Body, Time, Image: Amodal Sensation, Movement, and Inventive Mathematical Diagramming

• In Event: Theorizing Movement and Movement-Based Methods in Embodied Mathematics Learning

Abstract

Objectives
This paper focuses on the potentiality of the body, arguing that students' inventive mathematical diagrams can show perception and sensation operating amodally rather than multi-modally. We focus on students' diagramming of time and temporality in mathematical problems involving movement, in classroom episodes where students developed ways of decomposing movement into two dimensions using motion-visualizing software.

Theoretical Framework
Stevens (2012) suggests that we need to unravel the term embodiment, a term taken up in incommensurable ways by diverse authors to address the role of the body in teaching and learning mathematics. Sheets-Johnstone (2009) critiques the phrase “cognition is embodied” as demoting the body to a mere vessel for some higher act of cognition. We argue that body-assemblages are porous and partially closed, individuated only provisionally while their borders are sustained through various physical acts, and that mathematical concepts partake in this ongoing process. Learners' bodies are always in the process of becoming assemblages of diverse and dynamic materialities, including mathematical concepts and external physical objects such as pencils, compasses and calculators.

Methods
We examine video data from a classroom-based intervention (Grades 2 - 5) aimed at approaching the concept of function using motion detectors. The software, Motion Visualizer (MV), uses a camera for real-time motion tracking in two dimensions of a student moving an object in a plane. As the object moves, the software displays two graphs of its position components against time, and students strive to make sense of these displays in relation to their movements.

Our analysis of this data uses theories from Châtelet (1993) and Massumi (1998), for whom diagrams are “cutting out gestures” by which new mathematical practices have emerged historically. We find evidence that students' mathematical diagrams involve action that literally breaks down previously taken-for-granted determinations of what is sensible or intelligible, carving up matter in new, unscripted ways, while reconfiguring the body of the learner.

Scholarly Significance
Accounts of diagramming drawing on theories of embodiment tend to frame the activity in terms of spatio-temporal or phenomenological experiences (Gallese & Lakoff 2005; Radford 2009). Research on students’ representations of space-time in (proto-)Cartesian graphs points to the need to attend more carefully to the way students' sense of embodiment affects their diagramming habits (Bakker & Hoffmann 2005; Radford & Sherin 2000; Sinclair & Armstrong 2011; Stylianou, Smith & Kaput 2005). Bremigan (2001) and Sinclair & Armstrong (2011) have shown how dynamic geometry software can be effective in developing awareness of the spatio-temporal aspects of diagramming. Other researchers have shown how different technologies both constrain and expand student diagramming habits (Jamnik, 2001; Sinclair & Gol Tabaghi, 2010). We contend that we need to give attention to the space-time structuring of sensation to better understand how students can make sense of time and temporality in experiments with motion. We argue that sensation has to be studied amodally, rather than multi-modally, so that the potentiality of the body can be granted space for inventiveness.

Authors

• Elizabeth De Freitas, Adelphi University

• Francesca Ferrara, University of Torino