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1. Categorizing student interactions with manipulatives in statics

   Rupe K.; Davishahl, E. (May 2022, 2022 ASEE Zone IV Conference)

   This work in progress paper describes ongoing work to understand the ways in which students make use of manipulatives to develop their representational competence and deepen their conceptual understanding of course content. Representational competence refers to the fluency with which a subject expert can move between different representations of a concept (e.g. mathematical, symbolic, graphical, 2D vs. 3D, pictorial) as appropriate for communication, reasoning, and problem solving. Several hands-on activities for engineering statics have been designed and implemented in face-to-face courses since fall 2016. In the transition to online learning in response to the COVID 19 pandemic, modeling kits were sent home to students so they could work on the activities at their own pace and complete the associated worksheets. An assignment following the vector activities required students to create videotaped or written reflections with annotated pictures using the models to explain their thinking around key concepts. Students made connections between abstract symbolic representations and their physical models to explain concepts such as a general 3D unit vector, the difference between spherical coordinate angles and coordinate direction angles, and the meaning of decomposing a vector into components perpendicular and parallel to a line. Thematic analysis of the video and written data was used to develop codes and identify themes in students’ use of the models as it relates to developing representational competence. The student submissions also informed the design of think-aloud exercises in one-on-one semi-structured interviews between researchers and students that are currently in progress. This paper presents initial work analyzing and discussing themes that emerged from the initial video and written analysis and plans for the subsequent think-aloud interviews, all focused on the specific attributes of the models that students use to make sense of course concepts. The ultimate goal of this work is to develop some general guidelines for the design of manipulatives to support student learning in a variety of STEM topics. 

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2. An Exploration of How Students Make Use of Hands-On Models to Learn Statics Concepts

   Rupe, Kathryn M; Davishahl, Eric; Singleton, Lee; Borowski, Rebecca S. (June 2023, 2023 ASEE National Conference and Exposition)

   This paper describes the results from an ongoing project where hands-on models and associated activities are integrated throughout an undergraduate statics course with the goal of deepening students’ conceptual understanding, scaffolding spatial skills, and therefore developing representational competence with foundational concepts such as vectors, forces, moments, and free-body diagrams. Representational competence refers to the fluency with which a subject expert can move between different representations of a concept (e.g. mathematical, symbolic, graphical, 2D vs. 3D, pictorial) as appropriate for communication, reasoning, and problem solving. This study sought to identify the characteristics of modeling activities that make them effective for all learners. Student volunteers engaged in individual interviews in which they solved problems that included 2D diagrams, 3D models, and worked calculations. Participating students had prior experience with the models and related activity sheets earlier in the course. Data was collected at the end of the quarter and the activities emphasized conceptual understanding. Thematic analysis was used to develop codes and identify themes in students’ use of the models as it relates to developing representational competence. Students used the models in a variety of ways. They wrote directly on the models, touched and gestured with the model, adjusted components, and observed the model from multiple orientations. They added new elements and deconstructed the models to feel the force or imagine how measurements would be impacted if one parameter was changed while all others held constant. In interviews students made connections to previous courses as well as previous activities and experiences with the models. In addition to using the 3D models, participants also used more than one representation (e.g. symbolic or 2D diagram) to solve problems and communicate thinking. While the use of models and manipulatives is commonplace in mechanics instruction, this work seeks to provide more nuanced information about how students use these learning aids to develop and reinforce their own understanding of key concepts. The authors hope these findings will be useful for others interested in designing and refining hands-on mechanics activities toward specific learning goals. 

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3. An Exploration of How Students Make Use of Hands-on Models to Learn Statics Concepts

   Rupe, K. M.; Davishahl, E.; Singleton, L.; Borowski, R. S. (June 2023, 2023 ASEE Annual Conference & Exposition)

   Abstract This paper describes the results from an ongoing project where hands-on models and associated activities are integrated throughout an undergraduate statics course with the goal of deepening students’ conceptual understanding, scaffolding spatial skills, and therefore developing representational competence with foundational concepts such as vectors, forces, moments, and free-body diagrams. Representational competence refers to the fluency with which a subject expert can move between different representations of a concept (e.g. mathematical, symbolic, graphical, 2D vs. 3D, pictorial) as appropriate for communication, reasoning, and problem solving. This study sought to identify the characteristics of modeling activities that make them effective for all learners. Student volunteers engaged in individual interviews in which they solved problems that included 2D diagrams, 3D models, and worked calculations. Participating students had prior experience with the models and related activity sheets earlier in the course. Data was collected at the end of the quarter and the activities emphasized conceptual understanding. Thematic analysis was used to develop codes and identify themes in students’ use of the models as it relates to developing representational competence. Students used the models in a variety of ways. They wrote directly on the models, touched and gestured with the model, adjusted components, and observed the model from multiple orientations. They added new elements and deconstructed the models to feel the force or imagine how measurements would be impacted if one parameter was changed while all others held constant. In interviews students made connections to previous courses as well as previous activities and experiences with the models. In addition to using the 3D models, participants also used more than one representation (e.g. symbolic or 2D diagram) to solve problems and communicate thinking. While the use of models and manipulatives is commonplace in mechanics instruction, this work seeks to provide more nuanced information about how students use these learning aids to develop and reinforce their own understanding of key concepts. The authors hope these findings will be useful for others interested in designing and refining hands-on mechanics activities toward specific learning goals. 

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4. 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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5. Engaging STEM Learners with Hands-on Models to Build Representational Competence

   https://doi.org/10.18260/1-2--34541  

   Davishahl, Eric; Singleton, Lee; Haskell, Todd (June 2020, 2020 ASEE Virtual Annual Conference Content Access)

   Modern 3D printing technology makes it relatively easy and affordable to produce physical models that offer learners concrete representations of otherwise abstract concepts and representations. We hypothesize that integrating hands-on learning with these models into traditionally lecture-dominant courses may help learners develop representational competence, the ability to interpret, switch between, and appropriately use multiple representations of a concept as appropriate for learning, communication and analysis. This approach also offers potential to mitigate difficulties that learners with lower spatial abilities may encounter in STEM courses. Spatial thinking connects to representational competence in that internal mental representations (i.e. visualizations) facilitate work using multiple external representations. A growing body of research indicates well-developed spatial skills are important to student success in many STEM majors, and that students can improve these skills through targeted training. This NSF-IUSE exploration and design project began in fall 2018 and features cross-disciplinary collaboration between engineering, math, and psychology faculty to develop learning activities with 3D-printed models, build the theoretical basis for how they support learning, and assess their effectiveness in the classroom. We are exploring how such models can support learners’ development of conceptual understanding and representational competence in calculus and engineering statics. We are also exploring how to leverage the model-based activities to embed spatial skills training into these courses. The project is addressing these questions through parallel work piloting model-based learning activities in the classroom and by investigating specific attributes of the activities in lab studies and focus groups. To date we have developed and piloted a mature suite of activities covering a variety of topics for both calculus and statics. Class observations and complementary studies in the psychology lab are helping us develop a theoretical framework for using the models in instruction. Close observation of how students use the models to solve problems and as communication tools helps identify effective design elements. We are administering two spatial skills assessments as pre/post instruments: the Purdue Spatial Visualizations Test: Rotations (PSVT:R) in calculus; and the Mental Cutting Test (MCT) in statics. We are also developing strategies and refining approaches for assessing representational competence in both subject areas. Moving forward we will be using these assessments in intervention and control sections of both courses to assess the effectiveness of the models for all learners and subgroups of learners. 

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