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1. Applying a mathematical sense-making framework to student work and its potential for curriculum design

   https://doi.org/17010138  

   Gifford, J.D. and (May 2021, Physical review)

   null (Ed.)

   This paper extends prior work establishing an operationalized framework of mathematical sense making (MSM) in physics. The framework differentiates between the object being understood (either physical or mathematical) and various tools (physical or mathematical) used to mediate the sense-making process. This results in four modes of MSM that can be coordinated and linked in various ways. Here, the framework is applied to novel modalities of student written work (both short answer and multiple choice). In detailed studies of student reasoning about the photoelectric effect, we associate these MSM modes with particular multiple choice answers, and substantiate this association by linking both the MSM modes and multiple choice answers with finer-grained reasoning elements that students use in solving a specific problem. Through the multiple associations between MSM mode, distributions of reasoning elements, and multiple- choice answers, we confirm the applicability of this framework to analyzing these sparser modalities of student work and its utility for analyzing larger-scale (N > 100) datasets. The association between individual reasoning elements and both MSM modes and MC answers suggest that it is possible to cue particular modes of student reasoning and answer selection. Such findings suggest potential for this framework to be applicable to the analysis and design of curriculum. 

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

   Andrews-Larson, C. (January 2022, Proceedings of the 24th conference on research in undergraduate mathematics education)

   S. S. Karunakaran; A. Higgins (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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5. A Decomposition of the Teaching Practice of Building

   Leatham, K. R.; Van Zoest, L. R.; Peterson, B. E.; Stockero, S. L. (September 2022, NCTM Research Conference)

   We share a decomposition of building on MOSTs—a teaching practice that takes advantage of high-leverage instances of student mathematical contributions made during whole-class interaction. This decomposition resulted from an iterative process of teacher-researchers enacting conceptions of the building teaching practice that were refined based on our study of their enactments. We elaborate the four elements of building: (a) Establish the student mathematics of the MOST as the object to be discussed; (b) Grapple Toss that object in a way that positions the class to make sense of it; (c) Conduct a whole-class discussion that supports the students in making sense of the student mathematics of the MOST; and (d) Make Explicit the important mathematical idea from the discussion. We argue for the value of this practice in improving in-the-moment use of high-leverage student mathematical thinking during instruction. 

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