Drawing space: mathematicians’ kinetic conceptions of eigenvectors

Abstract

This paper explores how mathematicians build meaning through communicative activity involving talk, gesture and diagram. In the course of describing mathematical concepts, mathematicians use these semiotic resources in ways that blur the distinction between the mathematical and physical world. We shall argue that mathematical meaning of eigenvectors depends strongly on both time and motion-hence, on physical interpretations of mathematical abstractions-which are dimensions of thinking that are typically deliberately absent from formal, written definition of the concept. We shall also show how gesture and talk contribute differently and uniquely to mathematical conceptualisation and further elaborate the claim that diagrams provide an essential mediating role between the two.

Comparing Calculus Communication across Static and Dynamic Environments Using a Multimodal Approach

“…Furthermore, this article extends Sfard's notion of visual mediators by distinguishing two kinds of visual mediation, dynamic and static. The distinction was important for this study because of the potential for the dynamic visual mediators such as gestures and DGEs to evoke temporal and mathematical relations (Ng and Sinclair 2013;Sinclair and Gol Tabaghi 2010), particular for the study of calculus (Núñez 2006). It also helped guide the analysis in terms of how Bchange^was conveyed in students' discourse, by distinguishing deictic gestures from gestures (or dragging) that conveyed temporal relationships.…”

“…In the latter case, the gestures communicate temporal relationships of the linear function as opposed to the offering the shape of the linear function statically. A few studies have shown that temporality can be evoked by the use of dynamic visual mediators like gestures (Ng and Sinclair 2013;Sinclair and Gol Tabaghi 2010), especially in the study of calculus (Núñez 2006). These studies point to the dynamic and temporal aspects of mathematicians' thinking; they also reveal that mobile hand movements are important features of this type of mathematical thinking.…”

The Interplay Between Mathematicians’ Conceptual and Ideational Mathematics about Continuity of Complex-Valued Functions

“…Some of these IM explanations were domain-first conceptualizations, containing inconsistencies with the CM epsilon-delta definition of continuity. We believe that this research contributes to the growing body of literature on mathematicians' reasoning about mathematical topics (Anderson and Leinhardt 2002;Inglis and Alcock 2012;Martin 2013;Nemirovsky and Smith 2011;Sinclair and Tabaghi 2010;Weber 2008;Weber and Mejia-Ramos 2011;WilkersonJerde and Wilensky 2011), and may inform the teaching of continuity of real-valued functions, for which there is minimal research (Fisher 2010;Oehrtman 2009). Finally, our research may inform the teaching and learning of complex analysis.…”

“…Other researchers have explored how mathematicians convey mathematical notions such as projections (Anderson and Leinhardt 2002), eigenvectors (Sinclair and Tabaghi 2010), pointwise convergence in a Taylor series (Martin 2013), and topological notions (Nemirovsky and Smith 2011;Wilkerson-Jerde and Wilensky 2011). As with other research that compares mathematicians' and students' reasoning, Martin also showed that mathematicians tend to integrate conceptual images, while novices concentrate on surface level features.…”

Diagram, gesture, agency: theorizing embodiment in the mathematics classroom

“…In other words, he uses these historical episodes to explore ontological questions about the relationship between the virtual and the actual, as well as psychological questions about what it means to do mathematics. He argues that the study of such gestures can help us undo some of the troubling consequences of the Aristotelian division between movable matter and immovable mathematics (see also Núñez, 2006 andSinclair &Gol Tabaghi, 2010). The fear and loathing expressed by Bertrand Russell for the very idea of the motion of a point in space is an obvious expression of this tradition.…”