A Team-Taught Interdisciplinary Approach to Physics and Calculus Education

Purpose

Even the most gifted and dedicated students in beginning physics and engineering courses usually lack the mathematical skills required to obtain the maximum benefit from these courses. This is because the design of most physics and calculus sequences makes it difficult for students to learn the mathematics they need for physics in a timely fashion.

For example, Physics I students must learn mechanics without knowing vector and multivariable calculus, an experience that the director of this project compares to carving wood with a screwdriver rather than a chisel. The next course in the physics sequence uses mathematical material-including divergence, curl, Green's Theorem, Stokes's Theorem and Gauss's Theorem-which is only taught at the end of the mathematics sequence, if at all.

To be able to cover the beginning physics syllabi, most instructors teach a bare minimum of mathematics, which must compete for space with the physics material. The mathematical deficits that students incur in these early semesters worsen as they progress to upper level physics courses. Unfortunately, many students never do.

The project sprang from the collaboration of a mathematician and a physicist who noted excessive failure rates among students in the engineering calculus and physics sequences. Even those who passed the courses encountered difficulties when they transferred to a four-year institution and many changed majors.

The project's creators wanted not only to raise the success rate in the calculus and physics sequences without decreasing content or expectations, but to increase the number of students, especially underrepresented minorities and women, who transfer to four-year institutions and complete their degrees in science and engineering.

Innovative Features

To achieve these goals, faculty integrated the curricular content of beginning calculus and physics courses and developed a unified approach to education in these disciplines. The curriculum they created, the Special Intensive Program for Scientists and Engineers (SIPSE), was purposely designed to be adaptable to a variety of institutions.

The integrated curriculum involved the re-sequencing of physics and calculus courses so that they complemented each other and consisted of two courses, each combining calculus and physics, team-taught by a mathematician and a physicist. Thus, SIPSE's first semester combined Calculus I and Introduction to Engineering Physics, a new course designed to augment the physics background of beginning students. The second semester combined Calculus III (vector calculus) and Engineering Physics I (mechanics). Calculus II was reserved for the third semester.

SIPSE's I-III-II calculus sequence placed vector calculus immediately prior to linear algebra, which made the more abstract linear algebra treatment of vectors easier to understand than the traditional sequence, where students often take linear algebra before vector calculus. The new arrangement also placed the Taylor series just before differential equations, so that students would remember the series well enough to use it in differential equations.

SIPSE also altered the sequence of calculus and physics topics within a given semester, so that the mathematics was covered before it was needed. This aided progress in physics because the student had the appropriate tools, and progress in mathematics because mathematical concepts were put into context right away. Students became familiar simultaneously with mathematical and physics notations, terminologies, and approaches.

The new arrangement allowed sufficient time to cover divergence, curl, Green's, Stokes's and Gauss's Theorems, the statistics needed for physics, and the applications of calculus to physics. The integration-as opposed to the simple juxtaposition-of calculus and physics allowed the necessary flexibility to cover calculus concepts before they were required by the physics sections of the course. For example, most of the first weeks of SIPSE's second semester were spent on vector algebra and vector calculus, with little coverage of physics material until the students had mastered vectors.

Students were required to attend cooperative study groups patterned after those in the Professional Development Program designed by Uri Treisman at the University of California, Berkeley (see Lessons Learned II). The groups met for two two-hour sessions every week. They consisted of four to six students and a group leader-either an advanced SIPSE graduate or an upper-division student from Berkeley. The study groups met at one of two different times (but could have met simultaneously), under the supervision of a faculty member. The study group leader provided both individual tutoring and, most often, group instruction. The leader also served as a role model, discussing his or her experiences in more advanced coursework and in transferring to a four-year institution. Group assignments, which covered applications of mathematics to physics or subsets of homework assignments, took less than two hours to complete, leaving time for discussions and help sessions.

After SIPSE, students returned to the traditional curriculum to take Calculus II, which covers integration techniques and applications, the Taylor series and differential equations. They also took Engineering Physics II (electricity and magnetism), which uses material from the end of Calculus II, covered the previous semester. SIPSE's flexible format allowed students to transfer into it, or out of it and into the regular curriculum, at any point in their progress-a necessity given the college's student population, whose work and family responsibilities often disrupt their academic programs.

Team teaching not only allowed faculty to adjust the ratio of time spent on mathematics versus time spent on physics on a daily basis but also made it possible for the physicist to augment the mathematician's presentation with an explanation of the applications of the current topic to physics. The mathematician in turn contributed to the physicist's presentations by reviewing pertinent mathematical concepts and discussing differences in notation, terminology and style. And both instructors jointly discussed such topics as conservative vector fields or centers of mass and moments of inertia with single and multiple integration. In addition, students were able to witness discussions between faculty which arose spontaneously in or outside of class or in the study groups, and this encouraged them to become actively involved with the material.

