Mathematics

University of Missouri, Columbia
NeXTSTEP brings calculus to life with Mathematica

Reproduced with permission from NeXT Computer, Inc.
A Reference Guide to NeXT in Higher Education, Fall 1992
ยช 1992 NeXT Computer, Inc


These days, Professor Elias Saab's Calculus II class at the University of Missouri, Columbia, seems more like a think tank for experienced mathematicians than a lower-division college course. During a recent class, 25 students learned about Fourier Analysis, used series expansion techniques to solve differential equations, and solved a system of 24 simultaneous equations-problems typically reserved for their upper-division classmates.

When Saab pops into the computer lab, most evenings he finds a group of his Calc II students hovered around NeXT computers interpreting the results of their homework problems. "Students are actually discussing calculus theory with one another," says Saab. "Thanks to the computer, undergraduate students are talking to one another about mathematics, and they have a greater understanding of what calculus is all about."

To concentrate more heavily on calculus theory and less on technique, Saab decided last year to teach Calculus II as a NeXT computer-based course. Saab replaced his traditional textbook with Calculus&Mathematica, an innovative curriculum developed at the University of Illinois. Students now solve all homework problems and submit all assignments to Saab via e-mail, something he never thought he'd be doing.

Saab says he opted to teach on NeXT machines for several reasons: "The biggest plus is that the NeXT machine comes equipped with Mathematica at no extra cost. The speed of the machine is also an important factor. Other platforms, running at the speed of NeXT, would have been much more expensive and would have lacked a good user interface for students. NeXT is an easy machine to learn because of its user interface; students grasp it quickly."

Saab adds that because NeXT machines can easily compute solutions for a variety of calculus problems, his students can dedicate more time to learning about calculus concepts than students enrolled in traditional mathematics courses. "It is no longer necessary for me to go into lengthy detail about the technique of integration because the computer can compute such problems faster and more accurately than students can by hand. Instead, we have time to emphasize the theory."

Saab also spends a good deal of class time focusing on practical-albeit complicated-applications of calculus theory. "There is a lot more opportunity for students to solve interesting problems that pertain to real-life situations. I wanted to get away from textbook examples, most of which are dead examples. Examples on the computer are alive. One can change them over and over to develop an infinite number or examples. Students now solve problems with 100 unknowns. They can't do that using the tables in a textbook, but they can do it with a NeXT machine running Mathematica."

Saab says the computer-based course is initially "more work" for the students, but that their effort is paying off. To test current students' knowledge against former students, Saab gave one pencil-and-paper test during the semester. In past semesters, students earned an average score of 65 percent on the test while this class averaged 80 percent.

The winter of 1992, Nakhle Asmar, an assistant professor of mathematics at Missouri, also tried Mathematica and NeXT machines in his Applied Analysis course. Although his students still solved problems with pencil and paper, he held several computer-lab sessions during which he solved partial differential equations, for example, using Mathematica on a NeXT machine.

"I was skeptical at first about using computers to teach the class," Asmar says. "That's why I didn't have the students use Mathematica to solve equations. But in the end, I was amazed at what Mathematica could do in terms of graphing and how quickly equations could be solved. Consequently, I've decided to incorporate NeXT computers and Mathematica into the class next fall."

For more information, please contact:

Elias Saab
Chair and Professor Mathematics
University of Missouri
Columbia, MO 65211
(314) 882-0635
elias@esaab.cs.missouri.edu