Abstract
This study investigated introductory calculus students' spontaneous reasoning about limit concepts guided by an interactionist theory of metaphorical reasoning developed by Max Black. In this perspective, strong metaphors are ontologically creative by virtue of their emphasis (commitment by the producer) and resonance (support for high degrees of elaborative implication). Analysis of 120 students' written and verbal descriptions of their thinking about challenging limit concepts resulted in the characterization of 5 clusters of strong metaphors. These clusters were based on the objects, relationships, and logic related to intuitions about (a) a collapse in dimension, (b) approximation and error analyses, (c) proximity in a space of point-locations, (d) a small physical scale beyond which nothing exists, and (e) the treatment of infinity as a number. Students' reasoning with these metaphors had significant implications for the images they formed and the claims and justifications they provided about multiple limit concepts.
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