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1. Closed Loops and RREF as Conceptual Resources for Reasoning about Null Spaces

   Andrews-Larson, C.; Mauntel, M.; Plaxco, D.; Watford, M.; Smith, J.; Kim, M. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Higgins, A. (Ed.)

   Vector spaces are often taught with an axiomatic focus, but this has been shown to rely on knowledge many students have not yet developed. In this paper, we examine two students’ conceptual resources for reasoning about null spaces drawing on data from a paired teaching experiment. The task sequence is set in the context of a school with one directional hallways. Students’ informal reasoning about paths that leave the room populations unchanged supported more formal reasoning about null spaces. We found that one student used context-based resources (such as ‘loops’ in hallway) to reason about null spaces, while the other student drew largely on previously formalized mathematical resources (e.g. free variables, linear dependence). The use of formal resources sometimes required recontextualization, which may function to constrain student sense-making or afford opportunities for broadening students’ formal prior knowledge. 

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2. Contextual and Mathematical Conceptual Resources for Reasoning about Null Spaces

   Andrews-Larson, C.; Mauntel, M.; Plaxco, D.; Watford, M.; Smith, J.; Kim, M. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S.; Higgins, A. (Ed.)

   Vector spaces are often taught with an axiomatic focus, but this has been shown to rely on knowledge many students have not yet developed. In this paper, we examine two students’ conceptual resources for reasoning about null spaces drawing on data from a paired teaching experiment. The task sequence is set in the context of a school with one directional hallways. Students’ informal reasoning about paths that leave the room populations unchanged supported more formal reasoning about null spaces. We found that one student used context-based resources (such as ‘loops’ in hallway) to reason about null spaces, while the other student drew largely on previously formalized mathematical resources (e.g. free variables, linear dependence). The use of formal resources sometimes required recontextualization, which may function to constrain student sense-making or afford opportunities for broadening students’ formal prior knowledge. 

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3. What Epistemological Resources Affect Chemistry Teachers’ Sense of "What Worked"

   Schafer, AGL; Stowe, RL. (January 2021, 2021 NARST International Meeting)

   null (Ed.)

   Teachers’ sense of “what is taking place with respect to knowledge” drives their perspective on “what works” and “what is likely to work” in their classroom context. Scholarship by Hammer, Russ and many others indicates that this “sense” is very often context-sensitive and may be productively modeled as a local coalescence of small-grained epistemological resources. Presented here is an investigation of the epistemological resources contributing to high school chemistry teachers’ framing of “what works” in their learning environment. Teacher reflections are unpacked and characterized for classroom information noticed and responded to when considering “what worked” during the 2019-2020 school year. Preliminary findings suggest epistemological resources guiding “what worked” often align with a view of knowledge as propagated stuff. Thus, implying that teachers’ reasoning about “what works” is guided by how well knowledge is transferred to students. Also present was evidence that epistemological resources aligned to views of knowledge as fabricated stuff were activated. The perspective that knowledge is inferred or developed from other knowledge, rather than passed from an authority figure, aligns well with reform efforts that emphasize student sensemaking. This study is part of a larger program in which a teacher-researcher collaborative adapts and refines evidence-based curricular materials for an undergraduate chemistry course for use in high school. These materials are structured around scaffolded progressions of big ideas (e.g., energy, electrostatic and bonding interactions) that build in complexity as students make sense of increasingly complex phenomena. Ongoing improvement of transformed materials is dependent upon the ability to initiate and stabilize a sense of “what works” consistent with sensemaking aims. 

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4. Introducing Coding through Tabletop Board Games and Their Digital Instantiations across Elementary Classrooms and School Libraries

   https://doi.org/10.1145/3328778.3366917  

   Lee, Victor R.; Poole, Frederick; Clarke-Midura, Jody; Recker, Mimi; Rasmussen, Melissa (February 2020, SIGCSE '20: Proceedings of the 51st ACM Technical Symposium on Computer Science Education)

   This experience report describes an approach for helping elementary schools integrate computational thinking and coding by leveraging existing resources and infrastructure that do not rely on 1-1 computing. A particular focus is using the school library and media center as a site to complement and enhance classroom instruction on coding. Further, our approach builds upon "unplugged" knowledge and practices that are already familiar to and motivating for students, in this case tabletop board games. Through these games, students can use their prior knowledge and ease with tabletop gaming mechanics to cue relevant ideas for core computational concepts. We describe a model and an instructional unit spanning across classroom and school library settings that builds upon board game play as a source domain for computing knowledge. Building on expansive framing, the model emphasizes instructional linkages being made between one domain (the tabletop board game) and another (specially designed Scratch project shells with partially complete code blocks) such that the reasoning activities and different contexts are seen as instantiations of the same encompassing context. We present the experiences of three elementary school teachers as they implemented the unit in their classrooms and with their school librarian. We also show initial findings on the impact of the unit on student interest (N=87), as measured by pre- and post- surveys. We conclude with lessons learned about ways to improve the unit and future classroom implementations. 

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5. “More conceptual than actual”: Epistemic metacognition in response to a non‐numerical statics question

   https://doi.org/10.1002/jee.70035  

   Grundy, Lorena S; Koretsky, Milo D (September 2025, Journal of Engineering Education)

   Abstract BackgroundMetacognitive processes have been linked to the development of conceptual knowledge in STEM courses, but previous work has centered on the regulatory aspects of metacognition. PurposeWe interrogated the relationship between epistemic metacognition and conceptual knowledge in engineering statics courses across six universities by asking students a difficult concept question with concurrent reflection prompts that elicited their metacognitive thinking. MethodWe used a mixed‐methods design containing an embedded phase followed by an explanatory phase. This design allowed us to both prompt and measure student epistemic metacognition within the learning context. The embedded phase consisted of quantitative and qualitative analyses of student responses. The explanatory phase consisted of an analysis of six instructor interviews. ResultsAnalysis of 267 student responses showed greater variation in students' epistemic metacognition than in their ability to answer correctly. Students used different kinds of epistemic metacognitive resources about the nature and origin of knowledge, epistemological forms, epistemological activities, and stances toward knowledge. These resources generally assembled into one of two frames: aconstructed knowledge framingvaluing conceptual knowledge and sense‐making, and anauthoritative knowledge framingforegrounding numerical, algorithmic problem‐solving. All six instructors interviewed described resources that align with both frames, and none explicitly considered student epistemic metacognition. ConclusionsInstructors' explicit attention to epistemic metacognition can potentially shift students to more productive frames for engineering learning. Findings here also inform two broader issues in STEM instruction: student resistance to active learning, and the direct instruction versus inquiry‐based learning debate. 

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