Saturday, May 21, 2011

Thoughts about a class about proofs (Part 3)

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

one comment from a student evaluation was rather poignant:
apparently my lectures were very clear,
they could follow my proofs easily,
perhaps a little less so with the textbook ..

but when it came time to prove things by themselves,
they quickly became stuck.
"stuck." i don't know why, but that really bothers me ..

.. it's not in anger or resentment, but close;
it's the exasperation of being at my wit's end.

having structured the course as much as i could [2], i'm at a loss of what more i could have done, barring the extreme:
if i had enough time and willpower, i suppose i could have resorted to the socratic method, that is:

nudge them along with directed questions,
convince them to open their books, revisit previous examples,
guide them in building their own argument.

then again, i had 25-30 students. in addition to time and willpower, i'd have needed a few clones of myself to cover all of them.
so, at other times, i wondered if i was doing too much .. and paradoxically, if i still was doing enough.

sure, a student has the responsibility to learn,
endure the hard work if it becomes necessary for success ..

.. but it's still not clear to me if my students learned any mathematical self-sufficiency:
learning how to read a maths textbook [3],
dissecting proofs on their own;

developing a memory for definitions and theorems,
along with a sense of what is relevant, what's not;

looking out for tricks that recur,
learning how to be relentless in their own ideas to try.

as i said before though, maths isn't something that one learns immediately. after my own first course in proofs, it took many more years for me to become competent at maths ..

.. and even now, i sometimes think i have a proof,
only to realize that i made a gap in reasoning:
"wait: that one little lemma .. isn't so little. in fact, i don't know how to prove it after all.

argh .. is there a counter-example?
do i just need another hypothesis ..?"

teaching this proofs course has been an exercise in patience. not being a naturally patient person, i guess i did all right.

[2] .. in light of cutting aside time for research and writing and conferences and the like. it would be different if i were hired purely as an instructor.

[3] i wouldn't be surprised if someone could write an entire lesson about such a skill, equipped with worksheets in a 6-week tutorial. this sounds glib, i know, but if it means that students become good at it ..

No comments: