BRIDGING THE VECTOR CALCULUS GAP
Tevian Dray & Corinne A. Manogue
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. This material is essential for physicists and some engineers due to its central role in the description of electricity and magnetism. It is the goal of this proposal to bridge this gap.
The vector calculus 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. This leads to the all-important dot and cross products being memorized as algebraic formulas, rather than 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.
The traditional approach has one big advantage: It provides a single framework for handling quite general problems, the classic example being problems involving a paraboloid. But most practical applications, including virtually all at the undergraduate level, fall into a small number of special cases, such as those with spherical or cylindrical symmetry. There are no paraboloids in undergraduate physics! Problems with a high degree of symmetry become much more intuitive when the computations are not only done in appropriate coordinates, but also using a vector basis adapted to those coordinates. This emphasizes the geometry of the particular problem, rather than a brute force algebraic computation which many students fail to find illuminating.
We propose to develop supplemental materials, especially small group activities, which emphasize the geometry of highly symmetric situations, some of which are intended for use with an otherwise traditional vector calculus course, and some of which are intended for use in a new, upper-division physics course on related material. Such activities will introduce students to the types of problems -- and methods of solution -- which 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.
BRIDGING THE VECTOR CALCULUS GAP: EPISODE II
Tevian Dray & Corinne A. Manogue
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. This material is essential for physicists and some engineers due to its central role in the description of electricity and magnetism.
The two basic underpinnings of this proposal are the use of geometric reasoning rather than algorithmic computation -- a new emphasis for lectures -- and the use of open-ended small group activities -- a new emphasis for recitations. We believe that our major success so far has been the identification of geometric reasoning, using the vector differential, as the common theme underlying all of vector calculus. In the Proof-of-Concept phase of this project, we developed small group activities based on this approach, some intended for use in a vector calculus course, and some for use in upper-division physics courses on related material. These activities have been used successfully, by us and others, at several institutions.
It is the goal of this Full Development proposal to "bottle" our success by training other faculty in the use of our materials. We intend to offer workshops for those using these materials, and to write an Instructor's Guide containing information about this geometric approach to vector calculus, advice on using small group activities effectively, and tips on the individual activities. Four institutions have so far agreed to beta test these materials.
Enhancing students' geometric understanding of vector calculus will help to bridge the "vector calculus gap".
PARADIGMS IN PHYSICS: MULTIPLE ENTRY POINTS
Corinne A. Manogue, Tevian Dray, Barbara S. Edwards, David H. McIntyre,
& Emily H. van Zee
This proposal merges two very successful projects: The Paradigms in Physics Project, a complete redesign of the physics major, now in its ninth year, and the Vector Calculus Bridge Project, an effort to "bridge the gap" between the mathematics and physics of vector calculus, now in its fifth year. The merged project will be run by an established team, with two new members in education research, appropriate to its expanded role.
The primary thrust of this proposal is to design materials that provide multiple entry points to our successful curriculum, aimed not only at encouraging full adoption of our 18 redesigned courses, but also at supporting faculty teaching more traditional courses who may wish to experiment with one or more pieces, be it a single activity or an entire course. We have identified four main strands:
In addition to the impact on students, faculty, TAs, and visitors directly involved in the project, the primary goal of this project is to make what we have learned available to as wide an audience as possible. We expect to see impacts as a formal part of the project, but also in other, perhaps surprising ways, due to the use of multiple forms of dissemination. Each strand has the potential to reach beyond the boundaries of the project. We anticipate for example that the textbooks we develop will be used by many students and faculty beyond the immediate adopters of the Paradigms program. And the case studies on the website might be used for training TAs and other teachers. Our visitors will surely infuse our vision with unexpected insights and knowledge that will spin off in new directions. And the information gained by our research into student learning will be available to the entire education research community.
PARADIGMS IN PHYSICS:
INTERACTIVE ELECTROMAGNETIC CURRICULAR MATERIALS
Tevian Dray, Corinne A. Manogue & Emily H. van Zee
This project builds on the joint work of two projects: The Paradigms in Physics Project, a complete redesign of the physics major, and the Vector Calculus Bridge Project, an effort to bridge the gap between the mathematics and physics of vector calculus.
The focus of this project is on the upper-division content in the area of electromagnetism. The goal is to increase the usability of the materials in four distinct ways: improving the effectiveness of the classroom materials; continuing development of a resource wiki, including descriptions of sequence of activities; adding narratives and video of classroom practice; and creating a modular online text. Both the text and the wiki are designed to be modular, allowing maximum flexibility in use. Both also contain a "meta" layer extensively documenting multiple pathways through the individual modules. The wiki further encourages faculty users to design and document alternatives, tailored to the needs of their own students. Pilot versions of all four pieces have been tested by instructors both at Oregon State University and elsewhere; extensive feedback is guiding further development.
This project includes an established team, an experienced science education researcher, recent adopters of the materials, a National Advisory Panel, and an external evaluator.
The primary goal of this project is to provide online resources to a large audience, with most of the resources freely accessible to the general public. In the long run, the materials generated by this project can be used by many students and faculty well beyond the immediate adopters, both in the classroom and for professional development of TAs and other teachers. Furthermore, project research results and case studies are being disseminated to the education research community, not only on the project website, but also through presentations at conferences and publication in appropriate refereed journals.