Once upon a time, any serious student of mathematics had a slide rule. Then the development of electronic calculators made the slide rule obsolete, and the slip-stick took its place in museums alongside other evolutionary dead ends, like Neanderthal man.

The calculator became less important with the development of desktop computers and sophisticated software capable of producing graphical representations of equations. Unlike the slide rule, however, the calculator continued to evolve.

Today, mathematics instructors at colleges and universities across the country are requiring their students to use calculators that can produce graphs on their screens.

In some cases the graphical calculators, which look like pocket calculators with larger display screens, are used in conjunction with desktop computers. In others, professors teach classes with no computer assistance at all, relying on a textbook and the calculator.

Professors say the calculators are more affordable than a desktop computer equipped with mathematics software and offer a portability that computers cannot. With computer laboratories on many campuses either overbooked or too small for large mathematics classes, the calculators enable an entire class to work together in a regular classroom for an investment of about $80 each.

Students rave about the devices. Tony L. Draper, a junior majoring in criminal justice at Sam Houston State University, first took calculus as a sophomore, without a graphical calculator. “I’ve always been a good student, but I didn’t get anything out of that class,” he says. “I tried to just memorize formulas and plug in numbers, but that didn’t work.”

This year Mr. Draper is taking the same class, but it is being taught with a calculator. “Now I understand what you do to get a derivative, and why it’s important. I’ve gone from an F to an A,” he says. “I wouldn’t advise any of my friends to take calculus any other way. Being able to see pictures of what’s going on really makes a difference.”

A graphing calculator lets students quickly plot graphs, which can be critical to understanding calculus. The calculator also lets the student do statistics, numerical manipulation, and programming.

Professors can attach their calculators to special display devices used with a standard overhead projector. This lets students follow along as the instructor illustrates specific points.

Mercedes A. McGowen, a mathematics instructor at William Rainey Harper College, says one of the most important benefits of the calculators is their portability. “We’re a commuter school,” she says, “and a lot of our students don’t have computers at home.” Calculators are required in all alge bra and trigonometry classes, and next fall they will be required in all calculus classes, she says.

At Harvard University, the calculators have been required in beginning calculus courses for the past three years. Deborah Hughes-Hallett, a professor of mathematics at Harvard, says: “For us it’s very much easier to use a calculator than a computer because it’s not very easy to get all the same software loaded on all the different computer platforms for all the students. That is my idea of an administrative nightmare.”

Oregon State University requires a calculator in all its entry-level mathematics courses, including calculus.

Howard L. Wilson, a mathematics professor, says portability is important, and the devices are cheaper to use than computers. “We have classroom computer labs, but they’re very small,” he says. “We teach thousands of calculus students. We can’t teach those classes in the labs. That’s a money problem.”

Just as some feared that pocket calculators would cause schoolchildren to forget their multiplication tables, some professors worry that students will learn how to use graphical calculators without learning the concepts of mathematics.

“We clearly have to be careful to not have students do stuff on the calculator that they were actually doing correctly without it before, and there’s a line to be drawn somewhere,” says Harvard’s Ms. Hughes-Hallett.

“In an ideal world one might say people should plot graphs by hand and look at what they’re doing algebraically by checking the graphs that they have drawn,” she says. “Fat chance,” she adds with a laugh.

“If you look at how students were doing stuff, at least in the last 20 years, which is how long I’ve been teaching it, they weren’t drawing the graphs. Every faculty member I know would wring their hands and pull their hair out over the fact that students would write algebraic nonsense, never having checked it against a graph because they never drew any graphs.”

Ms. Hughes-Hallett says, and students agree, that most concerns about students using the calculator as a crutch may be unfounded, because students must understand the material before the calculator can be useful to them. “The calculator is not automatic,” she says. “If we just passed out free calculators to all the kids, they wouldn’t all suddenly start getting 90 per cent on their exams.”

Ray Cannon, a professor of mathematics at Baylor University, says calculators let students tackle problems that are more relevant to real life than traditional examples. “Part of what we want to do is knock down this idea that mathematics is something nobody likes, nobody understands, has no real meaning, and is just this esoteric drill that people have to go through to get their degree,” he says.

“The calculator changes the kinds of questions that you can ask students. A lot of problems we used to assign were very artificial, so the numbers would come out nicely. Today we don’t need to worry about that so much. The problems aren’t harder, but they’re not as neat.”

Despite such benefits, some professors, particularly those who teach higher-level mathematics, have been reluctant to use computers and graphic calculators. Some observers suggest that the higher levels haven’t adopted new technologies because they have not been through the curriculum revision that calculus has undergone in the last decade.

Gregory D. Foley, assistant professor in the division of mathematical and information sciences at Sam Houston State, expects calculators to become more popular in upper-division courses, but never to be used in every course.

“A lot of advanced mathematics is highly proof-oriented” and has no need for the devices, he says. “When you get into abstract analysis and abstract algebra, you might see some use of computers, but the graphing calculators are again probably going to be too limited. But for linear algebra, differential equations, and applied mathematics like tensor calculus, a souped-up graphing calculator will make a big impact.”

Mr. Wilson of Oregon State says many higher-level mathematics studies will continue to require very sophisticated computer workstations, rather than calculators. “It’s only at the conceptual-development level -- the freshman and sophomore level -- that the graphical calculators are really pedagogical tools for understanding mathematics,” he says.

But Donald R. LaTorre, director of undergraduate studies in mathematical sciences at Clemson University, says his institution has used the devices in a wide variety of classes. “Calculus and pre-calculus have a very strong graphical emphasis, so it’s a natural to use the calculators there. But we’ve been very successful in using them in differential equations and linear algebra,” he says. “The calculators are technically inferior, but not pedagogically inferior, to the computer.”

While some professors worry that teaching students to use the calculator will take up too much class time, Mr. Wilson says that two decades ago professors had to show students how to use a slide rule. “Compared to a slide rule, learning a calculator is a snap,” he says. “These kids are computer literate; they pick it up fast.”

Mr. Cannon of Baylor, who has been known to bring his slide rule to class to show his students what life was like in the Dark Ages, agrees. He says that many professors may have been frightened off by the difficulty of learning how to use desktop-computer programs.

“There’s a steep learning curve with computers, even with nice, powerful software,” he says. “The graphical calculators are easy to get into.”

And, he says, when students use a computer they lose something, because it’s almost as though the computer has done the work for them. “When they work a problem on a calculator, it’s *their* work, and they say, `Look what I did.’ There’s a sense of ownership. And the calculator is much more than a slide rule. It’s a mini-computer that they can stuff in their backpack.”