Date: Sep 26, 2011 11:20 PM
Author: kirby urner
Subject: Re: Continuing Education for Math Teachers (gnu math)
On Mon, Sep 26, 2011 at 4:38 PM, Haim <firstname.lastname@example.org> wrote:
<< snip >>
> 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
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
> 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
> 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
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
Should we claim Economics for the DM side of the fence?