In article <[EMAIL PROTECTED]>, <[EMAIL PROTECTED]> wrote:
>It isn't only in math that this situation exists.
>In that final years of my teaching science I was constantly faced with
>administrators who wanted the "hands-on-approach" used to teach science.
>I watched as teachers struggled to try to have students come up with the law
>of levers, (usually they guessed the correct simple relationship of two
>weights on sides of the fulcrum but it took large amounts of time), the
>inclinded plane (using a rolling car down an incline, with undiscovered and
>unknown friction being involved in each trial), the law of floating objects,
>etc. Some, very few and I mean VERY few students, ever 'discovered' the
>rules for themselves. All that happened was that students became bored,
>frustrated, turned-off to science, and generally confused. Of course, it did
>give them a good amount of time to talk with others (social interaction),
>fool around (leisure activity utilization), and generally allow the brighter
>or more industrious students to do the work (apportionment of talent).
>I guess I just didn't have Bohr, Edison, Heisenburg, Ampere, Volta, Curie,
>Galileo, or Einstein in any of my classes.
They had to be at the right place at the right time, and still be
lucky. Also, they did not try to get results in a class period or
two. Sometimes, they discovered something by a chance observation,
but it is still the prepared mind which can do it.
Galileo questioned the conclusions which had been obtained
from Aristotle's philosophizing. He probably rolled balls
down an inclined plane to slow the acceleration to the point
that he could observe the time.
But one of Galileo's observations shows how easy it is to miss
the obvious; that is the essential constancy of the period of
a pendulum with respect to its arc. Now swinging lamps were
used for a long time before Galileo; why did nobody find it
earlier, or at least nobody who made it public knowledge?
It can be easy to understand what has been learned, but not to
discover it; Euclid would have relished the Peano Postulates,
first found by Dedekind, but even leaving out the Middle Ages,
centuries were required before anyone hit on the simple idea.
Even the use of "algebraic" notation escaped the people who would
have done much more if they had it. It was a half millennium after
Archimedes and Euclid before Diophantus introduced the idea of *a*
symbol for an unknown quantity, and not until roughly 1600 that
Viete came up with multiple symbols. Mathematics took off shortly
after that. Now how is a child going to come up with that?
>Having to 'rediscover' everything that has been discovered before, seems to
>be quite a waste of time. I'm not sure that the 'discovery' of the theory of
>Pythagoras, makes it any more meaningful, more easily remembered or useful.
>I tend to think that the only triangle that most individuals remember when
>doing this kind of 'find it for yourself' activity is the 3-4-5 right
>triangle. Those taught in the 'traditional' way, come to understand that
>3-4-5 is only 1 of a series of this type of triangle.
The 3-4-5 right triangle was known long before anyone knew how
to prove that it was exactly a right triangle. This had to
wait for the axiomatic approach to geometry, which reduced
thousands of observations, all made with some error, to logical
development from a small number of axioms.
There is a saying,
Any fool can learn from his own experience,
The wise man learns from the experience of others.
Too many students show that the first line is not correct.
But the second is very important, and is the basis of
the setting up of an educational program.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
