- Bridging the Vector Calculus Gap
- Bridging the Vector Calculus Gap: Episode II
- Paradigms in Physics: Multiple Entry Points
- Paradigms in Physics: Interactive Electromagnetic Curricular Materials
- Paradigms in Physics: Representations of Partial Derivatives

DUE-0088901

**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.

DUE-0231032

**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".

DUE-0618877

**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:

- New content: We plan to develop textbooks for quantum mechanics and for vector calculus, emphasizing our nonstandard approach to these topics, while encouraging, but not requiring, the use of active engagement.
- Case studies: We plan to expand our existing websites to provide the information necessary for successful adoption of one or more of our activities, showing how to combine lectures and active engagement in a coherent way.
- Community of scholars: We plan to host a small number of visitors, who would be immersed in, and contribute to, the entire Paradigms package.
- Education Research: We plan to do research into students' ability to reason harmonically and metacognitively, and how these skills are affected by our materials.

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.

DUE-1023120

**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.

DUE-1323800

**PARADIGMS IN PHYSICS: REPRESENTATIONS OF PARTIAL DERIVATIVES**

*
Corinne A. Manogue, Tevian Dray, David Roundy, Eric Weber, & Emily H. van Zee
*

This proposal continues the joint work of two very successful projects: The Paradigms in Physics Project, a complete redesign of the physics major, now in its sixteenth year, and the Vector Calculus Bridge Project, an effort to ``bridge the gap'' between the mathematics and physics of vector calculus, now in its twelfth year. Curricular materials produced by these projects, including group activities, instructor's materials, and three published and one online textbook are currently in use at OSU and a number of other institutions.

The next phase of this project looks at representations of the quantification of change, particularly partial derivatives, across many STEM disciplines, with the goal of aiding students in moving toward the robust and multi-faceted understandings typical of STEM professionals. The project will include strands that explore the ways in which STEM experts use and represent change, that develop and test curricular materials for middle-division math and physics courses, that establish students' initial and ongoing levels of understanding as they progress through the curricular materials, and that make these curricular materials freely available online to the education community.

This project will advance knowledge within physics and mathematics education as well as across other science, technology, and engineering fields that engage undergraduates in learning how to use partial derivatives to model changing quantities in complex environments. Success in upper-level undergraduate and graduate courses in these fields requires understanding what partial derivatives are and how to use them. Drawing upon expertise in mathematics, physics, and education, the team is tracing learning trajectories from what novice students write, draw, and say when encountering partial derivatives in upper-level courses through various representations experts use as they identify and interpret ways that variables change under different circumstances. In analyzing such data, the team is extending and adapting ways of thinking from other fields, such as identifying the different ``epistemic games'' students and experts ``play'' when solving problems involving partial derivatives. Based on such research, the curricular materials will include prompts for encouraging metacognition, ways to help students become aware of their own thought processes while transferring their emerging expertise from one context to another.

Led by the PIs of the Paradigms and Bridge projects, the team includes curriculum developers, education researchers, and recent adopters of curriculum materials from previous projects. This team has published 29 papers and 3 books based on previous grants in this ongoing project.

This project will directly impact mathematics and physics education at the middle-division undergraduate level by providing classroom-tested curricular materials and associated instructor resources to the education community through existing, proven online resources (an activities wiki and textbook). Mathematics materials will support learning trajectories in multiple STEM disciplines, not just mathematics and physics. The addition of the new materials will make the existing resources easier to adopt by providing more complete coverage, in line with most common course structures. The project structure itself provides a model of how to advance STEM education holistically, combining an influential national advisory committee with a local interdisciplinary panel of experts drawn from affiliates in OSU's new Center for Research in Lifelong STEM Learning. All of these experts were chosen in part because of their potential to use the intellectual results of the work synergistically in their own related projects.