Re: Calculus texts

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

On Sat, 28 Dec 1996, Edwin R. Schweber wrote:

>     I think the problem applies to all H.S. math texts. There seems to
> be a
> tendency to assume that using pseudo-mathematical language is a
> substitute
> for explaining a concept. This semester, besides physics, I am teaching
> trig to some of our lowest level seniors. The text introduces the
> concept 
> of radian measure without any attempt at explaining why it is a more
> natural measure than degreees or without explaining why taking the
> ratio of arc length to the radius of a circle inscribed around the
> angle is independent of the circle's radius.

I thought that would be obvious! :-) It's indepedent of the circumference,
too. And the area...

Seriously, it's not obvious to students at first encounter, nor are a lot
of other things we take for granted. My wife tells of a math prof she had
in college who told her "If you can't see these things right away, without
effort, then you shouldn't be a math major." Indeed that was the
prevailing attitude of both math and physics profs in my experience. If
you have to sweat these 'trivial' things every time you encounter them,
you'll never have enough lifetimes to learn physics or math, they said.

So, granted that it's not obvious to many students; what do we say to them
to make it obvious? We must be careful not so say something they will
simply use as a memorized crutch, but something which will generate *in
them* the kind of thought habits and insights which might transfer to
other situations later. Too often we explain things so well that the
student never sweats the difficult process of learning how one learns
these things for oneself. There's an even earlier step: realizing that the
radian measure is independent of circle size, and then figuring out a
general proof of that theorem. I.e., *seeing* that there's a question to
be answered. 

I think it would be most instructive if folks here would post their own
answers to the proposition: "The radian measure of angle is independent of
the size of the circle." Then tell us how you'd get that across in an
insightful way to math-challenged students. Be careful what you assume as
premises. Be careful not to get into questions of whether the universe is
really Euclidean. Be careful not to generalize from a finite number of
specific cases. 

One problem today is that students are innocent of Euclidean geometry,
having not had a course in it, as we did back in the 50s. They are also
not accustomed to understanding proofs of general propositions, or
constructing their own. Back in high school we used to ask our teacher in
Euclidean geometry "Why do we have to do all these theorems and proofs?"
Our teachers replied "It teaches you to think." I hate to admit it, but I
think they were right after all. 

	-- Donald

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
Dr. Donald E. Simanek                            Office: 717-893-2079
Prof. of Physics                    Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745          CIS: 73147,2166 
Home page: http://www.lhup.edu/~dsimanek            FAX: 717-893-2047
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