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1. Designing Activities to Support Prospective High School Teachers’ Proofs of Congruence from a Transformation Perspective

   https://doi.org/10.1080/10511970.2021.1940403  

   St. Goar, Julia; Lai, Yvonne (July 2021, PRIMUS)

   null (Ed.)

   Undergraduate mathematics instructors are called by many cur- rent standards to promote prospective teachers’ learning of geometry from a transformation perspective, marking a change from previous standards. The novelty of this situation means it is unclear what is involved in undergraduate learning and teaching of geometry from a transformation perspective. To approach this problem, we illustrate how specific in-class activities and design principles might help prospective teachers make conceptual links between congruence proofs and a transformation approach to geometry. Additionally, to illustrate these activities for instruc- tors, we provide examples of prospective teachers’ work on some of these problems. 

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2. 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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3. 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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4. 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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5. A Modular Associative Commutative (AC) Congruence Closure Algorithm

   https://doi.org/10.4230/lipics.fscd.2021.15  

   Kapur, Deepak (January 2021, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kobayashi, Naoki (Ed.)

   Algorithms for computing congruence closure of ground equations over uninterpreted symbols and interpreted symbols satisfying associativity and commutativity (AC) properties are proposed. The algorithms are based on a framework for computing the congruence closure by abstracting nonflat terms by constants as proposed first in Kapur’s congruence closure algorithm (RTA97). The framework is general, flexible, and has been extended also to develop congruence closure algorithms for the cases when associative-commutative function symbols can have additional properties including idempotency, nilpotency and/or have identities, as well as their various combinations. The algorithms are modular; their correctness and termination proofs are simple, exploiting modularity. Unlike earlier algorithms, the proposed algorithms neither rely on complex AC compatible well-founded orderings on nonvariable terms nor need to use the associative-commutative unification and extension rules in completion for generating canonical rewrite systems for congruence closures. They are particularly suited for integrating into Satisfiability modulo Theories (SMT) solvers. 

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