Coming to Understand the Formal Definition of Limit: Insights Gained From Engaging Students in Reinvention

Abstract

The purpose of this article is to elaborate Cottrill et al.'s (1996) conceptual framework of limit, an explanatory model of how students might come to understand the limit concept. Drawing on a retrospective analysis of 2 teaching experiments, we propose 2 theoretical constructs to account for the students' success in formulating and understanding a definition of limit. The 1st construct relates to the need for students to move away from their tendency to attend first to the input variable of the function. The 2nd construct relates to the need for students to overcome the practical impossibility of completing an infinite process. Together, these 2 theoretical constructs build on Cottrill et al.'s work, resulting in a revised conceptual framework of limit.

Reinventing the formal definition of limit: The case of Amy and Mike

“…For discussion of the conceptual differences between the initial and latter steps of the genetic decomposition, seeSwinyard and Larsen (2010).…”

Teaching and Learning of Calculus

“…In the same spirit, Swinyard (2011) discovered that one of the greatest obstacles to the correct mathematical understanding of limit of function was student preference to focus first on the change in the independent variable, understanding the limit as x approaches c of f(x) to mean that as x gets closer to c, f(x) gets closer to the limiting value, a formulation that can be useful for finding limits but that is rife with opportunities for misleading concept images. Furthermore, Swinyard is representative of a growing body of research into how students use their understanding of limits to reason about limits (Swinyard and Larsen 2012). According to Swinyard (2011), students are able to reinvent a coherent definition of limit of function with a high level of significance.…”

“…This study provides a detail account of how students might think about limit formally. These details are encapsulated in an exploratory model of several levels of students understanding (Swinyard and Larsen 2012).…”

The Interplay Between Mathematicians’ Conceptual and Ideational Mathematics about Continuity of Complex-Valued Functions

“…In this sense the definition is a codomaindomain-domain-codomain statement. Previous research identifying students' tendency toward domain-first reasoning for limits (Cottrill et al 1996;Swinyard and Larsen 2012) refers to the final domain-codomain pair; given z in the domain, one examines the distance between f(z) and f(z 0 ) in the codomain. Codomain-first reasoning refers to the initial codomain-domain pair of quantifiers for ε and δ, the dependence of δ on ε, and their roles in framing the remainder of the definition.…”