I'm going to pick apart your post a little bit mathwonk; hope you don't mind :)
mathwonk said:
The top schools have much better qualified professors and much harder courses and much higher standards, and much stronger student bodies, than do state schools in general.
I'll assume that "top school" means "top private school in your major" based on most of the posts in this thread.
Professors at such a top school are "better qualified" for what? Teaching undergrads? I disagree. Teaching grads? Perhaps. Mentoring you as a postdoc and getting you an academic job? Probably.
"Much harder courses with higher standards?" Ivy league schools (I went to one) are known for grade inflation and hand holding. Based only my experience, which is in engineering, the course material was the same. Additionally, there was no noticeable difference in undergraduate and graduate student "strength" among the three schools that I attended (big private, small private, big state).
mathwonk said:
(google math 55, the "hardest course in the country").
Is it really that "hard" though? You have a PhD (I assume in math), so I'll take your word for it. It surprises me. However, it might be a misunderstanding on my part of what you mean by "hard." It would surprise me if the course is indeed that advanced and students at that level are getting something out of it -- the key word there is "advanced."
mathwonk said:
At Stanford more recently, they start the honors students out with volume 2 of Apostol. Which means you need to have studied volume 1 on your own, since Stanford (and Harvard) no longer offer such an elementary honors level course. I do not even know where one can find a vol 1 Apostol course in high school, certainly not here in Georgia.
The really good students were starting off freshman year taking graduate courses at Harvard. and some of them were only 15-17 years old.
How do those students do?
They sound really impressive. I have a lot of experience with Stanford as well and advanced engineering courses are definitely not taught to freshman.
mathwonk said:
If you are a bigger fish as a good student at a state school where there are relatively few good students, you may get special attention from professors who are probably students of the professors at the top schools.
In my experience, at a "state school" the general student population is large and the population of motivated and capable students is proportionally large. In my experience it is easier to obtain research opportunities at a smaller school ("top school"=smaller school by our assumed definition).
mathwonk said:
When I interviewed for honors calc and presented my all A's transcript, my 800 SAT's and my state championship awards in math (from Tennessee), I was told "those are a dime a dozen here". And they were. In my freshman honors math class they said half the class were either valedictorians or salutatorians in high school.
No doubt the students who enter a top school worked harder, academically, in K-12. They have a big head start but many factors come into play once they leave home and enter college.
mathwonk said:
half also dropped out after the first semester.
I don't understand this statement. Half of the students in your freshman math class at Harvard dropped out of Harvard? Or, only half of the students in the first part of the course (math 55a maybe?) moved on to the second part of the course?
Anyway, I just wanted to give a different perspective so that those attending the non-elite schools are not discouraged by the tone of this thread.
Based on your post, it sounds like mathematics students at Harvard and Stanford are a different breed. Makes me think of "good will hunting" -- teenagers stumping the professor :)