Even though I've been teaching myself math for a couple of years now I only just started (a month ago) at the university. My experience is rather mixed.

For starters, I'd like to mention that I'm 21 years old. As I understand it, this is not too young and not too old. Having said that, I can't help but feel jealous of all the young people who populate my courses. In one of my courses, the percentage of students under 18 is around 40% (haven't checked rigorously). The thing is I don't think that it would have bothered me so much if I hadn't felt like the academy is holding me back.

When I learn by myself from books, I just go from one thing I didn't understand to the next and no minute felt wasted. Ignoring the fact that I can already solve the test of three courses I'm in (I tried to avoid them but I have to do them), I find myself more often than not writing down homework solutions to problems I wouldn't have spent a minute on since I knew they weren't an issue. The pace is so slow that I regret having started at the university in the first place (with the exception of one course).

I understand that in some way this is something I did to myself (by teaching myself these things beforehand). I'd like advice or some support since I'm pretty close to taking back my decision to learn at the university. So far it feels like half the fun of math for twice the time (and all those little kids have the time since by my age they'll be doctoral students...). I really miss those self teaching days.

I feel like if I would only be given a chance to study at my pace, I could finish the degree in a year and a half and have much more fun doing it...

What am I to do?

EDIT: It may be worth mentioning that I got exemption from several courses due to past studies I did in an open university (one where you learn alone from books).

  • 7,933
  • 11
  • 42
  • 81
  • 11
    I think it may be better suited for http://academia.stackexchange.com/ :) Nice question though! – Ant Nov 26 '14 at 22:03
  • 9
    I am self-learned as well. This strikes far too close to home. – Eoin Nov 26 '14 at 22:04
  • 19
    Talking to the department chair about this couldn't hurt; the worst they can do is say "no" – Milo Brandt Nov 26 '14 at 22:05
  • stop whining and do your homework :) ... just kidding! I am sorry, I feel that what you are saying is right, though I can't really relate myself: I enjoyed reading and doing mathematics but more or less I am the product of a "standard education, regular paced, non-exceptional" approach. (And the advice to talk to the math Chair certainly sounds right, I have never been a Chair, but I have taught math, and I have seen students that do not seem to easily fit in a straightjacket). – Mirko Nov 27 '14 at 00:46
  • 1
    I find it difficult to believe that you cannot find stimulating topics at a university. Find out what kind of research your professors are doing and see if they are looking for summer internships. – snar Nov 29 '14 at 00:53
  • 18
    "BOHOO! Every things is too easy!" Really? Just do your part in university and keep studying by your own. =) – Pedro Nov 29 '14 at 03:35
  • 12
    A partial answer addressing the complaint "i find myself more often than not writing down homework solutions to problems i wouldn't have spent a minute on since i knew they weren't an issue": Perhaps you should use LaTeX to typeset your homework if you don't do this already. That way you will at least be getting useful practice in mathematical writing. – Trevor Wilson Nov 29 '14 at 03:38
  • 3
    Understandable situation. I've felt the same way in engineering before. I would recommend finding someone in the class who is struggling, and coming alongside that person to help them. It helps someone else do well (which is good for them), and the best way to learn something is to teach it (which is good for you). That's what I find myself doing. I spend almost as much time teaching classmates as I do studying. It's actually a great way to learn, since it forces me to think through things in different ways. Oh, and if you are planning on teaching in the future, it's a great practice. :) – FundThmCalculus Nov 29 '14 at 18:05
  • 1
    I am just like you, except i am a senior in high school. In my case I am in calculus BC, or the 2nd semester of college calculus. But the class is a joke, i already know what i will be learning for the whole year! I have yet to study or learn anything from the homework that they make me do, yet while i am in this class i am currently trying to prove my conjecture i made as a junior that involves combinatorics. But if you want my word on it, at least you are around people who can help you with the math you are self teaching. I have been looking for a someone to converse with for nearly 2 years. – Eric L Nov 30 '14 at 03:30
  • 1
    I get the sense that you feel you know every aspect of the subjects you are taking. There is absolutely no way that is true; you simply haven't looked hard enough to find the interesting stuff. Often times the book will have exercises geared toward the more advanced reader at the end of its exercise sets, see if that is true for your textbooks. Also, look for patterns in your problems sets or see if there is some abstract result that solves your problems more efficiently or gives you deeper understanding of the answers. – Karl Kroningfeld Dec 03 '14 at 07:18
  • 3
    Here in Brazil, if you feel confident enough, you can ask a special treatment. What we call "Exame de proeficiência". If you already know some course and you don't want to take classes, you can do just one exam covering the entire course. If you pass they approve you. So you could ask for the exam of all courses. If you pass then you receive the title of Bachelor in Mathematics. Theoretically you could do the graduation in just one week... But all the guys who tried this way failed... – Rodrigo Ribeiro Dec 03 '14 at 13:52
  • In what way do you think self teaching is damaging? What way are you teaching yourself and what are the results you observed that you claim are damaging? In the past, scientists thought people's brains became hard wired after they became old enough but it was later discovered that that's not the case and people don't become too old to learn. Maybe you thought you were going to become too old to learn and were trying to hurry up and cram in a lot of self teaching before you become too old to learn but I think it's probably better to take as much time as you need to learn what you're trying to – Timothy Aug 24 '19 at 04:39
  • teach yourself. When you try and hurry it, you use the panick approach and don't learn it properly because you've given yourself less time than you need to learn it whereas when you give yourself as much time as you need, you use the creative thinking approach. Maybe there is a really long limit in how much time you have to learn so I think you will learn better if you learn fewer concepts in depth than if you learn quite a large number of them spending very little time to study each one. – Timothy Aug 24 '19 at 04:44
  • I wonder what ever happened to this too-bright student. – rajb245 Jul 27 '20 at 12:22

