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I am currently a 15-year-old in the 9th grade, and I am studying to test out all of my high school math classes, which is something I didn't realize I could do in the 8th grade and would have if I had. I contacted faculty, and if all goes well, I'll test out of Geometry, Algebra II, Precalculus, and AP Calculus BC. I also hope that once at college, I can test out of Linear Algebra.

I have also been studying Japanese for a few years, and know the meaning of all the required Jōyō Kanji, although I lack vocabulary otherwise. Because of this background, I plan to Major in Pure Mathematics, and Minor in Japanese Studies.

If I end up taking classes at Delta, (a nearby community college) I would not be able to take Topology, Complex Analysis, Real Analysis, or Abstract Algebra. Instead, I would take Multivariable Calculus, Linear Algebra (if I fail to test out of it), and an ODE course, as Delta only has a Transfer Math program, and no Japanese language program.

I've been told that my high school will also pay for my college classes while enrolled in high school, although the only people they have ever done this for went to Delta during high school, and they were seniors in the 12th grade.

That being said, I have a few general questions about college that fall under the umbrella of the title.

  • Would it make sense to apply for colleges as anything other than a Senior, or duel enroll in non-community colleges?
  • What will writing a thesis be like for someone with no Math paper writing experience?
  • I have below-average grades in everything but Math, how much will this affect me?
  • As a high school freshman, I've yet to take the SAT, how much will this affect me?
  • For the math people reading this, how do you recommend studying math on my own?
  • My mother doesn't want me to test out of my math classes, because "they teach me hard work," and because she wants me to raise my other grades first before doing something "fun". How do I deal with that?

"Calc 3, ODEs, and LA are all solid classes to take. Nothing wrong with them."

I am currently studying these (and enjoying them, no less!) but the main reason these deterred me was that there are many online resources to self-study these with, whereas there are significantly fewer resources for the more pure subjects.


This post was moved from Academia, as Undergraduate admissions are not allowed there. However, the mods who closed my question directed me here. I wasn't sure if it would be appropriate, but because this site allows posts about "the process of learning mathematics," I thought it worth a try.

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    $\begingroup$ This is a fantastic time of your life to practice some of the skills your future self will be glad you have—one of which is the work ethic, and really the outlook on life, to excel at the things that don't interest you as much. I was "forced" to take an entire year of philosophy in college, and I ended up doing very little of the reading; now, I look back and say "I wish I hadn't passed up the best opportunity I'll ever have to read a lot of classic philosophy!" Given how bright you obviously are, your future self will be glad to have the most well-rounded knowledge and exposure possible. $\endgroup$ – Greg Martin May 1 at 16:35
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    $\begingroup$ By the same token: don't rush to finish college as fast as possible. College is also the most fantastic opportunity we'll ever have to develop our social skills, and broaden our familiarity with people different from ourselves, as we become the adult person we want to be. For me, I wasn't even all that great at doing so—but I'm grateful every single day of my adult life that I used that time to try. $\endgroup$ – Greg Martin May 1 at 16:37
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    $\begingroup$ What you really want is to strike it rich somehow, so you can "test out" of working. (You test the value of your portfolio, and find it to be far greater than the maximum value that still requires holding down a job.) Once you test out of work, you will find yourself with free time and may end up doing things like taking university math courses. $\endgroup$ – 509-249-3447 May 2 at 18:07
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So you dual enroll at the community college while you're in high school, and you finish up in two more years. While you're at it, consider some computer science courses. Also, physics is a lovely course. (I was put into chemistry and hated it. I would expect almost anyone who loves math to at least like physics.) Do you have a high school counselor you trust for non-math recommendations? If you figure out which teachers you'd like best, I bet you could do well in other courses.

For other higher-level math courses before you leave high school, consider Group Theory and Number Theory at Art of Problem Solving, which offers these classes online. Art of Problem Solving also has materials for studying on your own.

In two years, you should be ready for the college of your choice.

[Note to mom: People work hardest at what they love. I am a community college teacher, and I've published a book. I'm also a single mom. I have never gotten good at doing the things I don't like to do. I do grade my students' homework, which is my least favorite task. If your son is this advanced in math, he's been working hard at it. He will slowly get better over his lifetime at doing the parts he doesn't like to do. If he's passing his other courses, and excelling at math, that sounds reasonable to me.]

