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I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics that are covered in university, just the ones that happen to interest me (elements of differential and integral calculus, complex analysis, etc. You can see what I am interested in by looking at the questions I've asked and answered).

Now, a few months ago, back in school, I asked my math teacher for help on a differential calculus question whose solution I did not understand. I was told by this teacher that I should not be doing calculus and I should wait until I learn it in school. Other math teachers either did not understand what I was asking or shared the same view as my math teacher. For awhile this had distressed me very much, because some of my own math teachers were telling me to stop learning math and to wait three or four years to continue! Should I stop learning math by myself? I decided that I would keep going, because this is a hobby and interest of mine and I didn't think teachers should have the right to stop me from learning.

I find it more and more difficult to proceed learning on my own without a mentor who can and will help me, and I don't know what to do. I went to my school's math club, but alas, no one there was that interested in doing math for fun like me, and no one was interested in answering or helping me with my questions. This website has proven very helpful to me, however, it is not like talking ans asking a person face-to-face.

What should I do? Am I learning math too early? Should I wait until university to continue learning calculus? If not, how should I get a teacher or continue to learn on my own?

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Most definitely not! Can you participate in some math program, like training for the math olympics (?) at your school? Try and ask your teacher about it. Have you tried looking up online courses? Some lectures from very prestigious universities are available on youtube or on their main site. –  Olivier Bégassat Jul 6 '12 at 1:30
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Ignore your teachers. If you like doing it, it is not too early. Is there a college or university near you? If so, call up the mathematics department (or have your parents do it for you) and explain your situation and ask for permission to sit in on a class. There is a good chance that they will say yes. –  MJD Jul 6 '12 at 1:32
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If your teachers told you not to do calculus because it's 'too early' then they're idiots. The math I've learned from stumbling around above my level is stuff I remember much better than the things I waited to learn through class. It's more difficult, but it's also much more valuable. –  Robert Mastragostino Jul 6 '12 at 2:29
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You should be aware of calculus trap, but learning an introductory course in calculus at your age is, in my opinion, perfectly ok. (If you have no problems with the math you're being taught in class.) –  Martin Sleziak Jul 6 '12 at 5:55
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Sorry to say this, but your teacher is an idiot. –  Michael Jul 6 '12 at 10:04

7 Answers 7

up vote 34 down vote accepted

I was recently in a similar situation. After finishing precalculus at my high school, when I was 15 I started taking calculus at my local university and studying higher mathematics on my own (out of the book "Modern Algebra: An Introduction" by John Durbin, which in retrospect seems laughably basic but at the time blew my mind). Three years later, I can say without a doubt that it is the best decision I've ever made. I ended up learning mathematics through a combination of taking classes at university, talking with students/professors, reading textbooks, and using this site. I did have one major advantage over you though, as my parents are both professors (although neither of them math professors) which made it easier for me to get into classes. However, I know of other people doing the same thing without any connection to the university. Here are some things I would recommend based on my experience:

  • Get an introductory textbook for some relatively advanced subject, such as Calculus, Linear Algebra, or Abstract Algebra. Read reviews online before choosing one to find one that is both rigorous and easy enough for beginners. I'd recommend Spivak for Calculus (take this with a grain of salt though, as I never read it but have heard good things about it) or Durbin for Abstract Algebra. Make sure it comes with plenty of exercises, and DO THEM. If you don't know how to do a problem, or if you've done it correctly, ask here!

  • If you have a university nearby, take advantage of it. Email a professor teaching an upcoming introductory course and explain your situation to him/her, and ask if you can sit in on the class. You might even be able to enroll in classes as a non-degree-seeking student, if the university allows this (most do) and you can afford it (if it's a state school, the tuition for a single course might not be too bad). Don't be afraid to talk about math with professors. It can be intimidating, but remember, these people have dedicated their lives to math. Almost uniformly, they LOVE it. Half of the time I had to find a way to break off a conversation with a professor because they were so engrossed in the math at hand.

  • Find something specific you don't understand. It may be a theorem, a proof, a concept, or even an unsolved problem, so long as it fascinates you. Figure out what you need to know in order to understand it, and start down the rabbit hole. The experience of coming to understand something like this can be very rewarding in addition to teaching you a great deal of mathematics. I've had several of these over the past few years, most recently an unsolved problem known as the Triangular Billiards Conjecture which I'm studying right now.

If you have any questions about my experience, feel free to ask. Good luck!

