Note that this is more about K-12 than about 13-16, but of=20
course there's overlap, and many of the same arguments/attitudes=20
would apply. Any feedback welcome. For long time subscribers=20
to this list, you'll see the below as consistent with my earlier=20
posted views...
=3D=3D=3D=3D=3D=3D
Three threads above:
a. NCTM Standards - attention Herman and Alberto (Apr 21 99)
b. Re: NCTM Standards (Apr 22 99)
c. Java as Mathematics (Apr 26 99)
initiate some useful conversations regarding the place of
computer languages in math class. From my point of view,
the perennial debate about the appropriateness of graphics
calculators in math class/on tests is a dress rehearsal for=20
a larger, deeper discussion involving the place/use of more
advanced technologies in general: in particular computers
and computerized simulators (VR tech).
My contention is that "math class" in K-12 was never intended=20
as the sole property of specialized math professionals to make=20
do with as they pleased, i.e. in a democratic culture, we're=20
not to insist that students "track" themselves too early and=20
study math with an eye towards becoming indoctrinated in the
same way those on a doctoral path in that subject.
A given student has x hours per week of "chair time" in which=20
to concentrate on a spectrum of inter-related topics. So what=20
do we include and what do we drop? We need to consider the
world in which the students will actually live, today and=20
in the future -- and not just the rarified world of an Ivory=20
Towerite tenured math professor (always a tiny minority and=20
not necessarily the cookie cutter we're using when doing our=20
jobs as math teachers in the earlier grades).
I've been arguing for a remix wherein computer programming
and related patterns/styles of thought be phased in much
earlier, extending a megatrend already well promulgated by=20
Seymour Papert and his LOGO language for kids (based on LISP).
So how does this go, beyond the early grades?
What computers do is take a lot of the drudgery out of=20
repetitive tasks. Exercises establish the algorithms in our
minds, give students insights into the nuts and bolts, but=20
the programs then reiterate these computations way more times=20
than we'd want to do in drills -- e.g. graph thousands and=20
thousands of points whereas, by hand, we'd maybe settle=20
for 10 and consider the skill established. So one thing=20
computers provide (and this is obvious) is raw computing=20
power. But to tap this power, you need to write in a=20
language those computers will understand. What's shown
in math books is two often three or four steps removed=20
-- so better bridges need to be built.
Why is this important?
Because a lot of the math principles become a lot more=20
accessible if (a) you know the algorithms which show them=20
off and (b) you have the computing power to drive high=20
powered displays of these algorithms at work, drawing from=20
data sets much larger than you'd want to tackle with paper=20
and pencil alone.
Of course fractals and strange attractors come to mind as=20
examples of this approach. Indeed dynamical systems theory=20
as a branch of mathematics owes a lot to computers for its=20
very existence. I also think of STRUCK, where dynamical=20
systems meets up with tensegrity via "elastic interval=20
geometry" (EIG).
Beyond points (a) and (b), computers provide useful grist
for the mill in a more philosophical context. Kids see
movies like '2001' (we're almost there), containing really
smart computers like HAL, see other AI creatures in science=20
fiction (most recently 'The Matrix'). A common theme in=20
these works is human intelligence (humint) being superceded=20
somehow by AI. =20
Stephen Hawking is one who believes that intelligence will=20
somehow leap across the gap, from a less biochemical to a=20
more electrometalic hardware, the better to probe more deeply=20
into space my dear. However, Stephen is also one who thinks
there's a profound difference between biochemical and electro-
metalic circuitry today, which is maybe why humint taps the
Platonic realm (outside mere rule-following) whereas computers
remain "artificial" in their intelligence (bureaucrats, good
doobies when it comes to going through the motions, but=20
incapable of true genius).
With this thread already prevalent in the culture (and lets
not forget this Y2K business), I believe it's essential that=20
computers be probed, scoped out, disassembled, programmed --=20
approached from many directions -- in order to remove whatever=20
shrouds them in mystery (e.g. ignorance and fear) and makes=20
them targets for phobias and paranoias of various kinds=20
('The Net' and 'Enemy of the State' both play up high tech-
nology as instumental to a heavy handed monitoring regime=20
-- and I noticed the USA State Department was quick to=20
reinforce those perceptions when suggesting such means were=20
behind Turkey's nabbing of the Kurdish resistence movement=20
leader (one of them)).
Sure, teachers in other classrooms will be expected to interpret=20
these memes-patterns of pop culture to students with eager=20
minds, wanting to know, but it makes sense that math teachers=20
especially would be looked to for insights and role model=20
attitudes. Math teachers should be able to carry on intell-
igent conversations around such generic topics as (examples):
* Shall we fear a computerized Big Brother? =20
* What is GIS/GPS and how do satellites both show me=20
where I'm driving (when lost) and guide B2 bombs to=20
their targets? =20
As long as both civilian and terrorist applications of the=20
same technologies get funding, we'll need to instruct our=20
students in the meaning of "dual use" -- an important concept=20
in all of today's negotiations about peacekeeping, sanctions=20
and trade. Math classes should prompt intelligent chatter
re "dual use" technologies -- don't just leave this to=20
science and history teachers, who maybe haven't such a=20
strong grasp of the basic principles driving the show.
I think math teachers need to get their heads out of their
text books, look around, realize what a scary and puzzling
world this is to children and, as Keith Devlin puts it,
"make the invisible visible" -- let kids know what's going
on under the hood, from a mathematical point of view.
Tell us about the logic of it all, the reasoning driving the
policy making, the game theories involved, the simulations=20
and the assumptions behind the simulations -- and do this=20
now, please.
So that's why I'm advocating more "Java jive" (and the like)
in our math classes today. Visit any bookstore, and you'll=20
see an exploding number of titles about Java, Linux, Corba,=20
OOP, XML... don't make kids wait until college, or make them=20
surf, untutored, to decipher what all this means. =20
Math class is the place to fit all these puzzle pieces together=20
(regardless of whether specialized PhD mathematicians consider
this their business or not) -- AND to do lots of exercises, AND=20
to write programs AND to collect lots of data in real time.
Does any one else out there share this vision?
Kirby
****************************************************************************
* To post to the list: email mathedcc@archives.math.utk.edu *
* To unsubscribe, send mail to: majordomo@archives.math.utk.edu *
* In the mail message, enter ONLY the words: unsubscribe mathedcc *
* Words in the Subject: line are NOT processed! *
* Archives at http://archives.math.utk.edu/hypermail/mathedcc/ *
****************************************************************************