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Here we investigate how college students may conceptualize symbolic algebraic properties. This work uses the theory of Grundvorstellungen (GVs) to analyze how learners’ conceptions may or may not align with some desired goals of instruction. Through the analysis of interviews with students across a variety of courses, we describe several categories of conceptions, or descriptive GVs, that emerged in the data. We expect these categorizations to be a helpful first step in understanding learners’ thinking and improving instruction on algebraic properties. 

Here we explore how college students across different courses may conceptualize symbolic algebraic properties. This work draws on the theory of Grundvorstellungen (GVs) as a tool to analyse how learner conceptions do or do not align with some desired goals of instruction. In analysing interviews, several categories of conceptions, or descriptive GVs, emerged, which may be a helpful first step in understanding learners’ thinking and improving instruction on algebraic properties. 

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. 

In the context of proofs, researchers have distinguished between syntactic reasoning and semantic reasoning; however, this distinction has not been well-explored in areas of mathematics education below formal proof, where student reasoning and justification are also important. In this paper we draw on theories of cognitive load and syntactic versus semantic proof-production to explicate a definition for syntactic reasoning outside the context of formal proof, using illustrative examples from algebra. 

In this paper we explore how college students across different courses appeared to interpret the meaning of parentheses or brackets in the context of algebraic syntax. This work was influenced by theories of computational vs structural thinking, and also considered the extent to which students’ definitions, computational work, and explanations appeared to be consistent with specific normative definitions of parentheses. In analyzing student work, several categories of students’ conceptions emerged, which may be helpful in diagnosing which conceptions may be more productive or problematic as students progress through algebra. For students who appear to conceptualize parentheses as a cue to non-normative procedures, several categories of procedures were found, which could have implications for instruction. 

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. 

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. 

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. 

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. 

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.