14 Answers14


There are many things you can do.

  1. If you do understand the homework then it won't take you very long to write down the solutions. Go ahead and do it. Try to do it elegantly.

  2. Look for the subtleties, I tell my undergraduate students that they will really understand the material of course X after they teach the course.

  3. In your text there may be more advanced problems, try them or try looking in another book/source.

  4. If you feel that you can handle more advanced material, try getting permission to sit in a course that interests you, try doing the work from that.

  5. It is good advice to talk to the chairperson or a faculty member. Do make sure that your tests/homework are coming back as correct. If they aren't, it is still a good idea to talk to someone (it always is!) but then you might want to present your situation differently.

Milo Brandt
  • 58,703
  • 5
  • 98
  • 184
Aaron Meyerowitz
  • 1,553
  • 8
  • 12
  • 2
    I think in many places one can even officially take graduate classes while being an undergraduate, which will certainly provide some of what the asker is looking for. – user21820 May 07 '16 at 06:36

Eventually, all Mathematicians are self-teaching. Hopefully, on the course from Elementary School to Graduate School, we get weaned off learning from our teachers and more on learning for ourselves.

There are people who are comfortable within the Academy and many people who are suspicious of it.

Oh, yes. I’m very proud of not having a Ph.D. I think the Ph.D. system is an abomination. It was invented as a system for educating German professors in the 19th century, and it works well under those conditions. It’s good for a very small number of people who are going to spend their lives being professors. But it has become now a kind of union card that you have to have in order to have a job, whether it’s being a professor or other things, and it’s quite inappropriate for that. It forces people to waste years and years of their lives sort of pretending to do research for which they’re not at all well-suited. In the end, they have this piece of paper which says they’re qualified, but it really doesn’t mean anything. The Ph.D. takes far too long and discourages women from becoming scientists, which I consider a great tragedy. So I have opposed it all my life without any success at all. . . -- Freeman Dyson

Any discussion of (successful) autodidacts has to include Srinivasa Ramanujan. From what I understand he read his textbooks very carefully and built from that. His isolation from the Mathematical community meant although he reproduced old results, he did so independently and offered a new substantial point of view.

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary Results in Pure Mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs... The book, published in 1856, was of course well out of date by the time Ramanujan used it.

His acceptance into the mathematical community was not instant, but gradually with his correspondence to mathematician GH Hardy at Cambridge. From his first letter:

I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.

John Kingman known for the his paintbox process for Random Paritions (see also here on paintboxes). He only has a Master's Degree and he currently advises students at Oxford.

