On Fri, Dec 18, 2009 at 9:06 PM, J DeMouy <
jpdemouy@rocketmail.com> wrote:
> Kirby,
>
> First, thanks for sharing this. I'd be very interested in a progress
> report as things develop.
>
> Second, could you specify more clearly the nature of the obstacles as you
> see them? I'm hearing a lot of intriguing ideas, but having difficulty
> figuring out precisely what you're seeking.
>
>
Here's a real situation that's happened more than once.
Lets imagine a short video clip (a composed scene):
<take_one>
A guest math teacher comes into the classroom.
She or he is prepared with a set of polyhedra into which liquid or dry
grains (sand, beans) might be poured.
She holds up the regular tetrahedron and says "this is my unit cup measure,
the cup I will use to fill these other mixing bowls" (we're extending the
cooking analogy). These "mixing bowls" are other polyhedra with one face
open, making them easy to pour into and out of.
She then has students come up in turn and start pouring colored water from
this unit cup (a tetrahedron) into the mixing bowls, filling them according
to various ratios (what this lesson is about). Other students might fill
out a volumes table on the white board. We'll take a discovery approach.
The unit cup fills the octahedron exactly 4 times, the cube 3 times, the
rhombic dodecahedron, a space-filler, 6 times...
Of course these definite numbers require specific proportions in edge
lengths as well -- more of what this lesson is about (this nested
arrangement of concentric polyhedra is what we call the "concentric
hierarchy" in some lesson plans).
</take_one>
<analysis>
In what way guest math teacher has disturbed an equilibrium?
Suppose there's not much follow-up in the rest of the curriculum. This was
some out of the blue, one shot deal and now it's over.
The case could be made that the guest has weakened the game of the host
teacher, who will have the same students tomorrow but is not prepared to
follow up or provide context.
Apparently this idea that volumes always need to be cubes is incorrect, but
what are the implications? The whole thing dead ends, unresolved.
Nothing in the text book mentions using the tetrahedron in this way.
The guest teacher is like a visitor from Mars, is talking about some alien
way of doing geometry that Earthlings simply do not do.
</analysis>
Alternatively, lets script this a whole different way.
Fast forward and take two:
no guest teacher, just the original teacher, newly equipped with a set of
lesson plans, all interconnecting.
Tomorrow's lesson will feature a whole hour on the rhombic dodecahedron,
with stories about Kepler (twas one of his favorites).
Its role as a space-filler in a lattice we wanna yak about, well known to
chemists as the FCC, will help tie us back to the lesson on the day before.
That cube of volume 3 (remember pouring water) was embedded in said
dodecahedron as face diagonals.
The Portland World Trade center uses the FCC lattice (connecting a dot --
place based education means consistently referring to local geometry,
wherever one is).
Yes, these used to be college topics, but with the Obama administration is
pumping stimulus money into making the USA education system more world
class, there's pressure to come up with something more worldly and world
class. More use of a computer programming language will be another feature.
Of course I'm not sure how familiar you are with adventures in math reform,
in the USA or anywhere else.
My goal in this thread is not to focus on money and politics so much as the
philosophical difficulties some teachers may have with these interesting new
lesson plans coming from various corners (e.g. Oregon).
> With those caveats, I'll offer some thoughts which may or may not be
> relevant.
>
> My suspicion is that any discussion of, e.g. the arbitrariness of grammar
> might be counterproductive, perhaps even becoming another skirmish in the
> "culture wars" where "liberals", "secularists"
, or whatever else you may be
> called are accused now of seeking to even undermine mathematics.
>
>
Yes. Your comments show me you're quite aware of how debates quickly
deteriorate away from any philosophical content.
People get into food fights backed by pseudo-science, with precisious little
logic or even rules of road for debating a topic (the media has encouraged
this).
Getting back to some elementary philosophy of mathematics: the picture I
paint is of sandcastles on the beach, each one an "axiomatic system" with
foundations holding up some superstructure of proved theorems.
However the sandy beach itself is "neutral" enough to host these multiple
sandcastles, i.e. at "the level below axioms" there's simply the "natural
grain" of ordinary language and reasoning. Nor do you need a full-fledge
sandcastle with formal axioms to get what we might call a mathematical
language game, a gizmo with tight rules, such as chess or Chinese checkers.
Philosophers sometimes like to construct and/or get inside axiomatic systems
(like sand castles), and that's more like being a theorizer. You might even
be a mathematician some days of the week, working on some project with
polytopes, in the lineage of Coxeter (a student of Wittgenstein'
s -- see
'The King of Infinite Space').
I think those practicing a more Wittgensteinian brand of philosophy are less
likely to see themselves as system builders than as system brokers, i.e.
people comfortable going between and among systems, finding the raw material
of philosophy (language games, rule following, forms of life) everywhere one
visits.
That's partly why I turn to Wittgensteinians here, as I think the situation
requires ambassadors or at least mediators between the sandcastles, offering
reassurance that collaboration, not enmity, is the name of the game.
