Re: Pedagogical uses of a CAS with high school or undergraduate students


"Raymond Toy" <raymond.toy@xxxxxxxxxxxx> wrote in message
news:sxdveyvp9or.fsf@xxxxxxxxxxxxxxxxxx
>
> Perhaps students are different today, but I don't think knowing a good
> CAS (or any CAS) when I was in high school would have made a single
> bit of difference to me. (Well, homework would have been easier, but
> if a CAS were available, homework questions would have been quite a
> different.)
>
> It's not a replacement for knowing how to do things.
>
> Ray

Ray,

It's a way to get to know how to do things.

Your experience is not my experience. When I went to MIT in the late 1950's
there were many times I thought to myself that I could understand things
better if only I could do more calculations and examples. A good CAS would
have made a big difference to me, especially if I had known the basic CAS
usage before I got to MIT. Now that I am retired and have Mathematica I am
trying to learn some modern math and physics. I am especially trying to
learn some differential geometry, general relativity and differential forms.
I would like to explain how I do this because I think the approach would be
useful to other people, including high school students.

First, I try to do everything by calculation, including symbolic
calculation. I don't do any calculations or derivations by 'word processing
mode' and only a little (messy) squibbling with pen and paper. This means I
have to write a fair number of definitions and rules so I can implement the
calculations. But doing this is a test of my understanding of the
principles. It's difficult to write routines to do things if you don't know
how the things work. Often I learn how they work by the challenge of writing
the routines. When I finish I have a bonus. I actually have a collection of
useful routines that I can use to push further ahead.

Next, I try to write 'tutorial' notebooks. This is, of course, mostly for my
own benefit, but by making believe I am writing them for others I have to
clarify the material in my own mind. It's difficult to simply explain
something if it is not clear in your own mind. (That's why it's a good rule
to get doctors to explain what their doing if you can!) I use plenty of text
cells to give explanatory material - that's where the word processing is.
And I also try to use graphics and animation where it is useful. Many time
when I am doing derivations and calculations I put a number of steps in one
cell and intersperse the steps with Print statements that annotate and
explain what is being done at each step. Of course, it often takes me some
time to get these the way I want. (Would that qualify as knowing how to do
things?)

This whole paradigm uses the old standard 'textbook' or 'research paper'
style in that it mixes textual explanation with calculations and graphics.
But it is much better because the calculations are active and we can add
animation to the regular graphics. With this mode the student can do
whatever he wants. He does not have to learn some new interface restricted
to a limited type of problem, where perhaps he has a limited choice of
active buttons to push. It is the oldest, most general learning format we
have - only improved.

There is no reason that this paradigm can't be used at any level from high
school to advanced research. It's thousands of years old - and so is
learning by teaching.

So, as I said, for students who are interested and self-motivated, this is
the path.

I also think young students should learn a general purpose CAS that they can
carry forward, and not some special program designed for high school. They
certainly will not learn the entire CAS, but then nobody ever learns the
entire CAS.

David Park
djmp@xxxxxxxxxxxxx
http://home.earthlink.net/~djmp/







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