[This paper is part of the Focused Collection on Upper Division Physics Courses.] The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals. In this study, we focus on three notational systems: Dirac notation, algebraic wavefunction notation, and matrix notation. These quantum notations must include information about basis states and their associated complex probability amplitudes. In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules. We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system. We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation. These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.
Students in introductory physics courses are likely to have views about physics that differ from those of experts. However, students who continue to study physics eventually become experts themselves. Presumably these students either possess or develop more expertlike views. To investigate this process, the views of introductory physics students majoring in physics are compared with the views of introductory physics students majoring in engineering. In addition, the views of physics majors are assessed at various stages of degree progress. The Colorado learning attitudes about science survey is used to evaluate students’ views about physics, and students’ overall survey scores and responses to individual survey items are analyzed. Beginning physics majors are significantly more expertlike than nonmajors in introductory physics courses, and this high level of sophistication is consistent for most of undergraduate study
Representations in physics possess both physical and conceptual aspects that are fundamentally intertwined and can interact to support or hinder sense making and computation. We use distributed cognition and the theory of conceptual blending with material anchors to interpret the roles of conceptual and material features of representations in students' use of representations for computation. We focus on the vector-arrows representation of electric fields and describe this representation as a conceptual blend of electric field concepts, physical space, and the material features of the representation (i.e., the physical writing and the surface upon which it is drawn). In this representation, spatial extent (e.g., distance on paper) is used to represent both distances in coordinate space and magnitudes of electric field vectors. In conceptual blending theory, this conflation is described as a clash between the input spaces in the blend. We explore the benefits and drawbacks of this clash, as well as other features of this representation. This analysis is illustrated with examples from clinical problem-solving interviews with upper-division physics majors. We see that while these intermediate physics students make a variety of errors using this representation, they also use the geometric features of the representation to add electric field contributions and to organize the problem situation productively.
Recent research results have failed to support the conventionally held belief that students learn physics best from hands-on experiences with physical equipment. Rather, studies have found that students who perform similar experiments with computer simulations perform as well or better on measures of conceptual understanding than their peers who used physical equipment. In this study, we explored how university-level nonscience majors' understanding of the physics concepts related to pulleys was supported by experimentation with real pulleys and a computer simulation of pulleys. We report that when students use one type of manipulative (physical or virtual), the comparison is influenced both by the concept studied and the timing of the post-test. Students performed similarly on questions related to force and mechanical advantage regardless of the type of equipment used. On the other hand, students who used the computer simulation performed better on questions related to work immediately after completing the activities; however, the two groups performed similarly on the work questions on a test given one week later. Additionally, both sequences of experimentation (physical-virtual and virtual-physical) equally supported students' understanding of all of the concepts. These results suggest that both the concept learned and the stability of learning gains should continue to be explored to improve educators' ability to select the best learning experience for a given topic.
A hallmark of physics is its rich use of representations. The most common types used by physicists are mathematical representations such as equations, but many problems are rendered more tractable through the use of other representations such as diagrams or graphs. Examples of representations include force diagrams in mechanics, state diagrams in thermodynamics, and motion graphs in kinematics. Most introductory physics courses teach students to use these representations as they apply physical models to problems. But does student representation use correlate with problem-solving success? In this paper we address this question by analyzing student representation usage during the first semester of an introductory physics course for biologists taught in an active-learning setting.
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