Most very strong students are shut out from “top 20” universities and the Ivy League schools.
However, as @tsbna44 has correctly pointed out, graduate students at top law schools and at top medical schools come from “all over the place” (this is an exact quote from a doctor I know who I discussed this issue with). I have three close family members who did some form of biomedical-related graduate program (eg, one daughter is a DVM) and I work (part time now) with lawyers and they have all said the same thing — the other students in their graduate program came from a huge range of undergraduate schools.
Highly ranked graduate programs know that very strong students attend a very wide range of undergraduate programs for a very wide range of reasons.
I would not do this. Calculus depends a LOT on its prerequisites. I have consistently heard that students who are the least bit shaky on ANY of the prerequisites can find calculus to be very tough. I took the opposite approach of first making sure that I was very solid on the prerequisites, did not take calculus until my freshman year of university, and found it to be quite straightforward. Waiting until university to study calculus did not stop me from graduating from MIT with a bachelor’s degree in mathematics. I think that it is worth spending a year in high school on trigonometry and another year in high school on precalculus (I did take both classes at the same time during my senior year of high school).
And the only students who could possibly get away with rushing through any of the prerequisites for calculus are students who are VERY strong in math, and these are specifically the same students who are likely to be using calculus a lot in future classes, and possibly on the job. Calculus really is something that is very useful in a lot of ways, particularly for someone who is good at math.
Exactly! It is not a race.
One daughter attended a public university that is ranked roughly in the 100-120 range (one ranking had it slightly lower). She had a boyfriend who was premed at the same school who had never had a B in his life. He is an MD now (from a highly ranked medical school). He is one example of a student attending a “relatively average but very good” public university who would have been very comfortable at an Ivy League university. She had another good friend who had only had one or two B’s in their life, would also have been a solid student for an Ivy League school, and who is also now an MD. However, now as an MD (or DVM in this daughter’s case) their total education debt is lower because they all attended an affordable undergraduate university.
There are lots and lots and lots of other examples. At one point I made a list of the 10 smartest people I have ever met (eg, this included a couple of MIT professors, a Canadian speed chess champion, and a supreme court justice). I ended up with 14 people on the list, and the 14 of them had attended 14 different undergraduate schools. I recently added two more to the list, which added two more undergraduate schools. Smart students go all over the place, and they find some other very smart students wherever they go.
Anyone who thinks about it, including university admissions staff, understand that switching high schools multiple times is tough. I think that they will take this into account in considering your applications. Having some experience living outside the US can also be a valuable experience.
I think that you should take classes that you are solidly ready to take, do very well in your classes, treat people well, and do not worry about it. There are hundreds of very good colleges and universities in the US. When the time comes, as long as you make sure to apply to safeties you will get accepted to some very good universities.