On Mon, Sep 26, 2011 at 4:38 PM, Haim <email@example.com> wrote:
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> Kirby, I mean no disrespect, but whatever value there > may be to your "digital math", it cannot substitute for > the calculus which is rightly viewed as one of the great > achievements of Western Civilization. >
Many of your concerns seem somewhat tangential to mine, or shall we say complementary. In any case, correct, the DM track does not replace the AM track (digital does not replace analog math). You still have the problem of getting through on the AM track, if you want certain career advantages, no doubt you're right.
I'm just saying, STEM is a big place and there's plenty of room for DM track students wanting a future. And in the real world it's not either / or. We already have a continuous and a discontinuous metaphysics, tracing back to Euclid and Democritus respectively (one could say).
I know Hansen doesn't think philosophy matters, but I think it does in this case. If the universe is ultimately quantum, then real numbers don't exist, so we should have a curriculum that doesn't need them. I know that sounds crazy to some ears, but there ya go, it's a cultural divide of longstanding, already here when I got here. "We didn't start the fire..." etc.
A bridging figure might be Knuth. He's really into discrete math but isn't shy about the calculus. He thinks so-called O-notation, prevalent in computer science, used to measure the relative efficiencies of algorithms in general terms (e.g. "polynomial time" vs. "logarithmic time" kinds of curves) might rear its Gamera-like head in K-16 more consistently.
There's some calculus involved. Integration and like that. Good to learn about. AM is a great track.
> Jerry Uhl was one such person, and my respect for > him is boundless. But, did he change the nature of > the calculus curriculum in any large and important > way? I am guessing "no", for two reasons. > > First, had Uhl succeeded, I think we would have > heard a lot more about him and his program. > Second, there is the basic question of what is > fundamentally wrong with the calculus curriculum. >
Looking at history, a lot of good deeds and bold ideas get insufficient press at the time. People just don't have the context for it. Some little-known thinker published in Japanese about the possibility of buckminsterfullerene, I think it was. But it took the actual isolation of the molecule, followed by the buckytube explosion (nanotubes) to really make hexagon-pentagon cages (hexapents) a household concept in STEM. The Japanese study remains obscure.
Hard to guess in advance, another's legacy or one's own. We just don't have that luxury, of being able to judge our own time. We should look more at who won the Monkey Trial or something, as at least there we have perspective.
> I predicted fifteen years ago that whatever good Jerry Uhl > and his Calculus Reform Movement colleagues might > achieve---and I think they have achieve much good---they > would not be able to solve the motivating problem of the > Reform Movement, i.e., they would have little or no > impact on the attrition rate. And my thinking on this > question is exceedingly simple (I am an exceedingly > simply guy). > > There never was anything wrong with the calculus or > with how it is generally taught. For complicated > historical reasons, more and more students start the > calculus curriculum less and less well prepared. That's > all. So, to solve the problem of calculus attrition, you > have to solve the problem of the mathematical > preparation of the calculus students. >
I have not met Uhl personally. I do know more of Scott Gray's story, plus he's blogged about it here:
Basically: what was passing for "knowing calculus" in the USA didn't cut it in Russia, and this was apparently (by his analysis) because the requirement for oral exams elicited a different quality of comprehension in students.
His school in the States, where he was teaching, didn't have those.
To make a long story short, he came back to the US with a hope of instituting some reforms.
His particular program is not highly publicized but then education philosophy is an esoteric subject and relatively few will ever read math-teach for the same reason.
> Uhl and his colleagues took the attitude that they have > to teach their students as they find them. That is a > noble attitude, but wholly insufficient for the task at > hand. I think they would have done much better to > bring unbearable pressure, on the people responsible, > to produce better prepared students. Instead, by > their efforts to reform the calculus, they have > allowed the problem to fester. >
Scott did some other things after instituting oral exams. Distance education was just cranking up, and that's when he got more involved with Mathematica and Uhl's experiments. However, like I said above, I'm not an expert in this regard. I'm teaching on the DM side of the fence, doing mechanized logic, with exercises about adding coconuts to inventory, feeding all the animals in a relational database, painting GUIs on the screen. Lots of boolean algebra, finite sets, no real numbers.
> To put it another way, attrition in the calculus > curriculum is a political problem, not a > pedagogical problem, and I cannot imagine > another group of people more poorly suited to > a political fight than college math professors. Sad > to say, the Calculus Reform Movement is one of > the great missed opportunities in modern education > history. > > Haim > Shovel ready? What shovel ready? >
You may find it easier to get back to calculus via some much longer detour into a host of other math topics, say in number and group theory.
It's not like we're dealing with a strictly linear terrain.
If you don't want to climb calculus mountain right away, fine. Or take the cable car to the top, tour some spots, before deciding if you want to take her from the ground up.
That's the kind of stuff we might say along the DM track. We're not "anti calculus". However, we do see ourselves as having a responsibility to justify a discrete math approach, and that does involve tapping in to some of these western civ tensions, namely the difference between perfect continua (Euclidean planes) and non-continuous models of res extensa (common in physics, with its notion of atoms, energy quanta of various flavors).
Statistics is a another good example of a branch where differential equations may be used (stochastics, thermodynamics) and yet the underlying phenomena are adjudged to have discrete properties. We talk about the "flow" of people through an airport, but then people have their individual trajectories, also of interest.
You'll find a lot of the DM track topics in departments that talk about "artificial life", by which they mean something like Conway's Game of Life in the sense of rule based growth patterns with unanticipated (by closed form equations) outcomes.
You mentioned Economics above, as if economics had no calculus, but then of course it may have, depending on whether a fluid mechanics metaphor is being used (velocity of money etc.). Or am I mistaken about that?