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A funny thing happened to 1,700 first-year students who took Math 115 Calculus this term. Along with derivatives, integrals and functions, they learned something even more valuable. They learned how to think.

The learning process wasn’t always easy, according to Pat Shure, lecturer in mathematics. Shure is part of a team of U-M educators who have spent the past three years redesigning the Math 115 program.

With its emphasis on open-ended problems, written and oral communication, cooperative learning and group homework, Math 115 forces first-year students to learn calculus in a radically different way than the “plug-numbers-into-formulas” approach taught in most high schools.

Shure says she has come to expect a litany of student complaints at the beginning of each term, including:

-- “This takes too much time.”

-- “If there’s no right answer, how are we supposed to know if we solved the problem?”

-- “Why do we have to write complete sentences? This is a math class, not English!”

-- “I know what calculus is and this isn’t calculus.”

“The first two months are difficult for everyone,” agrees David Burkam, who teaches Math 115 in the Residential College. “It’s especially tough for students who realize that their high school study patterns aren’t working anymore.”

Burkam and Shure don’t enjoy making life tougher for first-year students. They, along with others at the U-M and many U.S. colleges and universities today, just believe the old ways of teaching calculus don’t work.

“The traditional approach to calculus instruction was based on lecture and formula memorization,” says Beverly Black, from the Center for Research on Learning and Teaching, who is the program’s instructional consultant.

“Students worked in isolation plugging in numbers that produced the ‘right’ answer, but with no understanding of why it was correct,” Black says. “Now we’re focusing on learning how to think through and solve problems that connect calculus to everyday life.”

During the first two years of reform, Math 115 was taught by a small group of instructors who volunteered for the job. This year, however, Math 115 is being taught by a wide cross-section of faculty and teaching assistants.

Since teaching Math 115 is very different than teaching a traditional calculus class, Black worked with Shure to improve the Department of Mathematics’ one-week instructional development program. All those teaching Math 115 are required to attend. Instructors also meet in small groups throughout the term to discuss teaching issues and midterm feedback from students.

“The adjustment is often more difficult for the instructor than the student,” Shure says. “It’s hard to give up the control of traditional teaching and make the transition to cooperative learning, where the instructor’s role is more like a coach.”

Ever since Math 115 began changing in the fall of 1992, Burkam has been analyzing student performance and compiling data for a formal evaluation of the new program’s effectiveness. His report for LS&A and the National Science Foundation (NSF), the primary funding source for the program, is due this summer. Morton Brown, professor of mathematics, is NSF principal investigator for the overall project.

Burkam says the biggest change in students who take Math 115 is in their attitude toward mathematics. “Students have a deeper understanding of the diversity of mathematical solutions and see math as more relevant to other

classes and to ‘real world’ problems,” Burkam observes. “They express a greater interest in mathematics and have more confidence in their own mathematical abilities.”

Burkam also points out the nonquantifiable benefits students mention in essays they complete at the end of the term, such as:

-- “Group homework forced me to open myself to others’ methods and develop friendships with people I would have never met outside of class.”

-- “The most important thing I learned in calculus was how to work as a member of a team.”

NSF is watching the U-M calculus program closely, according to Shure, to see whether the cooperative learning approach can work with large groups of students and instructors of diverse backgrounds and educational levels.

“Everyone’s looking at Michigan now,” Burkam says. “I don’t believe there’s any other institution this size attempting to implement such an innovative program that reaches all first-year students. We’re setting precedents for future reform—not just in calculus, but in larger questions of emphasis on undergraduate education and new roles for faculty at major research institutions.”