This doesn't preducle friendly intramural competition, as in speech and
debate, athletic events -- traditional inter-school rivalry, rah rah
(sandcastle = school of thought, wrapping some formal or core system e.g. a
Euclidean geometry).
What seems more likely to persuade are demonstrations of consistency (or
> rather, relative consistency: consistency with Euclidean geometry and
> familiar systems of measure) and applicability (which your remarks seem to
> suggest are available).
>
> A Platonist who is troubled is a candidate for Wittgensteinian therapy.
> Someone who simply accepts Platonist-sounding assumptions without ever
> making them explicit let alone being troubled by them is likely not.
>
> But please elaborate if these observations completely miss the point. as I
> fear they very well may.
>
> JPDeMouy
>
>
Here's an excerpt from a recent exchange with a USA midwesterner. He is not
a classroom math teacher, and the midwest is not a target demographic for me
(because I'm not there), however I think his disquietude is characteristic
of what I'm up against even around Oregon:
[ If wanting to follow closely, this would best be understood by consulting
these pictures first:
http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html (Fig 990.01)
]
<exchange>
Midwesterner:
That's three 100s, one for each edge, right?
Kirby:
Yes. The grid of triangles inside (on the grid paper) are 100 x 100 in
number, just like with squares. We're getting all the right answers with our
triangles. Is this what Klingons use? No wonder their spaceships are weird.
Midwesterner:
Sorry, violates my excellent (highest grades of any class in g-school)
geometric sensibilities (and triggers my anticreationism alarms) : "100x100"
makes no sense as you use it above ("...just like with squares"). *Not*
"like with squares." With squares "100" is the number of divisions along one
edge, and "100" is the number along a second edge at right angles to it.
"100x100" is the number of little squares that this situation *creates*; it
is not the number(*) of squares one merely counts up, by putting finger on
little square and saying "one," then another and "two," and so on, which
*is* what you are doing when you say "100x100" is the number of little
triangles you made with your 100x100x100 edges.
It's either speciousness or that personally despised Sophistry for you to
above say "..are 100 x 100 in number,..." instead of saying "are 10000 in
number, just like with squares" (which would not have raised my 'Warning!:
Sophist at Work!' red flags).
<exchange>
You see he mentions I'm triggering his "anticreationism alarms" meaning I'm
coming across as a threat, am being protrayed as anti-scientific or somehow
assaulting rationality.
In the more local context, what's at issue here is a new Digital Mathematics
curriculum for Oregon, a course that will count towards the required three
years Oregon requires for its high school graduates.
Lots of people besides me are working on this (per planning meeting Aug 7)
and here I'm just looking at some of the more problematic puzzle pieces,
ones engendering the most disquietude.
We're expecting a surge into spatial geometry topics simply because we're
moving to bigger and brighter screens and our students already have well
developed spatial sensibilities. Also, some of the industries around here
would like students to have well-developed spatial imaginations for economic
reasons, so a win there as well.
Spatial geometry means more GIS/GPS, more Google Earth type stuff, but it
also means doing more with polyhedra, not resting content with just the
flatlander stuff (plane geometry, everything flat). Going spatial means
doing more with these lattices, such as CCP (FCC) -- also known as the
octet-truss in architecture.
"Lattices are like the 3D correlate of 2D tessellations, or tiling, and we
already do quite a bit with the latter." Saying that is reassuring to
teachers i.e. explaining "this is nothing new" seems to be the best way to
appeal to the conservative element -- math being one of the more
conservative of all disciplines.
So now a real question (one with time pressure behind it) is whether to
include or bleep over the "whole number volumes" lesson plan above, and
those that connect to it (lots of lessons about number sequences, sphere
packing, geodesic spheres, space frames, some virology and crystalography
etc.).
My camp says "yes, we should include them, to do otherwise is to deprive our
students of important mathematical heritage they'll likely need in their
future, could use right now today".
Some other camps say "no, it's too alien, rocks the boat, and is therefore
wrong to include" (e.g. more such passages from our midwesterner on request
-- he's thankfully verbose, gives plenty of insight into his thinking).
Others will be on the fence whereas still others aren't yet aware of any of
these issues, don't track curriculum concerns closely enough to really care.
Those who've been through earlier chapters involving swift innovation within
the curriculum (e.g. "New Math" in the 1960s) know it matters a lot how we
communicate with teachers on the front lines. Our goal is to empower them,
back them up, not weaken their game. Giving them better mathcasts to
project (better math cartoons) is at the top of my agenda.
So how might philosophy, Wittgenstein style, reduce the level of xenophobia
around somewhat alien content? We need to help math teachers integrate the
new material without too much stress and anxiety.
Some schools will go ahead with piloting and training teachers in this
material (I have some gigs scheduled), others will not. My challenge, given
the role I'm playing, is to increase acceptance by keeping philosophical
confusions to a minimum. I could probably use some coaching, take advantage
of more field experience. Just letting my peers know of this challenge is a
positive step.
Kirby
--
>>> from mars import math
http://www.wikieducator.org/Digital_Math