Re: motivating a kid into analysis

James Dolan wrote:
> 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".

That brings back fond memories. Of the floating point math chip
I got for my Apple ][ along with the Forth programming language.

And a program called "Cavorite" that tried to solve the problem:

If you could turn off the Earth's gravity and fling an
object at the Moon, could you (at an appropriate time)
turn on the Moon's gravity and capture the object in an
orbit around the Moon?

That was a lot of fun, plotting the orbits of the Earth and Moon,
tracking the object between the planets, zooming in to a closeup
view of the Moon to glimpse the fly-by.

I never could get the object to orbit the Moon. I either crashed
into the surface or the object was flung off into space.


>
> 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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