Re: Programming languages for the very young

From: Brian Harvey (bh_at_cs.berkeley.edu)
Date: 01/19/04 

Date: Mon, 19 Jan 2004 18:34:53 +0000 (UTC)

<news-reply{at}stirfried.vegetable.org.uk> writes:
>Brian Mastenbrook <NOSPAMbmastenbNOSPAM@cs.indiana.edu> writes:
>
>[snip]
>> Critical thinking requires a solid, intuitive understanding of the
>> basics - you can't possibly visualize algebra, or do simple
>> trigonometry, if you can't rattle off what six times seven is without
>> thinking.
>
>Disagree. At the time I was studying maths in school and university, as
>now, I still find it a heck of a lot easier to picture a sine graph than to
>rattle off mere numbers, having had to think about 6*7 just there.

My understanding of this issue changed dramatically when I adopted a
(then) 12-year-old, very bright, but with learning disabilities, one of
which is a short-term memory problem. (It's rather ironic, because his
*long* term memory is better than that of anyone else I know.) I'm often
skeptical about this whole business of learning disabilities, and the fact
that the experts can't agree about whether he's ADD or not isn't helping
my attitude, but they all *do* agree about this memory business.

He has an especially hard time memorizing arbitrary facts that don't mean
anything to him, sich as what six times seven is.

Because of this, and because of the fact that schools make memorization
a prerequisite to understanding anything about math, Heath is convinced
that he's stupid about math, and won't even try algebra (which, of course,
is also taught in a way that puts memorization before understanding, but
this time what you're supposed to memorize is that you can subtract the
same thing from both sides of the equation).

But he's a terrific problem-solver, has great geometric intuition (much
better than mine, for example), and could really, imho, enjoy math if only
he'd seen some before he was allowed to founder on the reefs of arithmetic.

My own school experience was quite different; I had no problem doing
arithmetic. I went through elementary school before the New Math, so
nobody taught me words like "commutative" or "distributive" until 7th
grade algebra, but I was lucky enough to understand those concepts anyway,
so I understood place value, so arithmetic *wasn't* meaningless to me.
But I skipped a grade in elementary school (midyear, moving from 2nd to
3rd), and as a result I missed the whole process of memorizing the times
table. This was a slight handicap to my schoolwork for a month or two,
during which I did a lot of finger-counting, but after a while I found that
I'd somehow learned the times table without ever trying to. (But, I do
understand that I wouldn't have learned it without a lot of drill. It's
just the the drill was on multiplying two-digit numbers, with the assumption
that I already had the one-digit numbers down cold.)

But, even though I had no trouble doing it, I *hated* arithmetic. It was
boring! And so I thought I hated math until that 7th grade algebra class.
As soon as I opened the book, I was hooked, and I read the whole thing
over the first weekend of school. (Then I got bored in class, so my
teacher told me to stop coming, and instead gave me advanced math books
to read. At one point that year I "proved" that all sets are denumerable,
starting from the well-ordering principle, and none of the teachers knew
enough math to be able to show me why I was wrong.)

It's very likely that Heath and I are both, in very different ways,
exceptional, and that many, maybe most, other kids learn math best if
they start by memorizing stuff. But I don't think we're as exceptional
as, say, people with type AB blood, and the medical profession doesn't
pretend those people don't exist. I happened to *love* school as a kid;
it wasn't until much later that I learned to hate it. :-) Because school
worked fine for me, my instinct is to be impatient when people talk about
"different learning styles" and suggest giving kids big digits made of
sandpaper. But, in fact, Heath and I both have different learning styles
from the one that traditional schooling is meant to serve. For both of
us, and I bet for quite a lot of other kids, understanding has to come
before memorization. (Of course I don't mean that you have to understand
all of mathematics before you memorize anything.)

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