Re: Programming languages for the very young

From: Samuel Walters (swalters_usenet_at_yahoo.com)
Date: 01/20/04 

Date: Tue, 20 Jan 2004 01:45:04 GMT

| Tim Haynes said |

> Well, the details of the visualization are significantly different. The
> transformation "split it into two easy things and add" was almost
> pictured, but not in terms of countable bricks, rather more like `(10
> 8)' .... `(2 8)'. I saw the sub-expressions, not piles of bricks. Still,
> there's only one of me, be forever glad... :)

You are obviously well educated. Over the process of that education, you
built many strong metaphors, disengaged them from their real domain of
objects and then reapplied them to other domains. Some of those metaphors
were even changed slightly and then reapplied to the domain that they the
came from to change your perception of that domain.

And keep in mind. We're talking about how children learn to cope with
simple arithmetic and the step of abstraction from which calculus springs.
 If you've had 10 years (Grades 2-12) of reasonable math education, there's
a lot of accretion of knowledge and metaphors. Just because you are no
longer aware of how that came about in yourself, doesn't mean you started
that way. In fact, it's a lot like moving to a new town. At first you
have no clue where you are, or where you're going. Eventually, you don't
even think about it. You just get in a car and go. On a long enough
timeline, you will even forget all the times you got lost early on.

>> "Where Mathematics Comes From" is an interesting book. The first half
>> is, in my experience, spot on about how people relate to math. The
>> second half is a bit more tenuous. YMMV. Overall, well worth the read
>> if you ever wondered about the mechanisms by which one connects
>> real-life to that abstract crystalline space of mathematics.
>
> Yeah, I'll go dig out a copy. Thanks for the recommendation. :)

It's written by a linguist and a cognitive psychologist. Very interesting
to see the tools of those fields applied to math. My reservations about
the second half of the book have a lot to do with the shifting sands on
which they stand.

(speaking of functions, tables and graphs)
> Maybe if you ask what they had to put into it, in order to get <that
> line> out of it, that would help? <uninformed guess> :)
 
I use the machine metaphor of functions. (Put a number in, gets chewed up
by mathematical machinery and spits out another number.) While it's most
useful for composition of functions, I try to use it sparingly because it
breaks the notion of continuity. If anything I try to lead from machines
to tables because tables are less strongly removed from continuous lines.
One can imagine "slipping" an extra row into the table.

Someone out there right now is thinking "But a function is a set of
ordered pairs." Yes, it is a set of ordered pairs, and continuity is a
property of the aggregate set, but this idea is not useful until advanced
calculus or discrete mathematics. Until then, continuity is a damn useful
notion.

> I think you have to draw your sin, cos and tan graphs, at least once in
> a lifetime. However, to get the feel for the thing, varying the
> parameters (like I suggested above, maybe), I think you need a quicker
> plotter - so's not to waste time drawing, but to see a sucession of
> smooth changes. Just like you wouldn't sit around watching a rose
> opening after dawn because the movement is too slow to notice, you'd
> point a time-lapse camera at it and come back to see the compressed
> results a few hours later.

I think graphing aids are a big help in this area. (By graphing aids, I
mean calculators and other computer graphing utilities) However, they
don't come close to the experience the average student gains from having
to personally decide the appropriate scale of the axis, then physically
locate those points on a planar surface. Were I teaching a class of
gifted students I might reconsider my opinion, but I'm more concerned
about how Joe Sixpack can come to grips with trig functions. In fact,
imagine a truck driver. Chances are, the person you imagined is one of my
students. Now, picture that truck driver as a teenager. That teenager is
also one of my students.

Sam Walters. 

-- 
Never forget the halloween documents.
http://www.opensource.org/halloween/
""" Where will Microsoft try to drag you today?
    Do you really want to go there?"""

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