I’m currently a sophomore interested in pursuing a PhD in mathematics. I’ve learned a lot of calculus already and plan on learning almost all of the undergraduate level math courses before I finish high school. Someone recommended that I instead take the courses online and get the credits for them. Does any one have any thoughts or advice on the matter, or better yet, experience with this?
I do, :). Though I only really did classes until Calculus (through AoPS). Which website do you use?
For undergraduate math, students from my hs go to the nearby community college (or even better, I go to UCLA, which doesn’t give credits, but is really useful). CC’s give credits pretty easily. If you’re going to be a junior next year, it should probably be easy to fit CC courses into junior block schedules.
Perhaps do BYU’s math courses. They have everything from Accounting, to College Algebra, to Physics.
I haven’t picked a site yet, I’m just looking for information. I want to do these courses on my own time, preferably over the summer as my junior year schedule looks quite daunting. Does anyone know a particularly good place or have other advice? Also, I want to do multiple courses beyond Calculus I, namely Calc II, III, IV, Analysis, Abstract Algebra, Differential Equations and such.
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I get the feeling that you don’t quite know what you are in for with this plan. If you only can work over the summers, then it is very unrealistic to expect to finish an entire undergraduate math curriculum in two years no matter how talented you are. My advice would be to focus on the courses that your high school offers first. If you want to do some reading (without mastering) about other topics, then that is useful and something that you could do over two summers. There are also a (very) few research level problems that can be understood by motivated, bright students with a relatively minimal background. And by motivated, I mean with a capital M. Don’t be discouraged if you can’t understand in a week or a month or a year. Perhaps you could look into understanding one of these problems as better preparation for what getting a PhD in mathematics would be about.
My high school only goes one course above multivariable calculus, which I’ve already covered pretty well. This is pretty much the only thing I can say I want to focus on, even if it means no free time or doing it above my coursework. If you could point me in any particular direction that would be nice 
First of all, it’s fantastic that your high school has a course above multivariable calculus. I still say to exhaust that first.
Is there a teacher at your school you feel comfortable with? It is really important when doing courses online or self-study that you have someone you can talk to about things when you get confused or stuck. The last thing I want to see is someone who is so motivated in math to get discouraged.
Beyond that, I can only give a list of topics: you need to be able to prove things, linear algebra, discrete math, probability and statistics, differential equations, complex analysis, real analysis, algebra. If you are super motivated, the American Mathematical Society has a series of books called the Student Mathematical Library, which is absolutely fantastic. You could hardly do better than to find a library which has these books in them, and look through until you find a book that catches your eye and study it. I have my own favorites, but I haven’t read them all and think you should find something that intrigues you. They are typically aimed at undergraudate juniors or seniors, so don’t feel bad if you don’t understand everything. I am a research mathematician and don’t understand everything in those books.
Sorry I don’t have specific advice re: your original question, but hope this is at least somehwat helpful.
Thank you for the response dude. Yeah, there is a teacher here who can help me out. His classes are mostly self-studied (multivariable calc and differential equations), but I’m sure he’ll help me if I ask. I’m going to take his class next year for sure after taking the BC exam, so I’ll have ample access to him. I don’t think my public library will have those books but I’ll check just to make sure. I guess they definitely should have something of mathematical interest. Thanks again.