One of my most disappointing moments in life was when I'd spent weeks
tutoring a Vietnam Vet who realized after 4 years in college that he
didn't have the emotional temperament to be a teacher and decided he did
have the temperament/aptitude for Engineering. Uncle Sugar bought him a
nice big pile of books and a calculator and a tutor (me).
I fired him after several weeks of his not being able to get past the
idea that his brandy-new calculator was *right* and I was *wrong* when I
did decimal arithmetic (divide by 100, multiply by 10, etc.) in my head
and came up with different answers (my method didn't have roundoff
errors, go figure!) from those displayed on the brightly glowing LED
display of his $200 calculator. Sadly, I think he was like this
*before* he want off to the jungles of SE Asia to play kill or be killed
for 3 years. But the demands of the military machine, I don't think
improved his ability to think for himself, even given the stakes often
involved.
I spent my career developing tools and techniques for using computers to
aid our intuition, to help people understand the implications of their
data and their theories more viscerally, in the pursuit of improving
existing insight, gaining new insight and communicating that insight to
peers and public alike. I don't doubt that demonstrations (whether
pencil and paper, real-world physical setups, or computer simulations)
are key to helping people learn. I think computers can be magnificent
tools for leveraging and augmenting human cognition and understanding.
But you just can't help those who are insistent on being dumb about how
they use them.
Thomas Friedman wrote an article in the NYT titled "Dumb as we Wanna
Be"... about our blind self-destructive behaviour in the first 8 years
of this new millenium (and beyond that in both directions, but acutely
centered around that rather unfortunately acute period)... and I guess
I feel the same way about people who simply *won't* approach problem
solving intuitively.
I *do* correct clerks who punch the wrong numbers into their cash
registers and try to give me back more change than the original bill
tendered, but I suppose if it weren't for my Calvanist ethics, I'd
happily pocket the profits and cackle at the irony of it. Before the
industrial revolution, I'm not sure we had any choice BUT to do
*everything* with our own intuition and the extended intuition of the
work animals (hunting, herding dogs, draft animals, etc.) we lived in
intimate contact with and the tools we often built ourselves to provide
various forms of leverage.
Now, we live in intimate contact with mechanical tools usually not of
our own making, and often made by tools made by tools that maybe someone
made themselves. Even the pre-industrial technologists (craftsmen)
had deep and intimate, intuitive relations with their tools... the
smith's forge, anvil and hammer where her friends, the cobbler and her
needles and lasts, the weaver and his loom, his spinning wheel, his
combs. They lived, breathed and tasted their tools and materials.
Programmers today (well, some today such as the many of those here who
grew up barehanding machine code, implementing algorithms from first
principles, extending the limited paradigms offered (say... procedural
unto OO) to improve their intuition, their leverage) sometimes are as
intuitive as the craftsmen of yore. Like the modern labor saving
devices we can't imagine doing without, our computing tools are
outrageously leveraged, to the point that they can obscure our
intuition. How many of us use the compiler to find our errors rather
than exercising the tiny bit of extra care it takes to get the syntax
right the first time? How many of us don't just use debuggers and
memory management tools to help us code, rather than help us
troubleshoot the rare occasions when we make a misstep?
I still heat and cook on a woodstove. I occasionally take out my
two-man hand saw to cut down a tree, fire my forge with charcoal and
coal to heat iron to white hot and beat it into useful shapes, or turn
over my garden with a spade instead of my tractor... and sometimes I do
arithmetic in my head or longhand or pull out my old slipstick and
remind myself of the relations between logarithms and
multiplication/division. I owe myself a 10 mile walk into Santa Fe (20
by highway!) some day soon... just to be reminded it can be done. Up
hill both ways, in the snow!
Many might think this quaint or luddite or Calvanist... and maybe it is
all three... but for me, it is important to reground everything I do in
a visceral, intuitive experience now and again. Formalisms
(mathematical and computational) are incredibly leveraging and
utilitarian, just like the black-box control system controlling the
spark and fuel timing and mixtures into your car engine... just like the
automatic washing machine that fills, sudses, drains, rinses, spins at
the push of a button... But at some point, we need to get out and
ambulate far enough to realize the car is just "convenient leverage",
wash our own clothes by hand in a bucket or a river, do our own
arithmetic, conjure, implement and evaluate our own algorithms... or
else we will lose all connection with what we are doing, why we are
doing it.
I approached this talk on Math Education as a skeptic --I have always
thought that the idea of letting the computer do all the work sounds
great but is flawed. Of course, I don't like the idea of presenting
math in the schools as mainly rules of calculation, but I feared that
using calculators wouldn't be much better. If a student is asked to
find out how much it costs in all to buy a hammer for $23 and a pair
of pliers for $17 and if he hits the multiply button instead of the
add button will he realize that $391 is simply impossible? Most people
with a traditional math education will realize this immediately (I
hope) but if a person taught that numbers are nothing more than things
that come out of a calculator might not see any problem with an answer
of $391. The traditional approach does provide some intuition about
numbers. Even worse, the student who pushes the wrong button and gets
marked wrong, might see math as a boring subject where you have to be
so very careful about pushing the right buttons. Not unlike the
student who only learns meaningless rules of calculation who is bored
by the need to be so very careful about using the rules precisely.
I was somewhat relieved when I actually listened to the talk. He
actually said that mental arithmetic could be useful and he outlined a
bold approach to math that I would applaud. My fear remains that if
his program were adopted, the ambitious parts of it, which I like,
would only be given lip service, while the message would get through
to the schools that math is the same except without the drudgery.
On 11/28/10 2:06 AM, "Pieter Steenekamp" <piet...@randcontrols.co.za>
wrote:
I found the TED talk on math education at
http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html
very interesting.
In summary, this guy says that our math education is wrong. He
defines math broadly as the the process of (1) translating a
problem to a mathematical form; (2) deciding what result is
required mathematically; (3) doing the computation; and (4)
interpreting the result. His point is that math education focuses
on doing the computation by hand whilst that could be done very
easily by computer. He reckons math education should focus on the
points 1,2 & 4 and let the computer do the computation.
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
