Wednesday, October 21, 2009

sequences, familiar and strange [a teaching post]

for this university, it's that time of the semester when we work towards Taylor serιes expansions. i gave my first lecture today, in this direction; sequences.

almost every calculus textbook i've read will give the Fibonacci sequence as a standard example.

apparently it's "not clear" how to write down the general term.

really- come on. i understand that textbook authors don't want to explain the formula, but don't say it as if a formula were impossible!

i get a kick out of showing them the general formula involving the goldεn ratio. it especially surprises the computer programmers in the audience, who have been indoctrinated that recursιon is holy.

the formula is simple .. just square roots and exponentials.

then again, i don't explain its origins -- maybe that one needs linear algebra. a few students, the curious ones, ask about it. if i had to guess, the other students probably chalk it up to my being over-excited about obscure abstractions (again).

this year i added a new "weird" example to the lesson: "the see-it-say-it sequence." [1]

the students seemed to receive it well, but only because:
1. i told them they don't need to know it for homework or for the exam (so that they can relax).

2. i tell them that they can use the sequence to stump their friends, drive them nuts with a good riddle. honestly, it's not the sort of pattern that's obvious to guess:

1
11
21
1211
111221
312211
13112221
1113213211...

indeed, who doesn't like a good riddle? (-:
all that said, they seemed to tolerate me when i describe that it actually has good limit behavior (but not prove, of course). the same curious students get shocked, which is good.

too often a calculus student thinks that everything has been done, that there is nothing interesting and new in mathematics.

i think it does them good to see that there are always new(ish), unexpected directions. there is something new under the sun, something possible.

as related to a general theme of calculus:

if you can convince the students that they can "do" mathematics, then up to laziness, they will. this is not to say that you must build up their self-esteem, for that must be earned, but to get them to try -- do that, and that is worth something.