Re: motivating a kid into analysis
- From: jdolan@xxxxxxxxxxxxxxxxxxxxxxxx (James Dolan)
- Date: Wed, 5 Oct 2005 23:32:16 +0000 (UTC)
in article <1128528758.370918.176310@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
amanda <sca18@xxxxxxxxxxx> wrote:
|I'd like some hints from the experienced mathematicians on how I can
|motivate my 14-year-old son into Analysis. I'm an engineer and like
|Math, especially Analysis, and whenever I can I try to study in on my
|own. My boy loves math, but he's into number theory. He loves those
|problems like proving an integer number is prime, for example. He
|likes to deal with things like congruences, Fermat's Little Theorem
|and groups (according to his age, of course). I tried to introduce
|him to Analysis, but I noticed he had some difficulty to understand
|the epsilon-delta definitions of limit and continuity. He also showed
|some difficulty with the concepts of infimum and supremum.
|
|I'm not sure, but I think people who specializes in Number Theory are
|not into Analysis that much and vice versa. I've noticed that
|mathematicians that post about Analysis usually don't post about
|number theory, and the converse seems to be true.
the suggestion to let him do whatever he wants is of course a good
one, but one thing you can try is to encourage him to re-invent the
basic ideas of finite-difference calculus for himself, after which it
would probably be easy for him to understand the motivation behind
differential calculus (and its accompanying apparatus of "analysis"
ideas) as simply an idealized version of finite-difference calculus.
this approach tends to work iff the kid enjoys programming computers
for fun. you could try encouraging him to write from scratch (_don't_
download any astronomy or physics or calculus software or anything
like that) a simple solar system simulator. getting the earth to go
around the sun by obeying the law of universal gravitation is probably
the work of a pleasant week or two, and once that's accomplished,
almost the whole conceptual picture of calculus (and classical
mechanics and so forth) becomes clearly visible. somewhere inside the
program there is (or the programmer can put in) a variable that acts
as a sort of "accuracy dial" specifying how finely time is subdivided
in the simulation, and differential calculus and analysis are
basically about what happens in the idealized limit where you "turn
the accuracy dial up all the way".
warning: this approach tends to promote a healthy tendency _not_ to
get obsessed with the fine details of infinitesimal analysis at the
expense of the broad conceptual picture that's already visible
_before_ you "turn the accuracy dial up all the way".
it's hard to tell at a distance though whether the particular flavor
of "discrete mathematics" that your son seems to like is close enough
to the flavor of "discrete mathematics" that i'm talking about here
for the approach that i'm describing to work in his case.
--
[e-mail address jdolan@xxxxxxxxxxxx]
.
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