However, in this note, I'd like to discuss a mathematical issue that
comes to mind in considering the problem first posed.
I've forgotten (if it was stated) what math course the problem was
supporting. But, in this day and age, I would hope that anyone
teaching linear algebra would try to impart a little "cultural
background" on the likelihood of the applicability of the mathematical
techniques. In particular, I think any algebra-ician who tries
to extrapolate using linear approximations needs to attach a strong dose
of realism to the interpretation of the extrapolation!
How good is the linear approximation over the range
of available historical data? How large is the historical span compared
with the extrapolation? How nonlinear is the underlying system?
Is there any reason to suppose that a linear extrapolation would
have some validity? What might be a reasonable expectation for the
error? How can one get a quantitative handle on the likely error?
Persons who are learning linear algebra (and especially those who
will not go farther in math) need to develop some "wisdom" to
be able to avoid making big blunders in applying the methods to
areas where they might fail: particularly economics, sociology,
and other highly nonlinear systems applications. Similarly, in
constructing linear algebra problems and courses, I think it's
important that teachers employ similar common sense and
self-discipline.
I realize this is a difficult goal, since students
who won't go farther in math may already be stretching just to learn
the linear algebra. Further, the course is probably already
over-filled with linear algebra techniques alone! Still, I think it
needs some careful consideration and attention.
Here are a few ideas on how to construct a set of problems that might
illustrate the mathematical issues involved. Choose a physical system
that behaves nonlinearly: e.g., beaker(s) of hot
liquid cooling by conduction in a vat of water, or a pair of capacitors
in separate circuits discharging through their "bleeder" resistors.
Or maybe the periodic motion of a pendulum or a mass on a spring.
Or, if you want to be really "in" with current fads, choose a simple
formula that displays chaotic behavior. Whatever nonlinear system you
take, choose two sets of parameters, one set where linear extrapolation
works, and another set where it fails miserably. Give the students
some first hand exposure to success and failure through (in)applicability!
On Sat, 14 Sep 1996 07:38:39 EDT Phil Mahler wrote:
...
>The purpose of a mathematics exam is to test mathematics. Mathematics
educators
>need to provide the appropriate environment to accomplish this. Therefore,
>questions which are very likely to present strong, negative, outside
influences
>should be avoided whenever possible.
>
>Conclusion: The question presented should be rephrased, as already suggested
on
>this list, to deal with a more neutral topic.
>
...
>Philip Mahler
>Middlesex CC
>Bedford, Massachusetts
>
Regards,
William C. Mead
wcm@ansr.com
Visit "Adaptive Network Solutions Research, Inc." on the web
at http://www.ansr.com !