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A vanishing breed...
Posted:
Dec 19, 2000 11:07 AM


Jodi Cotten recently asked on this forum:
> Please give an example of what you consider to be a nonroutine problem.
The problem with sharing nonroutine problems is that they tend, with distribution, to start moving toward the routine! What you really need, in order to get a handle on this concept, is some characterization of what makes a problem nonroutine (or recognizable as such), and some nontransitory source of inspiration for generating more, as the ones you put into use start getting "tired" (which is not to say that they necessarily become bad problems, just that they lose that fresh edge for the poser and the solver alike).
The ultimate source of nonroutine problems is human sweat and ingenuity. I've always been a believer in the value of nonroutine problems, and have worked hard over the years to come up with new and different problems to include on tests and homework assignments. Of course, this is all relevant to the level of the subject matter, too. I've been fortunate to see in the calculator/computer revolution, in the course of which (logic dictates) all practicing and aspiring mathematicians have needed to change their computational ways, and during which our practical computational grasp has been hugely extended. The fact that not everyone has been willing to move in the indicated direction (including many textbook authors) has made it easy to puzzle together problems that are now easy at the introductory college level but which would have challenged the computational abilities of experts in the precalculator days, and which are also not yet to be found in standard textbooks.
Let me cite just one example of the sort of problem that became feasible (and posable) with appropriate calculator use. This is to show that the triangle with vertices (396,0), (0,297), and (219.03,4.96) is a right triangle. Setting these pairs up as lists on a standard graphing calculator and showing that the sums of the squares of their differences add up (satisfy the Pythagorean relationship) is the work of only a minute or two. Of course, these numbers were just thrown together on the spot here  with a little searching a more attractive and less forbidding (but still interesting) set of number pairs could easily be located. And of course we could ask the same sort of problem with pairs like (9,6), (0,6) and (16,6) where hand computation is feasible. Does a problem change its _nature_ when the calculations involved are drastically simplified or are, in contrast, put out of reach of accustomed computational practice?
RWW Taylor National Technical Institute for the Deaf Rochester Institute of Technology Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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