Mathematics

University of Iowa
Curriculum offers new approach to accelerated calculus

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


"Calculus is beautifully coherent and profoundly useful," says Keith Stroyan, professor of mathematics at the University Iowa. "It is one of the great achievements of the human intellect; it has served as the language of science for 300 years."

Stroyan notes, however, that calculus is not typically presented to beginning students as a great achievement and a powerful form of expression. "Development of skills and attention to technical details have grown out of proportion in beginning calculus courses to the point that students spend little time learning concepts or thinking about the power of the skills they are developing," he says.

Since the early 1980s, Stroyan has been working to rectify this situation, focusing his efforts on the department's Accelerated Calculus course in which students cover the contents of three semesters in two, using NeXT computers and Mathematica. During summer 1992, Academic Press will publish his textbook and software, Calculus using Mathematica, which includes the innovative materials he has developed for the course. Plans are under way to expand the approach to all Iowa University calculus students.

"We wanted to revise our accelerated calculus curriculum to present calculus as the language of science and show students real-world applications," says Stroyan. "As part of that revision, we needed up-to-date scientific computing."

He continues, "We use NeXT machines and Mathematica because I believe Mathematica with the Notebook front end offers the best available combination of well-integrated, high-power scientific computation and ease of use. Our aim is to make serious use of computing in the course, but to focus on calculus. Consequently, ease of use was a very important consideration for us. We wanted graphics, numerics, and symbolics, probably in that order of importance; and these three aspects of computing are well integrated in Mathematica."

Although the department received quotes from IBM, Apple, Hewlett-Packard, Silicon Graphics, and Apollo, Stroyan limited his hardware choices to Macintosh and NeXT-at the time, the only platforms that ran Mathematica with the Notebook front end. After carefully evaluating both platforms, the department purchased a lab of NeXT machines with cost-shared funds from the National Science Foundation. Stroyan says the new equipment was up and running within a week of delivery.

"We decided on the NeXT machines," he says, "and that has turned out to be an excellent decision. First, the NeXT machine has high screen resolution–about four times the pixel size of my Mac II color screen. With the NeXT, I can lay out two Notebooks side by side, compare them, and go back and forth. For doing this work, the big, high-resolution screen is important. Color is much less important.

"NeXT computers offer Macintosh-like ease of use and Mathematica Notebooks, but run a genuine UNIX network and have a host of bundled software besides Mathematica. UNIX gives us file security and convenient individual accounts which we use to submit and grade the electronic homework. Plus, as UNIX systems go, NeXT is quite easy to set up, use, and maintain. Our NeXT network has much lower system maintenance than our research networks," says Stroyan.

For the accelerated course, he has created exercises dealing with real-life situations. For example, students are asked to determine a sustainable harvest level for Sei whales, why polio, but not measles, was eradicated by vaccination; and the resonant frequency of an electrical circuit.

Mathematica Notebooks play two roles in the class: to help students overcome technical difficulties and to illustrate and reinforce basic mathematical concepts. "Mathematica helps students overcome messy computations of scientific problems," says Stroyan. "For instance, our students can easily compute 100 years of simulated whale harvesting given by a complicated recursive sequence. Such computations are intractably difficult by hand, so computing opens new possibilities for applications in our course."

He continues, "Our students learn the basic skills of differentiation and integration, but also learn how to use Mathematica to perform very elaborate symbolic and numerical computations. We don't labor some of the esoteric techniques of integration, or even practice complicated differentiations at length. If students' basic skills are backed up with modern computing, it is not necessary to drill them ad nauseum to make them proficient mathematical thinkers and users of calculus. "

Students attend two lectures each week with Stroyan in which he frequently uses slides prepared from Mathematica Notebooks, and attend small-group discussion sessions led by graduate teaching assistants. Students are assigned weekly electronic homework, which they work on after class in the department's dedicated NeXT computer lab. At least twice during the course, Stroyan assigns students two-week projects to solve in teams.

"Students have become very excited and involved in the projects," says Stroyan. "Faculty members have walked past our lab and asked me, `What are you doing in there? I've never seen that much excitement in a math course!'"

For more information, please contact:

Keith Stroyan
Professor of Mathematics
University of Iowa
Iowa City, IA 52242
(319) 335-0789
stroyan@math.uiowa.edu