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1. A model of students’ conceptions of equivalence.

   Wladis, C.; Offenholley, K.; Sencindiver, B. (July 2021, Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education)

   Inprasitha, Maitree; Changsri, Narumon; Boonsena, Nisakorn (Ed.)

   In mathematics education, much research has focused on studying how students think about the equals sign, but equality is just one example of the larger concept of equivalence, which occurs extensively throughout the K-16 mathematics curriculum. Yet research on how students think about broader notions of equivalence is limited. We present a model of students’ thinking that is informed by Sfard’s theories of the Genesis of Mathematical Objects, in which she distinguishes between operational versus structural thinking (e.g., 1995), which we conceptualize as a continuum rather than a binary categorization. Sfard also describes a pseudostructural conception, in which the objects that a student conceptualizes are not the reification of a process. We combine Sfard’s theory with a categorization of the source of students’ definitions, where stipulated definitions are given a priori and can be explicitly consulted when determining whether something fits the definitions, while extracted definitions are constructed from repeated observation of usage (Edwards & Ward, 2004). We combine these theories with inductive coding of data (open-ended questions, multiple-choice questions, and cognitive interviews) collected from thousands of students enrolled in a range of mathematics classes in college in the US, to generate categories of students’ thinking around equivalence. We see this model as a tool for analysing students’ work to better understand how students conceptualize equivalence. With this model we hope to begin a conversation about how students tend to conceptualize equivalence at various levels, as well as the ways in which equivalence is or is not explicitly addressed currently in curricula and instruction, and what consequences this might have for students’ conceptions of equivalence. 

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2. 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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3. 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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4. Reconceptualizing Algebraic Transformation as a Process of Substitution Equivalence

   Wladis, C.; Sencindiver, B.; Offenholley, K. (February 2023, Proceedings for the 25th Annual Conference on Research in Undergraduate Mathematics Education)

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

   In this theoretical paper, we describe how algebraic transformation could be reconceptualized as a process of substitution equivalence, and we discuss how this conceptualization affords mathematical justification of transformation processes. In particular, we describe a model which deconstructs the process of substitution equivalence into core subdomains which could be learned serially and then re-integrated, in order to make them accessible to students with lower prior knowledge in syntactic reasoning. Our aim in presenting this model is to start a conversation about what the core components of knowledge might be in order for students to reason about and justify algebraic transformation using symbolic representations. 

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5. Harmony and Dissonance: An Enactivist Analysis of the Struggle for Sense Making in Problem Solving

   Greenstein, S.; Pomponio, E.; & Akuom, D. (January 2021, Proceedings of the 43rd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education)

   Olanoff, D.; Johnson, K.; & Spitzer, S.M. (Ed.)

   This work seeks to understand the emergent nature of mathematical activity mediated by learners’ engagement with multiple artifacts. We explored the problem solving of two learners as they aimed to make sense of fraction division by coordinating meanings across two artifacts, one being a physical manipulative and the other a written expression of the standard algorithm. In addressing the question, “How do learners make sense of and coordinate meanings across multiple representations of mathematical ideas?” we took an enactivist perspective and used tools of semiotics to analyze the ways they navigated the dissonance that arose as they sought to achieve harmony in meanings across multiple representations of ideas. Our findings reveal the value of such tool-mediated engagement as well as the complexity of problem solving more broadly. Implications for learning mathematics with multiple artifacts are discussed. 

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