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Re: MATHEMATICS AND LEARNING: POLYA
Posted:
Aug 22, 1994 5:07 PM


On Mon, 22 Aug 1994, Michael Keyton wrote:
> With regard to J. Conway's article about lamenting having to strip off > the fluff to get to the mathematics. Isn't this one of the strengths of > mathematics, and one of those traits that one would like to inculcate in > a student? I can not imagine anyone advocating a complete separation of > mathematics from worldly aspects, though this is the argument of Hardy, > was it not? The purity and sanctity of mathmatics, beauty in its self?
There is some truth about it, but everything is good in moderate quantities. We have no time to go and check every geographical fact with our own eyes; in most cases we have to believe geographers. Same with `real life' in every school subject: a small dose of `reality' is refreshing, more is too cumbersome. It is not a matter of `purity and sanctity' it is an inevitable consequence of our mortality: we are short of time, because in a few scores of years we have to die.
> Having learned most of my geometry from Wentworth & Smith, I remember > still many examples mostly of militaristic applications, that great > source of much of mathematics, and of agricultural applications, that > primary phaenomenon of rural America, and of sailing problems.
Reread this book and you will see that all the `militaristic' etc. applications are schematized and simplified; if you will ever have to plan a REAL military operation, it will be quite a different story.
> As I recall there were no applications until the late editions of their > geometry that incorporated examples from sports, whihc might have become > the defining element of modern America. > > I can not imagine teaching trigonometry without discussing at some point > indirect measurement and giving examples, sometimes with appropriate > equipment having students make the measurements in surveying, in > determining within measurement of error confines distances and heights. > Nor can I imagine teaching Diff. Eq. without discussing applications to > growthdecay, springs, reasonance, etc.
Certainly. But you will discuss idealized springs, not real ones.
> However, thank goodness one does not still have to include "fabricated > examples" such as "Mary is now twice as old as her sister Beth, prove > that if they are twins then ... " or however that problem goes.
Problems about Mary's age are as good as others.
> I can give many examples that I have used and seen used that have gone > awry because of cultural ignorance or linguistic lacunae. I will never > use the word "wench" again in a problem, I promise. I still use in > precaluclus an example of a piston, because it is a good problem that > most students make an initial incorrect hypothesis concerning the nature > of the defining curve. And I expect my students to know the dimensions of > a bsaeball field, and in the words of Robin Williams that "three up and
I have no idea about dimensions of a baseball field.
> three down" means half an inning, not a corporal or whatever. > > With more and more students having English as their nonprimary language, > I find "reality problems" to be more a hindrance when lack of time is > significant.
Lack of time is ALWAYS significant.
> (Isn't this a common malady?) But then I do wish for them to > know that most of the world, if not all, can be thought of > mathematically, from looking at music using groups and geometry, to > analyzing the structure of poetry in through form (see Bronowski and > Scott Buchanan). After all doesn't mathematics come from the greek word > for "thought." > > Enough, let me return to the green fields of paradise. > > MICHAEL KEYTON > St. Mark's School of Texas > Dallas



