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1. Conceptualizing Mathematical Transformation as Substitution Equivalence: The Critical Role of Student Definitions.

   Wladis, C.; Sencindiver, B.; Offenholley, K.; Jaffe, E.; Taton, J. (January 2022, Proceedings for the 24th Annual Conference on Research in Undergraduate Mathematics Education)

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

   This theoretical paper explores student conceptions of transformation as substitution equivalence by linking it to their definitions of substitution and equivalence. This work draws on the work of Sfard (1995) to conceptualize substitution equivalence and its components, equivalence and substitution, as a spectrum from computational to structural. We provide examples of students’ work to illustrate how student notions of substitution, equivalence, and substitution equivalence as an approach to justifying transformation may related to one another. 

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2. SE3: Sequential Equivalence Checking for Non-Cycle-Accurate Design Transformations

   Li, You; Zhao, Guannan; He, Yunqi; Zhou, Hai (July 2023, Proceedings ACM IEEE Design Automation Conference)

   In high-level design explorations, many useful optimizations transform a circuit into another with different operating cycles for a better trade-off between performance and resource usage. How to efficiently check their equivalence is critical and challenging since most existing equivalence checkers are designed for cycle-accurate circuits. This paper presents SE3, an efficient sequential equivalence checker without assumption on cycle-accuracy, latch mapping, or I/O interface of the checked circuits. It proves the equivalence of two circuits by computing an equivalence relation between the states of the two circuits and utilizes syntax abstraction to accelerate this process. Experimental results show that SE3 is significantly faster than state-of-the-art sequential equivalence checking algorithms. 

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3. A Model of How Student Definitions of Substitution and Equivalence May Relate to Their Conceptualizations of Algebraic Transformation.

   Wladis, C.; Sencindiver, B.; Offenholley, K.; Jaffe, E.; & Taton, J. (January 2022, Proceedings for the 12th Congress of the European Society for Research in Mathematics Education (CERME12))

   Cho, Sung Je (Ed.)

   This empirical paper explores students’ conceptions of transformation as substitution equivalence by linking it to their definitions of substitution and equivalence. This work draws on Sfard’s (1995) framework to conceptualize conceptions of substitution equivalence and its components, equivalence and substitution, each on a spectrum from computational to structural. We provide examples of student work to illustrate how students’ understandings of substitution, equivalence, and substitution equivalence as an approach to justifying transformation may relate to one another. 

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4. Student Definitions of Equivalence: Structural vs. Operational Conceptions, and Extracted vs. Stipulated Definition Construction.

   Wladis, C.; Sencindiver, B.; Offenholley, K.; Jaffe, E.; & Taton, J. (January 2022, Proceedings for the 12th Congress of the European Society for Research in Mathematics Education (CERME12))

   Cho, Sung Je (Ed.)

   In this study, we describe a model of student thinking around equivalence (conceptualized as any type of equivalence relation), presenting vignettes from student conceptions from various college courses ranging from developmental to linear algebra. In this model, we conceptualize student definitions along a continuous plane with two-dimensions: the extent to which definitions are extracted vs. stipulated; and the extent to which conceptions of equivalence are operational or structural. We present examples to illustrate how this model may help us to recognize ill-defined or operational thinking on the part of students even when they appear to be able to provide “standard” definitions of equivalence, as well as to highlight cases in which students are providing mathematically valid, if non-standard, definitions of equivalence. We hope that this framework will serve as a useful tool for analyzing student work and exploring instructional and curricular handling of equivalence. 

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5. Modeling Student Definitions of Equivalence: Operational vs. Structural Views and Extracted vs. Stipulated Definitions.

   Wladis, C.; Sencindiver, B.; Offenholley, K.; E., & Taton (February 2022, Proceedings for the 24th Annual Conference on Research in Undergraduate Mathematics Education)

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

   This paper describes a model of student thinking around equivalence (conceptualized as any type of equivalence relation), presenting vignettes from student conceptions from various college courses ranging from developmental to linear algebra, and courses in between (e.g., calculus). In this model, we conceptualize student definitions along a continuous plane with two dimensions: the extent to which definitions are extracted vs. stipulated; and the extent to which conceptions of equivalence are operational or structural. We present examples to illustrate how this model may help us to recognize ill-defined or limited thinking on the part of students even when they appear to be able to provide “standard” definitions of equivalence, as well as to highlight cases in which students are providing mathematically valid, if non-standard, definitions of equivalence. We hope that this framework will serve as a useful tool for analyzing student work, as well as exploring instructional and curricular handling of equivalence. 

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