On Fri, Mar 5, 2010 at 1:00 PM, Dave L. Renfro <firstname.lastname@example.org> wrote: >
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> The only quote I found on the internet about this is > the following , 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." >
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> 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.
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).
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).
** 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.