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…

Isomorphism and homomorphism are topics central to abstract algebra, but research on mathematicians' views of these topics, especially with respect to sameness, remains limited. This study examines 197 mathematicians' views of how sameness could be helpful or harmful when studying isomorphism and homomorphism. Instructors saw benefits to…

This paper is a part of a larger research study that we are conducting to develop a framework consisting of the data-driven best practices around teaching and learning algebra in K-12 classrooms. The purpose of this paper is to develop a usable definition for algebra interventions for the second stage of screening within the larger study, which…

Research on dyscalculia has focused almost exclusively on elementary-aged students' deficits in speed and accuracy in arithmetic calculation. This case study expands our understanding of dyscalculia by documenting how one college student with dyscalculia understood algebra during a one-on-one design experiment. A detailed case study of 19 video…

Units coordination, defined by Steffe (1992) as the mental distribution of one composite unit (i.e., a unit of units) "over the elements of another composite unit" (p. 264) is a powerful tool for modeling students' mathematical thinking in the context of whole number and fractional reasoning. This paper proposes extending the idea of a…

Worked examples have been shown to improve student learning in algebra. However, less is known about how to design worked examples to support student learning in online settings. We explore how college students react to worked examples that vary in their degree of extensiveness and dynamicness. In an online, within-subjects study, 109 college…

In this paper, I propose a new construct named "analytic equation sense" to conceptually model a desired way of reasoning that involves students' algebraic manipulations and use of equivalent expressions. Building from the analysis of two existing models in the field, I argue for the need for a new model and use empirical evidence to…

Emergent graphical shape thinking (EGST) entails conceiving a graph as being dynamically generated via the trace of a moving point constrained by two changing quantities. As such, Paoletti et al. (2023) argue that meanings for quantities within a situation and meanings for graphical representations must be connected, or bridged, to engage in EGST.…

The study discussed here aims to describe students' understandings of the definition of a mathematical function, which was achieved through a pilot case study of clinical interviews with four participants -- two ninth graders and two twelfth graders. The participants were recruited from the same urban public high school in the northeast of the…

By highlighting the curriculum modifications that lead to maintaining, or enhancing, the mathematical quality of an algebra lesson introducing the substitution method for solving systems of equations from an algebra textbook, we present an analysis of how a teacher and her students impact how the mathematical ideas unfold across the lesson and how…

Units Coordination and Covariational Reasoning are powerful frameworks for modeling students' mathematics in arithmetic reasoning and construction of relationships between changing quantities, respectively. This case study of an advanced stage 2, 8th-grade algebra student, Daniel, investigated connections between his units coordination and…

This study aims to describe how preservice secondary mathematics teachers (PSMTs) reason about different function representations. The study focuses on two PSMTs' reasonings across static and dynamic representations of functions. Sfard's (2008) Theory of Commognition guided our analysis. Findings indicate that while static representations restrict…

The positive relationship between students' attitudes toward mathematics and mathematics achievement is well documented. Yet there is a worsening problem of low appeal of mathematics especially at the secondary level. Therefore, in this paper we focus on three high school students who report a strong dislike of mathematics. By analyzing student…

This paper provides a case study account of a preservice secondary mathematics teacher's thinking while engaging in slope tasks using dynagraphs. The data included audio recordings and screen captures of a small group of preservice teachers engaging with these tasks, with our analysis focusing on the case of Robin. Despite familiarity using slope…

This report proposes a framework for describing student analogical reasoning activities in abstract algebra that moves beyond the traditional literature-based treatment of analogical mapping. The Analogical Reasoning in Mathematics (ARM) framework captures the activities that students engage in when anticipating, creating, and reasoning from…