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Topic: A vanishing breed...
Replies: 7   Last Post: Jan 5, 2001 7:59 AM

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RWW Taylor

Posts: 30
Registered: 12/6/04
A vanishing breed...
Posted: Dec 19, 2000 11:07 AM
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Jodi Cotten recently asked on this forum:

> Please give an example of what you consider to be a non-routine
problem.

The problem with sharing non-routine 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 non-routine (or recognizable as such), and some
non-transitory 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 non-routine problems is human sweat and
ingenuity. I've always been a believer in the value of non-routine
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 pre-calculator 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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