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*From*: Leigh Palmer <palmer@sfu.ca>*Date*: Mon, 24 Feb 1997 20:35:52 -0800

Here's the piece I threatened you all with. I posted it to Physhare, which

promptly croaked. Trusting that it was not the lethal agent, I offer this

to you all. The material is, perhaps, less suited to this group than to

Physhare, but I pose a numerical problem at the end which is quite relevant

to the group.

Donald Simanek (who is always worth listening to) observes [on Physhare]:

What ever happened to Piaget? He documented how minds work through stages

of conceptual development, reaching the formal operational abstract level

last. Have we forgotten the lessons learned from his (and others')

research into how we learn? What you say above is right in line with

Piaget's observations, but, curiously, one seldom hears Piaget's name

mentioned in these discussions these days.

I noted that, too, and since the things that Piaget observed and advocated

always tended to be congruent with my own feelings I felt he had succeeded

in his analysis of cognitive development, at least on a phenomenological

level. A natural history of learning was well synthesized by Piaget.

I feel that somehow the credibility of the field of educational philosophy

(or at least the credibility of its contemporary practitioners) is greatly

diminished by their eveident disregard of the work of their antecedents.

Modern educationists seem not to feel the need of the shoulders of others

to stand on, and that suggests to me that they are still down there

grovelling in the primordial mud. It is sad that one hears the names of

Foucault and Derida from those folks more often than one hears of Piaget.

No one makes progress in physics by ignoring the works of Galileo, Newton,

Einstein, Maxwell... you get the idea.

And now for something completely different!

The Fermi question thread seems to have unravelled to a large numbers

strand. I'd like to contribute a large numbers example problem I use in

my statistical physics course. It is not original; it comes from

scientific American a long time back, likely from Martin Gardner's

excellent "Mathematical Games" column, which some of the younger readers

in this list may not recall. If anyone can find a citation for this I

would be grateful.

I call this (inappropriately) "The Rubber Worm Problem": You are provided

with an elastic rope exactly one kilometer in length. At time t=0 a worm

starts to crawl from one end of the rope to the other at a speed of one

centimeter per second. The rope itself is attached to a tractor which

is stretching the rope at a rate of one kilometer per second. The problem

as it was originally stated specifies that the rope lengthens by one

kilometer at the end of each second, and it is this discrete version of

the problem I will continue with here. At the end of one second the worm

has marched one centimeter toward the other end of the rope. At that

instant he is instantaneously transported to the 2 centimeter mark on a

rope which is now two kilometers in length. By calculation you will see

that at the end of the second second he will be three centimeters along

his way on the two kilometer rope, which will then lengthen to three

kilometers, taking the worm to the 4.5 cm point, etc.

First question: Has our worm embarked on a Sisyphusian journey; will he

ever reach the end of his rope? If not, prove that.

Evidently the answer to the first question is "No" as the sophisticate

will recognize. Otherwise, what's the point of the question? The second

question is: How long will it take the worm to reach the end of his rope?

Now I expect the students to approach the discrete problem by making some

computationally less challenging continuous model like the one suggested

above, and to estimate the magnitude of the error introduced by doing so.

This is a third year university course, so of course the problem is quite

challenging. The solution, which several students in the class always

succeed in finding, has two important points relevant to the interpretation

of statistical mechanical calculations, one regarding large numbers and

the other regarding the units in which quantities are measured. Statistical

mechanics as it is taught in Fred Reif's excellent textbook (and in many

others as well) plays fast and loose with the stylistic matter of using the

log function on arguments which are quite evidently not pure numbers, and

this problem is a good physical justification for such mathematical heresy.

Have fun with the problem. I guess now that I've posed it I'm going to have

to work it out again myself - later - unless one of you posts an elegant

solution. I realize that "High school physics" appears in the name of this

group, but I think that the answer to this question alone, being as it is

such a large number, may be of some interest. To many high school teachers,

in my experience, a challenging problem (which is not really a physics

problem, but rather a mathematical puzzle) is welcome exercise. If good for

nothing else, it can be used to stimulate that bright student who is always

looking for a new challenge.

Donald is almost as cynical as I about the deteriorating state of the

ability of students to use the tools of mathematics and formal logic. I

will leave you with one related thought which should send a shiver up the

spine of any aging professor who teaches premeds:: do you know what the guy

who finishes last in his class at medical school is called?

"Doctor."

Leigh

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