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    $\begingroup$ I think this is horrible teaching advice. At 15 one should not study all day long, but enjoy his childhood. When I first read your answer I thought you were being sarcastic. How is this guy going to handle a course in physics, chemistry, computer science, group theory and number theory at the same time? And much more importantly: How is he going to stay in touch with his friends? At this age building personality is much more important than some math skills. And this comes from someone who devotes much of his life to maths and physics. I mean come one.. $\endgroup$ – TheoreticalMinimum May 1 at 18:49
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    $\begingroup$ @TheoreticalMinimum I am 16 years old and already in my second year of university, majoring in physics and chemistry (but thoroughly enjoying my maths as well). If I had not been accelerated, I would still be in school, probably totally disengaged due to the content being far too easy for me. I didn’t have any friends either (not in any bad way, I just didn’t see the need for them at that point). (1/2) $\endgroup$ – bradrn May 2 at 13:10
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    $\begingroup$ @TheoreticalMinimum (2/2) But when I was accelerated, I found that I could easily handle the more advanced content — in fact more easily, since it wasn’t so boring — I was more engaged, and for the first time in my life I finally made some social connections (it’s a lot easier to connect with people when they’re interested in the same things as you!). Advice needs to be given individually for each student; for me acceleration worked well, although I can easily see a gifted student with lots of friends finding it hard to accelerate, for exactly the reasons you describe. $\endgroup$ – bradrn May 2 at 13:10
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    $\begingroup$ Enjoying your childhood is a foolish concept. You're in fact best off if you can conduct your life in which a way that you can enjoy yourself all life long as if it were one long childhood. To some kids, academics is child's play, moreover. $\endgroup$ – 509-249-3447 May 2 at 18:10
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    $\begingroup$ The "don't pursue what you love, do normal things instead" in the comment is really epically bad advice. If someone has a drive and passion to work at something, they're among the fortunate people to have that plan and focus. Your life is your life (all of it), at no point should you be put in a waiting room for it. $\endgroup$ – Daniel R. Collins May 4 at 5:07
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Calc 3, ODEs, and LA are all solid classes to take. Nothing wrong with them. Completely useful even if you move into pure math. For one thing a decent chance you may teach them! Also, it is good to build your manipulative skills. Finally, it keeps options open in case, you end up going into other fields (as those are common prereqs for most STEM majors).

Get the rest of your grades up. Colleges care about that stuff. It shows work ethic.

Take the SATs.

You may end up going to college early, although I wouldn't go immediately. A little physical maturing wouldn't hurt you either. Huge difference of 16 yo versus 14 yo.

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    $\begingroup$ As a professor in pure mathematics, I can emphasize that Calc 3 and ODEs are tremendously useful and indeed indispensible for many fields of research; and in my opinion, linear algebra is 100% indispensible for any mathematician. $\endgroup$ – Greg Martin May 1 at 16:32
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    $\begingroup$ Calc 3, ODEs, and LA would also be beneficial if you end up pivoting from math to science or engineering. $\endgroup$ – Dan Is Fiddling By Firelight May 1 at 18:44
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The Color of Grass

My guess is that your excitement to race through school is partly due to a low valuation of your current high school experience, including boredom and social isolation. College and the real world surely look like places where "the grass is definitely greener." It's possible that you will never value other human relationships as highly as the average person, and thus, racing through the phases of life is an optimal strategy for you. But I would bet that at some point, you will value those relationships much more than you do now, and find yourself in an awkward position. You see, high school and college are stressful and chaotic times, socially speaking. But they are also essential to building your social skills, because you will likely never be in a situation where you have so much close contact with people your age. Once you get into the working world, the spread of ages increases, and the level of social connection decreases. It gets harder to meet people the older you get. If you compress this time by 50% or more, you will likely look back on it one day and think: "Perhaps I should have spent more time meeting people and socializing back then. There were so many opportunities."

Completeness

It's easy to focus on the things we're good at, and avoid the things we are not. That is human nature. But if I had to guess, your mom is quite possibly worried about the scenario I'm describing above, but not explaining it as clearly as she could. Can you survive as a pure math prodigy? Well, there's Grigori Perelman (I did enjoy his biography). But unless you want to become a hermit, you will need to become somewhat well-rounded. At the very least, you will need to be able to communicate clearly with others, especially in written form. And if you choose to go into industry rather than academia, you will almost certainly need to learn some application of mathematics. I strongly second Sue's suggestion to study some Computer Science. You may find that programming suits your personality well (and gives you a lot of career choices later on). Odds are, no matter what path you choose, having some programming skill will be useful to indispensable in the future, so this effort will certainly reward dividends.

It's good that you mentioned the SAT. Obviously, having poor grades in non-math subjects does not bode well for potential scores. You don't need a 1600 to get into good schools, but performing poorly on the reading/writing section will not work in your favor. As I said before, communication will be an essential skill for the rest of your life. Spending an adequate effort developing this skill by reading broadly and practicing your writing is an investment that will only yield strong positive returns. There is nothing more frustrating than having a brilliant idea, but not being able to explain it to others. If you're slacking in your English classes, just pretend that English is a somewhat broken formal language, and use computer science skills to turn it into a math problem. That might help with your motivation. But reading outside of math is also important to become a complete human being. It's good to go deep (learning as much about one topic as possible), but it's also important to go broad (learning a little bit about as many topics as possible). Yes, you can do both; and there is no right answer about the optimal balance. But you will find that the most successful people in any field (including pure mathematics) almost all have a broad range of interests and pursuits, even if they have gone very deeply into one of them.