  • 23,583
  • 4
  • 35
  • 88
  • 2
    Interesting quote! Thanks for sharing – Flounderer Nov 29 '14 at 02:45
  • But it should be noted that his concept of Mathematics was not rigorous before he came to England because he only had acquaintance with Carr's *Synopsis*. –  Nov 30 '14 at 07:29
  • It's no coincidence I pick Freeman Dyson and Ramanujan since they occasionally worked on the same subject. In addition to QED, Dyson also worked in number theory https://en.wikipedia.org/wiki/Ramanujan%27s_congruences – cactus314 Nov 30 '14 at 13:38

Why not both? My Suggestions;

For each course you do, try and find a higher level book to complement it.
Many first and second-year course introduce concepts which you will generalize later in your studies. Find an excellent book that covers the course material in a more general setting. For example, if you're studying linear algebra, maybe try a book on functional analysis, for multivariate calculus, try a book on differential geometry. Do your homework and tests, but try and see how your course content relates to the bigger picture.

Learn to program.
It doesn't matter what language you start in (i prefer object orientated languages), but learn the fundamentals of programming and learn how to use it to solve mathematical problems. Use it to aid visualization of assignment problems. Check the software output against your intuition and try to describe any significant numerical errors mathematically. I would argue that programming is an invaluable tool for research and an extremely important skill for math graduates looking for work outside academia.

Learn latex, use it for all your assignments.
I found that learning to typeset my work forced me to think hard about the communication of a mathematical argument; to identify the important steps and to omit the trivial stuff. But most importantly, it pushed me out of the high-school (early undergrad) idea that maths is a sequence of equations, and English only appears at the start and the end of a piece.

Focus on developing understanding and intuition
This has been mentioned in other answers and I would say that my other suggestions all lead towards this. Make sure you're learning the ideas not just the procedure.

Hope this is useful.

  • 1,014
  • 1
  • 9
  • 26
  • 1,273
  • 7
  • 17

I think the problem you have is your mentality toward academia.

I have criticized for years that people treat academia with the wrong attitude. "Back in the day", people went to school to learn. Now, people go to school because they want to go through the motions, blindly often times, and essentially buy themselves a degree... they think they are entitled to college, entitled to a degree, and entitled to the better jobs and better pay that they automatically assume comes with the degree. To hell with the learning. Students are always dumping what they learn after an exam, or after passing a course, or after getting a degree. They barely study, and they play too much, and then they cram. It's not about learning for most of the people who actually want to go to school. And those that don't want to go to school are equally opposed to learning. It's the American way, unfortunately. "Geek" is supposedly a derogatory word, after all.

See, students go to college despite the debt they get into because they believe it will entitle them to better pay. It's a capitalistic game they play. Its a corruption of good moral sense, contrary to the ideal student and the ideal academy. A poor incentive to self-improvement measured not by intellect or skill but by career and income.

Everyone wants that chance in our money-based culture. Society feels morally blackmailed and compelled to give everyone "that chance". And so what happens? Colleges become flooded with applicants, and the quality of the graduate drops. The teachers become overwhelmed and their teaching quality drops further, and the quality of the graduate drops even more. It's an endless cycle.

In order to keep quality high, colleges keep the demand as low as possible... how do they do that? Like any supply-demand system in an economy, they jack up the prices. Students treat academia as a commodity. Schools are forced to do the same over time. We love to criticize colleges for being profiteering, but in reality it is the students fault, it is societies fault... and it is the fault of the employers who would rather hire someone with a degree they don't need, implying skills they don't need, and skills they honestly probably don't even have, instead of the degree-less applicant that actually knows what he is doing.

Here you are. You think college is obligated to give you something... something you're entitled to. But may I point out that whether you're a paying student or just a public library regular, you still have to crack open those books and study on your own either way. The school doesn't teach. The school gives you the means to learn for yourself. Then they grade you and measure your worthiness to move forward. Nothing is stopping you from continuing your education on your own outside of the lesson plan. Why do YOU hold back your own learning? If learning were truly your desire, you wouldn't stop on account of slow classmates. You admit to learning on your own already, but you stop simply because you're in college? That seems like the exact opposite of what you're supposed to be doing.

Here is an important question. Do you actually understand everything you taught yourself? Have you proven the math? Or do you just have a list or procedures memorized? Rote memorization won't get you far, especially in math. Being the best at following the solution processes doesn't make you a good problem solver. Being skilled at solving problems doesn't mean you understand the math behind it, or why the theorems you evoke even work in the first place. You might be one of the top students in some of your classes, but you are certainly not as bright as you think you are. Keep your pride and arrogance in check. No matter what you know or what you think you know, there is always more to know.

