On Mon, 03 Oct 2005 14:08:18 GMT, bh@[EMAIL PROTECTED]
(Brian
Harvey) wrote:
>Consider teaching them algebra.
>
>Elementary school math (arithmetic) depends mainly on two skills:
>(1) memorizing arbitrary stuff; (2) tolerating the arbitrariness of the
>arbitrary stuff. [Yes, I know that once you understand a lot of math,
>both the number facts and the multi-digit algorithms become
non-arbitrary.
>But there isn't one kid in 20 who understands all that.]
You are right. I've asked adults, even some teachers, to divide one
fraction by another. They get it right. They invert and multiply
with no errors. Then I ask them *why* they did it that way. However,
it should not be surprising. If you learn to play the guitar or any
other musical instrument, you are first taught how to place your hands
and how to move them. You will practice scales on the piano, not
knowing why at the time, but that practice lays a firm foundation even
though the understanding is missing initially. It is the same with
all study, and is to be expected. There are always exceptions, of
course, but genius is rare.
>Algebra is completely different. There are *reasons* for things. The
main
>skill is logical reasoning. If you think your kids don't have that,
watch
>them playing computer games.
Not entirely completely different. Also, "logical reasoning" is also
applicable to any other study. The main present skill is in learning
the skills necessary for later realisation of the connections between
ideas. Those connections can not be made immediately, since they
might [and do] cover a lot of ground, and create a maze too deep for
the beginner. Algebra generalises the rules of arithmetic, ["x"
stands for anything"], but the rules are the same. What algebra does
do, in the beginning, is to allow the person to see more clearly what
is happening through observing [the key part of the process] how
things change and move around or stay the same. So the observer, the
"student", can see that a system not only works every time, but all of
the time, even for problems not yet done. Then, of course, it becomes
a study in itself, leaving arithmetic in the background, but still
being a basis for that understanding.
The key is still observation by the individual. There are always
those who see only x's and y's all over a page, with no apparent
connection.
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