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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. 

Substitution is a key idea that is woven throughout the mathematics curriculum. In secondary school, substitution is described as an interchangeability of equal numbers, and then as a method for finding solutions of systems of equations. In university, substitution is used as a means to recognize familiar structures in Integral Calculus. Despite its prevalence in mathematics, there is little research on substitution, especially on students’ understanding of substitution. This work aims to investigate students’ meanings for substitution, and how they use it. We draw on Tall and Vinner’s (1981) ideas of concept definition and concept image to explore students’ meanings of substitution through their personal definitions of substitution, what they identify as substitution, and how they perform substitution. In this presentation, we report on elementary algebra students’ responses to questions about substitution. Data comes includes written responses to multiple-choice and open-ended questions and transcripts from clinical interviews across multiple semesters at a community college. Through a combination of thematic and conceptual analysis, we categorized students’ thinking about substitution and what features appeared to impact how they enact it. We found that students often identify substitution as a process of replacement of one mathematical object for another but differ in the generality of the mathematical objects that they consider (e.g., strictly as the replacement of a number for a variable versus replacement of any expression for another expression). Students further differed in whether or not they thought that substitution entailed equivalence of the objects being replaced. When performing substitution (e.g., substituting x+1 for y in 2y^2), we found that students’ activity was heavily based on their understanding of the structure of the expression where the substitution is taking place (the unified ‘pieces’ of 2y^2). In addition to other findings, we elaborate on the mental processes that students engage in when performing substitution and synthesize our findings with the notion of substitution equivalence (Wladis et al., 2020).