You are too focused on who will become a doctorate at a younger age. Why compete like that? It's not about the doctorate. Who cares if they earn their doctorate before you're even done with your bachelors? The fact is, if you are nearly as bright as you claim to be, you will know the material that much better and be that much more prestigious with a doctorate. Rodents with degrees on their walls are still just rodent.

I am like you in many ways. I am an autodidact. I learn a hell of a lot about a hell of a lot. Why? Not because I want a degree but because I want to learn about the world. I am in no rush to graduate. The more time I spend in school, the better..., the more I will learn and the better I will know it. I don't care if people graduate before me. I am still more confident in my skills than they will ever be.

And frankly, I don't mind an easy grade. I get straight A's across the board in nearly every subject. So what if classes are a breeze? All that means is my GPA will be high. And it also means that I have freed up more time during my college semesters to study ahead. Why learn next quarters material next quarter when I could learn it this quarter instead? Why put the burden on myself next quarter to study concepts I don't yet understand when I could just review instead? I for one don't like the challenge created by being pressed for time. I do enjoy the challenge of understanding new concepts though.

  • 1,014
  • 1
  • 9
  • 26
  • 3,263
  • 1
  • 17
  • 32
  • I don't think this gives the OP enough credit making your last four paragraphs not all that useful. "Freeing up time" can still mean 20+hr/wk work for lecture/homework if you have four classes, which is half as much as 40hr/wk but way more than 0hr/wk! +1 anyways because the cynical viewpoint is what's getting me through uni with As. –  Dec 04 '14 at 02:17
  • My viewpoint isnt cynical. Its how it should be. Everyone else in uni are the cynical ones. Think about that. Whether or not my last four paragraphs are useful to you really isnt the point. I dont see why youd complain about freeing up 20 hours per week for getting ahead next quarter... or excelling this quarter... or doing anything else you might want to do with your time. – CogitoErgoCogitoSum Dec 06 '14 at 23:19
  • I didn't say "useful to me". –  Dec 07 '14 at 02:03
  • I didnt say "freeing up this time for this reason". – CogitoErgoCogitoSum Nov 23 '17 at 21:18

I can just feel your frustration! You are in a turbulent spot but you can turn things around for yourself and come out much further ahead. Right now you say

I feel like if i would only be given a chance to study at my pace i could finish the degree in a year and a half and have much more fun doing it...

and I think this is the most important point of your whole post. You just want a faster pace and the environment doesn't seem to provide it. Your whole post reads like you are somewhat defeated but you don't have to concede defeat yet! You could turn things around because a lot more is possible once you put it as a goal for yourself. The clearer you are when it comes to your goals the better results you'll have. Setting a good goal is not difficult. Setting a great goal is somewhat difficult mentally. If it makes you blush a bit thinking about it, then you are on the right track. Let's re-write the complaints as a big goal. You can write it in your own words but here's one way to write it:

I want to finish my degree in a year and a half and have much more fun doing it!

I don't know how far into your program you are but if you are at the start, doing a 4 year degree in 1.5 sounds quite challenging. So depending on where you are in your program, this could either be a good achievable goal, or a great-somewhat-out-there goal. The important thing is that this is something clear that you can work towards and you are no longer a victim. With a goal like this, you are taking control of your life and your career. Then every situation lends itself to problem-solving strategies. When a course is boring and slow you will no longer think about how much this sucks but rather you'll think about how much more you can achieve, and the steps you need to take to get there.

If you think 1.5 years is quite doable then you can make it challenging for yourself by shortening it to 1 (and this is only an example). You might need to convince yourself that it is something you can achieve. Except for any technical difficulties from the department (those shouldn't be so difficult to deal with) everything else pretty much depends on you and how you go about your studies. To achieve your goal fast you need to be industrious. You need to become more systematic in your studying. If you already find yourself to be a good learner, being more systematic will only propel you further.

To be more concrete I'll give you an example. Let's say you usually take 3 courses. Let's see what kind of problems you might face

  • The tempo is too slow.

You can do a few things. You can take more courses for starters. You can take 4, 5, 6 courses - however many you want really. You just might need to ask for permission from the department. Or you could go deeper into each course. I find that the latter options is ... healthier.

