Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

Abstract

Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student metaphorical reasoning, we examined 11 college instructors' metaphorical reasoning on limit concepts. This paper focused on previous research of metaphor clusters observed among students to answer the following: (a) Do college instructors use metaphorical reasoning to conceptualize the meaning of a limit? (b) Can we characterize instructor metaphorical reasoning similar to those observed among students? (c) Will an instructor's self‐identification of metaphor clusters be consistent with our metaphor coding? We found that college instructors' perspectives vary, either graphical or algebraic, in their explanations of limit items. All the instructors used metaphors, and instructor metaphorical reasoning aligned with student metaphor clusters. Instructors tended to change their metaphors with respect to the limit item. Instructors were not aware of their use of metaphors, nor were they aware of their inconsistency in their choice of metaphor. We believe that instructor awareness of their own distinct perspectives and metaphors would assist students' understanding of limit concepts.

University students’ mental construction when learning the Convergence of a Series concept

“…It is known from the researchers' experiences that a source of cognitive difficulties for students is the limit concept (Denbel, 2014). Many students struggle with limits (Cappetta & Zollman, 2013;McCombs, 2014;Patel, McCombs & Zollman, 2014). Those researchers identified specific student difficulties with limits.…”

“…Cappetta and Zollman (2013) found that students have difficulties with understanding limits, involving the infinite processes of limits and the value of limits. Additionally, metaphorical reasoning, a way of understanding a situation to resolve disequilibrium that students encounter when faced with a new problem situation (Patel et al, 2014), develops without the knowledge of the lecturer and impedes students' understanding of mathematical concepts. A student who understands a mathematical concept can move between numerical, algebraic, graphical and application representations.…”