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1. 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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2. 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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3. Development and validation of a scientific (formal) reasoning test for college students

   https://doi.org/10.1002/tea.21555  

   Kalinowski, Steven T. ; Willoughby, Shannon ( April 2019 , Journal of Research in Science Teaching)

   We present a multiple‐choice test, the Montana State University Formal Reasoning Test (FORT), to assess college students' scientific reasoning ability. The test defines scientific reasoning to be equivalent to formal operational reasoning. It contains 20 questions divided evenly among five types of problems: control of variables, hypothesis testing, correlational reasoning, proportional reasoning, and probability. The test development process included the drafting and psychometric analysis of 23 instruments related to formal operational reasoning. These instruments were administered to almost 10,000 students enrolled in introductory science courses at American universities. Questions with high discrimination were identified and assembled into an instrument that was intended to measure the reasoning ability of students across the entire spectrum of abilities in college science courses. We present four types of validity evidence for the FORT. (a) The test has a one‐dimensional psychometric structure consistent with its design. (b) Test scores in an introductory biology course had an empirical reliability of 0.82. (c) Student interviews confirmed responses to the FORT were accurate indications of student thinking. (d) A regression analysis of student learning in an introductory biology course showed that scores on the FORT predicted how well students learned one of the most challenging concepts in biology, natural selection.

    

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4. Syntactic Reasoning and Cognitive Load in Algebra

   Wladis, C. ; Sencindiver, B. ; Offenholley, K. ( February 2023 , Proceedings for the 25th Annual Conference on Research in Undergraduate Mathematics Education, RUME, http://sigmaa.maa.org/rume/Site/Proceedings.html)

   Cook, S. ; Infante, N. (Ed.)

   In the context of proofs, researchers have distinguished between syntactic reasoning and semantic reasoning; however, this distinction has not been well-explored in areas of mathematics education below formal proof, where student reasoning and justification are also important. In this paper we draw on theories of cognitive load and syntactic versus semantic proof-production to explicate a definition for syntactic reasoning outside the context of formal proof, using illustrative examples from algebra. 

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5. 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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