A decade after mathematicians began a crusade to make calculus more relevant to undergraduate students, a backlash threatens to derail the effort and divide the profession.

A growing number of professors believe that the movement to reform the teaching of calculus has been a disaster. Critics say that courses have been watered down since the effort began in 1987 with money from the federal government. Students, they say, are coming out of introductory calculus courses with no idea how to solve complicated mathematical problems.

“This approach really shies away from anything but superficial use of skills,” says Ralph L. Cohen, a professor of mathematics at Stanford University, which after seven years has decided to stop teaching the"reform calculus” and to move back to something more traditional."For students who really need to know math and use it, this wasn’t nearly sophisticated or rigorous enough.”

How to teach calculus might not seem to be an emotional question, but the controversy about it has been as intense as any of the recent curricular battles over the literary canon. Professors who have tried to change the way calculus is taught tell of being harassed -- sometimes physically -- by colleagues who oppose the changes. Other reformers say they have been pushed out of their departments by traditionalists.

“The debates are as deep as those between two different religious groups,” says Ronald G. Douglas, provost at Texas A&M University, who is considered the father of the reform movement. He was chairman of the mathematics department at the State University of New York at Stony Brook a decade ago when he began looking for ways to change calculus.

The stakes are high because about 40 per cent of America’s undergraduates take at least one calculus course. According to the Mathematical Association of America, about 500,000 students were enrolled in calculus nationwide in spring 1994. It is the cash cow of mathematics departments, bringing in more tuition dollars than anything else they teach.

The reform movement got under way after a meeting in Washington in 1987 at which mathematicians announced a crisis in calculus education: As many as 40 per cent of undergraduates were failing introductory calculus, and even those who passed did not appreciate the subject’s relevance. Courses often consisted of bland lectures in which students learned how to calculate derivatives and integrals. Students practiced the calculations at home, and on exams professors asked similar problems with different numbers. Students, professors recall, were bored and unengaged.

“They learned they could stick in a couple of key symbols, statements, and equations and put forward what were found to be acceptable solutions, even though they had no idea what was going on,” says Morton Brown, a professor of mathematics at the University of Michigan.

The National Science Foundation agreed there was a problem and spent $35-million from 1987 to 1995 on dozens of projects to update the teaching of calculus. The reforms have caught on. In its 1994 report, the mathematical association found that nearly 70 per cent of 1,048 institutions had made"modest” or “major” changes in the teaching of calculus. It also estimated that in spring 1994, about 150,000 students, or a third of those taking calculus in the United States, were enrolled in “reform” courses.

The new classes have changed how professors teach and how students learn. Professors still lecture, but they also are likely to hold discussion sessions during class to get students talking about concepts. Students often work in small groups to apply what they have learned to multi-step problems that involve real-life issues. The problems mix in material from a variety of disciplines, including ecology, physics, and engineering.

The reform courses encourage students to rely on computers and graphing calculators to get past paper-and-pencil computation. Rather than asking students to produce numbers and symbols to answer problems, undergraduates must often write lengthy explanations of their solutions.

“Very few students are going to be math majors, and a lot of algebraic skills have only minimal applications elsewhere,” says Thomas W. Tucker, a professor at Colgate University who has adopted some of the reforms in his teaching."The new problems are much more open-ended than the old ones.”

At least 10 new calculus textbooks have been published in the last few years, fueling the reform. Each puts its own spin on how calculus should be taught, with some making heavier use of computers and calculators than others. But they agree that students should spend less time on computation and more on solving problems that have some application to things they see in daily life.

The most popular reform textbook is *Calculus* (John Wiley & Sons, 1994), by Deborah Hughes Hallett and Andrew M. Gleason. The two are mathematics professors at Harvard University and worked on the book with faculty members from 11 colleges and high schools. Their book has been adopted by about 500 institutions.

Known as the Harvard book, it presents intricate calculus problems that can take up to a page each to outline. One shows a graph that expresses the time it takes a mother starling to collect worms for her nestlings. Students are asked how many worms the mother should collect on each trip if she wants to maximize the rate at which she brings food back to the nest.

The textbook asks students not only to come up with answers, but to explain how they arrived at their answers and why they are correct."This establishes that mathematics courses are ones that involve thinking and talking about mathematical ideas, not just memorizing four types of problems,” says Ms. Hughes Hallett.

