|
|
Re: Fractals
Posted:
May 7, 1995 4:34 PM
|
|
Thank you, Han, for this very interesting message.
I must apoligize: I wrote something which seemed to
mean that I am against speaking about fractals in
middle school. Of course, not. Let me make a comparison.
Suppose somebody asks: Does it make sense to put
on bosoms of cheerleaders' T-shirts a sign: `E=MC^2'.
Why not ? After all, it is not obscene.
But, PLEASE, do not call it physics.
This is the point. You may discuss quantum mechanics
with small children and it may make some sense.
But it is NOT a study of quantum mechanics.
You may (perhaps, should) some time show middle school
students fractal pictures and in may have a positive
effect on their development. (Everything what intelligent
adults do WITH children has a positive effect on their development,
including, for example, practical joking.)
But, PLEASE, do not call it mathematics.
And do not do it INSTEAD of a regular course.
Andrei Toom
On Sun, 7 May 1995, Chih-Han sah wrote:
> There appears to be two major points about wanting to introduce
> fractals into the early grades. Namely:
> 1) To get the students to become more familiar with 'infinity'.
> 2) To get the students to become more familiar with
> 'self-similarity'.
>
> Let me suggest that both of these can be accomplished already
> in the 'traditional setting' without necessarily getting involved with
> 'fractals'.
>
> a) Arithmetics. Consider the example of expressing the reciprocal
> of *one digit* numbers as decimals. (For the history buffs, one can
> discuss Egyptian unit fractions.)
>
> 1/9 = 0.111111...., 2/9 = 0.222222.... etc. The pattern is
> clear until one reaches:
>
> 1 = 9/9 = 0.999999....
>
> To resolve this apparent puzzle, one may ask the student:
>
> Do you know how to subtract one decimal from another?
> If so, subtract 0.999999.... from 1.
>
> It clearly provides a heuristic feeling that somehow the *remaining
> digit* has receded to infinity. At the same time, the student may
> get a feeling of understanding something that is *infinitely small*.
>
> In particular, the *smallness* is connected to the number of
> 0's needed before one can write down the first non-zero digit.
>
> In this context, 1/7 = 0.142857142857...... is especially
> interesting. One can easily extend the discussion to more complicated
> fractions.
>
> b) Geometry. Start with regular pentagon. Draw the five
> diagonals and one sees the appearance of a smaller regular pentagon
> on the inside. This clearly illustrates self-similarity as well as
> the infinite iteration process. The more advanced students may be
> asked to compute the side ratio for the two regular pentagon. As
> this point, one meets the Golden ratio.
>
> In both cases, one does not have to face with the problem of
> confusing the issues because the more advanced techniques are not
> available.
>
> Han Sah, sah@math.sunysb.edu
|
|
