Component-Wise Reasoning as a Mechanism of Sense-Making in Real Analysis

Undergraduate concepts are often first introduced in a single-dimensional setting and then extended to multiple dimensions. For instance, many undergraduate real analysis students will first learn of the metric topology on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} before being exposed to more general metric spaces. I conducted a paired teaching experiment (Steffe & Thompson, 2000) with introductory real analysis students that explored their learning of the general metric function. In this experiment, I was able to observe the students’ mathematical activity as they made sense of analytic ideas in increasingly general settings. I present a construct, called component-wise reasoning, that offers an explanatory mechanism for the ways that the students leveraged their understandings of phenomena on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} to construct new schemes for similar phenomena in other metric spaces. I discuss how component-wise reasoning can offer explanatory power for students’ sense-making in abstract spaces across the undergraduate curricula.