The Interaction Between Embodied and Symbolic Knowledge in Learning Calculus

• In Event: 69.045 - On the Interaction Between Embodied and Symbolic Mathematics Knowledge: Implications for Instruction

Abstract

Title: The Interaction between Embodied and Symbolic Knowledge in Learning Calculus

Many theorists have highlighted the role of embodiment in understanding mathematics (e.g., Barsalou, 1999; Lakoff & Nunez, 2000; Nemirovsky & Ferrara, 2009). Research on conceptual change suggests that reconstruction of prior embodied knowledge is necessary for coordination of symbolic and embodied knowledge (diSessa, 1993; Tall, 2004). We examined how embodied knowledge change in the process of coordination, by comparing the different types of embodied knowledge used by beginning an advanced learners of calculus in thinking about rate of change.

We recruited ten pairs of undergraduates at a mid-western university. Each pair consisted of an advanced student who had taken at least three calculus related courses, and a beginner who had taken one college-level calculus prior to recruitment. We provided a diagram of a funnel-shaped container and asked the pairs to imagine that water was poured into the container at a constant rate. They were asked to graph the height of water and its rate of change as functions of time, and they were asked to think aloud along the way. We video recorded all interactions and analyzed the participants’ language and gestures.

The most common difference was in identifying what determines the rate of change. The advanced students consistently used a “slice” imagery to describe what variable of the container determines the rate of change. For example, one said "since the slices are getting bigger and bigger, it takes more water to fill each height." The advanced students’ representations were guided by their formal understanding of integrals: they saw the water-adding event as a slice-by-slice accumulation of volume. This is a visualization of the symbolic operator of integrals, which implies that formal, symbolic knowledge of calculus may influence what learners see and how they interpret a physical phenomenon. In contrast, the beginners used either undifferentiated terms such as “it”, or multiple contextualized terms (e.g. “container”, “funnel”, “shape”, “opening”, etc.) interchangeably to describe the variable of concern. Likewise, the advanced students’ gestures often represented the "slice" imagery, but the beginners’ gestures were connected to the surface features of the problem.

This study contributes to our understanding of the relation between embodied and symbolic knowledge in learning advanced mathematics. We showed how embodiment of calculus differ among beginners and advanced students in application, which implied a trajectory of change in embodiment as learners coordinate their embodied and symbolic math knowledge.

References：

Barsalou, L. W. (1999). Perceptual symbol systems. Behavioral and Brain Sciences, 22, 577–
660.

diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10 (2&3),
105–225.

Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind
brings mathematics into being. New York: Basic Books.

Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159–174.

Tall, D. O. (2004), The three worlds of mathematics. For the Learning of Mathematics, 23 (3),
29–33.

Authors

• Yanning Yu, Northwestern University

• David Henry Uttal, Northwestern University