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?