  • The tempo is too slow but there is a lot of course-work.

In this situation you probably won't do too well to take more courses. The work still needs to be done. This is where you can get more industrious. Maybe there is a way for you to write out the exercises and problems faster. Maybe the calculations are tedious but you can automate them. The worst feeling is when you know the material well but you haven't mastered it to the point where you are correct on all the exercises. Self-deception is a real danger when you self-study.

Last but not least, I'm sure that a lot of teachers would be willing to let you work on later assignments earlier in the course. Imagine you know the material well enough to do the later assignments. Imagine that you do all the assignments in your first month and only show up for the exams. At such a point it might even be a waste of time to take the course. Sometimes you can challenge a course and write an exam where if you pass you get credit.

I think it's safe to say that if you go high enough in the course-ladder you'll meet your match. Eventually you can get to difficult courses that will be all the fun and take all your time. At this point you would really lower the number of courses that you take. If you run out of undergraduate courses, there's always a ton of grad courses. I doubt you'll run out of those while you are still an undergrad but that's a possibility you shouldn't limit yourself from.

So, to put it shortly - look at all the courses you need for your degree and see just how many you can do fast by taking more courses in every semester. If you think you'll run out of courses, take grad courses as well.

Alexander Vlasev
  • 3,080
  • 14
  • 21

Let us analyze your situation part by part.

...The thing is I don't think that it would have bothered me so much if I hadn't felt like the academy is holding me back. ...

Are you sure that except the one course that you have mentioned, there is nothing in other courses that stimulates? I am not saying that you are wrong or something but have you done a careful analysis? Because often it happens that when we are frustrated or annoyed, our mind stops working rationally and logically and the opinions we make reflects our impulse. Remember that I am not saying that it is the same case here also, but it may be a possibility. However if you are sure that you have done enough analysis then you may be right. From this point on I'll assume that you have analyzed your situation carefully yourself.

...When I learned by myself from books I just went from one thing I didn't understand to the next and no minute felt wasted. Ignoring the fact that I can already solve the test of three courses I'm in (I tried to avoid them but I have to do them), I find myself more often than not writing down homework solutions to problems I wouldn't have spent a minute on since I knew they weren't an issue. The pace is so slow that I regret having started at the university in the first place (with the exception of one course). ...

The way you have presented your problem, it seems to me that you are much more annoyed than frustrated. To be frank, it really becomes difficult if you have to do something you don't like. Especially in case of Mathematics, it becomes even more difficult. But, if you are annoyed because the homework problems weren't an issue, I would suggest you to think again. It often happens that a particular problems seems easy to deal with but a lot harder to express it formally. This being said, I don't think that it is the reason of your annoyance. It may be due to the reason that you have to spent time on seemingly trivial problems against your will. In that case, it would have been very hard on you but believe me, in this respect no expert can help you and to be frank it seems to me that if this is your problem then it would be better to leave the University because if you force your tormented mind to adapt to something you don't like, the result would be counterproductive.

...So far it feels like half the fun of math for twice the time (and all those little kids have the time since by my age they'll be doctoral students...). I really miss those self teaching days.

I feel like if i would only be given a chance to study at my pace i could finish the degree in a year and a half and have much more fun doing it...

If you don't have fun in doing what you are doing, I suggest you to leave doing that. Don't be confused. If you don't enjoy the Math, it will be very difficult for you to actually learn it. And from your words I do think that you really want to learn Mathematics. So, my only suggestion would be to go for what you like. Enjoy learning!


I don’t feel that self-teaching is damaging at all. Rather I think that it gives you some advantages.

Having taught myself calculus 1 and part of calculus 2, then having to retake calculus 2 in college, I felt that way at first. As time went on, though, I found that my ability to help others learn was reciprocated to my benefit. And you can shorten your time with extra effort. I am on course to finish 4 years in 3 and my brother to finish in 2.

How you respond partially depends on the atmosphere at your school, and the class sizes. I am also self-taught and currently have some of the same challenges you do. Here are some things you can try that I have tried.

