Some 300 years ago Isaac Newton and Gottfried Leibniz independently developed it as a technique to help explain how the world works, yet in the ensuing centuries their invention has probably caused more anguish among aspiring students that any other subject.
The technique developed in the 17th century is calculus, a form of mathematics that students across the United States are flunking at a remarkably high rate. Of approximately 300,000 students who start a calculus course each year, only 140,000 complete it with a grade of "D" or better.The high failure rate has raised concerns among top mathematicians, who say the fault is as much with how calculus is taught as it is with the subject's inherent difficulty.
About 100 of those mathematicians met at the University of Minnesota recently to coordinate 25 projects designed to reform the teaching of calculus. The projects are backed by a $1.29 million grant from the National Science Foundation.
Calculus is important because it is the essential mathematics not only for all of the sciences, but also for economics, engineering, business and even some of the arts, said John Kenally, a foundation program director who is coordinating the reform effort.
"Calculus is the gate to everything," he said. He added that, in modern mathematics education, calculus "sits there and chokes up the system."
Calculus, simply defined by several mathematicians at the meeting, is the "mathematics of continuous change."
Measuring the rate of change of anything from a chemical reaction to the stock market involves calculus. That it is taught poorly in the United States means not only that fewer students go into the sciences, but also that fewer business managers are able to figure precisely such things as supply and demand.
Wayne Roberts, chairman of the mathematics department at Macalester College in St. Paul, Minn., described the failure rate in calculus as disgraceful and said there are a host of reasons the subject is in trouble in the United States.
Calculus courses have become bloated with new demands from a wide variety of fields, he said. As new problems are added to textbooks, none of the old, outdated problems are taken out. And with this expansion, the demands on students are so great that they spend their time learning formulas, not the underlying theories and ideas of calculus.
Drawing the most fire from the mathematicians are the calculus textbooks used in high schools and colleges.
"We have to get those awful books out of the schools," said New York University mathematician Peter Lax.
"If you look through one of those big textbooks you find it has nothing to do with the way a mathematician thinks of the problems."
Lax said that two-thirds of the contents of calculus books could be "washed away" and that students would be better off for it.
Developing a textbook that would teach calculus in a more thoughtful yet efficient manner is a worthwhile goal, Lax and others said, but it will be difficult.
With so many fields using calculus in different ways, it is difficult to come up with one text that is suited for both a business major and a chemistry student.
Beyond the problem of developing a text that a majority of mathematicians can agree on, Lax said, is the difficulty of convincing textbook publishers to take on such a project.
"It now costs a half a million dollars and five years to get a (calculus) book to the market," he said, and publishers are reluctant to take on a new, experimental text that doesn't include every conceivable aspect of calculus.