I have always loved to teach at all levels; but in recent years I have discovered that I have a special interest in helping students make the difficult transition from introductory to upper-division physics. Introductory students typically know how to solve short problems by choosing the equation with the right symbols and plugging in numbers. In the upper division, students must learn how to read equations as easily as they read English, they must see the relationships amongst equations, concepts, and a wide variety of graphical representations, and they must master the complexity of problems that may take several days or even weeks to solve. Most students are intimidated by this transition, referred to locally as ``the junior-year brick wall.'' For under-prepared students, especially some transfer students who must also adjust to life at the big university, and for non-traditional students (whether older-than-average, women, minorities, etc.), who may distrust that they ``belong'' in physics, the transition can be overwhelming. A large part of my teaching, and, more recently, also of my scholarly research activity, has been devoted to finding ways of helping as many students as possible through this transition successfully.

**The Paradigms in
Physics Project**

In 1997, I became PI (with co-PI's Janet Tate and Philip Siemens) on a major NSF grant to create and implement a ground-breaking new upper-division curriculum designed to improve students' analytical and problem-solving skills. The NSF indicated that our program was one of their top priorities for funding; they are extremely excited about the possibilities of our vision for reforming the teaching of physics majors nationwide. Our development team has involved 11 faculty members from two departments, 7 graduate students, and an external evaluation team. The traditional order of material has been completely revised: the junior year features intensive, three-week modules each focusing on a single, paradigmatic example in physics; the senior year features courses which systematically present the traditional subdisciplines of physics, as well as courses describing the phenomena and methodology of modern research areas. Our pedagogical approaches include interactive small-group learning, technology-based visualization activities, and project-based classes. Now in its ninth year, the Paradigms project has attracted attention at national meetings of the AAPT and APS in a sessions on ``Revitalizing Undergraduate Physics'' and international attention, especially in Canada and Sweden. Articles on the Paradigms project has been published by the American Journal of Physics and Physics Today. A second grant from the NSF was awarded in the spring of 2003 to use workshops to begin the process of national dissemination of this novel curriculum.

As well as heading the overall project, I have developed three of the intensive junior-year modules (Symmetries and Idealizations, Static Vector Fields, and Central Forces) and the senior-year course, Capstone on Mathematical Methods. My own contributions continue to lie at the nexus of physics, mathematics, visualization, and problem-solving and I have begun a small research group to explore this area. Katherine Meyer completed a Master's Project studying the characteristics of successful interactive activities at the upper-division level. Kerry Browne has completed the first Ph.D. at Oregon State University in Physics Education, ``Student Use of Visualization in Upper-Division Problem-Solving.'' Insights from these research projects have influenced my teaching style and strategies. More information is available on the Paradigms website.

**Bridging the Vector
Calculus Gap**

While working with Tevian Dray, from the Mathematics Department at Oregon State, on the Paradigms Project, we became aware that there is a "vector calculus gap" between the way vector calculus is usually taught by mathematicians and the way it is used by other scientists. The gap goes much deeper than a difference in emphasis. Ask a physicist or engineer what topics should be covered in vector calculus, and the answer will pretty much agree with the existing syllabus used by mathematicians. But the traditional language used by mathematicians to teach this material is so different from the way it is used in applications that students are often unable to translate.

A major part of the problem is the traditional mathematics emphasis on Cartesian coordinates to describe vectors as triples of numbers, rather than emphasizing that vectors are arrows in space. Students then memorize the all-important dot and cross products as algebraic formulas, rather than comprehend them as geometric statements about projections and areas, respectively. It is hardly surprising that many students are then barely able to compute line and surface integrals, or the divergence and curl of a vector field, let alone understand their geometric interpretation.

We have received funding from the National Science Foundation to develop and disseminate supplemental materials, especially interactive small-group activities, which emphasize geometry. Some of these materials are intended for use in an otherwise traditional vector calculus course, and some are intended for use in transition courses to upper-division physical science or engineering classes (such as the Paradigm on Symmetries and Idealizations or an introductory Mathematical Methods class). Activities introduce students to the types of problems - and methods of solution - that they will encounter in their chosen specialization, while at the same time increasing their understanding of traditional vector calculus and its applications, thus bridging the gap. Research papers, instructor's materials, and information about faculty development workshops are all available on the Bridge website.

- ©
*Corinne Manogue, 2005.*

*Last Update: 12/5/05*