Bulletin of the American Physical Society
APS April Meeting 2023
Volume 68, Number 6
Minneapolis, Minnesota (Apr 15-18)
Virtual (Apr 24-26); Time Zone: Central Time
Session H17: Research on Mathematical Reasoning in PhysicsEducation
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Sponsoring Units: GPER Chair: Eric Kuo, University of Illinois Urbana-Champaign Room: Marquette VIII - 2nd Floor |
Sunday, April 16, 2023 1:30PM - 1:42PM |
H17.00001: Developing and Validating a Computational Thinking Assessment Instrument for Introductory Physics Justin Gambrell, Eric Brewe Computational thinking in physics (CTIP) is an essential skill contributing to physics literacy. In a previous study we interviewed physicists in industry and academia (N=26), all participants believed that computation should be integrated in introductory mechanics classes. Many institutions have integrated computation into their courses, but the curricula varies based on resources and instructor time among others. Ultimately, in order to support the integration of computation into introductory physics we will need means of assessing CTIP. Our goal is to develop an assessment of computational thinking in physics. Based on our previous work, we identified three approaches to assessing CTIP: comprehension- reading and commenting code; skills- giving students a program to edit or create; or attitudes- collecting data on how attitudes change over time regarding CTIP. We begin by developing an assessment that tests students' comprehension of CTIP. The VPython/Python computational language is used in the assessment as it is the most common for introductory physics. The assessment design is principally multiple choice methods to provide an easier analysis for users. We present preliminary results from giving the assessment to a pilot group of students with the intent of further validating the assessment tool. |
Sunday, April 16, 2023 1:42PM - 1:54PM |
H17.00002: Investigation into the mathematical preparation of introductory physics students David E Meltzer, Dakota H King, John D Byrd We summarize key findings of our six-year investigation into the mathematical preparation of students in introductory physics courses. After administering written and online diagnostic tests to over 7000 students at five campuses of four universities, and carrying out about 90 individual interviews, we find several consistent themes: (1) difficulties with basic pre-college operations involving trigonometry, algebra, and graphing are widespread; (2) replacing numbers with algebraic symbols significantly decreases students' problem-solving success rate; (3) most students lack familiarity with physical units or appreciation for their essential role in physics problems; ( 4) most students attempt to "arithmetize" algebraic operations by premature substitution of numerical values, decreasing their ability to check physical units or individual steps; (5) difficulties with different types of operations are highly correlated, in that difficulties with trigonometry imply difficulties with algebra, etc.; (6) evidence suggests that students' success on the mathematics diagnostic is closely linked to overall course success. |
Sunday, April 16, 2023 1:54PM - 2:06PM |
H17.00003: Comparison of evaluation strategies in physics problem solving between first- and third-year students. Abolaji Akinyemi, John R Thompson, Michael E Loverude One expected student outcome of physics instruction is a set of quantitative reasoning skills that include evaluation of problem solutions. As part of a larger project, we developed and administered tasks to physics students that probe their use of these kinds of evaluation strategies. In a pair interview setting, we asked first-year students and juniors to evaluate expressions for the final velocities of two skaters involved in a one-dimensional elastic collision. The techniques used by the two groups show the differences between novice and intermediate versions of certain evaluation strategies. To do this, we focus on the role of algebra and mathematical operations in the checking process, how the students seem to view equations, and the different ways numbers are plugged into the given equations. We observed a few characteristic differences between the groups which showed an evolution in strategy implementation. For instance, the first-year students plugged exact numbers into the given expression (e.g., m1=2kg, m2=5kg) while the juniors plugged in ratios of numbers (e.g., m1=100m2). Other differences include the way knowledge of physics is used, how students seem to view equations, and the role of algebra and mathematical operations while evaluating the given expression. |
Sunday, April 16, 2023 2:06PM - 2:18PM |
H17.00004: A Study of Physics Students' Interpretation of Matrix Multiplication Michael E Loverude, Pachi Her As part of a multi-year project to investigate student use of mathematics in upper-division physics, we have studied student reasoning with matrix multiplication. Matrix multiplication and eigentheory are used by physics students in courses including classical mechanics, optics, and quantum mechanics. While there have been recent investigations of student use of linear algebra in physics, prior work has focused largely on quantum mechanics. We have investigated student use of matrix multiplication as a modeling tool as well as the extent to which physics students' interpretations of matrix multiplication were consistent with research in mathematics courses. In particular, we used the categories developed by Larson and Zandieh (2013) to describe math students' interpretations of matrix multiplication: linear combination, system of equations, and transformation reasoning. We examined the responses of physics students to written questions and used the results to develop an interview protocol to examine how physics students interpret matrix equations using the three interpretations to classify their responses. Results are shown from an interview sample of six undergraduate physics majors. |
Sunday, April 16, 2023 2:18PM - 2:30PM |
