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1. Categorizing student interactions with manipulatives in statics

   Rupe, K. M. ; 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. 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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3. Cognitive agency and computer-based tasks

   Riling, M. ( January 2018 , Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education)

   As computer-focused policies and trends become more popular in schools, more students access math curriculum online. While computer-based programs may be responsive to some student input, their algorithmic basis can make it more difficult for them to be prepared for divergent student thinking, especially in comparison to a teacher. Consider programs that assess student work by judging how well it matches pre-set answers. Unless designed and enacted in classrooms with care, computer-based curriculum materials might encourage students to think about mathematics in pre-determined ways. How do students approach the process of mathematics while using online materials, especially in terms of engaging in original thought? Drawing on Pickering’s (1995) dance of agency and Sinclair’s (2001) conception of students as path-finders or track-takers, I define two modes of mathematical behavior: trail-taking and bushwhacking. While trail-taking, students follow an established approach, often relying on Pickering’s (1995) disciplinary agency, wherein the mathematics “leads [them] through a series of manipulations” (p. 115). The series of manipulations can be seen as a trail that a student may choose to follow. Bushwhacking, on the other hand, refers to actions a student takes of their own invention. It is possible that, unknown to the student, these actions have been taken before by others. In bushwhacking, the student possesses agency, which Pickering (1995) describes as active (rather than passive) and as hallmarked by “choice and discretion” (p. 117). In this study, students worked in several dynamic geometric environments (DGEs) during a geometry lesson about the midline theorem. The lesson was originally recorded as part of a larger study designing mathematically captivating lessons. Students accessed both problems and online addresses for corresponding DGEs via a printed packet. Students interacted with the DGEs on individual laptops, but were seated in groups of three or four. Passages of group conversations in which students transitioned between trail-taking and bushwhacking were selected for closer analysis, which involved identifying evidence of each mode and highlighting the curricular or social forces that may have contributed to shifts between modes. Of particular interest were episodes in which students asked one another to share results, which led to students reconsidering previously set approaches, and episodes in which students interacted with DGEs containing a relatively high proportion of drag-able components, which corresponded to some students working in bushwhacking mode, spontaneously suggesting and revising approaches for manipulating the DGE (e.g., “unless you make this parallel to the bottom, but I don’t think you... yes you can.”). Both types of episodes were found in multiple groups’ conversations. Further analysis of student interactions with tasks, especially with varying levels of student control and sharing, could serve to inform future computer-based task design aimed to encourage students to productively engage in bushwhacking while problem-solving. 

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4. 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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5. 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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