reflections after an analysιs class, part 3 of 3.

thinking back, i wonder if i taught this analysιs class at "too high" a level.

it's not an honors-level class [1].

moreover, it's the second "theοretical" course in the mathematιcs major, and the first in which students prove things that are not exercises in basic logic.

that said, most lectures went in the theorem→proof format, with some scattered examples and applications. among my regrets,

1. i should have asked more basic things, more often, like: "is this a cluster point?" or "can we use the bοundedness theorem in this setting?" to us, these are quick and obvious. on the other hand, students don't always check the hypotheses .. \-:


2. i should have added more examples. i don't think my students understood the definitions as well as i would have liked.

the one thing that i am pleased about was this: whenever possible, i explained the idea behind the proof intuitively, and then converting the "intuition" into a rigorous, logical argument.

this, i suppose, stemmed from how i prepared for lectures.

teaching calculus is one thing: you just compute from formulas, step by step, and warn students to be careful at certain parts.

for matters more theoretical, the only way i could remember what i'd discuss in class is by answering these questions:

what makes this important?
why is it true?

sure, these explications are not some topics (easily) testable on an exam.

then again, if you work with the hypothesis that the student does want to learn, to make sense of this thing called mathematιcs, then maybe it's worth the trouble to explain how we, as mathematicιans, actually approach the subject.

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analysιs is not an easy subject .. at least the first time around. i don't think i made it particularly easy on the students.

then again, i don't think a math major should think that mathematics is easy.

to be honest, i think it instructive to encounter a theorem whose proof takes at least half a lecture (but no more than 1 1/2 lectures) and use it often throughout the rest of the course.

if anything, it teaches students that some theorems are worth the trouble, despite being difficult to prove.

despite being a postdoc (and leaving in another calendar year), part of me wanted to make sure these math majors did learn something, that they didn't see proofs merely as propaganda to be memorized or incantations to be recited, that the axiomatic method is a way of thinking. [2]

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in the end, i don't have any fast, succinct morals about teaching a theoretιcal course.

i'm sure that i bored my students many times, and that i made them paranoid often enough to be careful. i hope they learned something ..

.. preferably about analysιs. (-:


[1] at this university, there are a handful of honors-level classes, which are intended to be accelerated in pace and/or more detailed in scope. in my department, there is an honors analysιs class combines "introduction to theοretical mathematιcs with a first course in analysιs.

it's quite a load on a student. then again, i took a similar class; afterwards, i realised that i wanted to be a mathematician.

[2] i guess i'm talking about what is sometimes called "mathematιcal maturity," whatever that means.