From: fateman@peoplesparc.Berkeley.EDU (Richard Fateman)
Newsgroups: sci.math.symbolic
Subject: Re: symbolic algebra and teaching math/sciences
Date: 5 Oct 1992 22:06:38 GMT
Organization: University of California at Berkeley
NNTP-Posting-Host: peoplesparc.berkeley.edu

There are undoubtedly good reasons to show people how to use computer
algebra systems, so they can use them to solve problems they later
encounter.  It is not obvious that the complexity and unreliability
of such a system contributes to the learning of (say) calculus.

I recently posted this note to "mathgroup" -- Mathematica users,
but it seems to be relevant to this group as well.
......

In the last few years there have been a number of texts and proposals
for combining computing and calculus.  There have been a number of
inquiries about how to best combine Mathematica (or Maple or ...) with
such courses, or with associated labs. 

It is sometimes hard for instructors to remember what is difficult or easy
from the students' point of view.  Here are some opinions.


NEGATIVES:

If you recall how much trouble you had on epsilon-delta proofs, assuming
they are still taught,  I doubt that current computer systems would help.

If you recall how much trouble you had (as a student) understanding
the meaning of the tall skinny S and the "d" in integral(f(x) dx), or
the x in f:x ->y, I doubt that current computer systems would help.

If you recall how rushed you were in covering the material, adding
new material on computer systems probably would not help.

Once you have taught students enough to understand they have to type
(in Mathematica syntax...)

f[x_]:=Sin[x],

some students will embarrass you with questions like
"Wouldn't  f=Sin@#&  or just f=Sin  be better?"   (Do you even know the answer?
Do you care? Is this relevant to calculus  [actually, it may be relevant
to "lambda calculus"  but you aren't teaching that, are you?])


POSITIVES:


Graphing math functions is very useful. Many programs can do graphing.
You can have students buy a graphing calculator, perhaps for less
money than the textbook.

Numerical integration ("quadrature") is also useful. The calculators
can do that too.  As well as solving for the intersection of curves.
These can help if you want to have a more numeric flavor to your
course.

If you really want to force students to learn to type and use computers
so they can pass calculus, then you are helping them learn life skills
that they may find more useful than epsilon-delta. 

THE BIG WIN, however, (in my opinion) is that you can ask your department
chair to set up a lab with NEAT TOYS and advertise that your section of
calculus will use SPANKING NEW COLOR COMPUTERS, and the students
will be ALL EXCITED and spend ALL THEIR SPARE TIME fiddling with the
lab equipment, and have LOTS O' FUN.   You will gain credit for
innovative teaching, and maybe be popular with the kids, who never
really wanted to learn calculus anyway.

Other comments:

Some people seem to think that because you can do some symbolic
(indefinite) integration that these programs would be especially
useful to courses that teach humans how to do symbolic indefinite
integration.  This is not at all obvious to me.  The primary use of
systems should be in advanced courses where the need for integration
(as well as ugly amounts of algebra) comes up.  We have such a course
at Berkeley (scientific computation for engineering) but it unfortunately
is still 100% numeric (using Fortran and Matlab).

I am not particularly defending the current calculus curriculum,
by the way.  But the calculus reform movement has its own large
literature.





-- 
Richard J. Fateman
fateman@cs.berkeley.edu   510 642-1879