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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. 

Symbolic representation is a cornerstone of mathematics and science. While much research has explored student understanding of various mathematical symbols, there is very little known about the way students con- ceptualize the use of Greek letters in scientific notation. In this work, we share excerpts from interviews with upper-division physics students, which illuminate their experience with Greek symbol use. The students re- ported frequently mistaking pairs of similar-looking Greek and Latin letters. They also felt that their texts and instructors had not sufficiently introduced novel Greek letters or explained what they represented in the equa- tions. The students also disliked how often a single Greek letter would be used in multiple contexts – or that different texts and instructors would not follow the same convention for which letter to use in a single context. These observations suggest that educators should devote additional time to introducing and discussing Greek letters in scientific contexts. 

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. 

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. 

Next Generation Science Standards foreground science practices as important goals of science education. In this paper, we discuss the design of block-based modeling environments for learning experiences that ask students to actively explore complex systems via computer programming. Specifically, we discuss the implications of the design and selection of the types of blocks given to learners in these environments and how they may affect students’ thinking about the process of modeling and theorizing. We conclude with a discussion of some preliminary findings in this design based research to inform design principles for block-based programming of science phenomena as a medium for learning to build theory.