H17.00005: Student Understanding of Eigenvalue Equations in Quantum Mechanics: Symbolic Forms Analysis Anthony Pina, Zeynep Topdemir, John R Thompson Prior research on the use of mathematics in physics, primarily at the introductory level, has demonstrated that students think about mathematical tools, operations, and structures differently in mathematics and physics problem solving. Quantum mechanics is flush with opportunities to observe physics students use of more advanced mathematics. Eigentheory is central to the mathematization of quantum mechanics. As part of an effort to examine students’ mathematical sensemaking in a “spins-first quantum” mechanics course, students at two institutions were asked to construct an eigenvalue equation for a one-dimensional position operator on an in class written assignment. Sherin’s symbolic forms framework guided the deconstruction of student responses into symbol templates and conceptual schemata. Analysis yielded three symbolic forms for an eigenvalue equation, all sharing a single symbol template but with unique conceptual schemata, as well as an unproductive application of Sherin’s parts-of-a-whole form. Interview data complements a subset of these findings. Our results corroborate prior literature on a construction task rather than a comparison or deconstruction task, and with a continuous variable after instruction on discrete systems. |
Sunday, April 16, 2023 2:30PM - 2:42PM |
H17.00006: Comparing student understanding of quantum mechanics notations and expressions across curricula William D Riihiluoma, Zeynep Topdemir, John R Thompson 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, being able to translate expressions between these notations is an important skill to develop, and the extent to which students are able to relate these expressions is crucial to understand. To investigate student understanding of the relationships between expressions used in these various notations, a survey was distributed to students in upper-division quantum mechanics courses at multiple institutions over two years. These courses included those structured as "spins-first" and "wave functions-first." Network analysis techniques were used to investigate student understanding of common expressions used in these courses, in both Dirac and wave function notation. Comparing results between students in spins-first and wave function-first courses suggests that the former appear to conceptualize Dirac expressions as more vector-like than the latter, consistent with common instructional emphasis. |
Sunday, April 16, 2023 2:42PM - 2:54PM |
H17.00007: Student Understanding of Constituent Derivatives of Divergence and Curl from Vector Field Diagrams Zeynep Topdemir, Michael E Loverude, John R Thompson Vector calculus is central to several physics topics; derivatives of vector fields represent meaningful quantities. Previous research explored student understanding of divergence and curl of vector field diagrams. In order to examine how student understanding of constituent derivatives of divergence and curl relates to the determination of divergence and curl, we designed tasks providing a two-dimensional vector field representation and asking about the signs of the divergence and curl as well as their constituent derivatives at one or more locations. Written data were collected in Mathematical Methods in Physics courses, after relevant vector calculus instruction, at two universities. We observed different patterns of responses for the derivatives comprising the divergence and those comprising the curl. For the latter, many students struggled to identify which vector field component was described by a given derivative. Similar confusion was observed in responses to tasks in which the vector field had a single component. |
Sunday, April 16, 2023 2:54PM - 3:06PM |
H17.00008: Student reasoning about the signs of backward definite integrals in mathematics and physics John R Thompson, Rabindra R Bajracharya, Vicky L Sealey Researchers have documented student difficulties with the signs of definite integrals in both physics and calculus, particularly negative integrals. Less studied are how students reason about the sign of a “backward definite integral”, i.e., taken right to left, which is important in several physical contexts. We report on two related studies that explore student reasoning about backward integrals in graphical and/or symbolic representations. Our analysis uses the concept image framework and a recent categorical framework for mathematical sense making. The concept images of dx and ?x affected student reasoning. We found four prevalent ways students made sense of the integral signs: invoking the Fundamental Theorem of Calculus, using macroscopic area under the curve (spatial or graphical), considering the Riemann sum (microscopic area, both types), and using a physical context. Students were better able to interpret backward integrals when they linked the mathematical concepts of integrals to some physical context beyond spatial area, whether self-generated or prompted. The context seemed to provide meaning to the difference represented by ?x or dx and thus to the sign of that difference and the definite integral. |
Sunday, April 16, 2023 3:06PM - 3:18PM |
H17.00009: Using Ego Network Analysis Techniques to Explore Student Mistakes in Vector Addition and Subtraction Nekeisha Johnson, John B Buncher Physics students of all levels need to be able to manipulate vectors to succeed in courses. Building on many prior studies that have looked at student ability to use vectors and vector skills like addition, subtraction, and multiplication, we use network analysis techniques to explore the relationship between students' performance on vector addition and vector subtraction questions. To this end, we surveyed introductory algebra-based students using a multiple-choice assessment that prompted them to do many addition and subtraction questions of vectors in different alignments. Using ego network analysis on this multiple-choice data, in conjunction with handwritten versions of the same questions, we investigate the intersection of different student mistakes. Results of this analysis will be discussed, as well as implications for teaching. |
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