SETTING THE STAGE
=================

I take some credit for sparking a lot of debate on
math-thinking-l this month. If you check the archive,
you'll see that's been a pretty quiet list for awhile.
We have something to talk about now: the move
to fold computer science into the regular math track
at the high school level.

The politics are as follows: students don't have a
lot of time for an elective computer science course
that doesn't fulfill basic requirements. In an
increasingly technological world, people are finally
starting to get wise to the fact that there's simply
no excuse for only using scientific calculators,
keeping real computers at bay, relegated to some
other course. Math teachers are starting to see
that they might want to including some computer
programming as a part of the mix.

A lot of us thought this was happening with the
first PC revolution. Logo and BASIC were big back
then and it looked like mathematics would be using
those. This was not to be in most schools however.
I was working at McGraw-Hill at the time, on
computer literacy materials, mostly for the junior
high age group and above.

Then came the second revolution, the open source
revolution, which made pretty much all the software
they'd need free of charge, even though schools
could still budget for commercial packages if they
wanted. Surely this relief from software costs would
motivate a major shift? Yes, in some schools it has,
but North America may not be the leader of this
trend, except in pockets like Portland perhaps, an
open source capital according to Christian Science
Monitor (2005). Linus Torvalds lives here, as does
Ward Cunningham, the inventor of the Wiki, etc.

LINK TO PHILOSOPHY
=================

So where does Wittgenstein and philosophy enter
this picture?

Imagine different subcultures practicing computer
skills in various walks of life, all wanting to have
some kind of footprint at the high school level.

In part it's a recruiting issue. That's why the Army
is always sniffing around the schools, looking for
kids who might want to join. College departments
need a next generation as well.

Speaking of which, my daughter is getting all these
mails from the colleges, even though she's only a
sophomore.

My gut reaction is why would we give these institutions
a lot of money when they're not sharing our heritage,
are bleeping over tetrahedral mensuration and all
that goes with it, which adds up to "another tomorrow"
(different from endless Orwellian war on terror).

http://controlroom.blogspot.com/2010/02/necklace-theory.html

Colleges don't teach my culture, and yet expect
me to pay them... looks ugly. What might change
in a year or two?

Anyway, the functional programming group is really
anxious to cash in on Computational Math or
whatever we call it. Army Math? Think of those
recruiting commercials showing young people
finally getting to learn with technology -- except
not bait and switch this time, not just using
enticing eye candy. Are we talking about a more
professional army then?

This new course, being brainstormed in Oregon
among other states (I presume other states are
paying attention), will give you a year of math
credit and yet will allow you, in my vision at least,
to learn about Google Earth, programming, SQL
(structured query language), Unicode, von Neumann
architecture, TCP/IP (watch 'Warriors of the Net')
along with a lot of math topics such as number
types, vectors, polynomials, polyhedra... all that
good stuff.

I've been running some pilots, writing them up,
posted some history, partly for for Maria's benefit
(she was asking if we had results on the ground):

http://mail.geneseo.edu/pipermail/math-thinking-l/2010-February/000543.html

My computer skills tribe is more object oriented
(OO) than functional (FP) and this is somewhat
disturbing of the equanimity in this particular archive,
as the FP camp is resolute in casting the OO people
as mathematically naive, full of mumbo jumbo, and
to blame for most of the bad software that's out
there (indeed, most of it is OO based I think that's
safe to say, once you count all that _javascript_ in
everyone's web browser).

My thread with Rex distills the issue. I want to explicate
the meaning of the capital greek letter Sigma, used
as a summation symbol in math, by putting some
computer code side by side.

The idea is: a Greek letter notation all by itself
may be intimidating, but if you see a simple
translation into running code, you'll be able to
go back and forth, like seeing in stereo. Both
notations (the greek one and the machine one)
become more readable in light of the other.

So far so good. But then I propose to use
something like this on the computer side, which
looks pretty straight forward. This is actual
Python code and the range function returns
integers between start and stop non-inclusive
of the upper bound, which is why the added 1
(to keep the stop value in the picture).

def sigma (function, start, stop):
total = 0
for index in range ( start, stop + 1):
total = total + function (index)
return total

This would correspond to

b

SIGMA [ f ( i ) ]

i = a

where f is some function and SIGMA is that
capital greek letter.

What the functional programmers tell me is that
for-loops of this kind, which feature a "mutable
variable" (one that changes), actually requires
very deep mathematics beyond the capacity of
anyone in high school to understand. You already
need a college degree, probably in mathematics,
to understand what's going on under the hood in
the above looping structure. Therefore using
for-loops of this kind should be inadmissible at
the high school level.

http://mail.geneseo.edu/pipermail/math-thinking-l/2010-February/000615.html

To me, this comes across like an absentee landlord,
in some high tower someplace, telling me I can't
share heritage with my students because I don't
"own the math" whereas the absentee landlord
understands it very well (is the owner-controller).

By monopolizing authority over what "math" is
(controlling that turf), the functional programmers plan
to shove aside the object oriented programmers.
Interesting strategy. Will it work?

They will not have such an easy time finding a toe
hold or footprint. Their languages will be discounted
as too "anti-mathematical".

http://mail.geneseo.edu/pipermail/math-thinking-l/2010-February/000553.html

To my ears, saying the math is "too deep" around
for-loops is like saying 1 + 1 = 2 is "too deep" for
high schoolers too, because they haven't understood
Principia Mathematica yet.

The idea of math being part of our ordinary language
and common heritage, not beholden to high tower
(ivory tower) formalists and their "rigor" (mortis) is
the idea Wittgenstein champions. He brings
mathematics back to its anthropological roots, where
it weaves seamlessly with the rest of culture, as a
socio-economic activity, a form of collaboration, a
set of rule-following language games (redundant
but why not spell it out) that bear a family resemblance
to one another.

http://mail.geneseo.edu/pipermail/math-thinking-l/2010-February/000592.html

CONCLUSIONS
============

This is actually a somewhat important debate with lots at
stake, in terms of how we design the next generation of
high school math course.

What irks me is how academic philosophers have let themselves
off the hook when it comes serving in an air traffic controller
role. We have issues regarding the philosophy here, and a
need for decision-making. I think philosophers should serve
more like referees, especially those trained in Wittgenstein's
methods of investigation.

In the debate above, I'm obviously in a biased role, am more
a player on the field than a referee. I post back to this
archive to leave some trace of what it looked like, in the
midst of wars....

Kirby
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