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More thoughts re early math ed
Posted:
Nov 18, 1999 3:26 AM
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I think mathematician Keith Devlin had a thoughtful notion at
the Oregon Math Summit in 1997: basic numeracy, which includes
knowing how to read graphs, charts, scientific instruments
(e.g. clocks, thermometers, scales, rulers), do basic arithmetic,
make change, is something all adults need, and all should pass
on to the next generation.
Every teacher is responsible for seeing to it that kids know
the basics. It's not the job of a "math teacher" per se, to
teach the multiplication tables, as basic numeracy is not the
same thing as "mathematics". It's a perversion of the culture
to think knowing how to multiply 13x45 is the sole province
of some specialist with all this training in a particular
discipline known as "mathematics". Nor should you have to be
a mathematician to know that 13 is a prime number, or that
all whole numbers factor uniquely into primes. That's just
basic knowledge.
We shouldn't allow kids to get stuck in the trap of thinking:
"I know I'll never be a mathematician, and a math teacher
is some kind of mathematician, ergo whatever they teach me
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
long as we encourage the illusion that basic numeracy is the
exclusive property of any one discipline, vs. the common
heritage of all.
So when learning history, one might learn about when the paper
and pencil algorithms we use for addition, subtraction, multi-
plication and division were invented or first introduced.
This is what school children had to learn then (check out
Roman numerals!), as well as now.
In the context of a history section, we could start building
basic competency in these algorithms. Really learn what it
was like before calculators -- in part because you want to
appreciate how school children before you learned about their
world. And really think about what it was like to live before
TV (in a lot of ways, curriculum writers are only just beginning
to discover this medium, even after all these decades, thanks
to DVD).
Mathematics is a discipline with its own heritage and heros.
We should definitely teach it. But it's not the same thing
as basic numeracy (which many disiciplines share) and I think
a lot of confusion arises from trying to make round pegs fit
in square holes, from confusing basic numeracy with mathematics
-- kind of like confusing "learning to read" with "doing literary
criticism"; although true enough that the one is a prerequisite
for the other.
What a lot of early education is about, or should be about, is
simply "how the world works". What do students need to know to
make sense of their environment? This indeed requires a focus
on "applications". I'd put computing and computers into the
mix, have exposure to programming concepts be a part of the
bigger picture of learning about operations, procedures,
processes -- whether these be strictly "mathematical" in
nature isn't supercritical (sometimes yes, sometimes no).
How do TVs and radios work? You need some math to appreciate
signal, how sine waves can be combined and separated. Trig
functions. How do we use binary numbers to signify 256 colors
on a screen? Permutations. What does an oscilloscope do?
This sounds like basic engineering, and in a lot of ways it
is -- but you can "tease the math" out of these applications
by focusing on what's common across the board. The concepts
of bits, variables, functions (added or composed) -- these
come up in many different contexts.
Given this kind of exposure, having seen math concepts used
in the context of explanations of "how things work", one thereby
develops an appreciation for mathematics as a "skeleton key".
It unlocks many doors, makes the content of many disiciplines
more understandable. Once this faith in the relevance of the
material is established, then (and only then) is it time to
introduce the kinds of formalisms which anchor mathematics to
pure principles, irrespective of special cases.
We do indeed want students to appreciate the "purity" of logic,
considered quasi-independently of history or circumstance. But
it's a dynamic interplay, a delicate balance -- professional
mathematicians need to be careful not to evidence disdain or
aloofness vis-a-vis the special case applications of their
discipline, since in the "How Things Work" context, our focus
is building confidance (no, I don't mean "self-esteem", I mean
respect for math itself as powerful and important -- an
attitude we need to cultivate, not simply presume as a given).
The danger, when we allow specialized mathematicians to steer
the early curriculum, as that they will be too interested in
the purity of their discipline to allow appreciation for it to
grow naturally in others, including among hardened skeptics
(which many young people are). But I know lots of pros who are
well aware of this danger, and compensate for it admirably.
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,
pilots, engineers, doctors, physicists, chemists, linguists,
stock brokers, bankers, actuaries, morticians, electricians,
musicians, advertizers, manufacturers... all get a chance to
introduce the mathematical aspects of their respective
disciplines. The math teacher then has the job of abstracting
the math from these diverse inputs and distilling this to
its essence. Then those students most inspired by the "purity"
of this subject will have the choice to pursue it further towards
its source -- including in the context of a more advance
curriculum.
Kirby
Curriculum Writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
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