Saturday, February 05, 2011

good teaching involves mentoring, but ..

teaching this theory course is like teaching for the first time again. it's hard to figure out what they don't know how to do.
for example, take definitions:

over several office hours, i think a handful of my students have started to appreciate them as a (necessary) starting point. for problems in naive set theory, sometimes it's best to consider an arbitrary point in a given set, walk through the logic, and slowly assemble our conclusion.

while grading quizzes, however, quite a few papers seem to demonstrate this arcane manipulation of symbols. admittedly, set operations and relations like
$\cup$, $\cap$, $\setminus$, $\subseteq$
are like arithmetic, but only after one proves theorems akin to rules of arithmetic!

i suspect that they are too used to computing without reflecting on their computations.
ideally, i'd sit each of them down for a "how to write a proof" tutorial.
we'd introduce ourselves. i'd figure out the student's background to determine how to fit my explanations in a form they'd most readily understand.

i'd choose a theorem. then we'd "go in reverse" -- dissect the proof, learn where it came from, how it was written.

then i'd choose a related theorem with a slightly different proof, and go forwards: think about why it's true, jot down some ideas, write down a good proof.
let me emphasize: ideally. i have 30 students in this course alone; in my other course, 64 students are learning linear algebra from me.

there's just no time.

i don't have the energy.

maybe some students don't want to learn how to write a good proof: they could be applied persons or future high school teachers [1] and to them, this theory course is just another hurdle between them and what they really want to do.

then again, for those (few?) potential mathematicians sitting in class:
shouldn't someone tell them that they can't learn everything just by sitting passively in lectures?

shouldn't someone tell them that there is a craft to proof,
that maths is a language that requires practice for fluency?

shouldn't someone tell them that one doesn't become a mathematician by accident, and that discipline is just as important as curiosity?
i've never been good at telling others what to do.

teaching this course involves, i think, an understated responsibility of mentoring. i used to say that i'm too young to be someone's mentor ..

.. but that's not it. forget age: i'm too impatient and reluctant to be anyone's mentor. so far i think i'm teaching this theory course competently, but not particularly well.


i guess, at some point in my life, i'd have had to teach a course like this. inevitably it'd be a learning process for me .. and, of course, it happens in my last semester as a postdoc at this university ..

[1] honestly, that's no excuse. the fact that one of my students might be training the next generation of young people is a reason for why he/she should know how to write a good proof .. not against.


Anonymous said...

I have learned that: 1. students don't know what a definition is. I told my proofs class to learn the defs. for the first exam and told them they are in the colored boxes in the text. Most chose to define by giving an example. I had to explain that I meant not just _read_ the box, but memorize and regurgitate the contents of the box. 2. Proofs are minimized in too many high school geometry classes, and few students take a critical thinking/logic class. Thus, you class is the first real contact with proofs that they have and so they are going to suck at it. Maybe try giving them some high school geometry problems to practice the "style" of proofs.

janus said...

Maybe try giving them some high school geometry problems to practice the "style" of proofs.

wouldn't that involve re-teaching them a little geometry? (-:

i wouldn't be opposed to that. however, the situation of the class is already a little tenuous. you see, we're using an intro to analysis textbook for the course (bartΙe-sherbert) which isn't really a good fit for a first course in proofs.

in it, though, there's an appendix A for logic and proofs. even though it's not actually on the syllabus, i covered it in detail anyway. my guess was that students would have seen a proof in a previous class, but they might not have understood what was going on (i.e. the aim of the proof: what do we actually conclude from an abstract computation or this series of sentences?).

at any rate, i'm 2 weeks behind the other theory class, now. let's see if i can catch up.