By contrast, problems in the best-selling"traditional” college textbook, *Calculus Early Transcendentals* (Brooks/Cole ITP, 1995), are more terse and ask students to express their answers in specific mathematical calculations.

Few data have been collected to prove whether students in the new courses learn more than students who are taught with the old methods, but those who back the reforms insist they are working. The University of Michigan began revamping its two-course sequence in introductory calculus about five years ago. More than 5,000 students a year take the courses, which are based on the Harvard textbook. The university previously used lecture sections that enrolled more than 200 students each. Now, no class has more than 30 students, and courses are held in special rooms outfitted for group work.

Back in the early 1980s, when Michigan was still using the traditional lecture-style format, about 30 per cent of its undergraduates received D’s and E’s in beginning calculus. Now that proportion is down to about 11 per cent, says Dr. Brown, the mathematics professor who led the reform effort. Students in the new courses are more likely than their predecessors to report that their interest in math has deepened.

Some institutions, like Michigan, have transformed all of their calculus courses, while others have just begun to do so. Last year the University of Minnesota’s Twin Cities campus began teaching reform courses that involved about 100 students. This year the number is up to 160, or about 15 per cent of all students who take calculus there."The main aspect of the reform here is to try to have students become much more actively involved and encourage group learning with technology,” says Harvey Keynes, a professor of mathematics.

Students in the new courses can visit sites on the World-Wide Web that pose mathematical problems based on scientific phenomena. For example, in one application, students are asked why a rainbow always seems to hit the horizon at the same angle -- about 42 degrees -- and why the color distribution is the same.

Critics say such tasks may be interesting, but they don’t teach students what they need to know. By relying on calculators and computers, they say, students may never learn to perform difficult calculations."The reformers believe that they will get around the roadblocks of basic arithmetic so students can get to higher-order skills,” says George E. Andrews, head of the math department at Pennsylvania State University and a prominent critic of calculus reform."But to learn piano, you must learn scales and chords before you move to the ‘Moonlight Sonata.’”

Han Sah, who teaches advanced calculus courses at the State University of New York at Stony Brook, says he sees the problems created by calculus reform first hand. He has noticed a difference between students who took traditional courses early on and those who took reform courses."Students coming in from the reform calculus, I have to fill in all the details and the missing topics,” he says.

At least one publisher, Brooks/Cole ITP, is capitalizing on the discontent. James Stewart, who wrote the most popular traditional textbook, is now working on a new one that will use some principles of reform. But it also will include many of the basics that critics say other reform books have left out.

Some institutions are in the process of moving away from reform courses. In 1989, the University of Iowa began developing a calculus course that made heavy use of computer assignments and even required a term paper -- something almost unheard of a decade ago. By fall 1994, 400 students were enrolled in the revamped courses. Two years later, that number had dropped to 120. The decline was the result of a battle between reformers and more-traditional professors, who won a return to the way calculus used to be taught.

Keith D. Stroyan, a professor of mathematics at Iowa who developed the reform courses, says he and other colleagues were attacked by those who opposed the new approach. The traditionalists, he says, would scour the new calculus materials, looking for errors."There was immense haggling,” he says.

Stories of physical attacks also float around the mathematics community, although they are hard to pin down. According to one tale, a reform-minded professor at a university in Pennsylvania was thrown up against the wall by a colleague who opposed the movement. Then there is the story, probably apocryphal, of a man who was teaching reform calculus at a Midwestern university where the debate over the new style had been particularly divisive. The professor reportedly wrote the names of two traditional professors on a piece of paper, gave it to his wife, and told her that if he were murdered, one of the professors would be the perpetrator.

At Stanford, Brad Osgood, a professor of mathematics, says he has been made to feel so uncomfortable in the mathematics department since it decided to abandon the reforms he helped create that he will probably move to the university’s School of Engineering."This is a battle for the soul of the profession,” he declares.

Ed Dubinsky, who has written a reform-calculus textbook, moved this year to Georgia State University, where he says his efforts have been"better received” than at Purdue University, where he had taught since 1987. He says he is"pessimistic” about the future of calculus reform, predicting the movement will fade away in the coming years because traditional, lecture-style courses are so much easier to teach.

“Except for a small number of isolated pockets, it will be hard to tell that there was a calculus reform,” Mr. Dubinsky predicts. But he also thinks criticism about the way calculus is being taught will resurface. In a few years, Mr. Dubinsky says,"we’ll become upset that very few people are really learning calculus and we’ll have another round of reforms. I hope that round survives.”