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Mathematical Representations at the Interface of the Body and Culture
Edited by Wolff-Michael Roth, University of Victoria, Canada
A volume in the series: International Perspectives on Mathematics Education - Cognition, Equity & Society
Series Editor(s): Bharath Sriraman, The University of Montana and Lyn English, Queensland University of Technology
Over the past two decades, the theoretical interests of mathematics educators have changed substantially-as any brief look at the titles and abstracts of articles shows. In the 1980s, mathematical knowing and learning were explained in psychological models, focusing on motivation, interest, abilities, information processing, and conceptual (including alternative) frameworks. Jean Piaget's ideas about accommodation and assimilation reigned as a pair key theoretical concepts that many drew on to explain observations made in individual interviews, teaching experiments, and quantitative classroom studies. From an a-posteriori perspective, however, one can now detect early signs of a way of thinking that has revolutionized mathematics education since. Early papers of Paul Cobb (e.g., 1985) show that while he attended to individual cognition, his or her motivation, and the (radical) construction of individual worlds, Cobb's reference list already shows a beginning to the social aspects of mathematical cognition, as he was well aware of the literature in the social studies of science (e.g., Karin Knorr-Cetina, Barry Barnes, Thomas Kuhn).
Largely through the work of Paul Cobb and his various collaborators, mathematics educators came to be attuned to the intricate relationship between individual and the social configuration of which she or he is part. That is, this body of work, running alongside more traditional constructivist and psychological approaches, showed that what happens at the collective level in a classroom both constrains and affords opportunities for what individuals do (their practices) (McClain & Cobb, 2001).
Increasingly, researchers focused on the mediational role of sociomathematical norms and how these emerged from the enacted lessons. This perspective often is artic ulated in terms of the relations between the inter- and intra-mental (or psychological) processes that Lev Vygotsky (1978) has articulated. Hereby, practices and processes in which a child participates with others (inter-psychological plane) are internalized and then operate on their own on an intra-psychological plane.
A second major shift in mathematical theorizing occurred during the past
decade: there is an increasing focus on the embodied and bodily manifestation of mathematical knowing (e.g., Lakoff & Núñez, 2000).
Mathematics educators now working from this perspective have come to their position from quite different bodies of literatures: for some, linguistic concerns and mathematics as material praxis lay at the origin for their concerns; others came to their position through the literature on the situated nature of cognition; and yet another line of thinking emerged from the work on embodiment that Humberto Maturana and Francisco Varela (1980) advanced. Whatever the historical origins of their thinking, mathematics educators taking an embodiment perspective presuppose that it is of little use to think of mathematical knowing in terms of transcendental concepts somehow recorded in the brain, but rather, that we need to conceptual knowing as mediated by the human body, which, because of its senses, is at the origin of sense.
One of the question seldom asked is how the two perspectives, one that focuses on the bodily, embodied nature of mathematical cognition and the other that focuses on its social nature, can be thought together.
Cultural-historical activity theory, with its simultaneous orientation toward embodied operations and social already integrates the two perspective, allowing us to theorize the intricate relationships not only between individual and social cognition but also other aspects of mathematical activity such as motivation and emotions (Roth, 2007). Another avenue to theorizing the material body and the social aspects of knowledge togethe r has emerged in a line of work that began with phenomenological sociology (e.g., Alfred Schutz) and, on the one hand, has developed into ethnomethodology and conversation analysis (e.g., Garfinkel, 1967), and, on the other hand, has found its expression in early forms of dialectical theories in which social fields and individual dispositions are complementary aspects of the same situations (e.g., Bourdieu, 1990). The first line of work has been realized, for example, in the ethnomethodological studies of mathematicians (Livingston, 1986) whereas the second line of work centrally influenced mathematical cognition in the everyday world (Lave, 1988). In both lines of work, great attention is paid to the various ways in which mathematical cognition is exhibited, which involves not only mind, but also and especially so the body; and because mathematical cognition is exhibited, it inherently is for others, and therefore social even though singularly expressed in and by the performance of individual bodies.
The past two decades also have seen an emergent interest in inscriptions, that is, forms of mathematical representations inscribed in some medium, including paper, computer monitors, chalkboards, and whiteboards (Latour, 1987). The term inscription was chosen to focus theorizing away from the mental representations that had been the concern of cognitive psychologists.
Inscriptions, because of their material nature easily inspectable by all parties in a communicative setting, play in important role in the expression of mathematical cognition-both directly (as instance of) and indirectly (mediating it)-in the articulation and development of individual and social aspects of knowing (Meira, 1995).
This edited volume situates itself at the intersection of these theoretical and focal concerns. All chapters deal with mathematical inscriptions as these are interpreted, produced, or both. The domains include geometry (Edwards, Hwang et al.) and data-representing inscriptions including line graphs, histog rams, and scatter plots (Cobb, Rasmussen, Roth, Krummheuer, Nemirovsky, Radford). In each case people collectives are involved, sometimes in whole-class settings (Cobb, Edwards, Hwang, Krummheuer), at other times in two- or three-person transactions (Nemirovsky, Rasmussen, Radford, Roth). In all papers, the current culture both at the classroom and at the societal level comes to be expressed and provides opportunities for expressing oneself in particular ways; and these expressions always are bodily expressions of body-minds (especially emphasized in Edwards, Hwang, Roth, Nemirovsky, Nuñez, Radford). As a collective, the papers focus on mathematical knowledge as an aspect or attribute of mathematical performance; that is, mathematical knowing is in the doing rather than attributable to some mental substrate structured in particular ways as conceived by conceptual change theorists or traditional cognitive psychologists. The ethnomethodologically and conversational analytically informed study by Roth brings out the simultaneously social and bodily embodied nature of mathematical cognition in situations that are intelligible to, and intelligibly created by, the co-participants to conversational transaction. The collection as a whole will show readers important aspects of mathematical cognition that are produced and observable at the interface between the body (both human and those of [inherently material] inscriptions) and culture.
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