I already replied to this once, howevr after a hiatus I have some
more thoughts, so I'll jump in again... Somewhat lengthy...
--- In
WittrsAMR@yahoogroups.com, "J" <wittrsamr@.
..> wrote:
>
> Kirby,
>
> (During our exchanges, I find myself picturing the videogame
character "Kirby" bouncing around tetrahedra - though that's not how
the game looked. I hope that observation doesn't offend. It's on no
way meant to trivialize your points.)
>
There is a Nintendo character named Kirby, whom I reference in my
slides at
http://www.myspace.com/4dstudios
No offense taken, I'm happy to be associated with something other
than a vacuum cleaner for a change (Kirby's are famously sold door
to door, a high pressure gig -- they tried to sell one to me in
fact, and I'm embarrassed to admit how close I came to succumbing).
> >
> > I think you were clear, and I'm encouraged you would spend
> > some energy
> > on this thread.
>
> I find the subject interesting, certainly. You've indicated that
I've been able to help and I don't feel I'm wasting my time as can
sometimes happen in these threads.
>
That's good.
I do think we're looking at real philosophy of mathematics here, and
that Wittgenstein'
s approach is undervalued. What is this approach?
We've talked about "assembling reminders".
"Triggering perceptual shifts that restore clarity" might be one way
of characterizing it. That's part of the inherent difficulty too, as
once in the realm of perceptual shifts, gestalts, how does one know
if this "triggering" has occurred?
"An ability to continue in some language game in accordance with its
rules (grammar)" might be construed as evidence (of understanding)
.
We're in familiar PI territory at this point.
> > You point out that a tabular arrangement of rows and
> > columns is easy
> > to read, whereas a grid of triangles is less familiar,
> > more
> > problematic, even if it provides a logically consistent
> > model of
> > multiplying lengths to get area. In making the
> > transition, one is
> > likely to slip up, make mistakes.
>
> There are mistakes one can imagine, such as failing to see that the
bowling pin arrangement doesn't fit what we're meant to do here, but
I wouldn't quite say that such mistakes are my principal concern.
>
> How "easy to read" one representations is compared to another is
closer to the point, but it might be closer to say that I'm not sure
how to read one as I would the other. Or that they don't say the
same things.
Yes, I understand your concerns. It helps me to regard you as a
delegate from the math teaching community (a kind of role playing)
and wondering if interjecting this particular language game is really
a good idea. You'll get these pedagogical concerns, that exposure to
an unfamiliar model of area and volume, not based on right angles,
will only interfere with student comprehension of traditional content.
The challenge as I see it is to phase in some of this new thinking in
a way that bolsters teaching what's already on the books. Teachers
feel on the ropes, in the trenches, beleaguered. Here's something
novel suggesting opportunities for more artistic treatment
(claymation?
). Build your portfolio of original works why not?
Do we teach this in art schools? In some maybe.
A part of meeting this challenge, though maybe not the most important
part, is to be clear that we're not breaking any deep rules of
mathematics. We're exploring our freedoms, our heritage, and thereby
gaining more insights into what mathematics is all about. There's
room for diversity.
You could see this as countering some unexpressed view that "math is
fascist" in the sense of "my way or the highway".
From the point of view of struggling students, sometimes finding math
oppressive, a little novelty might go a long way towards forestalling
a disconnect. I see premature disconnect is one of the chief
challenges to overcome, am hopeful that the new mnemonics, our little
package, will prove life supportive and forestall that sense of
suffocation and/or claustrophobia that students feel when the math
goes over their heads too quickly.
That's the thing though: mathematics as we teach it in schools is
conservative, change-averse. We might think of it as the one
discipline least likely to change. This has to do with math's
reputation for dealing with eternal verities, with nothing new under
the sun, at least where primitive geometry is concerned.
On the other hand, math teaching has already gone through many
changes in my lifetime, starting with New Math in the 1960s, followed
by the introduction of more calculators, some use of computer
languages (Logo, BASIC), followed by a lot of canned software since
the PC revolution, then the Internet, then open source... some
schools have adapted more than others to these changes.
What might be next? More spatial geometry thanks to more computer
power? A more sophisticated approach to spatial geometry? This is
what I've been anticipating, but then I thought we'd be somewhat
further along by now. That may be because we don't have enough
people attuned to the big picture. That's why I say math teachers
need help from philosophers, especially those in the Wittgenstein
camp, given their pragmatic operationalism (meaning through use).
>
> As an approach to length and area, the concerns I've raised are not
a problem at all, as far as I can see. If we are approaching this as
"magnitudes" rather than "multitudes"
, the differences between the
two uses of "n x n" are less significant. But the background of why
one might be more inclined to accept the squares as a picture of n x
n (multitudes)
, why this picture better reflects different aspects of
the calculation, might be something to acknowledge up front, to
emphasize the differences so that there is no charge of "sleight of
hand" or "sophistry".
