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I am a senior in high school and will be starting at the University of Washington in the fall. My goal is to become a successful research mathematician (in academia), and, from what I can gather, my chances at this goal are increased if I am admitted into a "top" math graduate school for my PhD studies. Hence I am trying to be as proactive as possible in preparing for graduate school.

Here is the deal: I have had a very different educational experience; namely two years of basically no math in 9th and 10th grade, and then math at community college in 11th, and finally an honors sequence at the University of Washington this year, in 12th grade. The class covers single variable calculus, some multivariate, ODEs, and linear algebra, all with rigor. Going in to college in the fall I will have over 100 quarter credits completed (so I will be at the level of a junior). Moreover, I will be in the senior level algebra sequence and in another honors sequence that covers some real analysis and complex analysis. After my freshman year I will only have a few undergrad classes left to take (namely one year long sequence). Lastly, I am currently self-studying (baby) Rudin.

So considering my background and current level, it seems I will be taking a large number of graduate courses in my college career if I stay for four years. (Indeed, theoretically at least, I could largely complete the coursework for a PhD by taking two sequences per year). I am worried that if I do stay for four years I will be looked upon negatively by graduate schools, since they might think I am being lazy. Moreover, what will they think when they see 6 years of college transcripts? Should I plan to finish my degree two years from now to avoid this situation? Or perhaps in 3 years?

Edit: Perhaps lazy wasn't the appropriate word. I get the intuition that if an applicant had taken 6 years to complete his/her degree, whilst taking graduate courses even, the admissions committee would think them strange. What I am wondering is, would my situation be any different?

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    Why would taking the usual amount of time post-high-school to complete a B.S., including lots of graduate classes, appear lazy? I would imagine it would give the opposite impression. – ff524 May 8 '14 at 6:15
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You are getting worried about nothing. You will not be judged negatively for having taken college classes before starting your BA, even a lot of them. Graduate committees want to have students coming into their program who are going to be successful, and they don't care much whether it took 6 years or 2 years to get that preparation. They are going to look at what you did with your time, not how long you took. As long as you're happy and it's not too much of a financial hardship, there's no reason to graduate in 2 or 3 years. Instead, take the time to concentrate on things that will strengthen your application:

  • Take the hardest courses available which you can perform well in.
  • Build relationships with faculty. Taking graduate courses can help with this, but you can also seek out research opportunities or get involve with things like math club.

Performing well in challenging classes and having stellar letters of recommendation are the basis of a strong application to graduate school. Issues of timing make absolutely no difference in comparison to them. Other considerations:

  • You could do a second major, if there's something else you really want to try. I think this has probably taken on a totemic significance with students beyond its real import (math grad schools are mostly concerned with how you've done in math; being well-rounded is something like a tertiary consideration), but it sounds like you have the room in your schedule, and certainly no one will think you're lazy.
  • Find good ways to occupy yourself over the summers. REUs are one good possibility. It sounds like you'll have a decent shot of getting into one after your freshman year. Another is if you can find a counselor position at summer program like PROMYS.
  • Consider spending a year or semester somewhere else. There are highly regarded study abroad programs specific to mathematics in Budapest and Moscow, and also programs like Penn State's MASS. There's also Part III in Cambridge, which perhaps you could do after graduating in 3 years (or in 4).
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    This is an excellent answer. Nobody will look down on a student for having started taking college courses as a HS student. In fact, if you're attending a different university than the one you did the work in HS in, the graduate school may not even see your HS transcript. – aeismail May 8 '14 at 11:48
  • Well, of course, the student needs to submit the transcript; I think probably you should (certainly I've seen examples of it). – Ben Webster May 8 '14 at 15:34

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