
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    $\begingroup$ "and am very fluent in various aspects of mathematics." $\endgroup$ – Mister Benjamin Dover Dec 14 '14 at 18:53
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    $\begingroup$ Calculus was the turning point for me as well. $\endgroup$ – Matthew Kvalheim Jul 15 '16 at 2:47
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When I learned number theory, I found that I had no intuition for anything about the proofs, where my classmates seemed to pull the things from thin air; in math, there's always that little (or not-so-little) brilliant leap required for the proof. When I just started learning number theory, I had no idea how people were figuring these things out - even knowing the proofs, I found them hard to follow. I don't really think I got up to speed with my intuition until months after initially seeing the material.

This would be because I was learning math and could, naturally, not do it very well. Nowadays, I see such proofs as entirely trivial and have no idea what was so hard for my past self to see in this. There's something intangible at work here and when you're just starting it can be easy to see these objective things - like other people being so much quicker to the proof than you - and to not know what you're missing, but, with practice in mathematics, it may come. If you like math, there's no reason to stop now.

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    $\begingroup$ Figuring out that missing "brilliant leap" in the thought process of students is a source of research in education right now. It has been found that, as you suggested, for someone who understands it very thoroughly, not seeing the leap is very difficult. This is one of the reason that some seasoned professors can have problems teaching very basic concepts: they can't see why people don't just get it. There have been some advancements though, specifically allowing open discussion between students has been shown to help, as other students are going through the same process (finding the leap) $\endgroup$ – Nate Diamond Oct 10 '14 at 1:03
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Listen! You don't need to ask others for whether you want to continue in math or have potential. Based on what you said, which is thinking about Math in all your spare time, I'd say you could be a genius already, just not fully developed! Take you time and enjoy the process. Many people seem to think they are good at math, but all they are doing is memorizing formulas and becoming human calculators!

Don't depend on other's opinions for such matters, this is about you! Only you can answer if whether or not you can do it. Don't feel pressured to contribute to math either. That is a very hard thing to do in this day an age where is's already so developed. Ancient Greeks had (IMHO) a much easier time coming up with new math concepts then we do today since it was much more foundation and basic then.

I'd hate to see someone with so much interest in math, give up because they don't think they are as good as others. I'm telling you don't give up unless you really don't enjoy it, but if you do, don't! The education system rewards those who are the human calculators the most IMO, but that shouldn't stop you from going beyond it and excelling to your dreams.

Personally, I was the type who hated the human calculator system, but didn't enjoy math enough to think about it as much as you. I thought I was stupid, but really I actually wanted to "understand" math at a deep level, and it just wasn't taught in my school. All I learned was formulas and little tricks such as FOIL and the like.

You will have to learn that you are your greatest teacher, and that you are the only one that can really speak "your" language. Teach yourself with "your" language.

I wish someone told me this when I was in your shoes. :) Good luck and just remember to enjoy the process. The most successful people aren't the most organized or talented, but those who love what they do the most because they are operating in their optimal mindset.

Also, I don't believe there are those that are good at Math and those that aren't!

Some have had better teachers than others, some had their parents help them with homework when they were young. Everyone had a different experience learning math, and some had more help, and more advantages. But that shouldn't stop you, in fact, the ones without all the advantages tend to succeed more. Just study the richest businessmen, they often came from poor backgrounds.

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    $\begingroup$ This is a great answer! Not being "smart enough" is a shame-based expression that plagues our society, and even some of our most brilliant people, yet it doesn't really apply, and certainly not to someone studying and enjoying number theory at a university! I agree that the OP has genius potential in math or a similar endeavor. Many geniuses have a similar always-questioning mind. Being faster may indicate less fascination, and thinking more shallowly, than the person who is questioning all day how fully they understand something! $\endgroup$ – Dronz Oct 9 '14 at 21:03
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I don't think the time it takes to learn a concept is necessarily an indication of ones intelligence. Some people are quick learners and some aren't. Of course there are advantages to learning quickly; however, perhaps when you do finally understand a concept, you understand it better than those quick learners! With this in mind, some people just aren't so strong mathematically. It's hard to say based on what you wrote if that applies to you, but I don't think so. Introductory concepts are often hard to learn at first. I think many people currently studying mathematics can sympathize with it taking a long time to understand a concept, or to forget something you recently learned.

I think if one isn't strong enough mathematically it wouldn't be something that they would be truly passionate about.

You should do math if it is something that excites you and something you get enjoyment out of. If it is your passion then you will be willing to put in the work it requires to understand concepts and make contributions.

