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Looking for a new soapbox

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?