Mathematics
Knox College
Mathematica and NeXTSTEP help lead calculus reform
Reproduced with permission from NeXT
Computer, Inc.
A Reference
Guide to NeXT in Higher Education, Fall 1992
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1992 NeXT Computer, Inc
"I think the major advantages of using technology to teach
mathematics are visualization and experimentation," says
Dennis Schneider, professor and chair of mathematics and
computer science at Knox College.
"NeXT computers are particularly well suited to my work
because of the quality of the graphics-that's related to
the visualization aspect of teaching and learning-and
speed, which is related to experimentation and being able
to quickly generate a quality image. Using NeXT computers
and Mathematica, students become involved in the
discovery of mathematical results rather than being
observers of pre-established fact."
Since fall 1988, Schneider has used Mathematica in
the three-course calculus sequence at Knox College. In
addition, the school recently began offering a pre-calculus
course in which students use Mathematica. With
grants from the National Science Foundation, Eli Lilly
Foundation, and Pew Charitable Trust to develop this
Mathematica-based curriculum, Schneider says his
aim is to "thoroughly exploit computer technology in the
teaching of calculus and multivariable calculus."
Before acquiring NeXT machines in fall 1991, the
Mathematics Department was running a lab of Macintosh
IIcis. According to Schneider, "The NeXT is up to 12 times
faster and cost us less to install and maintain than the
Macs. NeXTSTEP is also very stable. When running
Mathematica, it rarely crashes. The Macs crash
three to four times more frequently than the NeXTs."
For his calculus classes, Schneider has created a series of
Mathematica Notebooks. "With the Notebooks, I'm
able to cover topics more thoroughly and deeper than I ever
could before," he says. In one Notebook for his
multivariable calculus course, Schneider asks students
conceptual questions about functions of two
variables-problems that cannot be found in typical calculus
textbooks. "To do computations by hand to verify properties
of complicated functions would be horrendous," says
Schneider. "But with NeXT, students are able to take on
such difficult problems."
In another exercise, students are given parametric
equations of an epicycloid and asked to experiment with
different integer values of two variables (representing the
radii of the two circles) to obtain a variety of
interesting curves. They are also asked to conjecture about
the relationship between the two variables. In solving
these problems, students are typically asked to give
analytical as well as graphical evidence to support their
answers.
"Using NeXT and Mathematica, we're able to
introduce real-world applications that would be too
difficult to solve otherwise," says Schneider. "Plus,
students are able to confirm mathematical theory with the
analytical, numerical, and graphical evidence they obtain
with Mathematica. The graphs they're coming up
with on the NeXT machines are ones that could not be drawn
by hand. The NeXT has really increased students' ability to
visualize mathematical theory.
"Mathematica on the NeXT platform provides an
incredible advantage in teaching math," Schneider
concludes. "It used to be that students were exposed to
only one point of view. They had a template problem and
practiced with examples. NeXT machines and
Mathematica take calculus students far beyond what
they could learn in a traditional classroom environment."
For more information, please contact:
Dennis M. Schneider
Professor and Chair of Mathematics and Computer Science
Knox College
Galesburg, IL 61401
(309) 343-0112 x420
schneidr@knox.bitnet