Re: (WSJ) As Math Skills Slip, U.S. Schools Seek Answers From Asia

From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 01/02/05 

Date: 2 Jan 2005 04:27:09 GMT

Barb Knox wrote:
> I'm not so sure that it's such a basic abstraction.
> [...]
> Indeed, from a Representationalist viewpoint, one can consider a number to
> BE exactly the equivalence class of possible expressions for it.

Even there, the basic abstraction is present: that a number is a
conceptual entity distinct from any single representation of it.

For example, we were taught that we could count objects with marks
like "||||||||||||", and that we could write 1 for "|", 2 for "||" and
so on up to 9, and 0 for nothing. If we had more we could make groups
of ten and maybe have some left over, like "(||||||||||) ||". Then we
count the groups the same way. So we had two different ways to write
the same count. One was really easy to understand and use, and the
other was harder but shorter to write. That's what introduced me to
the idea that a number might be written different ways but still be
the same number.

The more difficult concept for me to grasp (long ago) was that
operations on representations could also form equivalence classes.
Obviously I didn't think in those terms. I was wondering how
multiplication of numbers by forming rectangles of objects with
certain side lengths and counting them could have anything to do with
the manipulation of numerals by means of a big ugly table and some
strange rules.

For a while I had a strong distrust of multiplication, thinking that
maybe if I multiplied big enough numbers then doing the two different
procedures might give me two different results. It had occurred to me
that the tables we were being taught to memorise could have been
chosen so their answers came out right, and so it worked for small
numbers. It was a fair bit longer before it occurred to me that the
two sets of rules might really be the same thing in different forms
like the numbers were the same thing in different forms.

I think almost all of mathematics boils down to trying to take some
representation or other, and find another form that represents the
same thing but is easier to use.

- Tim