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Time for thread #2 on the abelian group exercise. We will try to answer the question: is this an easy exercise?
I presented it as quick: it certainly was quick to write, and depending which proof you came up with, it was or wasn't quick to solve.
I presented it as quick: it certainly was quick to write, and depending which proof you came up with, it was or wasn't quick to solve.
But what about ease? Some found it easy, some knacked it after some work, some couldn't find their way and/or got bored or sad before they found one. This I expected.
What I hadn't expected, and should have, was the many people confused by a double standard in maths terminology.
What I hadn't expected, and should have, was the many people confused by a double standard in maths terminology.
I assumed, as I was taught, that a (by default, binary) operation on a set A is an internal operation, that is a map AxA—>A.
However, many wondered whether the target of the operation could have been a larger set, and how one would prove that A is closed under the operation.
However, many wondered whether the target of the operation could have been a larger set, and how one would prove that A is closed under the operation.
I should have remembered that this language exists, but didn't.
This isn't due to me picking an exceedingly unlucky case: inconsistent terminology is ubiquitous in maths, and the source of a lot of difficulty.
This isn't due to me picking an exceedingly unlucky case: inconsistent terminology is ubiquitous in maths, and the source of a lot of difficulty.
Finally, my choice required comparatively little knowledge (eg, no Sylow thms 
) but a very specific attitude from the solver. There was no easy example to start with, and trying to build one was cumbersome.
) but a very specific attitude from the solver. There was no easy example to start with, and trying to build one was cumbersome.
People motivated by a concrete reason to study a problem also came out empty: I gave you no intuition, no motivation, no context.
I learned informally what a group was at 14, then kept learning more until the full formal definition at 18.
I learned informally what a group was at 14, then kept learning more until the full formal definition at 18.
By then, I was also very familiar with proof techniques, quantifiers, and formality in general. Had you asked me in summer 1985, I would have said this was a challenging problem but not too hard.
But what if a class were composed, just like a Twitter audience, of people with very different backgrounds?
Some, like me, would have a huge advance over people who barely learned what a group is, with no exposition to proofs.
Some, like me, would have a huge advance over people who barely learned what a group is, with no exposition to proofs.
Does it mean those in difficulty are less "gifted" for maths?
It seems to me ridiculous to think so.
Similarly, all levels of mathematical ability can be equally found among those who are more motivated by concreteness, computability, and meaningful examples.
It seems to me ridiculous to think so.
Similarly, all levels of mathematical ability can be equally found among those who are more motivated by concreteness, computability, and meaningful examples.
At the other end of the spectrum, many people with a more advanced background than mine found the exercise standard, and possibly trivial or at least known.
This is relevant for thread #3, in which you will learn WHERE I got the problem from and WHY.
No spoilers, please!
This is relevant for thread #3, in which you will learn WHERE I got the problem from and WHY.
No spoilers, please!
TL;DR The exercise isn't in itself easy or hard; it becomes so depending on individual circumstances, and isn't a good proxy for mathematical "gift".
Whenever we insist that only people who can work their way through this kind of exercise without help should study mathematics, we are cutting off from the subject a huge pool of talent.
This _does_ happen in
(I haven't taught undergrads elsewhere).
This _does_ happen in
(I haven't taught undergrads elsewhere).
While I'm no sociologist, I suspect that who we cut out correlates with gender, race, disability status, social status, and other factors that should be irrelevant for access to maths education.
It's a question with no easy answers, but one we can't ignore.
/end
It's a question with no easy answers, but one we can't ignore.
/end
PS Thanks to everyone who keeps thinking about the problem.
I did find it rather interesting, and I'm glad to have been able to share.
I'm also impressed by the many generalizations people have come up with!
I did find it rather interesting, and I'm glad to have been able to share.
I'm also impressed by the many generalizations people have come up with!
