IGU said:
Sorry, I didn't make it clear. I'm in the US. My son is home schooled. No further maths or official classes at all (although he's audited some math classes at local universities). He mostly just learns math on his own. He hasn't taken STEP. Will that be a problem?
I'm most curious whether Cambridge is the right place to be for a total math kid who's quite good. Is it a "top school" for undergraduates in mathwonk's sense, and particularly suited to students who just want to do mathematics? Perhaps it's better than Harvard because at Harvard he could take Math 55 but he'd also have to take a bunch of things he isn't interested in and won't do.
Make a Google search for "Cambridge university home school". Refer to Aleph's post.
I can't speak for how well suited it would be for anyone, as I haven't been there myself. All I can say is that I've looked at their syllabus for the maths program a while back, and it is certainly intense.
I cannot say much about math 51 either, other than it being a bulky course. Much has been said about the course here on the forums. Mathwonk has written about it in his thread and elsewhere, I believe.
I recall that they require 4-5 AP courses (of course, he should do calculus!) with grades of 4 or 5. I don't know how they will use those grades, as A2 mathematics and further mathematics cover more material than AP calculus (complex numbers, differential equations, mechanics and/or statistics, some linear algebra too I think), but I have a feeling that he will need to do brilliantly at the interview. It would be much more straightforward to get into other great UK schools like Warwick, UCL, Imperial or King's, as they do not usually have an interview process. That said, I think that the AP exams are required in addition to an American high school diploma and the SAT reasoning test.
If your son wants to do maths and just maths, European universities are a good bet. Germany, Belgium, Switzerland, UK, Netherlands...3 year bachelor's too. General, the academic level is high, and entry is quite straightforward. i.e, proficiency in language and so and so grades achieved. I am not sure how AP exams are treated in other European countries. You will need to look that up. And outside the UK, European universities *tend* to be of a similar quality. So, apparently, it won't matter too much if one's bachelor's is from Munich, Zurich, Leuven, or Dresden.
Whereas in the US, studying at a flagship state U like Michigan is probably a better idea than studying at even a top liberal arts college like Williams, simply because the latter has a limited course offering. That said, at a smaller school, it may be easier to get research opportunities (but limited research areas, because of smaller faculty) than at a bigger school. But if you want more maths and less everything else, Europe (or a school with loose requirements like Brown) is a better option.
If your son wants to attend a top US university, he will need some kind of distinguishing trait (for e.g: writing a best selling book in a niche subject or co-author of some papers). Look up Cal Newport's "How to be a High School Superstar". I gather that the more impressive the applicant sounds, the higher the odds of him getting in *somewhere* in the top schools. But again, if one is going to try to be impressive just to get into a school, odds are one won't be too happy trying to be impressive, and odds are one won't end up doing something that is all that impressive anyway!
At least, this just my opinion. I will stop here, as I think others have tackled this very question far too many times already, and many of them are better qualified than me (former students, admissions officers, counselors, professors, etc) to write about this. Some have written books on the subject, i.e, Cal Newport.
Anyway, I think you should make another thread if you have further questions.
