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Re: sketching, qualitative reasoning



At 10:45 AM 5/30/99 -0400, Peter Schoch wrote:

It's a Pre-CalcII question, almost straight out of a textbook
on math -- y'know, the ones that are notorious for trying to write
"application" problems, but are always atrocious at it!

Yeah.

I like the revisions that hve been mentioned, except for the continual
references to "sketching". I presume that everyone here could take 10
minutes and perform the necessary calculations in a spreadsheet, and use
it to graph it as well.

I agree that computers have changed our lives, including the way we teach
and learn. I agree that there are many advantages to firing up a
spreadsheet when trying to figure out how a given function behaves. I
agree that sketching should not be over-emphasized.....

.... But on the other hand ......

We should not under-emphasize sketching in particular, and qualitative
reasoning in general. Feynman practically defined "understanding" an
equation as being able to describe the qualitative properties of the
solution, without actually grinding out the solution in detail.

Qualitative reasoning should not be taken as a euphemism for doing a sloppy
job on an intrinsically quantitative question. Instead, it is the skill of
giving an appropriately qualitative answer to a qualitative question.

I consider myself fortunate to have been given an unusually good training
in qualitative reasoning. The professors weren't at all subtle about it.
I vividly remember one problem from Leighton-Vogt: Given a bridge with a
lot of triangular trusswork, the question is, which girders could be
replaced by cables? There are two ways to answer the question: you could
spend hours quantitatively calculating the direction and magnitude of the
forces on every node (perhaps using AutoCAD), or you could spend about 5
minutes applying the principle of virtual work.

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This might be a good topic for this list: what do *YOU* do to teach your
students qualitative reasoning, scaling laws, dimensional analysis, et
cetera? Do you have any good examples? General principles?

Cheers --- jsd