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Exponential functions are foundational to modeling dynamic phenomena in physics, yet students often strug- gle to integrate their mathematical form with corresponding physical interpretations. This study reports on upper-division physics students’ reasoning about exponential decay in the context of projectile motion with drag. Using the knowledge in pieces framework, we analyze how students activate and coordinate mathematical and conceptual resources during problem-solving. Case studies reveal that while participants demonstrated pro- cedural fluency with exponential expressions, they did not construe these forms as meaningful representations of physical systems. In contrast, polynomial forms elicited stronger conceptual associations, suggesting that curricular familiarity plays a role in resource coordination. These findings underscore a persistent disconnect between symbolic manipulation and physical interpretation in students’ reasoning. We argue for instructional designs that explicitly foster connections between mathematical structure (e.g., ekt) and mechanistic models (e.g., velocity-dependent drag), thereby supporting more integrated and expert-like engagement with exponen- tial functions in physics contexts.
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Upper-division physics simulations with equation manipulation
Digital simulations are powerful instructional tools for physics education. They are often designed to visualize canonical physical phenomena, with adjustable parameters for influencing the system. While this is sufficient for developing conceptual and qualitative intuitions, it does little to help physics students build connections between physical systems and the mathematical models and equations that represent them. We present PhysMath, a suite of interactive physics simulations for use in upper-division courses. These simulations allow students to explore connections between mathematical equations and the phenomena they represent by inputting, modifying, and observing changes in system behavior. In this paper, we describe our first simulation—the Bead-On-Hoop for Classical Mechanics—and report findings from pilot interviews with intermediate physics students interacting with the simulation. Our findings validate the simulations’ design and highlight its potential for scaffolding students’ mathematical sensemaking.
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- Award ID(s):
- 2235569
- PAR ID:
- 10663857
- Publisher / Repository:
- APS
- Date Published:
- Journal Name:
- Physical Review Physics Education Research
- Volume:
- 21
- Issue:
- 1
- ISSN:
- 2469-9896
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Digital simulations are especially helpful in physics education, but most simulations provide only a visual- ization of a phenomenon while obscuring the mathematical relationships that model its behavior. Our team is developing a suite of online simulations called DynamicsLab, which combine visual representations with an ability to input and alter the governing physics equations. Here, we share excerpts from a group of clinical interviews, in which intermediate physics students explored the first iterations of a DynamicsLab simulation of a characteristic problem in Classical Mechanics: the bead on a spinning hoop. The students were given predict- observe-explain prompts to investigate the way they connected the mathematical representation to the physical phenomenon. We highlight three episodes in which students had to revise unsuccessful predictions, and how these instances indicate that engaging with the DynamicsLab simulation encouraged the students to draw upon a more diverse range of knowledge elements to support their physical and mathematical reasoning.more » « less
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Studies have shown that students often struggle to solve quantitative science problems because they fail to make connections between mathematical equations and associated scientific phenomena. This struggle has been attributed to instructors providing limited sensemaking opportunities that connect science and mathematics sensemaking. Our prior research on college instructors teaching population growth shows that the types of sensemaking elicited varied and were organized in different ways. This case study extends the research by exploring sensemaking opportunities about mathematical equations provided by three instructors teaching Gibbs free energy. This study also examines different levels of connection between sensemaking types within and across the science and mathematics dimensions and analyzes factors (i.e., pedagogical approaches, equation types) that might shape the types of sensemaking that were provided. The Sci-Math Sensemaking Framework was used to identify sensemaking opportunities provided by three instructors during the lessons and a comparative case study approach was employed. Findings showed that while each instructor provided both science and mathematics sensemaking opportunities, they had distinct ways of sequencing their sensemaking even when teaching the same scientific phenomenon using mathematical equations. Mathematics sensemaking was mostly presented separately from science sensemaking, with only a few instances of connected sensemaking occurring in instructors’ lessons, albeit at varying levels. When instructors expose students to connected science and mathematics sensemaking, they model for students how to use resources from two different disciplines to understand scientific concepts or solve quantitative problems. We provide evidence to show that levels of connection between sensemaking types within science and mathematics dimensions reflect distinct approaches to developing more robust scientific explanations or to working with equations. Based on these case studies, we suggest that instructors’ equation choices and pedagogical approaches support specific types of sensemaking and levels of connection between sensemaking opportunities.more » « less
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[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.more » « less
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Abstract: In science education, gestures have emerged as a valuable tool for both students to represent their reasoning and educators to assess learners’ developing knowledge. One reason is that gestures afford representing dynamic causal reasoning, a fundamental aspect of understanding various natural phenomena. However, gestures inherently lack feedback mechanisms and are possibly epiphenomenal to learning. We address these limitations by pairing haptic physical models with a computational NetLogo simulation designed to support students in comprehending dynamic interactions within complex systems. In this study, we investigated the impact of haptic learning experiences and computational environments on student-generated gestures. Our findings revealed that haptic learning experiences augment the computational environments by providing students with sensory feedback. This, in turn, leads to a shift in students' gestures, as they transition from representing aggregate patterns, such as quantity and motion, to individual interactions between physical entities and forces.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1103/PhysRevPhysEducRes.21.010151
