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1. Prospective High School Teachers’ Understanding and Application of the Connection Between Congruence and Transformation in Congruence Proofs

   St. Goar, J.; Lai, Y.; Funk, R. (January 2019, Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education)

   Undergraduate mathematics instructors are called by recent standards to promote prospective teachers’ learning of a transformation approach in geometry and its proofs. The novelty of this situation means it is unclear what is involved in prospective teachers’ learning of geometry from a transformation perspective, particularly if they learned geometry from an approach based on the Elements; hence undergraduate instructors may need support in this area. To begin to approach this problem, we analyze the prospective teachers’ use of the conceptual link between congruence and transformation in the context of congruence. We identify several key actions involved in using the definition of congruence in congruence proofs, and we look at ways in which several of these actions are independent of each other, hence pointing to concepts and actions that may need to be specifically addressed in instruction. 

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2. Teaching Geometry for Secondary Teachers: What are the Tensions Instructors Need to Manage?

   https://doi.org/10.1007/s40753-023-00216-0  

   Herbst, Patricio; Brown, Amanda M; Ion, Michael; Margolis, Claudine (August 2024, International Journal of Research in Undergraduate Mathematics Education)

   This paper contributes to understanding the work of teaching the university geometry courses that are taken by prospective secondary teachers. We ask what are the tensions that instructors need to manage as they plan and teach these courses. And we use these tensions to argue that mathematics instruction in geometry courses for secondary teachers includes complexities that go beyond those of other undergraduate mathematics courses–an argument that possibly applies to other mathematics courses for teachers. Building on the notion that the work of teaching involves managing tensions, and relying on interviews of 32 instructors, we characterize 5 tensions (content, experiences, students, instructor, and institutions) that instructors of geometry for teachers manage in their work. We interpret these tensions as emerging from a dialectic between two normative understandings of instruction in these courses, using the instructional triangle to represent these. 

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3. AGREEING ON OBJECTIVES OF GEOMETRY FOR TEACHERS’ COURSES: FEEDBACK FROM INSTRUCTORS ON AN INITIAL LIST

   Ion, M; Herbst, P; Ko, I; Hetrick, C (November 2023, University of Nevada, Reno.)

   Lamberg, T; Moss, D (Ed.)

   We report on an effort to vet a list of 10 student learning objectives (SLOs) for geometry courses taken by prospective geometry teachers. Members of a faculty online learning community, including mathematicians and mathematics educators who teach college geometry courses taken by prospective secondary teachers developed this list in an effort to reach a consensus that might satisfy various stakeholders. To provide feedback on the final list of 10 SLOs, we constructed and collected responses to a survey in which 121 college geometry instructors ranked a set of potential SLOs, including the 10 proposed SLOs as well as 11 distractors. The 10 SLOs were, for the most part, among the highest ranked by the sample. 

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4. Can AI Develop Curriculum? Integrated Computer Science As a Test Case (Research to Practice)

   https://doi.org/10.18260/1-2--56052  

   Smith, Julie; McGill, Monica; Koressel, Jacob; Twarek, Bryan (June 2025, ASEE Conferences)

   Introduction: Because developing integrated computer science (CS) curriculum is a resource-intensive process, there is interest in leveraging the capabilities of AI tools, including large language models (LLMs), to streamline this task. However, given the novelty of LLMs, little is known about their ability to generate appropriate curriculum content. Research Question: How do current LLMs perform on the task of creating appropriate learning activities for integrated computer science education? Methods: We tested two LLMs (Claude 3.5 Sonnet and ChatGPT 4-o) by providing them with a subset of national learning standards for both CS and language arts and asking them to generate a high-level description of learning activities that met standards for both disciplines. Four humans rated the LLM output – using an aggregate rating approach – in terms of (1) whether it met the CS learning standard, (2) whether it met the language arts learning standard, (3) whether it was equitable, and (4) its overall quality. Results: For Claude AI, 52% of the activities met language arts standards, 64% met CS standards, and the average quality rating was middling. For ChatGPT, 75% of the activities met language arts standards, 63% met CS standards, and the average quality rating was low. Virtually all activities from both LLMs were rated as neither actively promoting nor inhibiting equitable instruction. Discussion: Our results suggest that LLMs are not (yet) able to create appropriate learning activities from learning standards. The activities were generally not usable by classroom teachers without further elaboration and/or modification. There were also grammatical errors in the output, something not common with LLM-produced text. Further, standards in one or both disciplines were often not addressed, and the quality of the activities was often low. We conclude with recommendations for the use of LLMs in curriculum development in light of these findings. 

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5. DEFINING KEY DEVELOPMENTAL UNDERSTANDINGS IN CONGRUENCE PROOFS FROM A TRANSFORMATION APPROACH

   St. Goar, J.; Lai, Y. (October 2020, Psychology of Mathematics Education-North American Chapter)

   null (Ed.)

   Previous work by the authors (St. Goar et al., 2019) identified two potential key developmental understandings (KDUs) (Simon, 2006) in the construction of congruence proofs from a transformation perspective for pre-service secondary teachers in an undergraduate geometry course. We hypothesized the independence of the potential KDUs in previous work, meaning that students may have one potential KDU but not the other, and vice versa. We tested this hypothesis with analysis of an expanded data set and found that this hypothesis did not hold in general. We report on the expanded analysis and discuss implications for the scope and limitation of the potential KDUs. 

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