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From: rouben@math13.math.umbc.edu
Newsgroups: sci.math,comp.sys.handhelds
Subject: The use of calculators in teaching calculus
Keywords: Calculators, calculus
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Date: 4 Dec 90 05:05:57 GMT
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Organization: Mathematics Department University of Maryland, Baltimore County
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Here are a few thoughts and ideas on the role of calculators 
and computers in teaching freshman calculus.  I am interested
to find out if there are others who share these thought, or if
there are some who disagree with me.  Comments from both 
teachers and students of calculus are welcome.  

--
I have thought mathematics at several universities for the past fifteen
years.  It's no secret that there is a great deal of dissatisfaction among
the mathematics faculty of the American colleges and universities with the
traditional approach to teaching freshman calculus.  "Calculus reform" has
become a very trendy topic among the educators.  The National Science
Foundation has a special program with emphasis on developing new approaches
to teaching calculus.  Innovative instructional ideas have begun to emerge.
It seems certain that the future calculus syllabus will be substantially more
applications-oriented and will rely less on rote learning and drill-type
problems.  For the better or worse, the students may not be required to
remember the formula for the anti-derivative of 'sec x tan x'  but they will
be expected to set up solve, and interpret the equations of motion of a
planet around a sun.  (A full circle back to the roots of calculus!)

Certainly calculators and computers will play significantly more prominent
role in the teaching, learning, and use of calculus than what they have
today.  At one end of the spectrum of the advocates of the use of
"technology" in the classroom are the more conservative types who are content
with sprinkling the traditional calculus textbooks and courses with a
moderate amount of "calculator problems," e.g., problems dealing with
computing roots with Newton's method or approximating integrals with Riemann
sums.  At the other end of the spectrum are those who almost throw out the
baby with the bathwater and advocate doing away with the lecture format
altogether and teach calculus with the help of sophisticated software and
computer algebra systems (mathematica, derive, maple, etc.) in a computer
laboratory.

I tend to think that the student who gains proficiency with a computer
algebra systems begins to treat the computer as a vital link in performing
calculus-related functions, much in the same way that most of us treat a
calculator a vital link in performing mundane arithmetical tasks (When did
you last extract a non-trivial square root *without* using a calculator?)
Now this may or may not be such a great idea, but who knows, it may be the
wave of the future.

My personal preferences lie somewhere in between; I would not want to go as
far as to teach differentiation out of the mouth of mathematica, but I would
love to introduce a heavy dose of non-trivial calculator applications in the
courses I teach.  I would emphasize, for instance, the numerical computation
of areas in 2D and volumes in 3D.  I will include numerical solutions of
differential equations, both ordinary and partial.  I would do away with the
epsilon-delta definitions of continuity -- not because I don't like epsilons
and deltas, I am an analyst of sorts, but because at this level epsilons and
deltas obscure rather than illuminate -- I will approach limits numerically.
Programmable calculators are ideal for repetitive computations and
investigation of the limiting values of functions.  Computing the limit of a
difference quotient numerically does wonders in driving the point home that
not all functions are differentiable;  it is trivial to program a calculator
to evaluate the difference quotient for the functions |x|  or  x.sin(1/x) at
zero.  Among other things it becomes clear that the domain of a functions
should be carefully prescribed otherwise the program may flash an error
message and halt.

The latest advanced calculators, such as Hewlett-Packard's HP48sx, allow
computation of roots of functions, numerical integration, graphing, and even
some rudimentary symbolic algebra!  These capabilities, and their
programmability, makes these calculators more like computers than
calculators.  One great advantage over computers is that is their total
portability.  The student may carry the calculator to the classroom or to the
library or to the cafeteria or to the dormitory, and he/she can use it for
other courses too.  Another advantage is that the student, by the virtue of
being the owner of the calculator, has a vested interest in learning how to
use it efficiently and effectively.  In contrast, a computer at the calculus
laboratory or wherever, does not belong to him/her, the manuals for the
hardware, software, and peripherals are not generally easily accessible, and
it is not clear that any investment in time and effort to deeply familiarize
with the facilities will pay off.

The down-side of the ownership of calculators is their cost.  The basic HP48
retails for close to $300.  This may be a non-trivial amount in a freshman's
budget.  Do I, as the instructor of the calculus course, require everyone
in my class to buy the calculator?  Do I structure my course so that it
becomes difficult, if not impossible, to get by without a sophisticated
calculator?  What about those who absolutely cannot afford the expense?
What if some buy less expensive models with fewer features?  Doesn't it
put them in a sad disadvantage when doing homeworks and taking tests?

I would like to hear your thoughts and comments on this.  Specifically:

A - Is the traditional U.S. style of teaching freshman calculus
    in need of reform?
B - Will the use of computer algebra systems (mathematica, maple,
    derive, mascsyma, mu-math, etc.) enhance the learning of calculus?
C - Will the use of programmable calculators enhance the learning
    of calculus?
D - Should the cost of the calculator be a factor in deciding whether
    to prescribe it as a required tool for enrollment in a course?
E - Have you thought, or have you been a student in a "non-traditional"
    type calculus course?  What was your experience?
E - Other thoughts and comments.

--
Rouben Rostamian                            Telephone: (301) 455-2458
Department of Mathematics and Statistics    e-mail:
University of Maryland Baltimore County     bitnet: rostamian@umbc
Baltimore, MD 21228,  U.S.A.                internet: rostamian@umbc3.umbc.edu