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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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