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Abstract Title: Representation Issues: Using Mathematics in Upper-Division Physics
Abstract: Upper-division students must learn to apply sophisticated mathematics from algebra, limits, calculus, multi-variable and vector calculus, linear algebra, complex variables, and ordinary and partial differential equations. The presenters in this session will discuss how the representations that we choose may affect whether students are able to use this mathematics spontaneously and correctly, whether they can move smoothly between representations, and the extent to which their understanding of the mathematics enhances their understanding of the physics. The discussant will incorporate the perspective of research in undergraduate mathematics education as it applies to the representations that have been presented.
Abstract Type: Poster Gallery Session

Author/Organizer Information

Primary Contact: Corinne A. Manogue
Oregon State University
Department of Physics
Weniger 301
Corvallis, OR 97331-6507
Phone: 541-737-1695
Fax: 541-737-1683
and Co-Presenter(s)
John Thompson, University of Maine
Fr. Joseph Wagner, Xavier University

Poster Gallery Session Specific Information

Poster 1 Title: Graphical Representations of Vector Functions in Upper-division E&M
Poster 1 Authors: Edward Price, Cal. State Univ. San Marcos
Elizabeth Gire, University of Memphis
Poster 1 Abstract: In upper division electricity and magnetism, the manipulation and interpretation of vector functions is pervasive and a significant challenge to students. At CSU San Marcos, using in-class activities adapted from the Oregon State University Paradigms in Physics Curriculum, students' difficulties with vector functions become evident in two types of in-class activities: sketching vector functions and relating vector and scalar functions (e.g., electric field and electric potential). For many students, the cause of these difficulties is a failure to fully distinguish between the components of a vector function and its coordinate variables. To address this difficulty, we implement an additional in-class activity requiring students to translate between graphical and algebraic representations of vector functions. We present our experience with these issues, how to address them, and how in-class activities can provide evidence of student thinking that facilitates curricular refinement.
Poster 2 Title: What a Mathematical Representation Tells Us About Reasoning About Integrals
Poster 2 Authors: Michael C. Wittmann, University of Maine
Poster 2 Abstract: When solving two integrals arising from the separation of variables in a first order linear differential equation, students have multiple correct choices for how to proceed. They might set limits on both integrals, use integration constants on both, or on only one equation. In each case, the physical meaning of the mathematics is equivalent. But, how students choose to represent the mathematics can tell us much about what they are thinking. Typically, limits indicate more physical thinking, constants more mathematical thinking. Furthermore, evidence for the way they interpret the physical situation is found in choices of limits or constants. At the University of Maine, using a resources (knowledge-in-pieces) framework, we organize these results to show the many skills needed in solving seemingly simple integrals.
Poster 3 Title: Representations of Partial Derivatives in Thermodynamics
Poster 3 Authors: John R. Thompson (University of Maine)
David Roundy (Oregon State University)
Donald B. Mountcastle (University of Maine)
Poster 3 Abstract: One of the mathematical objects that students become familiar with in thermodynamics, often for the first time, is the partial derivative of a multivariable function. The symbolic representation of a partial derivative and related quantities present difficulties for students in both mathematical and physical contexts, most notably what it means to keep one or more variables fixed while taking the derivative with respect to a different variable. Material properties are themselves written as partial derivatives of various state functions (e.g., compressibility is a partial derivative of volume with respect to pressure).  Research in courses at the University of Maine and Oregon State University yields findings related to the many ways that partial derivatives can be represented and interpreted in thermodynamics. Research has informed curricular development that elicits many of the difficulties using different representations (e.g., geometric) and different contexts (e.g., connecting partial derivatives to specific experiments).
Poster 4 Title: Representations for a Spins-First Approach to Quantum Mechanics
Poster 4 Authors: Corinne Manogue, Oregon State University
Elizabeth Gire, University of Memphis
David McIntyre, Oregon State University
Janet Tate, Oregon State University
Poster 4 Abstract: In the Paradigms in Physics Curriculum at Oregon State University, we take a spins-first approach to quantum mechanics using a java simulation of successive Stern-Gerlach experiments to explore the postulates.  The experimental schematic is a diagrammatic representation that we use throughout our discussion of quantum measurements.  With a spins-first approach, it is natural to start with Dirac bra-ket language for states, observables, and projection operators.  We also use explicit matrix representations of operators and ask students to translate between the Dirac and matrix languages.  The projection of the state onto a basis is represented with a histogram.  When we subsequently introduce wave functions, the wave function attains a natural interpretation as the continuous limit of these discrete histograms or of a projection of a Dirac ket onto position or momentum eigenstates.  We are able to test the students' facility with moving between these representations in later modules.