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


David Park wrote:
> "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/

It sounds like a lot of work for one problem. Could be useful as a mini
project or one big question at the end of an assignment ***. However,
what you describe doesn't seem like it would be good for a drill
***.

.