Kirby
Curriculum writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
<SOAP BOX>
Some recent posts to this list have pointed out that the standard
math curriculum preps students for calculus, should their career
path require it, and provides a step-ladder to that overlook.
Other students, headed elsewhere, may feel their needs are
going unmet, given the Riemann symbol is not going to appear on
every other page in the documentation they have to wrestle with
on the job -- and yet the math content in such docs as they DO
encounter may nevertheless be considerable.
Case in point: where do the polyhedra come in? My diagnosis
is lots of students feel betrayed by the current flatlander
approach (everything planar), which dangles the polys at the
end of the textbook in high school geometry (only a few teachers
get to that chapter) and then drops visualization altogether,
except for f(x) in the xy plane (what graphing calculators can
handle), an occasional construct in an xyz cube (beyond what
the calculators can do), always in conjunction with a lot of
cryptic algebra (but very little re Euler's Law V + F= E + 2,
other simple rules -- like how many kids know that for omni-
triangulated polys the ratio of Vertices:Faces:Edges is 1:2:3,
after subtracting a pair vertices (derives directly from Euler's
law)?).
Whatever happened to all those fun shapes (stellates, duals,
zonohedra, space-fillers, truncations), so important to
Renaissance thinkers, appearing in the etchings of Durer, in
the notebooks of Leonardo and Kepler, and later in Escher (whose
work is considered emblematic of mathematics by so many)? You
can't say the polyhedra are amathematical -- they are quint-
essentially so. Nor are they inapplicable in the real world:
chemistry and crystallography (i.e. materials science) and
also biology in its primitive forms (e.g. the viral protein
sheath tends to be icosahedral), all find roots in polyhedral
geometry -- as does architecture. So when do we get to them,
in the standard, required curriculum? Answer: we never do.
Hence this feeling of betrayal. "Why did we skip the good stuff?"
many are thinking. Because we're still sacrificing to the calculus
gods, that's why (where we slice and dice a few shapes to get their
volume, and that's about it) -- at a very high opportunity cost
(all the stuff we COULD have been doing), and with lots of "math
curriculum casualties" in the process.
I think young students are especially drawn to the geometry of
polyhedra because of their computers: creating "game worlds"
or "virtual worlds" requires fluency in spatial relationships,
which means a lot more than just familiarity with algebraic
methods in the xyz coodinate system. For example, how many
students learn that 12 equi-diametered spheres pack around a
nuclear sphere to give the vertices of a cuboctahedron, that a
second layer of spheres gives 42 spheres, then 92, 162...
10L^2 + 2 where L = layer number, and the shape is always
cuboctahedral (or twist-cuboctahedral if that's how you start),
and that this is the so-called face-centered cubic lattice, and
that rhombic dodecahedra encasing these spheres fill space
without gaps (as per Kepler)?
Relevant graphics:
http://www.teleport.com/~pdx4d/images/vesphere.jpg
http://www.teleport.com/~pdx4d/quadray/ve3f.gif
http://www.teleport.com/~pdx4d/quadray/ve5f.gif
http://www.teleport.com/~pdx4d/images/rdpack.gif
Anytime I throw out a little item like this in a usenet newsgroup
(e.g. misc.education), I get pounced upon by pundits who "knows
it all" re math education (by Dr. Herman Rubin for example).
These are "isolated facts" and not worth dwelling upon. The
only worthwhile math curriculum points to a grand generalized
superstructure, from which all "isolated facts" may be logically
derived. Unless your classroom content is geared entirely towards
implanting this generalized superstructure in the minds of your
students, you're wasting their time. Or so they tell me.
And I happen to agree, in principle, but I don't think we share
the same superstructure. Theirs traces to roots in a philosophy
of mathematics owing to Bertrand Russell and 'Principia Mathematica'.
Mine traces to roots in a philosophy owing more to Ludwig Wittgenstein
and his 'Philosophical Investigations'. So what we're left with
are different schools of thought, based in different philosophies.
Mine promulgates polyhedra, front and center, theirs rarely mentions
them in the standard curriculum (how many grade schoolers get a
chance to build the Platonic Five, how many high schoolers to
render them in VRML, or as ray-tracings?). And so I wrote
to Dr. Rubin:
===
sci.math,misc.education 9-18-98
hrubin@b.stat.purdue.edu (Herman Rubin) wrote:
>nature. The type of mathematics needed to understand the
>foundations seems totally foreign to you, as evidenced by
>your postings.
I agree with this last statement -- that my school of thought
is utterly foreign to yours vis-a-vis our respective approaches
to low-level mathematics and the philosophies we bring to bear
in this domain.
But you should realize, as a scholar, that any invocation
of an authoritative "consensus" around this issue is bound
to be bogus, even if you succeed in recruiting a rag-tag
band of sneerer-jeerers ala Chapman, Rickert and Scott.
At best, you and I represent different schools of thought.
For you to claim access to the "one true teaching" re
mathematics and its relationship to science and the
"real world" is to come off as a some kind of dogmatist
or imperio-snoot or religious fanatic. We have different
and competing views.
Your patronizing tone lacks diplomacy and suggests an
unawareness of history.
Kirby
===
Perhaps I too lack diplomatic skills, but the fact remains that
I've done my homework and resent being patronized by "know
better" types who refuse to acknowledge that my direction has
any validity whatsoever. My curriculum is NOT irrelevant, and
to marginalize spatial geometry any further (or longer) is to
invite an even bigger student backlash, as it becomes ever
clearer to many of them that their math educations were
sacrificed to the calculus gods for no better reason than
textbook publishers are too slow to adapt to the times (have an
investment in recycling the same material with only cosmetic
changes -- very profitably -- because "math never really
changes" does it).
As I posted earlier, mathematics is vast and whereas we may
distill it to "pre-calc" essentials, this is more for
historico-political reasons than mathematical ones -- and the
job market has changed a lot since the advent of desktop
computers, to where all that pre-calc drill 'n kill has lost a
lot of its raison d'etre (something students intuitively sense,
even if many don't know how to verbalize it).
On Oct 2nd 1997, we had a 'Math Summit' here in Oregon with
some of the biggest names in the business flown in to share
their outlook. Sir Roger Penrose, Ivars Peterson (lots of
popular math books), Keith Devlin and Ralph Abraham were all
invited to address the assembled K-16 rank and file re their
hopes and dreams for mathematics teaching in the 21st century.
They advocated a Renaissance approach, diffusing the
"arithmetical stuff" throughout the curriculum (because you
need basic numeracy to understand presentations of data in
all disciplines -- so teachers of all subjects should be
sharing at least this part of it) and freeing math teachers
to get deeper into making "connections" (as per James Burke's
famous TV series by that title).
Mathematics is about "making the invisible visible" is the way
Keith Devlin puts it, and an invisible infrastructure, consisting
of multimedia, electronic banking, satellites, microwaves, permates
our reality. You need mathematics to make sense of it. Students
need our assistance in deciphering the real world, which, more
than ever, is NOT what you directly perceive with your five senses.
Mathematics is about developing that "sixth sense" required to
get a clue about what is actually going on aboard Spaceship Earth.
It's time we start giving students more clues and relying less
on what we ourselves were taught, at least in areas where this
has since been superceded. Life moves on.
</SOAP BOX>
Thanks for listening.
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