<p>CountingDown-I’m glad it worked for your son. I know some students here who had to (and failed) to jump through insurmountable hoops to take classes they wanted. (eg cs majors whose math background was deemed insufficient to take a class in logical model theory, who had to take the same class as “independent study” instead). The problem with the “advising” system being in place is that sometimes you have to follow their advice. Personally I was glad to be treated as an adult when choosing what courses to take…(on the other hand those who were looking for guidance at Caltech were often not able to find it or given explicitly bad advice–a friend of mine was told by his “advisor” that unless he took at least three math classes a term he would not get into grad school. You had to figure things out by talking to your peers or going with the flow).
I will mention that many students at Caltech and MIT take more graduate math classes as undergrads (I took
28–in retrospect many of them were so over my head I didn’t get anything out of them…as in didn’t understand a single word) than are OFFERED at Chicago (this is of course not reflected in any rankings since grad students don’t choose departments based on course offerings). OTOH Chicago makes up for it with the directed reading program which allows students to study any subject in math they want with grad student mentors, and is probably as good approximation as any to what grad school is really like. </p>
<p>So the environments in the math departments are very different (and have nothing to do with the research interests of the faculty, or even the specific course requirements for the major, which are not that different), some may prefer one over the other, and such a choice will not affect your chances of becoming an awesome mathematician except by affecting your motivation to become one.
(Of course if you’re choosing between Caltech, MIT, Cambridge, Chicago etc the differences in math departments shouldn’t be your principal criterion…)</p>
<p>Re:exams: I loved not having many exams in college (I NEVER had an in-class test) but definitely got more out of classes with more exams rather than more homework. Having exams encourages professors to teach a well-defined chunk of material and students know what they are getting out of the course–if instead students are given impossible homework problems every week…after a week of working on them they still might not learn anything except math is hard. I like Chicago’s approach here better…</p>