Middle school students' algebraic lettering through an enactive theory of cognition

Abstract

Within algebra education, students’ conceptions of “variables” (or letters applied with mathematical purpose, such as x in the expression 2x+3) have been a frequent topic of inquiry (e.g., Ellis & Özgür, 2024; Kieran, 2006; Küchemann, 1978; Trigueros & Jacobs, 2008). In this dissertation, I show how an alternative epistemological perspective on students’ activity--an enactive theory of cognition--can refocus inquiry on algebraic lettering, a term I introduce to refer to the generating of letters in the context of their algebra (e.g., writing x or saying y). Across two papers, I apply case study methodology to 1) examine the key features of three middle school students’ lettering processes over time (Chapter 2, “Identifying and Characterizing Algebraic Lettering in Middle-School Student Activity”) and 2) explore how it developed through students’ classroom interactions (Chapter 3, “The Emergence of Lettering in Classroom Interactions”). ☐ In my first paper, I investigate: What did three middle-school students learn with respect to lettering by the conclusion of an instructional unit in their classroom? I frame this inquiry as a multiple case study of the three students, drawing data from two task-based interviews conducted at the beginning and end of a unit of instruction about linear equations they experienced together. I applied qualitative techniques to construct cross-case data displays to compare and contrast the role of lettering across students and time points. I found that the students learned to letter in Process-Oriented, Representationally Connective, and Strategically Connective ways. I discuss implications for researchers and practitioners in designing tasks that can support future students’ emergent lettering processes in algebra. ☐ In my second paper, I explore the research question: How did students’ lettering emerge in classroom interactions during an instructional unit about linear equations? I investigated the three students’ activity as a collaborative group through five video-recorded classroom observations during their linear equations unit. Drawing from enactivism, I developed a technique called monologic centering analysis to examine the interdependent lettering trajectories of the students. Related to the three aforementioned features of lettering, I found that students’ lettering emerged within their classroom negotiations as they coordinated their lettering to achieve group goals and draw connections across representations and strategies in their work. I also found that the students drew upon five central environmental resources within these interactions—what tasks, the teacher, and groupmates said/wrote with respect to lettering, as well as lettering applied across representations of the same/different situations. I discuss how students’ development of algebraic literacy in classrooms can be supported as a result of these findings.