>
> (Apart from the mission of educating teachers, encouraging the
students to think of different ways we use pictures, to think about
concepts like "grouping" and "correlation"
, would be beneficial.)
>
Going back to my volumes table, I'm suggesting what's novel is our
ability to interject more whole number volumes and simple fractions
than previously, even though we're introducing such non-rectilinear
concepts as the rhombic dodecahedron, a space-filler. We assign it
a volume of six, slice and dice it to come up with a corresponding
cube, octahedron and tetrahedron of 1/2, 2/3 and 1/6 the volume
respectively (i.e. 3, 4 and 1) -- the beginning of a language game,
with more pieces to come later.
Having this more sophisticated visual vocabulary coupled with less
intimidating, more memorable whole number relationships, is something
new (since the 1960s). It's new because our Roman and Greek
forebearers considered right angles "normal" even though squares
have no inherent structural stability. When you do post and lintel
architecture, rest cross-beams atop columns, you get used to thinking
of structure as rectilinear. The tetrahedron (tetra for four),
although the minimum wire-frame enclosure, more primal than the cube,
didn't get as much focus in the early days of western civ. Those
mental habits are difficult to counter to this day. Yet what better
example of challenging a dominant paradigm? Math teachers need help.
However since the invention of microscopy, other more powerful
instrumentation, it has become apparent that nature is more
triangulated in her designs. Our more sophisticated visual vocabulary
is going to help us down the road, as future biologists, chemists,
engineers. The world of sphere packing, of lattices, will be more
front and center, thanks to our more 60-degree based approach.
Stabilize what we have, trail blaze new material. It's an exciting
time to be a math teacher.
That's the PR anyway.
In other words, we're hoping to excite teachers about these
developments, not trigger a "let's not rock the boat" backlash right
from the get go. Given widespread frustration with the current
fare and a paucity of polyhedra to begin with, it's not like we're
facing much organized competition. "Unconscious inertia" would seem
closer to what's in our path. We could use some help from philosophy
to at least turn unconscious inetia into conscious debate, which is
where Wittgenstein'
s "therapies" enter in.
Also, as soon as one touts something as "better" or "new and
improved", there's some resulting anxiety about introducing the
change, undermining the existing "music of authority" in some way.
This may account for why this material is still unfamiliar and not
widely discussed, even after a half century. There's an underlying
defensiveness perhaps?
People in my camp (which includes some math teachers) could benefit
from the focused attention of a few philosophers, especially those
trained to add clarity, reduce confusions.
> It isn't just a question of practicality. What I hope to convey is
that what the "90-degree-based conventions" show as a picture of
arithmetic operations is different. Yes, there is an analogous
"input" and "output" with the triangular case, but not all of the
same transformations.
>
> "Seeing as" is important here. An n x n square can be seen as n
columns or as n rows. And each column (or row) is the same
arrangement (squares in a lined up or stacked) making their grouping
perspicuous. Being the same in this sense, the connection between
multiplication and addition is clarified: n x n is a stack of n
height (or in a line, n length), n times ("times" as literal
repetition: the same again). And that they are the same is not just
a matter of their having the same arrangement (grouping) but their
lining up (correlation)
.
>
> What corresponds to grouping, to correlation, or to "the same
again" with the triangles?
>
I've been giving this more thought.
Maybe the following cartoon would serve:
Any square, subdivided into smaller squares, may be skewed (tilted)
to form a rhombus. Each smaller square then becomes a rhombus as
well. Each of these may be cut in half with a diagonal, to form
two equilateral triangles.
We will use this n x n rhombus (of n x n sub-rhombi) to display
area in the usual fashion, as an array of colored-in rhombi. 3 x 4
= 12 would be shown as the usual three by four array.
However, because each of the 12 rhombi is already subdivided into
two triangles, it's easy to simply take one triangle from each rhombus
and say this would be the area in an "alternative currency" where
little triangles are unit (a different "coin of the realm"). We
start thinking "exchange rate".
I'm seeing lesson plans with tessellations piling up in this region.
What shapes tile a surface? What complementary shapes? If we take
shape A as our unit, and know the ratio A:B, then this polygon of so
many As and so many Bs, has what area?
Portland's Math Learning Center already has some of this material
developed I'm pretty sure. Some of our public schools use it.
Tessellations (tilings) have been an important aspect of elementary
school, could be revisited in high school.
In volume or 3D or space, you have space-filling as a topic, using
either the same shape (e.g. rhombic dodecahedron) or complementary
shapes (e.g. tetrahedron + octahedron). As tessellations are to
area, so space-filling is to volume.
A point of these lessons is we're free to take any of these areal
or volumetric shapes as our unit. At that point, it's the fixed
ratio between our unit and other shapes that will allow us to compute
a specific area or volume made from these shapes (or fractional
parts thereof).
