You may recall that I posted some time ago about a book my eleventh grade math teacher gave me that covered, among other things, the relative sizes of various infinite sets. As it turns out, I knew something about this long before then. I first learned about it in a book called Realm of Numbers by Isaac Asimov. The Mountain View library has a copy of the book in the children's section. Reading parts of it brought back a few fond memories. I think I read that book when I was in either the second or third grade, not too long after my family moved into our first house. I don't remember what I thought of infinite sets; it is unlikely I attributed any significance to them. I sort of vaguely recall wanting to be an astronaut sometime around then, as many kids did who were in grade school during the 1960s, what with the Apollo space program and all. (I later found out I couldn't be an astronaut because of my nearsightedness.)
So, I'm still somewhat puzzled why classes like 6.045 gave me so much trouble, seeing as I was familiar with the basis for the classes already. Maybe it was because I went so long without thinking about those subjects that I had to relearn them in the classes, whereas other students had been thinking about them all along. I was reading soc.college.admissions, and there is an article where the original poster (OP) suggests that students who are interested in math (or subjects that use math) take some time off between high school and college to study it independently, because there isn't enough time in a normal curriculum to cover everything in depth for someone who isn't already mathematically mature. In my case, I doubt that would have made much difference. No one in my family is into higher math, so I wouldn't have had anyone to guide my learning. I would've had to go to some kind of college, and since MIT accepted me, it made sense to go there.
The person following up to the OP made an analogy between students studying math and music. The suggestion was that a student who is (already a) virtuoso is more likely to get a lot out of college than someone who goes to college to learn how to be a virtuoso. That got me to thinking: how does admission to a place like MIT differ from admission to a music college of the same caliber? One major difference is that you have to audition to get into a top music college, and the pieces you perform are part of a standard repertoire that is universally recognized by music academics. AFAIK, you don't have to audition to get into MIT or anyplace like it. Lots of students have already taken a fair amount of college (or at least AP) level math, biology, chemistry, etc., but it isn't a requirement to do so. The admissions committee will accept students who show the potential do well at MIT, even though they may not have the same background as students who've already satisfied several MIT requirements before they go there. This is a very controversial subject, especially among students who don't understand why they didn't get into MIT even though they have done well in many AP classes, in addition to having SATs, ACTs, etc. that are at least on par with other admits. But how would this work in music schools? Do music schools ever admit students who show the potential to be virtuosos, even though they may not have completed the same amount of music training as other students who already are? What kind of experience is a student likely to have who does not have that level of music training? At MIT, if you haven't taken AP classes, you can still take classes that do not assume AP-level knowledge, but there is a much steeper learning curve for these students than those who already come in with AP-level knowledge. Having the first semester as pass/no record helps somewhat, and there are some students who are able to make up the difference. How possible is this at a music school?
Speaking of music, time to go work on my sonatina.