Academic Set Theory

and as faculty are rational beings

Experience shows this to be false, alas. Besides, you're forgetting something: What about the students who can learn without attending or working hw problems, but do so anyway? Makes it even harder to (dis)prove.

For this is not actually a mathematical theorem, but rather a conjecture about the properties of real, non-spherical students, we have to treat it as any scientific theory: We cannot definitely be sure it is true, though the experimental evidence may be overwhelming. The statement surely is a good enough description of reality, as it makes reliable predictions.

A counterexample may easily be obtained by considering a class which has no students enrolled, in which case both sets are the empty set, and equal.

QED. =)

This seems to be constant over time. I last lectured 15 years ago (as a TA then a lecturer) and this was just as true then as now. I used to even tell the class that I never had failed anyone who basically showed up all the time. That meant they and took every test and assignment. Never less than a C+/B- I think. I never deliberately did this, it just happened.

1. Perhaps faculty are just trying to preserve their largely unnecessary jobs, because if people realized they could learn things by just reading books, the entire university system would collapse and they'd have to learn plumbing or some such useful trade.

2. Perhaps the set of people who can do that is simply not particularly overlapping with the set of people who think they can -- that is, a lot of people who think they can can't, but a lot of people who think they can't could.

In college, I wasn't always a perfect student, but I did figure out one thing: there was no more efficient way to learn that going to class. It may have been true that I could have figured shit out if I skipped, but it would have taken me two hours for every one having the material explained to me in class, so what would the point have been? It seems the laziest students should be the ones who never miss class!

The only folks I knew of who were consistently able to learn the material of a course without attending lectures or working homework problems, were almost always "closet" prodigy types. For a number of these "closet" prodigies when they were younger, they were the types who read old textbooks of university level math, physics, engineering, etc ... and worked out numerous problems on their own. (Frequently their parents were intellectual types who still had all their old university textbooks on the living room bookshelf).

So by the time they were in university, many of the courses they enrolled in were largely "easy credits" for them, such as courses in the generic lectures + assignments + exams format. Lab courses were an entirely different story. So it wasn't too surprising these folks ended up majoring in something that was theoretical and/or didn't require many mandatory lab courses. (ie. Typical majors: mathematics, math + physics, engineering physics, math + economics, etc ...).

For most of these "closet" prodigy types, they just "appeared" to be smart. They generally didn't tell anybody else about how they spent their childhood and/or teenage years, in reading textbooks and working out numerous problems.

What about the students who can learn the material without going to lecture but don't want to skip because they're afraid they can't? I have a feeling that a large portion of the high-As fit into that category.

These are the same people that ask the brilliant or leading question, or ruin the surprise of the result through their question or comment.

I did figure out one thing: there was no more efficient way to learn that going to class

Well, in some cases. But there are also professors who copy the textbook verbatim onto slides, then read the words off the screen in a slow monotone. If I wanted that, I'd have bought the audio version of the textbook.

My philosophy in undergrad was "if you make class worth going to, I'll show up". Most professors fulfilled their end of the bargain. A few did not. I passed all their classes anyway.

What about the subset of students who think they can learn by copying homework from someone else in class? Continuing to grade those papers also calls into question the rationality of the professor.

The weird phase transition, badly thought through in many schools, is this. In grades 9-16 (middle school, high school, community college, state college or private university) students are rewarded almost exclusively for individual effort, and punished for any collaborations not explicitly assigned. Then in grades 17-infinity (grad school, postdoc, instructor, adjunct, professor, chair, dean) the same people, now older, are rewarded almost exclusively for collaborations and relatively punished (Social Darwinism) for over-reliance on individual effort.

My friends in Europe, India, and China tell me that the USA is anomalously perverse in this regard.

A senior philosophy major expressed an interest in philosophy of science, and asked if it was OK for him to sign up for my introductory biology course. I said fine. A little later, collegues told me the guy was extremely bright but would never attend class. I encountered him on campus and explained to him that he did not have to attend class, but that lab attendance was manditory. So he attended lab, and was, so the TA said, an interesting student. He did not attend lecture except for the tests. He made the high grade in the class. One member of some very small set, I suppose.

Or the people who can (more-or-less) learn without the lectures, etc., attend classes, but learn something unrelated to the course in their own time? (So it'd be something you wouldn't be measuring.)

I think it's a bit relative to where you are in your education. After all, as research scientists we are forever learning for ourselves and the kids that grew up with science are just a bit further along than the rest. To my mind once you've done undergraduate classes to about the second year, you should all be able to learn for yourself, and the classes from third year on should have more of a focus on discussion, etc.

@6: Consider someone who is hard-of-hearing or deaf, and who has to learn a lot of the stuff "twice" in order to "get" what was trying to be communicated!

@11: I'd like to work in Europe, myself not that I can see anyone doing me any favours! I'd prefer systems that recognised individuals who want to plow on ahead on their own initiative, too. Our grant committees have a similar emphasis on collaborations, etc... Speaking of which, I'd better get back to that grant application...

A corollary: Any student,regardless of ability, is practically guaranteed to get at least a B or better simply by coming to all the classes, doing all the homework, and reading all the assignments.

My older brother was 2 years ahead of me in college, and noticed I was getting consistently better grades than he had gotten in the same classes. He asked me how come, and I told him just what I wrote above. He looked incredulous, and exclaimed, "That's *it*?! Aaaggh!" It was like the concept had never occurred to him before. Of course, by that time it was too late, although he did get better grades during his last quarter than he had during the entire previous time.

+1 to what Evan said.
You're probably undercounting the first set.