Here's a recent post of mine to the NCTM 2000 standards area=20
at the Math Forum. I think my position has a lot in common=20
with yours (too bad you think we're on the "out in left field=20
and on the lunatic fringe" huh?). =20
The examples I give re using math to explain "how the world=20
works" are less whole systems minded (planetary focus) than=20
yours.
Kirby
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
Subject: More thoughts re early math ed=20
Author: Kirby Urner <pdx4d@teleport.com >
Date Posted: 18 Nov 99 03:26:55 -0500 (EST)
I think mathematician Keith Devlin had a thoughtful notion at=20
the Oregon Math Summit in 1997: basic numeracy, which includes=20
knowing how to read graphs, charts, scientific instruments
(e.g. clocks, thermometers, scales, rulers), do basic arithmetic,=20
make change, is something all adults need, and all should pass=20
on to the next generation.
Every teacher is responsible for seeing to it that kids know=20
the basics. It's not the job of a "math teacher" per se, to=20
teach the multiplication tables, as basic numeracy is not the=20
same thing as "mathematics". It's a perversion of the culture=20
to think knowing how to multiply 13x45 is the sole province=20
of some specialist with all this training in a particular=20
discipline known as "mathematics". Nor should you have to be
a mathematician to know that 13 is a prime number, or that=20
all whole numbers factor uniquely into primes. That's just
basic knowledge. =20
We shouldn't allow kids to get stuck in the trap of thinking: =20
"I know I'll never be a mathematician, and a math teacher=20
is some kind of mathematician, ergo whatever they teach me=20
in math class is what I'll never need to know". That's bogus
reasoning of course, but there's a certain logic to it, so=20
long as we encourage the illusion that basic numeracy is the
exclusive property of any one discipline, vs. the common=20
heritage of all.
So when learning history, one might learn about when the paper=20
and pencil algorithms we use for addition, subtraction, multi-
plication and division were invented or first introduced. =20
This is what school children had to learn then (check out=20
Roman numerals!), as well as now. =20
In the context of a history section, we could start building=20
basic competency in these algorithms. Really learn what it=20
was like before calculators -- in part because you want to=20
appreciate how school children before you learned about their=20
world. And really think about what it was like to live before=20
TV (in a lot of ways, curriculum writers are only just beginning=20
to discover this medium, even after all these decades, thanks=20
to DVD).
Mathematics is a discipline with its own heritage and heros. =20
We should definitely teach it. But it's not the same thing=20
as basic numeracy (which many disiciplines share) and I think=20
a lot of confusion arises from trying to make round pegs fit=20
in square holes, from confusing basic numeracy with mathematics=20
-- kind of like confusing "learning to read" with "doing literary=20
criticism"; although true enough that the one is a prerequisite=20
for the other.
What a lot of early education is about, or should be about, is=20
simply "how the world works". What do students need to know to=20
make sense of their environment? This indeed requires a focus=20
on "applications". I'd put computing and computers into the=20
mix, have exposure to programming concepts be a part of the=20
bigger picture of learning about operations, procedures,=20
processes -- whether these be strictly "mathematical" in=20
nature isn't supercritical (sometimes yes, sometimes no).
How do TVs and radios work? You need some math to appreciate=20
signal, how sine waves can be combined and separated. Trig
functions. How do we use binary numbers to signify 256 colors=20
on a screen? Permutations. What does an oscilloscope do? =20
This sounds like basic engineering, and in a lot of ways it=20
is -- but you can "tease the math" out of these applications=20
by focusing on what's common across the board. The concepts=20
of bits, variables, functions (added or composed) -- these=20
come up in many different contexts.
Given this kind of exposure, having seen math concepts used=20
in the context of explanations of "how things work", one thereby=20
develops an appreciation for mathematics as a "skeleton key".=20
It unlocks many doors, makes the content of many disiciplines=20
more understandable. Once this faith in the relevance of the=20
material is established, then (and only then) is it time to=20
introduce the kinds of formalisms which anchor mathematics to=20
pure principles, irrespective of special cases. =20
We do indeed want students to appreciate the "purity" of logic,=20
considered quasi-independently of history or circumstance. But=20
it's a dynamic interplay, a delicate balance -- professional=20
mathematicians need to be careful not to evidence disdain or=20
aloofness vis-a-vis the special case applications of their=20
discipline, since in the "How Things Work" context, our focus=20
is building confidance (no, I don't mean "self-esteem", I mean
respect for math itself as powerful and important -- an=20
attitude we need to cultivate, not simply presume as a given).
The danger, when we allow specialized mathematicians to steer=20
the early curriculum, as that they will be too interested in=20
the purity of their discipline to allow appreciation for it to=20
grow naturally in others, including among hardened skeptics=20
(which many young people are). But I know lots of pros who are=20
well aware of this danger, and compensate for it admirably. =20
I appreciate their input and guidance (I am not a professional
mathematician myself).
In my own approach to early math ed, I prefer to let architects,=20
pilots, engineers, doctors, physicists, chemists, linguists,=20
stock brokers, bankers, actuaries, morticians, electricians,=20
musicians, advertizers, manufacturers... all get a chance to=20
introduce the mathematical aspects of their respective=20
disciplines. The math teacher then has the job of abstracting=20
the math from these diverse inputs and distilling this to=20
its essence. Then those students most inspired by the "purity"=20
of this subject will have the choice to pursue it further towards=20
its source -- including in the context of a more advance=20
curriculum.
Kirby
Curriculum Writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
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