A recent retooling of the way calculus is taught at the University of Rochester has left many students unusually excited about a subject often regarded as one of the toughest in the undergraduate curriculum.
In fact, one proctor at last semester's "Calculus for Understanding" final said the test-takers she oversaw were "relaxed," "excited," and "eager" -- hardly the words typically used to describe calculus students at a final exam.
While this observation might once have been dismissed as a fluke, mathematics faculty say these students' attitudes illustrate a new trend in thinking about calculus at Rochester. Undergraduates now have an unusually broad choice of calculus courses that allows them to find the best match with their own needs and skills:
• In some courses, students do individualized homework over the Internet on a new program dubbed "WeBWorK," with a professor's help on problems never more than a quick e-mail note away. • Other calculus classes see small groups of students working together over pizza on tougher problems. • A third option is a calculus course that's closely integrated with an introductory physics course. • And the most motivated of students may choose a rigorous sequence of four introductory calculus courses in which students stay with the same professor for a full two years.
The more traditional calculus classes that three-quarters of all Rochester freshmen normally take haven't been completely ousted by the new approaches to calculus, the wide-ranging study of derivatives and integrals regarded by many as the foundation of engineering and the natural sciences. Traditional courses remain available for those who prefer them.
"After its first two semesters, this arrangement seems to be both successful and popular with our students," says Professor Douglas Ravenel, chair of the math department. "In lieu of a one-size-fits-all calculus course, we now have options to suit students of widely varying interest levels and academic needs."
Fifteen calculus students wrote to President Thomas H. Jackson this year to tell him that learning calculus in their "Calculus for Understanding" class was "almost as easy as breathing." Their joint letter added that the class "fostered the learning process by nurturing instead of restricting our inquisitive natures ... the program's stress on the 'whys' of calculus helped us to develop a deeper understanding."
In keeping with the University's efforts to enhance undergraduate teaching, the math department has set out to devise a series of courses that are smaller and more attractive to students, yet still provide the rigorous calculus foundation needed by science and engineering students. It has implemented a new calculus track to fill the gap between the standard calculus sequence taken by most science and engineering students and its most rigorous calculus honors sequence, which treats calculus almost like one of the department's graduate-level offerings. (These undergraduate honors courses, which have been offered for some 35 years, have also been reworked and now attract record numbers of students.) The new courses are half the size of the mid-level calculus courses and feature more challenging problems that students attack in weekly work sessions over snacks and under the watchful eye of a graduate teaching assistant.
"These classes aren't better just because they're smaller," says Ravenel. "Rather, our new approach has created a climate of close intellectual collaboration between students, TAs, and faculty that we haven't experienced before in freshman calculus."
Rochester's math department has also launched an experimental program that uses computers to distribute homework assignments. The resulting use of the Web as a medium for homework has taken the form of WeBWorK, a popular Internet-based homework program formulated by Professors Arnold Pizer and Michael Gage and programmed by three Rochester students. WeBWorK gives students immediate feedback on whether their answers to individualized homework sets are correct. Pizer says this provides students with an incentive to continue working on problems until they've got them right, and makes it easier for professors to monitor student progress.
"When homework is graded by hand, a student often won't receive a corrected assignment back until a week or more after it was done," notes Gage, who recently won a regional teaching award from the Mathematical Association of America for his work on WeBWorK. "By then, the problems are no longer fresh in the student's mind, and there's little incentive to go back and figure out the errors. In fact, at that point the student is probably more concerned with the next homework assignment."
Using WeBWorK's feedback mechanisms to contact professors by e-mail, an impressive 80 percent of students get all of the answers correct eventually. Gage says that the program opens up the lines of communication between students and professors -- lines all too often closed in introductory college courses.
"There is no doubt that using WeBWorK changes the social workings of the classroom," Gage says. "The lively e-mail exchanges between students and faculty testify to a new dynamic in the learning process."