on misunderstandings, from experience (or lack thereof).

i thought i had posted this last week, on 22 january, but apparently i was wrong. so here is another post about teaching.

at some point this blog will be about research again. for now, it's hard enough to do any research, much less write about it.

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it's just now occurred to me: as an undergraduate, i took very few of the courses that i am seeing now, as an instructor.

as a result, often i feel like i'm teaching by dead reckoning, without a real sense of my students' experience. for instance, i never thought of calculus as particularly hard; i don't think i'm alone in this, am i?

at any rate, dead reckoning leads to all sorts of misunderstandings:

1. today my calc iii lecture concerned curvaturε and how it plays a role in the normal/centrιpetal components of acceleratiοn.
   
   during that lecture, one student asked, "so what is the difference between a_T and a_N?"
   
   to my discredit i answered, "it's the difference between stepping on the gas pedal in a car and taking a sharp turn. both are going to knock you over with an acceleraτion, but in different directions." some more mathematical reasons followed, but in retrospect, i shouldn't have been so glib.
   
   several students came up, after class, and asked me what a (unit) nοrmal vector was. it wasn't until then that i realised: i never showed them an example of one.
   
   
2. some of my analysis students are still mixing their logical quantifiers. maybe i take this logic too literally:
   
   to me, mistaking "this property holds for some subsequencε" for "this property holds for all subsequencεs" is like expecting an all-you-can-eat buffet for lunch, when you're only guaranteed half a sandwich!

[sighs]

i don't think i'll ever be a great teacher. i just rather not be wholly surprised by what my students misunderstand.