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Re: Peter Hilton on Math Anxiety
Posted:
Mar 5, 2010 6:35 PM
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On Fri, Mar 5, 2010 at 1:00 PM, Dave L. Renfro <renfr1dl@cmich.edu> wrote:
>
<< snip >>
> The only quote I found on the internet about this is
> the following [1], which I think isn't from the essay itself
> but from his autobiography:
>
> "After years of finding mathematics easy, I finally reached
> integral calculus and came up against a barrier. I realized
> that this was as far as I could go, and to this day I have
> never successfully gone beyond it in any but the most
> superficial way."
>
<< snip >>
> Dave L. Renfro
I'll respond to this in part because Isaac Asimov was a
big influence in my life. I was one of those kids who
"fell for" his trick, which was to get us hooked on science
and math through science fiction. That's exactly what
happened in my case. From reading his novels I developed
a healthy appetite for his non-fiction writings.
One of Hansen's recent posts was about the impossibility
of ever getting more than 25% through a Dolciani type
pre-calc / calc program, no matter how good the teaching.
We're going to lose 75% due to whatever factors ("home
life", "lack of aptitude", "math anxiety"... a long list of
factors take the blame).
My question is why assume the precalc / calc pipeline
as the one and only. That's like going to a water slide
park and having only the one water slide (typically a
pipe), or going to a roller coaster park (like that one
near LA) and finding only the one roller coaster.
Surely mathematics is a more varied playing field, and
if we're going to admit up front that 75% won't make it
through calculus, then why can't we offer other rides?
In my 2008 talk at Pycon, I inveighed against what I call
"Calculus Mountain", which is precisely this killer hill
that is used specifically and by design to "weed out"
those who can't hack it.
http://www.youtube.com/watch?v=hbeHdg8mtdc
(almost 2,500 views -- more than most journal articles).
Those making it over this mountain feel proud of
themselves, glad to be gifted in whatever way, but is
this really all that great a design in the first place?
Many posters here have questioned this status quo
over the years. It's not really heresy.
Enter discrete math and its relatives ("digital math"
still on the back burner as not well defined, except
maybe by me on Wikieducator). If we had the budget
(not saying we do, given economics is not well under-
stood), we could easily design a track for "calculus
refugees" that sampled a lot of other connected topics
and (drum roll) actually prepared students with
technical skills for careers that (drum roll) don't involve
much if any calculus. These career paths actually
exist.
Sometimes calculus gets to be more motivated when
there's real physics involved, and I've seen past
postings to this archive where the view was: lets
leave "calculus" to the physics department and
concentrate on something "more pure" for the
"real mathematicians" (segue to "real analysis" at
this point). More than just feeding the snobbery
of "pure math" aficionados, one could see this is
a useful concrete suggestion: leave calculus to
the physics department at the high school level.
Those wishing a "pure math" approach will elect
to pursue this later.
We have a lot of history explaining how the calculus
managed to insert itself as the road hog and singular
"gateway discipline" it has become in today's
(manifestly broken) design. It simplifies things for
the education industry, to just have this one pipeline,
never mind the 75%+ attrition rate. A vast army of
calculus teachers gets steady employment. It's a
known territory, well explored, a status quo, a
comfortable regime, for those on the inside. I've
taught it myself for pay, another calculus mercenary.
Granted all that, is it high time for a coup? Shall we
keep the precalc / calc track as an option, but not
pretend it's the one true realistic ticket to a technology
career?
Should we stop treating calculus the way some
people treat Christianity, as the one true straight and
narrow, as in "my way or the highway"? Why give
it such monopolistic status? Complacency breeds
corruption, we all know, and here we have a manifestly
broken system that rarely questions the self-interest
of those most vested in its perpetuation -- a classic
case of this very syndrome? Who dares to protest
the hegemony of AP Calc indulgences (= less time
in purgatory if you kowtow to the high priests)?
The competing track or tracks, which may contain
more group and number theory, more about primes
and composites, more about polyhedra and vectors,
Euclid's Algorithm for the GCD, more about crypto-
graphy and cartography, more on hexadecimals,
unicode, spherical trig, GPS/GIS, simulations, more
computer programming... might also contain some
calculus, just nowhere nearly as much. Some
stellar documentaries would cover the basics. Lots
would get previewed. We believe in whetting the
appetite more than drilling or killing at this age.
We're more like Asimov in that way.
We'd even do some history, explain about the Newton-
Leibniz contention and how Bishop Berkeley, for
whom UC Berkeley was named, tried to blow
calculus out of the water because of its sloppy
proofs (in those days, pre lots of fancy formalisms).
Telling that story (among others):
http://mathforum.org/kb/message.jspa?messageID=6990127&tstart=0
Sharing history is in keeping with our more general
philosophy of sharing lore, telling stories, not keeping math
teaching bereft of an historical dimension, a hold-over
from divide-and-conquer management strategies that
are anti-liberal arts in their objective: to keep everyone
so narrowly specialized so they won't be in a position to
question or second guess their orders (keep the overview,
the view from a height, for an inner circle of generalists **).
Hansen was agreeing with me that Mathematics for
the Digital Age by Gary and Maria Litvin was a
worthwhile text. There's a lot to pack in to just one
year in that book, and by adding topics, branching out,
including more polyhedra, even more truly object oriented
programming, one could imagine fleshing that out to
two, three, even four years of material. Might we
serve the calculus refugees in this way?
This gets back to my AM versus DM track proposal.
AM = analog math = pre-calc / calc. DM = digital
math = still mostly under-developed, and therefore a
potential growth industry, could be huge. I think it
works as shorthand. The Computational Thinking
course they want on math-thinking-l, edu-sig and
places would fall under the DM heading. They'd even
like a Discrete Math AP test, although Gary Litvin
explains why that's not happening (doesn't address
the real problem e.g. that 75% attrition rate with
disproportionate performance by zip code area).
Kirby
** per 'Operating Manual for Spaceship Earth' (recommended
earlier, on my reading list at Princeton), the great pirate
managers instituted this divide-and-conquer strategy,
making everyone under them extremely narrow, and then
were themselves unable to follow the action when the
sciences took off into the invisible realms (radio, quantum
physics). The great pirates died out and now we're stuck
with everyone too narrow to see a big picture, so no one
in charge, everyone semi-paralyzed, perhaps blaming
some non-existent cabal of supposed insiders. Asimov
wrote a similar essay entitled 'View from a Height' which
made similar points, about how challenging it is to develop
big picture views, given the knowledge explosion that
engulfs us, a confusing morass of information if we don't
keep hammering on those heuristics.
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