Nobody told me that self teaching could be so damaging... [closed]

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).

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.