MIXED METAPHORS: UNDERGRADUATES DO CALCULUS OUT LOUD

Eric Hsu

University of Texas-Austin

Erichsu@math.utexas.edu

Micheal Oehrtman

University of Texas-Austin

Oehrtman.utexas.edu

We describe our findings from an exploratory study on the use of metaphors in learning calculus among college freshmen. By metaphor, we mean any set of cognitive tools used as a referent system to assist in understanding and employing a mathematical concept. What are the core metaphors that students use to comprehend and work with essential calculus concepts? How are these metaphors used and how do they develop?

During the spring semester of 2000, we conducted task-based interviews with 15 students enrolled in a second semester calculus course at a major southwestern public university. Eight interviews, lasting two to two and a half hours each, were conducted with pairs of students. The students worked on a problem dealing with position, velocity and acceleration and reflected on their work and what difficulties other students of their level would have. The interviews were audio-taped, and transcripts were analyzed for emergent themes.

In our initial analysis, we identified the six most common uses of base metaphors labeling them as “slope,” “rate of change,” “automobile,” “motion detector,” “motion on the graph,” and “vertical motion,” and observed the following: first, most students alternately employed two or three of these metaphors but rarely attempted to do so at the same time; second, each metaphor came with a corresponding set of language use, signs, and images; third, their mode of analysis changed consistently with the metaphor they employed. Often, this would result in the students either unknowingly obtaining contradicting results or creating unnecessary confusion due to cross-talk between similar signs of different metaphors.

This was a planning study for a large-scale study in fall 2000 with over a hundred students in a first-year calculus course. We will track the development of their use of mathematical metaphors in weekly written work, tests, in-class “snapshots,” and interviews.