[Wittrs] [C] Re: help the math teachers?
- From: "J" <ubersicht@xxxxxxxxx>
- To: wittrsamr@xxxxxxxxxxxxx
- Date: Sun, 03 Jan 2010 08:55:48 -0000
Kirby,
My apologies for the delay. I've gotten distracted by some much less
worthwhile discussions as well as some worthwhile ones that simply required
less of my time and energy than this. I hadn't forgotten you.
Of course, I think I still have a couple of other posts out there waiting for
you, so perhaps apologies are unnecessary.
> There is a Nintendo character named Kirby, whom I reference
> in my
> slides at http://www.myspace.com/4dstudios
That's the one!
I'm happy to be associated with something
> other
> than a vacuum cleaner for a change
The Dysons are much better.
You ever heard of Jack Kirby? He co-created Captain America, The Fantastic
Four, The Incredible Hulk, The X-Men, and many others.
http://en.wikipedia.org/wiki/Jack_Kirby
> "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?
The discussions of "aspect blindness" in the later work (PI pt. II, RPP I and
II, and LWPP I & II passim.)
Often, in cases like we're discussing, it is evidenced in a capacity to put a
picture to a new use.
Be careful though. Sometimes we can confuse resistance to ideas with an
inability to understand them. Recall the "you just don't get it" of identity
politics.
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.
That's an empirical question of course. I don't know whether exposure to
variant schema would enrich comprehension or cause confusion. Likely, it
depends a lot on age.
> Do we teach this in art schools? In some maybe.
I think it definitely has a place in design schools!
> You could see this as countering some unexpressed view that
> "math is
> fascist" in the sense of "my way or the highway".
Again, I wouldn't call this a philosophical issue, per se, but if people misuse
the word "fascist" and if their notions of political liberty blind them to the
role that authority, unquestioning agreement, and strict training by repetition
play in the functioning of our understanding, our language, and our very form
of life, then perhaps they need a wake up call and perhaps their notions of
liberty are simply bosh.
Seeing freedom as well as constraint in all of this is certainly a good thing.
You're right about that.
But people who make such judgments as "math is fascist" are in my experience
among the worst dogmatists, obscurantists, and opponents of human freedom and
dignity going. As well as being total asshats.
That's not diplomatic though. Obviously.
>
> From the point of view of struggling students, sometimes
> finding math
> oppressive, a little novelty might go a long way towards
> forestalling
> a disconnect.
hmmm. Perhaps.
I wonder though if students like that whose interest one might hope to pique
with the new thing ("Ooooh! Shiny!") would have any follow through anyway.
That would be nice and I do not rule it out.
> 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.
"Has to do with"? Is the conservatism of mathematics explained by such
notions? Are such notions rooted in the conservatism? Or are such notions an
expression of the attitude with which we approach mathematics and of how we
treat mathematics? And isn't how we treat mathematics an important part of the
various roles it plays for us?
(Conservatism in mathematics and conservatism in the teaching of mathematics.
Conservatism in the teaching of fundamental mathematical concepts and
conservatism in the teaching of further extensions. Compare.)
That's
> why I say math teachers
> need help from philosophers, especially those in the
> Wittgenstein
> camp, given their pragmatic operationalism (meaning through
> use).
hmmm
Did you catch my remarks elsewhere regarding operationalism? (It was in a post
to JRStern "re: QM without consciousness" or something like that.) I'm
wondering if you mean "pragmatic" as a reference to Pragmatism or simply
colloquially. And if you mean "operationalism" in a more or less doctrinaire
sense?
I certainly would not be inclined to call myself a pragmatist or an
operationalist.
> 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.
I find this aspect very interesting.
I've shared some of this with my father, an engineer who, when I was a child,
made a hobby of welding models of Platonic solids and the like. He was
interested as well.
>
> 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.
I wonder here about the "even though"...
When you do
> post and lintel
> architecture, rest cross-beams atop columns, you get used
> to thinking
> of structure as rectilinear.
Is this meant to be an explanation? What sort of evidence would support it or
falsify it?
The tetrahedron (tetra
> for four),
> although the minimum wire-frame enclosure, more primal than
> the cube,
"More primal" in the sense of "minimal" or in some other sense?
> 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?
I mentioned in another post, I wonder about putting things in these terms.
Do you mean "paradigm" in anything like a Kuhnian sense? It doesn't seem to me
that this approach is incommensurable without the prevalent one in anything but
the familiar and epistemologically innocent mathematical sense.
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.
>
Ah.
I've never trusted PR.
> 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.
Okay, yes. Though I wonder if math teachers as a group are as persuaded by
such forms of persuasion as some others might be or if their liable to think,
"Sounds like snake oil. Too good to be true. Panacea. We've been burned
before."
I don't know enough about the attitudes of math teachers in various parts of
the country to do more than entertain possibilities that might warrant caution.
Take what you will.
> 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.
I think experiences with New Math might offer legitimate reasons caution
without needing to posit... um, I'm not quite sure what :music of authority" is
meant to suggest.
> 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?
I'd be reluctant to speculate about motives here.
The remarks about various transformations, comparisons with tessalations, and
so on, were quite interesting in their own right but seem to not connect with
my point. (I still found the material quite fascinating though.) I think I
made the point better in a subsequent post though, so I'll leave it.
> 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.
Why in heaven's name would math teachers give a gods damn about what
philosophers have to say and why would any philosopher presume to suppose she
could approve or disapprove here? (I know the answer to the latter: some
philosophers are a presumptuous lot. But I hope no Wittgensteinian would so
presume.)
A philosopher (perhaps a better one than myself) can help in exploring how we
relate these new concepts with the old, how we think about the transitions
between the different cases.
Whether the new approach is legitimate is a mathematical question and not a
philosophical one. It is a question of whether one can translate between the
systems, whether conversions between them preserve consistency. From what I
can tell, that's not a problem, but a mathematician is far more qualified to
answer that.
If the answer to the mathematical question is affirmative, then it may be best
not to emphasize talk of "paradigms" (unless the buzz word is just good for PR
rather than being offered as a philosophical insight) and such because the real
question is one of the allocation of educational resources (a procedural,
political question, not a philosophical one - save perhaps to a Rortian).
On that question, I could only entertain possibilities with you. If thinking
about mathematics philosophically gives me any particular insight into
reservations people might have, that would be a side effect. It wouldn't make
it a philosophical matter.
A philosopher might be able to assist in thinking and speaking more clearly
about the relationships between the different approaches and in helping others
to but - as the reference to PR may remind us - clarity may not be the winning
strategy in political matters anyway.
JPDeMouy
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