I've been browsing the NCTM 2000 documentation, plus engaging in some threads re math curriculum on the misc.education newsgroup (where I posted an earlier draft of the mini- manifesto below).
I find the NCTM stuff necessarily dry, because it's written in deliberately toned-down language, is not supposed to give offense. The problem with such writing is it conceals ideologies as much as reveals them. Since the internet permits more open format debate and gives voice to individuals (vs. committees), I'm taking the liberty of turning up the volume a bit, making it clearer where I'm coming from with my proposed direction for at least some schools of thought doing the K-12 thing.
But that doesn't mean what I'm proposing is inconsistent or entirely out of synch with what the NCTM is proposing. Indeed, the principles of educating for participation in democracy and accepting technology as a background premise are very much at the heart of what I'm pushing as well.
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 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 as the sole property of specialized math professionals to make do with as they pleased, i.e. in a democratic culture, we're not to insist that students "track" themselves too early and 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 to concentrate on a spectrum of inter-related topics. So what do we include and what do we drop? We need to consider the world in which the students will actually live, today and in the future -- and not just the rarified world of an Ivory Towerite tenured math professor (always a tiny minority and not necessarily the cookie cutter we're using when doing our 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 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 repetitive tasks. Exercises establish the algorithms in our minds, give students insights into the nuts and bolts, but the programs then reiterate these computations way more times than we'd want to do in drills -- e.g. graph thousands and thousands of points whereas, by hand, we'd maybe settle for 10 and consider the skill established. So one thing computers provide (and this is obvious) is raw computing power. But to tap this power, you need to write in a language those computers will understand. What's shown in math books is two often three or four steps removed -- so better bridges need to be built.
Why is this important?
Because a lot of the math principles become a lot more accessible if (a) you know the algorithms which show them off and (b) you have the computing power to drive high powered displays of these algorithms at work, drawing from data sets much larger than you'd want to tackle with paper and pencil alone.
Of course fractals and strange attractors come to mind as examples of this approach. Indeed dynamical systems theory as a branch of mathematics owes a lot to computers for its very existence. I also think of STRUCK, where dynamical systems meets up with tensegrity via "elastic interval 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 fiction (most recently 'The Matrix'). A common theme in these works is human intelligence (humint) being superceded somehow by AI.
Stephen Hawking is one who believes that intelligence will somehow leap across the gap, from a less biochemical to a more electrometalic hardware, the better to probe more deeply 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 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 computers be probed, scoped out, disassembled, programmed -- approached from many directions -- in order to remove whatever shrouds them in mystery (e.g. ignorance and fear) and makes them targets for phobias and paranoias of various kinds ('The Net' and 'Enemy of the State' both play up high tech- nology as instumental to a heavy handed monitoring regime -- and I noticed the USA State Department was quick to reinforce those perceptions when suggesting such means were behind Turkey's nabbing of the Kurdish resistence movement leader (one of them)).
Sure, teachers in other classrooms will be expected to interpret these memes-patterns of pop culture to students with eager minds, wanting to know, but it makes sense that math teachers especially would be looked to for insights and role model attitudes. Math teachers should be able to carry on intell- igent conversations around such generic topics as (examples):
* Shall we fear and/or fight and/or embrace a computerized Big Brother?
* What is GIS/GPS and how do satellites both show me where I'm driving (when lost) and guide B2 bombs to their targets?
As long as both civilian and terrorist applications of the same technologies get funding, we'll need to instruct our students in the meaning of "dual use" -- an important concept in all of today's negotiations about peacekeeping, sanctions and trade. Math classes should prompt intelligent chatter re "dual use" technologies -- don't just leave this to science and history teachers, who maybe haven't such a 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 and the assumptions behind the simulations -- and do this 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 see an exploding number of titles about Java, Linux, Corba, OOP, XML... don't make kids wait until college, or make them surf, untutored, to decipher what all this means.
Math class is the place to fit all these puzzle pieces together (regardless of whether specialized PhD mathematicians consider this their business or not) -- AND to do lots of exercises, AND to write programs AND to collect lots of data in real time.