Prof. Ivanov,
>
> The semester is over, and so here's my pass at giving feedback for the
> course:
>
> In studying up for the exam I got a much better perspective on why we
> looked at various constructs, ideas that hadn't been fully clear upon
> first inspection. For instance, reviewing the Sylow theorems made
> apparent just how many previously covered theorems were made trivial
> (easier, anyway) by these more central, more powerful theorems.
>
> The opacity of things like that is, in part, my own blindness in math --
> in particular with groups. The exam's uniqueness blunder
> notwithstanding, I had a much easier time seeing how the pieces fit
> together with fields (though this is perhaps related to the depth of the
> material as much as it is to the material itself). It was, however,
> something of a recurring external problem as well. Above all else, the
> segment of the course on the simplicity of A_5 felt particularly
> unmotivated (at best undermotivated), and on those grounds I suppose I
> agree with Adam that it should be either lengthened or cut short. The
> related symmetric group material was reasonably motivated (Cayley's
> Theorem is really all that's necessary), as were most of the other
> separate topics in the course, but simply being told that A_n was
> important to Galois theory without seeing any further evidence didn't
> satisfy. Going back to my blindness, a short amount of time spent at
> the end of each section revisiting past results with more modern tools
> (modern wrt the course progression anyway) might be a good idea; it
> would tie the package together more neatly.
>
> The amount of time we spent on group actions was fantastic, but I wish
> it had been treated slightly earlier; a lot of things that we did as
> homework problems (like #16, 17 on the review sheet) are viewed very
> clearly through the lens of group actions, and arguably less so as the
> more generic homomorphisms with which they are currently presented.
> Talking to other people about this, it turns out that this is a pretty
> common complaint about Herstein's two algebra texts, and while we did a
> reasonable job of seriously employing group actions later in the course
> (where Herstein himself avoided them) it would be even better if we had
> started well before that.
>
> I really appreciated the times we spent near beginning of the course
> when we went over homework problems. I understand that it became both
> less necessary since you were handing out solutions and less possible
> since we were running out of time to cover the necessary material, but
> between the more leveled interaction with the professor (cf. straight
> lecture) and the opportunity to see problems solved in front of us, I
> think that the few sessions helped more than a bit and would have been
> even more useful later in the course when the problems became actually
> difficult. There was a definite difference of tone when re-solving
> these problems; an exceptional amount of detail was used, and it helped
> to glue together the stray parts from the course material. I suppose
> weaning us off these hand-fed solutions could be seen as being pushed
> out of the 'algebraic nest' so to speak, forcing us to stand up on our
> own and solve the problems without a model to follow. I'm not sure
> which of the two experiences would be more valuable; if you get the
> chance and you can make the schedule work, it might be worth a shot just
> as a comparison.
>
> There was some confusion in studying for the first exam, when I
> discovered to my dismay that the numbers assigned to the theorems were
> actually important. Being told to state and prove "the theorem on
> factor groups" is pretty worthless without knowledge of what the
> attached identifier means -- at that point there were too many theorems
> on factor groups to have any idea which you meant without memorization
> of the number. While I started copying down the numbers immediately
> thereafter, I was at a loss for the first exam and for the important
> double quiz. Either more meaningful names or a heads-up to the students
> at the beginning of the course would be both courteous and helpful.
>
> The biggest trouble with making concrete suggestions is that the course
> as it stands is fairly streamlined and compressed, and altering any part
> of it significantly will cause trouble with that. It's possible to fit
> 417, 418, and a good portion of 500 into a single semester -- I mean, we
> did -- but the schedule really does have to be worked down to the day.
> Suggestions mess this careful planning up, and in the end if I had to
> choose between failing to sort out some minor issues with the
> presentation of the material and losing the material outright, I'd
> rather keep the course as it stands.
>
> And, really, the course worked remarkably well. I know I went from
> embarrassingly messing up complex multiplication to coming off as
> halfway competent by the end of the semester (modulo thinking x3 + 1
> was irreducible on the exam), and that counts for something.
>
> If I think of more, I'll send a second email. It was an enjoyable
> class, and thanks for dealing with us for a semester; you mentioned that
> it was no small task, and we appreciate it.
>
> -Eric