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  • $\begingroup$ What I am saying is that many times I just have the proof right off my head. I mean I read the book so many times that I can even write the whole proof about in my mind. But it just does not really satisfy me. I don't know how to express this feeling, but it just does not seem to be very intuitive like 1+1=2 $\endgroup$ – Kun Oct 8 '14 at 0:47
  • $\begingroup$ Some concepts, I asked my professor if he can come up with a intuitive way to understand this, he seems to be stunned too. I mean, he can prove it and reason it to me very well, but an inuitive way is just not always available. Is that true that even very complicated results can be intuitively understood? or not. I've seen a post claiming that a great mathematician knows something is true even they cannot prove it by intuition, $\endgroup$ – Kun Oct 8 '14 at 1:01
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    $\begingroup$ The theorems in current texts, such as on elementary number theory, are presented very clearly and in a way that makes the reader feel the concepts should be intuitive. But remember the proofs of these theorems took very large amounts of time to be discovered and furnished. In my opinion, once you become more familiar with the topic, you'll get the intuition. For example, when I was in high school, there was no real proof. I memorized how to differentiate and integrate. For me, a deep understanding/intuition of these concepts only became clear after 3rd year analysis. $\endgroup$ – JonHerman Oct 8 '14 at 4:30
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In my mind, "understanding intuitively" means that you understand something in terms of things you know really well from your past experiences. This means that you need to gather experiences to have a lot of intuition. That is, mathematical experiences to build mathematical intuition, and mathematical intuition to understand mathematics. Other kinds of experiences, such as knowing how physical objects behave, would help a lot but doing mathematics that really counts. Especially, there are a lot of structures in mathematics that do not even remind you of things you would experience in the real world. When you teach higher mathematics to relatively inexperienced audience, there is always a risk in trying to make everything "intuitive", because this can oversimplify or completely distort the real picture. In any case, what you are doing is good. Question everything, and move forward slowly but surely. One day, you will notice that a lot of things you thought were unintuitive became trivial matters.

On another note, people learn things differently, and it can happen that the current teacher's approach is basically the opposite of what would be the ideal approach for you. If you suspect if it might be the case, you can study from books independently and see if it helps.

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  • $\begingroup$ Thanks. I think it really helped! I actually love math and do not bad on contest math. I think I am overthinking sometimes. $\endgroup$ – Kun Oct 8 '14 at 2:08
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Think of mathematics as a language. Some people are native speakers or have a gift for speech, some have to work at it, some people stutter.

Replace "math" with "French" in your question above. It will get easier the more you do it, even if you don't become a fluent speaker.

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If you are smart enough to be in number theory, then you are smart enough to understand it. If you are smart enough to choose number theory as your field of study, then you are smart enough to find your way to your goal. I am not just saying that.

Considering the amount of time you have been spending studying number theory, your brain has been taking in much information and will process it subconsciously. Some time when you are not even thinking about mathematics or anything related to it, number theory related thoughts might come to mind. Years from now, you will understand things which presently you might think you cannot understand.

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The late great Paul Sally told me a story about his days as a post-doc. He was struggling with his research, and complained to his mentor, "I'm busting my a** and I still can't get a theorem!"

The mentor replied that yes, hard work is necessary for good results. In consolation, he replied that "In time, you'll find it's sufficient."

Apologies for the name drop. It's necessary to read the quote in his native Bostonian inflection.

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    $\begingroup$ Didn't know he had passed, sorry to hear. He is the first person I think of when I hear of people considering ( leaving ) a career in pure math. It seems to me, he was one of the few people that could put in to words the unique social and intellectual environment that you're signing up for, and in my experience that is much more a determinant of success and happiness in the mathematics world than raw talent. Really a great person(ality), and a shame future generations of mathematicians will only know him through his influence. $\endgroup$ – Callus Oct 9 '14 at 23:25
  • $\begingroup$ Yes, it was a big blow. He won an award from the AMS this year and the citation is a eulogy of sorts. I am a contributor to an upcoming memorial article in the Notices of the AMS; hopefully it will be out soon. $\endgroup$ – Matthew Leingang Oct 10 '14 at 11:54
  • $\begingroup$ Nearly four years later, the memorial article has been published. $\endgroup$ – Matthew Leingang Jun 5 '18 at 15:49
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Something to understand is that although it may all look the same from the outside, each branch of mathematics is quite different. They all have a different feeling, different ways of approaching problems, different levels of rigor. My personal story is that I've always been one of those people who just "gets" math. I'd be able to look at problems and either guess the answer or the right approach to the answer. I'd anticipate new ideas before the professor introduced them. Basically mathematics "fits" my brain.

And then I took a course in differential geometry, and it just didn't work. I couldn't "get" it. I understood the concepts from a definitional point of view, the same way a computer would if you taught it the axioms, but I had no intuition about how tensors work, what forms, bundles, etc. are, or how to approach even simple problems. The professor was excellent; I had great experiences with him from other courses. My conclusion was that unlike most (really all) other mathematics that I had encountered up to that point, differential geometry really did not fit my brain. It was a bit of a shock, but an important lesson.

So what I'd say to you is if you can't get number theory, don't worry about it. It might not be the math for you. There are lots of mathematicians who are terrible at number theory (there's a famous story about Grothendieck, the father of modern algebraic geometry, who thought that 57 was prime). Try other fields of mathematics. Maybe analysis is your cup of tea. Or maybe it's algebra. Or topology. Or logic. Or combinatorics. Or probability and statistics. Each has such a different feel to it that most mathematicians will hate or at least strongly dislike working in at least one subfield, if not most subfields other than their own.

It is good to know the basic concepts from each subfield of mathematics, just so that you are aware of what's out there. It can give you new insight to ways to solve problems, and perhaps most importantly it can show you what kinds of problems have already been solved in case you encounter them doing something else.

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  • $\begingroup$ By the way, another tough nut to swallow: you may find some areas of mathematics to be very interesting, but very difficult for your brain to actually work with. I find most mathematics to be at least moderately interesting, but there are only a few areas where I work the best. $\endgroup$ – asmeurer Oct 9 '14 at 23:17
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It is often a matter of technique. Once you master the technique you advance easy and fast, but before that it is many long stops. Those who dwells in this first state may also achieve more insight into the next stage? And previous knowledge really means a lot (that's why it take such a long time breaking new grounds).

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