2 ms·

> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved

> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. That was the bane of my University degree. "And, since our function f happens to be of this form, all the difficult stuff cancels out and we're left with this trivial stuff" and then none of the problems have these "happy accident" cancellations and you're none the wiser on how to proceed. The statistics book we used was an especially egregious offender in this regard.

dan-robertson 2y ago
I think often the reason this happens is that the chosen examples[1] are just more advanced topics in disguise. Eg maybe you are given some group with a weird operation and asked to prove something about it, and the hidden thing is that this is a well-known property of semi-direct products and that’s what the described group is. Two I remember were: - In an early geometry course there was a problem to prove/determine something described in terms of the Poincaré disc model of the hyperbolic plane. The trick was to convert to the upper half-plane model (where there was an obvious choice for which point on the boundary of the disc maps to infinity in the uhp). There I was annoyed because it felt like a trick question, but the lesson was probably useful. - in a topology course there was a problem like ‘find a space which deformation-retracts to a möbius strip and to an annulus. This is easy to imagine in your head: a solid torus = S1*D2 can contain an embedding of each of those spaces into R3. I ended up carefully writing those retractions by hand, but I think the better solution was to take the product space and apply some theorems (I think I’m misremembering this – product space works for an ordinary retraction but for the deformation retraction I don’t think it works. I guess both retract to S1 and you could glue the two spaces together along that, or use the proof that homotopy equivalence <=> deformation retracts from common space, but I don’t think we had that). I felt less annoyed at missing the trick there. [1] I’m really talking about exercises here. I don’t really recall having problems with the examples.

dan-robertson 2y ago

I think often the reason this happens is that the chosen examples[1] are just more advanced topics in disguise. Eg maybe you are given some group with a weird operation and asked to prove something about it, and the hidden thing is that this is a well-known property of semi-direct products and that’s what the described group is. Two I remember were: - In an early geometry course there was a problem to prove/determine something described in terms of the Poincaré disc model of the hyperbolic plane. The trick was to convert to the upper half-plane model (where there was an obvious choice for which point on the boundary of the disc maps to infinity in the uhp). There I was annoyed because it felt like a trick question, but the lesson was probably useful. - in a topology course there was a problem like ‘find a space which deformation-retracts to a möbius strip and to an annulus. This is easy to imagine in your head: a solid torus = S1*D2 can contain an embedding of each of those spaces into R3. I ended up carefully writing those retractions by hand, but I think the better solution was to take the product space and apply some theorems (I think I’m misremembering this – product space works for an ordinary retraction but for the deformation retraction I don’t think it works. I guess both retract to S1 and you could glue the two spaces together along that, or use the proof that homotopy equivalence <=> deformation retracts from common space, but I don’t think we had that). I felt less annoyed at missing the trick there. [1] I’m really talking about exercises here. I don’t really recall having problems with the examples.