on mentoring (read: NOT ME).

(at some point i'll write less about teaching and more about research, but as the semester is ending, teaching pains become more severe than research pains.)

i know many mathematicians who are nurturing and who promote young people to pursue mathematics. on the other hand, i suspect that my manner unconsciously discourages students to go any further in mathematics.

as for why, i have a few hunches, but they'll have to wait.
for now, i have a review class to teach.

[continued from earlier] maybe i shouldn't count teaching calculus as either nurturing or discouraging. i mean, it's .. calculus.

let me rephrase that: a calculus class is not the sort of place where you run into many excited, curious minds .. at least not with a curiosity towards maths. this is not to say that it is some sort of pur9at0ry or prison, either.

in fact, it's more like a long flight .. say, twice as long as the battery life of your laptop. there's nothing you can do as a passenger but wait out the trip, hope for the best, and if possible, make good use of your newfound "free" time. [1]

there's an apathy at work: a lot of academic majors require calculus and to pursue their own disciplines, students have no choice but to sit through the lectures and the problems with funny symbols and diagrams.

admittedly, i understand this.

as an undergraduate, for my own major i had to take a physics course. it wasn't bad and i do think physics is mildly interesting .. but nonetheless, i wouldn't have taken the course if i weren't forced to take it. in fact, i was perfectly happy to take (english) literature and sociology courses instead of physics courses.

but i digress.

as for why i discourage my students, my teaching manner bifurcates. i feel like whenever i explain something, then i receive one of two reactions from the crowd of student faces:

1. ye gods, this is boring. you just do this and that. didn't we go over this stuff earlier? i know how to take a gradient, already!
   
   
2. i have no idea what he's talking about. he seems to be speaking english, but .. what just happened? ¢rap: notes! i have to write this down, somehow.

as for why, i have my suspicions.

i hate starting a topic from scratch; it's like a "cold boot" for a computer, and for me it's disorienting. if mathematicians understand anything about the human condition, then above all else they understand what confusion feels like.

so i like to start a new lecture by reviewing a little of last lecture or a familiar topic from before: thus, reaction 1.

of course, "familiar" is a relative word. my students might remember that the computations will go roughly this or that way; to them, those computations are "familiar." [2]

remembering, however, a geometric fact and then using to explain this fancy new formula, may not be familiar. if they only remembered, then everything would make sense. instead, it has all suddenly become gibberish. hence, reaction 2.

i referred before to the demographic of my students, but i should clarify something.

where i was a ph.d. student, we would never get to teach any mathematics majors. they were tracked early to honors classes, leaving us graduate student instructors to teach the students who had no particular enjoyment for math and wanted their course requirement out of the way.

here at my postdoc institution, the culture is different. humble beginnings are fine. some of my students have told me that they are considering the mathematics major.

whether they will follow it through, i don't think i'll be of much help.

i'm too young, too pessimistic;
i even ask myself why i do mathematics, sometimes.

i don't think i have enough hope, now, to share any with the next generation.

[1] ah, who am i kidding? flights are their own share of he11.

[2] from experience, students love equals signs; everything is equal to everything else. they will say "equals" when they really mean "thus."

sometimes i fear that if i don't write a fact down as an equation, then half of my students may never remember it.