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Thanks a lot! If you don't mind, I may ask you a question in the future. –  Argon Jul 10 '12 at 19:50

You shouldn't listen to your teacher, except if he has a valid reason to tell you this. The only valid reason I can think of is that your grades in math aren't up to par (up to par in your case meaning #1 or close to #1). What is your class rank? Are you first in math (if not, close to first?). If you aren't, then the teacher has a point, if you are, disregard what he says, my physics teacher told me the same thing while I was #1 of my class by a mile, so I just did what I wanted to do.

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There is no reason to delay looking at whatever you want to look at. For that matter, I would say that you have no obligation to "systematically" read anything, or do exercises, unless you want to. In the U.S., not only is the high school math curriculum stultifying and anachronistic, but also most of the undergrad curriculum is the same. In particular, I'd recommend not being toooo trusting of "standard textbooks", because the design of conventionally-published textbooks is very often strongly influenced by pressures from publishers to make the things match the usual curricular structure.

Looking around either in physical libraries or on-line stuff at _non_textbook_ writing (at the very least as a supplement to textbooks) gives a much better perspective on "where math is going", and ideas, rather than "lists of required topics". Further, I think it is misguided to tooo strongly require "mastery" of a given thing before looking at "the next thing"... because the purpose of a thing/idea is often only revealed by what happens later, and that hindsight often makes many of the earlier details much more intelligible. It is important to allow oneself the flexibility to "skip forward to see what happens", all the more so given the tendency/tradition of mathematics writers to logically re-order things: instead of giving illuminating examples with observable properties, subsequently organized by choosing terminology, the style is to give definitions first, etc.

Also, with textbooks, beware that many of the exercises will be simply "made up" to meet expectations. In particular, they may not really be worthwhile, but just filler. Hard to know which is which, for a beginner, too. In extreme cases, "doing exercises" can actually be a strange avoidance mechanism that obstructs learning! :)

Also, the contest-math world is not very much about mathematics, although if you have a flair for this it can be a good PR vehicle for you, and a way to meet other people approximately interested in mathematics. The down-side is the obvious emphasis on speed and cleverness, rather than learning serious mathematics, so keep in mind that this is not at all "mathematics proper".

Again: trust your own interests, judgement, curiosity.

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Here is a link to the Berkeley Math Circle:

http://mathcircle.berkeley.edu/

It is replete with really outstanding teachers and offers lots of material for study at many levels. Maybe you're lucky enough to live near there.

Otherwise, if you go to the "Links" section, there is a further link to other math circles.

I would venture that through their generous endeavor you will find something.

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I would like to add to the comment by MJD that in most countries, the participation on lectures (not seminars and excercises) is free to anybody, by the Academic Freedom.

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In other words: math lectures don't have bouncers. –  Peter Sheldrick Dec 2 '12 at 19:13

I want to sound a note of caution. When one has a talent for mathematics its easy to get thoroughly absorbed by it.

It may look to you that a teacher is stopping you from learning, and it may be that he is; but a more charitable view is that he is concerned about emotional development: there are more sides to a person than the purely intellectual. And there is much more to intellectual development than pure mathematics.

Having said that, it is true that school curricula is stultifying, and its hard not to get irritated and annoyed and want to move beyond it. Once you get to university, you'll find this less of a problem, but it is difficult in other ways. (You'll see how when you go to seminars and everyone is just copying what is being said on the board). One has to learn how to handle the outside demands on ones time.

There is a romantic myth that the best mathematicians do their work when they're young - this isn't true. A good look at scientific history will show that it is not. This was definitely a problem for me when I was your age.

You seem to be already well aware that it's possible to learn outside of the core curriculum at school. I don't doubt that you will continue, but it's also important to find a mentor and peers.

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I hate to chime in so late, but I could have been in a similar situation in high school. Fortunately, I did have a supportive teacher (who passed away soon after I took her precalc class, what a shame) who dis answer my calculus questions. That said, she introduced me to her math teacher's journal (this is 1987, I cannot recall now) and showed me some extra hard problems that used elementary math. I also made sure I participated in all of the competitive math leagues and events that were feasible, whether or not sanctioned by the school. (I went to the ARML in 1988 despite nobody from my school ever even having heard of it.)

That said, of course I went and found a book from which to learn calculus on my own: "Calculus the Easy Way." Excellent excellent book, I got it in a snap and by the time I was formally taking calculus, I was on differential equations and Taylor series. The school was a bit bemused, but not discouraging, not at all. I am so sorry that you met that attitude, I really cannot condone it.

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