  • Ask the professor and other students interesting questions – questions which stimulate you and the other students. Many professors are experts in the fields of which they are teaching you only a small amount. They appreciate if you want to learn more because that attitude is rare. One student that is excited about the subject can motivate an entire class to go deeper. Ask with humility though, or else it will backfire.
  • Research any additional branches of mathematics that you come across while studying. Often when you bring up a difficult extension to what you are learning a teacher will suggest that you research it on your own. Step up to the challenge. Share your results with the class.
  • Create a website. Add a blog with in-depth articles relating to the subject you are working on. Share your blog posts with your professors and see if you can get extra credit for doing them.
  • Create assistive handouts for the teacher with concepts or formulas from the subject you are studying.
  • Try tutoring. Many universities are glad to find students who are willing to share their knowledge with others and the university will often pay you for it.
  • If you feel that the teacher is going slowly because certain students can’t go any faster try helping those students to understand the material. You will do the student, the teacher, and the whole class a favor. And you will understand the topic much more thoroughly yourself.

Although you feel like you are in a hard spot there are some privileges to your position. Look at how you can help others, rather than how slow it is for you and you will find that it comes back to help you after all.

It is impossible to change all of academia but you can encourage a teacher and inspire a class if you encourage a passion for numbers.

Math Man
  • 722
  • 5
  • 17

In my experience, it can be greatly beneficial to get to know your instructors. The fact is that if you can convince your instructors that you could or ought to be in a higher course, then all it takes is for them to write an e-mail. Want to take a graduate analysis course? Want to attend a reading course? If you can convince other professors that you're ready, then you're fine. Also, there's often a credit-by-examination system for a number of things, e.g. your calculus sequence.

Further, think about looking into other areas. Not as an alternative to math necessarily, but as other interesting areas, e.g. CS.

  • 7,909
  • 2
  • 15
  • 39

One nice thing that school brings is interaction with your peers, interaction with your professors, and structure. The biggest danger on self-studying is having holes in your learning. Classes are great to fill in these holes. This is particularly pronounced in CS/IT programs with the computer programming coursework.

Even if you don't have holes, you get to explain things to others. That helps you cement your understanding of the material and helps you make friends. If you end up in a required physics class and don't get physics, maybe one of your engineering friends can help you there.

It sounds like you're in freshman calculus or second year calculus. Once you hit the proof-writing courses, thing will change and get better. I would also suggest looking at relevant courses in other departments. Economics and Computer Science are great departments to explore. Start with intermediate microeconomics and macroeconomics. Micro will give you a feel for behavioral econ and some optimization. Macro has more of a diff eqs and dynamical systems feel. From there, maybe take a game theory course. Econ also has graduate coursework that is much more technical than the undergrad. The mathematics for economists course is a nice grad course that partially duplicates the math curriculum, and partially covers new material you probably won't see (hyperplane geometry, more thorough treatment of linear programming and optimization, correspondences and continuity). The graduate microeconomics and macroeconomics courses are also very technical and mathematical. Micro I at the grad level is generally game theory, industry structure (monopolies, oligopolies, social welfare), and consumer theory (preference relations, utility functions, ordering, topology).

CS also has a number of relevant courses like algorithm analysis, theory of computation, machine learning, and AI. Algorithm analysis is generally algorithmic graph theory, with some additions like Big-O, NP-Completeness, String matching, and maybe some computational geometry. Theory of Computation gets into things like formalizing a computation machine mathematically, and then answering questions about what can and cannot be computed. The Undecidability of the Halting Problem is proven with a Cantor Diagonal Argument, which you see come up to show that $\mathbb{R}$ is uncountable.

Machine Learning is an application of statistics and linear algebra. AI is a lot of stats, game theory, and graph theory.

Having perspectives from these different departments will help you stand out in a very good way. Plus you'll get more interesting material than the math folks simply doing math.

  • 13,668
  • 2
  • 23
  • 45

Go teach yourself as much as you can, as fast as you can. Your homework and tests are a breeze, so finish them as quickly as possible and begin teaching yourself more advanced mathematics. Your university library is FULL of amazing textbooks. Do more mathematical instruction on your own. You have plenty of time since it takes you little effort to do what is required of you in classes. Start learning Masters/PhD level mathematics on your own. Find syllabi and graduate level curricula for top programs. MIT OpenCourseware is a great place to start. It should only take you a year (two max) to complete a self-taught masters curriculum. Once you've learned all the mathematics for a Master of Science/Arts in mathematics, use your university library to start reading peer-reviewed literature in mathematics. Being at the university is the only reasonable way to access the broad array of peer-reviewed journals. You have online access to many of these through the university as well! Pick a topic you learned about, and read the latest research papers. All of them. Track down every reference in every paper, and then track down the references therein. Read them all. Understand them all. Teach yourself the techniques they use, all the background, etc. When you find something you don't know yet, just go find textbooks or the original papers on the topic from your library. Verify every theorem and proof you find.