Evaluation and Project Impact

The external evaluator's study showed that SIPSE students, although at the outset they possessed lower mathematical skills than students in the traditional sequence, had significantly higher course success (a final grade of C or above) and retention rates than their peers in the control group. Keeping in mind that students frequently withdraw because of poor grades, if more traditional students had completed the courses, the differences in success rates between the traditional and the SIPSE courses would have been even larger.

In three of the four courses tested, SIPSE students earned significantly higher grades than the controls. In the fourth course, there was no significant difference between the two groups.

Women and minorities were more likely both to persist and to succeed in SIPSE than in the traditional program. Over the course of the project, the withdrawal rate differences were especially remarkable-a mere six percent for women and minorities enrolled in SIPSE versus 24 percent for all students in the control group. A comparison of the final grades of women and minorities in SIPSE courses versus women and minorities in traditional courses shows that the SIPSE group earned significantly higher grades for all courses except beginning physics.

According to the instructors, SIPSE Physics I students performed better than the controls on common final examination questions. The scores of SIPSE students on the Wells-Swackhamer Force Concept Inventory and Mechanics Baseline tests compared favorably with those of students at Harvard and Arizona State.

Instructors also observed that SIPSE students seemed stronger than their peers in post-SIPSE lower division mathematics and physics courses, more confident in their abilities, and more likely to transfer to a top-level four-year institution and to graduate in engineering or science. SIPSE students reported significantly more self-perceived positive changes in their study habits and in the amount of time they spent studying than their peers in traditional courses.

The project's most profound effects were on Physics I, which is not surprising because both the creation of "Introduction to Engineering Physics" and the rearrangement of the calculus sequence had been intended to prepare students to function well in this course. SIPSE's effects on performance in calculus, though significant, were not as dramatic. This too was to be expected, because students benefit more in physics from knowing the necessary mathematics than in mathematics from learning its applications to physics.

In the opinion of project faculty, SIPSE courses-with the exception of the new course, Introduction to Engineering Physics-were taught at a more rigorous level than traditional courses. And SIPSE students had significantly higher grades in Calculus III despite the fact that, unlike students in the control group, they had not taken Calculus II.

Overall, there were few failures among SIPSE students, and many of these were due to family or financial problems. Often, these students returned to SIPSE and later did well.

Lessons Learned

For successful team-teaching in SIPSE-style sequences, the physics instructor must have a strong background in mathematics. The traditional approach to physics is dictated by the students' inadequate mathematics background. But because SIPSE students do not have this deficit, the physics instructor must be prepared to use more advanced mathematical tools than those covered in the text. Although it is less important that the mathematician have a strong physics background, both faculty members must be flexible and willing to relinquish absolute control over the class.

It appears that combining Calculus III and Engineering Physics I in the second semester is essential for the success of the revised calculus sequence. Faculty allowed some non-SIPSE students who were not enrolled in Engineering Physics I to take calculus in the I-III-II sequence. The students did quite poorly.

Study groups were also crucial in the transition from Calculus I to Calculus III. In addition to instruction, they gave students support and a sense of community, something not often encountered on two-year college campuses. Frequently the groups endured beyond SIPSE into more advanced courses and sometimes even into the four-year institutions to which former SIPSE students transferred.

Because instructors team-teaching in SIPSE spent twice as much time in the classroom as they would have in an ordinary course, they received credit for two courses, plus two credits for the study group. (For every SIPSE course students received credit for a mathematics course, a physics course, and an additional unit for the study group.) Although the mathematics department was able to absorb this change in load, the much smaller physics department found it difficult to adjust, especially when several physics faculty retired and were not replaced.

Project Continuation

With the exception of Introduction to Engineering Physics, which became the prerequisite for all sections of Physics I at the college, SIPSE, despite its success, was not institutionalized. Because SIPSE classes were somewhat smaller than usual, the institution deemed the program not cost-effective, and it ended with the expiration of FIPSE funding.

Dissemination and Recognition

Project staff presented SIPSE at many meetings across the United States. In addition, the project director organized the Consortium for the Combined Instruction of Mathematics and Physics, consisting of Adirondack Community College, N.Y.; Auburn University, Ala.; Diablo Valley College, Calif.; Dutchess Community College, N.Y.; North Seattle Community College, Wash.; Rose Hulman Institute of Technology, Ind.; SUNY/Binghamton, N.Y.; and the University of Puget Sound, Wash. All these institutions, some of which have grants from FIPSE and the National Science Foundation, either have in place or are working toward a combined calculus/physics program. The consortium's mission is to disseminate findings about these programs through interdisciplinary presentations, publications, and Web pages.

Available Information

Further information may be obtained from:

David B. Johnson
Department of Mathematics
Diablo Valley College
Pleasant Hill, CA 94549
Telephone: 510-689-1230 ext. 489

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