Students' conceptualizations of multivariable limits

Abstract

Scope and Method of Study:

The objective of this study was to describe how students with a dynamic view of limit generalize their understanding of the limit concept in a multivariable setting. This description emphasizes the type of generalization that takes place among the students (Harel and Tall, 1989) and the role of motion among students' conceptualizations.

To achieve these goals, a series of task-based interviews were conducted with seven students enrolled in multivariable calculus. These interviews were analyzed and a coding scheme was developed to describe the data. This coding scheme arose from analysis of the data combined with the role of limit in formal mathematics. It emphasizes three models for understanding the limit concept, the dynamic model, the neighborhood model, and the topographical model.

Findings and Conclusions:

After analyzing the coded data, two important interactions between the three models of limit were described. First, it was found that students superimposed dynamic imagery on top of existing topographical structures in order to understand multivariable limits, and a weak topographical understanding of multivariable limits contributed to students struggling to understand the multivariable limit concept. Second, it was found that students implementing dynamic imagery in the context of multivariable limits confronted an infinite process of analyzing motion along an infinite number of paths. It was found that students' struggles to understand the multivariable limit were connected to their struggles to understand this infinite process. Additionally, it was found that the condensation of this infinite process led several students towards the neighborhood model of limit.