Learning

I find that the best way to learn something new is to pick a goal and work towards it. Try to find some concept that interests you, and learn everything you need to understand the state of the art in that area. Sometimes you will encounter an enticing side trail, which you should follow. Sometimes, you will discover an even more interesting problem that you didn't know about when you started, and you will shift your focus. The important point is keeping yourself motivated. If you're getting stuck on a concept, it means you have gone too far, and need to go back and study the fundamentals. Nailing the fundamentals is essential in every field and skill. But practicing them for their own sake is not terribly fun. That's why it's helpful to "go deep" first, and find a problem that motivates the need for stronger fundamentals. This then gives you the motivation to back and learn something better that may have seemed boring at first glance.

Where to get ideas? Well, StackExchange is a pretty decent place to start. Just browse random questions on the Mathematics SE (which I assume you already do), and look for ones where the answer involves concepts you haven't encountered yet. Chase the ones which sound most interesting, and your plate will be full in no time. It's not unusual to find math-heavy questions on Physics or Computer Science or [Electrical] Engineering or Aviation, so it's worthwhile checking those out as well (and helps with the "going broad" strategy).

One of the reasons that going broad is such a valuable strategy is in the nature of learning itself. Ultimately, the abstract reasoning which separates humans from most other animals comes down to generalizing across distinct ideas. It's about finding connections. It's about looking at an apple and a tennis ball and a rock and thinking about what makes them different, but also what makes them the same. Your brain will do this for you all on its own, as long as you feed it lots of data. And the more diverse the data, the more connections you will discover. Many fruitful ideas come from cross-field collaborations, where people studying ostensibly unrelated topics end up finding an unexpected connection that helps them understand better. It seems hard to believe that your pure mathematics skills could be improved by studying biology or aviation or economics, until you realize that partial differential equations well describe many phenomena in all those fields, and having physical instances to visualize and reason over strengthens your mathematical intuition about the equations themselves.

Conclusion

If I were you, I'd make this deal with your mom:

Mom, I'm learning math much faster than my high school classes can possibly teach me. Sitting through them is a waste of my time when I could be productively learning maths several years ahead. On the other hand, I recognize that I have not been as diligent in my other subjects as I could be, but that these subjects will be more important in the future than I have given them credit for.

Thus, I propose this bargain: you let me test out of all the math I can and take whatever college math courses are available to me, and I will raise all my other grades to at least a B. If, at any point, my grades fall below a B, I will withdraw from my college math courses. Is that fair?

Finally, I would recommend that you try to socialize with a broad distribution of your high school peers. You surely find that you have little in common with many of them, but learning the strengths and shortcomings of your fellow humans and members of society is most certainly a skill that you will use for the rest of your life; and the better you are at it, the happier you will be, even if you end up cloistered in an ivory tower inscribed with Greek letters. That includes playing a sport, if you don't already. "Balance" is about developing the body as well as the mind (and physical fitness improves your brain function). Team sports are especially good, because few activities teach you how to cooperate with others (including exercising and following leadership) than playing on a team.

IMO, your best strategy is to learn as much math as you can, anywhere you can, while still spending 4 years in high school and 4 years in an undergrad program where you can fully develop your non-math skills and exploit the wealth of social opportunities that you will almost certainly never again encounter in later life. Good luck!

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  • $\begingroup$ Spending thirteen years six hours a day in a fenced-off gated compound is not the best way to hone one's social skills. If he can get out quickly and move on then kudos to him. $\endgroup$ – Rusty Core May 2 at 2:09
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Would it make sense to apply for colleges as anything other than a Senior, or duel enroll in non-community colleges?

yeah you can definitely look into applying for colleges a year early. I've heard of it and I think some people had good experiences doing it.

As for dual-enrolling in non-community colleges: what do you mean by dual enroll? You could definitely take classes at non-community colleges, but you probably don't have to go through an application process for it. See if your high school can help set it up.

For the math people reading this, how do you recommend studying math on my own?

Not a math person (CS), but you should do shit that you like. Do you like math competitions, like math olympiads? If so, do those. They're nice in that a lot of high school+ math is about learning procedures to solve math problems, but competitions and puzzles may involve more creative thinking, which is closer to what the fun part of math is: figuring out how to solve new problems.

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