For example, this shape would have a volume of 2 in our canonical
system, as it's made from the volume 4 octahedron, less what we call
A-modules, six per each face (6 x 8 = 48, 48 * (1/24) = 2, 4 - 2 = 2).
http://www.flickr.com/photos/17157315@N00/4207163375/in/set-72157622797118549/
The analogy with currencies, other units of measure, is apropos. We
might speak of conversion factors, going from dollars to yen, or
from acres to hectares.
I've not given myself a lot of time to work on these kinds of lessons.
My focus has been the end goal, a volumes table that triggers new
gestalts, new ways of looking, mainly by streamlining and
concentrating information into newly memorable forms (what a well
designed curriculum is supposed to do).
If the philosophers think what we're doing is OK, i.e. passes enough
tests for being coherent, not breaking the rules, then math teachers
might feel comfortable enough to develop a lot more bridging
exercises. This means unleashing their creativity and going to town
(idiom) on some interesting material. But we won't get much of this
help without buy in.
I'm not certain that philosophy is the bottleneck here, but it might
be. The philosophy of mathematics intersects with the history of
ideas. Lots of ideas connect here. I'd argue it's difficult to tell
the story of recent intellectual history without touching on these
threads. Hugh Kenner's 'The Pound Era' is a case in point, a work in
the humanities connecting writers around Ezra Pound. Norman O. Brown
also patches in here. Both authors admire and quote Bucky Fuller.
Actually, Hugh wrote a whole biography ('Bucky') plus 'Geodesic Math
and How to Use It'. The recent 'King of Infinite Space' by Siobhan
Roberts advances the narrative further. I think we're talking about
somewhat unavoidable content for many brands of literature majors.
This will likely make waves in philosophy too then (already has).
Of course this all might be a complete misreading of the intellectual
landscape on my part. That's another conversation I yearn to have
with history of ideas people, even here on this list. Let's compare
notes and see what holds water...
FYI here's a Youtube showing some action:
http://www.youtube.com/watch?v=yWANBOvq8kw
The theme here is going from 90-degree to 60-degree based modeling
based on the analogy of 90 and 60 degree clock relationships, like
going from 3 o'clock (90) to 2 o'clock (60).
> Also, with squares, we also have the ability to form rectangles, n
x m, which obey the same rule of multiplication and exhibit the other
aspects mentioned above. What is an "n x m" triangle, where n does
not equal m? And can n x m still be multiplied?
>
> One of the colored graphics you showed previously does show how
this works, but it is not really analogous to the "n x n" triangle in
the way that the n x m rectangle is analogous to the n x n square.
>
It sort of is if you think of a parallelogram and how it has the
same triangular area on both sides of a diagonal. We have two
transformations to consider: slanting a rectangle, slicing off one
of the mirror images. This is our "currency conversion" operation
(slant, slice).
> To put it in your terms (?), the "n x n" when speaking of triangles
belongs to a different "namespace". Or rather, the relationship
between the arithmetic operation and the pictures is not the same
here as it is with the square. That's not because the later is the
"correct" picture or because the triangles are "sophistry", as your
Midwestern interlocutor might have it, but just because the
connection between the operation and the square involves more than
just the fact that you can count the squares along each edge and
multiply them to arrive at the same total as counting all of the
squares within the larger square.
>
> He is right to object to "it means the same here", though he offers
no clear grounds that I can see.
>
What I hear you saying is it's important to acknowledge a "change in
meaning" here and not simply say "these two pictures communicate the
same thing". Neither picture makes the other wrong, it's possible to
go back and forth, and mathematics is open to both.
Your use of "namespace" seems apropos to me, is how I would use it
as well. These are different namespaces, like different countries,
and we'll want to keep them distinct, somewhat apart. Trying to
play by both sets of rules at the same time is unnecessarily
confusing.
Your advise matches my own intuitions and explains why I've recently
chosen to package my volumes table as "Martian Mathematics" (i.e.
math from Mars). That's supposed to add to the fun quotient, but not
in a cheap and silly way. We've identified some good reasons to put
some distance between the paradigms.
Speaking of which, another aspect of recent intellectual history is
all this talk of "paradigms" (since Thomas Kuhn's 'Structure of
Scientific Revolutions' especially). One could argue that having
these two "paradigms" (or "namespaces"
) smack dab in the middle of
some math classes is going to give students better purchase, an
easier handle on, the cultural discourse of our day.
Does this sound like a lot of empty hype I wonder? To me it's like
a no brainer that this is true.
> But if I've helped you see what would be good grounds for that
objection, perhaps by pointing out the differences as well as the
similarities, such conflicts can be averted.
>
> I have no objection, provided
> > we're free to
> > establish / explore a logically permitted alternative in
> > some
> > lessons. We want to keep using that concentric
> > hierarchy with its
> > volumes table.
>
> Certainly! I have no objections to any of this!
> >
>
> JPDeMouy
>
Kirby
============
=========
=========
=========
==
Need Something? Check here:
http://ludwig.squarespace.com/wittrslinks/
