# Collapsing dimensions, physical limitation, and other student metaphors for limit concepts : an instrumentalist investigation into calculus students' spontaneous reasoning

## Abstract

This research is an investigation of first-year calculus students’ spontaneous reasoning about limit concepts. The central theoretical perspective guiding the study design and the data analysis is based on an interaction theory of metaphorical reasoning first proposed by Ivor Richards and later developed by Max Black. From this perspective, strong metaphors are characterized as those that both support high degrees of elaborative implication and are ontologically creative. This study investigates students’ spontaneous reasoning about limits that may be described as metaphorical in this sense and are thus implicative for the students’ emerging understandings. John Dewey’s instrumentalism, a theory of inquiry as the application of cognitive tools against problematic situations, provides a focus on functional as well as structural aspects of students’ metaphors. Descriptive answers to the following questions are sought: 1) What conceptual metaphors do students use to reason about limit concepts, and how are they applied in specific problem contexts? 2) How do these metaphors affect students’ interpretations of content presented in class? 3) What are the implications of trying to directly influence students’ metaphorical reasoning? The methodology is a micro-ethnographic study of students’ problem solving through clinical interviews, writing assignments, and classroom observation. The main result is the characterization of five metaphor clusters for limits that were used in a variety of problem contexts by several students. These metaphors involved reasoning about limits in terms of a collapse in dimension, approximation, closeness in a spatial domain, a physical limit for which nothing smaller could exist, and the treatment of infinity as a number. Contrary to the implications of much of the informal language associated with limits, students were not observed to use motion imagery in significant ways to reason about limit concepts. While many aspects of the students’ metaphors generated mathematically incorrect entailments for limits, most students were still able to use them to great conceptual advantage. Two important factors were whether students critically reflected on their own reasoning to make refinements and whether they attempted to make connections to relevant mathematical structures.