The list of topics you can study is staggering...the world is your oyster, but you'll only be able to read the research (without paying a fortune) while you're at the university. Then, start writing research papers of your own and publishing in the peer-reviewed literature. By this point, I imagine you will be in your second or third year at university, so you might be able to partner with a senior faculty member just for name recognition. All the ideas and hard work will come from you, but the "system" generally requires a well known name attached to the paper. With your newly acquired, expert level knowledge of mathematics, deriving novel results, conjectures, and proofs should be straightforward. By the time those young kids are just beginning their dissertations, you'll have a solid publication record, and you'll effectively have a Master's level education before finishing your undergrad. Now, apply for Master's programs, and when you're bored in those classes, blow through the homework and tests, and do this whole thing over again! Write more papers! You'll be a world renowned mathematician with an excellent publication record in no time.

  • 4,550
  • 14
  • 30

Spend more time going after what your gut says and less after the curriculum... In a university you have more choice of books and courses for wider exposure, ... that should not be ignored, and use it to sharpen your instincts.

  • 36,354
  • 7
  • 34
  • 88

I hope this isn't a repeat but from skimming the other answers, I didn't see it mentioned. Many US universities allow you to CLEP out of courses you know, not to sure about foreign schools though. It cost less than $\$100$ I think which is significantly cheaper than the actually course. So just CLEP out of all the courses you know really well.

Here is an excerpt from the wikipedia page:

The College Level Examination Program (CLEP) is a group of standardized tests that assess college-level knowledge in several subject areas that are administered at more than 1,700 colleges and universities across the United States created by College Board.[1] There are 2,900 colleges which grant CLEP credit.[2] Each institution awards credit to students who meet the college's minimum qualifying score for that exam, which is typically 50, but it does vary by school and exam.[3] The tests are useful for students who have obtained knowledge outside the classroom, such as through independent study, job experience, or cultural interaction.[4] CLEP also offers students (including international and homeschool students) the opportunity to demonstrate their proficiency in subject areas and bypass undergraduate coursework.

  • 7,933
  • 11
  • 42
  • 81

Remember that life is stochastic and never linear. Learn about the true eigenvector driving research - ego. Then you may stand a better chance to understand the difference between math as poetry, and the stark reality of basic arithmetic determining your chances for retirement. Good luck, and happy math dreams!

Antoni Parellada
  • 7,767
  • 5
  • 30
  • 100
  • So, if the eigen*value* driving research is in $(0,1)$, your ego vanishes as you do more, if it's in $(1,\infty)$, it grows arbitrarily large? (And if it's in $(-\infty,-1)$ you have crazy mood swings) – Milo Brandt Dec 03 '14 at 02:13

I myself am starting university. Since you clearly are ahead of the game, it would be wise to join clubs that interest you and/or are related to your program. My instructor has told me this before, and I will reiterate what he said: "A degree will only get you so far. Its the connections you make that can make your degree worthwhile". So go out there and get involved, I'm sure something good will present itself to you. Also, look into applying for co-op, because some experience in where ever you want to work would be handy as well. Hopefully, this helps.

  • 1,014
  • 1
  • 9
  • 26
  • 213
  • 2
  • 11
  • How does this answer the question? – Eric Stucky Dec 06 '14 at 02:40
  • @EricStucky I think they are saying the connections at the school are more important then the content in it. At an Ivy (also lump in Stanford, MIT, CalTech, ...) league, the connection could well be worth more than the education, but in general, probably not so much. – dustin Dec 06 '14 at 03:17
  • @EricStucky this answers the question as OP wants to know on how to progress for the next few years in University. Joining clubs can make connections to people in the industry as well as spend time doing something productive. The amount of things you can potentially learn from clubs can be invaluable later on too . I should also add that extra-curricular does look good on a resume, so I've been told. Being productive in something new (and learning something new) AND scoring some potential sweet connections, how is this a bad thing? – user3170251 Dec 06 '14 at 19:51