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1. Symbolic forms analysis of expressions for probability in Dirac and wave function notations for spins-first students

   https://doi.org/10.1103/PhysRevPhysEducRes.21.010105  

   Riihiluoma, William D; Topdemir, Zeynep; Thompson, John R (January 2025, Physical Review Physics Education Research)

   [This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] The ability to relate physical concepts and phenomena to multiple mathematical representations—and to move fluidly between these representations—is a critical outcome expected of physics instruction. In upper-division quantum mechanics, students must work with multiple symbolic notations, including some that they have not previously encountered. Thus, developing the ability to generate and translate expressions in these notations is of great importance, and the extent to which students can relate these expressions to physical quantities and phenomena is crucial to understand. To investigate student understanding of the expressions used in these notations and the ways they relate, clinical think-aloud interviews were conducted with students enrolled in an upper-division quantum mechanics course. Analysis of these interviews used the symbolic forms framework to determine the ways that participants interpret and reason about these expressions. Multiple symbolic forms—internalized connections between symbolic templates and their conceptual interpretations—were identified in both Dirac and wave function notations, suggesting that students develop an understanding of expressions for probability both in terms of their constituent pieces and as larger composite expressions. 

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2. Network analysis of students’ conceptual understanding of mathematical expressions for probability in upper-division quantum mechanics

   https://doi.org/10.1103/PhysRevPhysEducRes.20.020102  

   Riihiluoma, William D; Topdemir, Zeynep; Thompson, John R (July 2024, Physical Review Physics Education Research)

   One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well. 

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3. Using Network Analysis Techniques to Probe Student Understanding of Expressions Across Notations in Quantum Mechanics

   Riihiluoma, William; Topdemir, Zeynep; Thompson, John R. (January 2022, Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Higgins, A (Ed.)

   One important outcome of physics instruction is for students to be capable of relating physical concepts and phenomena to multiple mathematical representations. In quantum mechanics (QM), students are asked to work between multiple symbolic notations, some not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, many of which describe analogous physical concepts, a survey was distributed to students enrolled in upper-division QM courses at multiple institutions. Network analysis techniques were shown to be useful for gaining information about how students relate these expressions. Preliminary analysis suggests that students view Dirac bras and kets as more similar to generic vectors than to their physically analogous wave function counterparts, and that Dirac bras and kets serve as a bridge between vector and wave function expressions. 

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4. Student Understanding of Eigenvalue Equations in Quantum Mechanics: Symbolic Forms Analysis

   Pina, Anthony; Topedmir, Zeynep; Thompson, John R (August 2023, Mathematical Association of America)

   As part of an effort to examine students’ mathematical sensemaking (MSM) in a spins-first quantum mechanics (QM) course, students were asked to construct an eigenvalue equation (EE) for a one-dimensional position operator. Sherin’s symbolic forms were used in analysis. The data suggest three symbolic forms for an EE, all sharing a single symbol template but with unique conceptual schemata: a transformation which reproduces the original, an operation taking a measurement of state, and a statement about the potential results of measurement. These findings corroborate prior literature on a construction task rather than a comparison or deconstruction task, and with a continuous variable after instruction on discrete variables. 

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5. Implementation of a Specifications Grading System in Four Upper Division Chemistry Courses

   https://doi.org/10.1021/acs.jchemed.4c00953  

   Drummond, Michael J; Sager-Smith, LeeAnn M; Fishovitz, Jennifer (January 2025, Journal of Chemical Education)

   Specifications (specs) grading systems use a “checklist” approach to assessing students that asks them to demonstrate a high level of proficiency in course content, often coupled with multiple attempts at revision. Students also must demonstrate mastery in some specs to earn high letter grades. There have been several reports in lower division college chemistry courses that use specs grading systems (e.g., general and organic chemistries), but there remains a dearth of accounts of specs grading systems in upper division courses. In this manuscript, we report on the use of specs grading systems at a primarily undergraduate women’s college in four upper division chemistry courses: biochemistry, inorganic chemistry, thermodynamics, and quantum mechanics. The conceptual framework for designing specs tailored to upper division chemistry courses and their use to assess student understanding of course content are shared along with student outcomes and feedback. The upper division students generally had a positive view of the specs grading system with students viewing themselves as working hard on assessments that were tough but fair. Finally, instructor comments are presented in an effort to highlight the perceived benefits and challenges of specs grading to future adopters. 

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