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Of Sequence and Success
Posted:
Nov 2, 2012 3:19 PM
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In another thread, Lou Talman and Robert Hansen are discussing sequence.
Lou brings up the point that the ancient Greeks invented much of what
we call mathematics without being especially proficient in arithmetic
in the way our own subcultures were recently trained. We aren't the
same way about arithmetic today, since calculators and other devices
made everything so easy, although we still school in "the algorithms"
in ways the Greeks never did. Were they "unsuccessful in math"?
Quoting Lou:
"""
And the ancient Greeks---who invented modern mathematics---are
certainly a counterexample to your "natural progression". They
accomplished a great deal without beginning with the algorithms we ask
kids to study today. Indeed, it's likely that they weren't very good
at arithmetic at all. So their "progression", if there was such a
thing, was entirely different from the one you think you've
identified.
This last example suggests very strongly that arithmetic, while it may
be *an* entry into mathematics, is not the *only* entry. Your "natural
progression" completely ignores a significant possibility: The
primacy of arithmetic is simply an artifact of a curriculum that
denies entry to those who haven't acquired proficiency at arithmetic.
(A curriculum, moreover, that's now strongly distorted by the effects
of fifty years of standardized, multiple-guess, truth-or-consequences,
mis-matching tests.)
"""
Of course once you get through whatever sequence growing up, you're
not done with sequences. It's always "one damn thing after another"
(Henry Ford, on history). So even if we argue about the "one right
sequence" (I'm against the notion) for child-to-adult math learning,
we're not done. What else might we try with adults?
I'm probably more into adult guinea pigging than most. For me, it's
not all about what we teach to 10 year olds.
Even with kids though, I emphasize V + F == E + 2. I don't worry
about what if it has more holes too much. It's a wire frame to begin
with probably and a simple convexity, a polyhedron for which we have a
name probably, or part of a series.
[ Waterman Polyhedra for example, I worked on them, with Steve, their
definer and early sculptor of their form (he used spreadsheets and
actual physical models made of little balls) -- our team on
Synergetics-L (e.g. Gerald de Jong, myself) did a bunch of the
computer graphics, using Qhull. Other yet more expert implementations
by other collaborators (with Steve Waterman) came later. Google and
ye shall find. ]
Lou again:
"""
And consider the popularity of puzzles like sudoku---which are based
on very mathematical, but non-arithmetic, reasoning---in a nation that
despises mathematics. Where do such phenomena fit in your "natural
progression"?
"""
Kirby
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