afrieden, top schools means Harvard , Stanford, MIT, CalTEch, Michigan, Chicago, the same schools that are usually considered the best on any top 10 or so list. I agree UCSD is a good math school, I have friends there whom I respect highly, but I do not think they would place it on a level with Harvard.
The half class of dropouts did not drop out of Harvard, but from the honors course back to the non honors course, or a different major entirely. Of course some did later drop out of Harvard.
I don't know what state school you are talking about, but I taught at one for over 30 years, and the number of motivated students and well prepared in an average calculus class of 35 was often as few as 1 or 2. After a scholarship program enticed more good students to stay in state, it went to about 5 per class. The most motivated and talented students at our school are encouraged to take the special honors class, taught from Spivak, which I never managed to snag. For these students our school may be a better place than Harvard, where they either might not get into this class, or might be overwhelmed by the competition. At our school they get good individual treatment.
In the year I personally know about, the Stanford students did as poorly as the Harvard students did in the honors class I myself took, i.e. they dropped like flies. In both cases after the second semester only a handful were still in math, as I recall.
But if you are asking about the 15 year olds, they were the best in the course and maybe in the school. I got help from one 15 year old on the most difficult elite honors calculus problems freshman year. He was a 15 year old freshman but he had had the elite honors level course while in high school at the Bronx high school of science. The 17 year old I met was a senior honors student, in philosophy I believe. There is a student at Harvard now who enrolled I think as a 14 year old and is doing fine, in math.
I am talking about elite honors classes here, not just regular honors classes. and yes, math 55 is that hard. The first day I was there, the professor came in rubbing his hands together, and said "this is math 55, it's hard, hard, hard; it makes strong men weep and women cry", and a guy in the front row got straight up and walked out for good, breaking the tension with laughter. I do not think he was a plant, as the professor looked so surprised.
More than half the class was freshmen, and we did calculus in infinite dimensional Banach space, the way it is done in graduate analysis. That was the fall, in the spring we did spectral theory of compact hermitian operators, as in a graduate functional analysis course, and applied it to sturm liouville systems of differential equations. We finished up with differential manifolds. The text was Loomis and Sternberg, in preprint form, as it had not yet been published.
More recently they covered also tensor products as I do when I teach graduate algebra. So it is a hard graduate level course taught to freshmen and sophomores.When you speak about grade inflation it reminds I went there in 1960, and you probably went to an ivy much later. So indeed my information may be obsolete to some extent, as possibly it's a different world there now.
There was very little if any grade inflation visible to me in 1960. The average grade was a C of some stripe. On my first writing assignment in philosophy or maybe English comp, the instructor gave our section 38 C's one D and one B. My C- comment was something like: "unoriginal and dull, you're lucky it wasn't a D."
The only break I ever got on a grade was in an advanced graduate functional analysis class I took as a senior. It had both real analysis and complex analysis as prerequisites, I had only reals, and complex was not being offered that semester. So I took it with the instructor's encouragement that all I had to do was read Knopp the first week. He said it was easy. I was unable to master that book on my own in a week, and I tanked, but and he let me off with a passing grade.
5 or 10 years ago I read in the Harvard alumni magazine that the average grade had gone up to A- or something like that. When asked, the current students said this trend reflected that they were smarter than we were, but oddly enough their average SAT scores had gone down since the 1960's.
My Stanford experience is also dated, being from 20 years ago. Things have indeed changed, but i suspect Harvard and Stanford are still miles ahead of most state schools. A quick look at the senior faculty list at Harvard shows at least two Fields medalists known to me, (McMullen and Mumford). The people I know at Stanford are also amazing.
When I was a freshman my beginning calculus class was taught by John Tate, a famous arithmetic geometer. Oh and in looking up the following link on him, I see John Tate has an Abel prize, which you probably know is an award which is given for a distinguished career, as opposed to a spectacular start to a career by someone under 35, which the Fields medal recognizes. According to wikipedia, he also has a Cole prize, Wolf prize, and Steele prize, if you want to look those up.
http://en.wikipedia.org/wiki/John_Tate
I could be wrong, but I do believe the closest you can likely come to that experience at most state schools, is to be taught by a good student of such a person. Or at least it is much less likely.
Going to a top school offers the possibility of encounters with the best people in the world in any field. But you may have to work hard at getting that contact, since they are often careful of their time.
Of course i don't know what goes on at every state school, nor many schools at all now. Even my own former school is changing and getting better. But I have been convinced for a long time that the "top" schools, i.e. Harvard, MIT, and ones on that level, are very different environments indeed from most other places.
I am still trying to make or agree with the point that all this may not matter to you. As the students in our state school Spivak class know, you may well get better, more personal instruction at the state school, but the depth of knowledge of the professor, the challenge, competition, and stimulation is unmatched at a top school, maybe too much so.
