Sports

A natural history of mathematical gesture

Description
A natural history of mathematical gesture
Categories
Published
of 21
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  A Natural History of Mathematical Gesture Laurie D. EdwardsSt. Mary’s College of California The spontaneous gestures exhibited by fourteen undergraduate studentstaking a mathematics course for prospective elementary school teacherswere analyzed, utilizing the work of David McNeill as a starting point.The analysis determined that the gestures fell into four types: iconic- physical, iconic-symbolic, metaphoric, and deictic. McNeill’s hypothesisthat mathematical gestures have distinctive, semi-conventional formsreceived mixed support, with gestures related to specific fractions havingthe most structure in common. Introduction Within the field of cognitive science, the pursuit of the understanding of human naturewas initially based on certain assumptions. These included the assumption that cognitionconsists of the manipulation of symbols according to formal rules, that the mind isfundamentally a computational entity, and that the surest path to understanding humanthinking is to build computational models that also process symbols according to formalrules. Underlying these assumptions was a more foundational one, inherited fromDescartes: that the mind is separable from the body. That is, in order to understandhuman cognition, the fact that human brains reside in bodies is irrelevant. Equallyunimportant is the fact that these bodies have evolved over hundreds of thousands of years, by coping with a range of environmental challenges within certain fixed physicalconstraints (the existence of gravity, the basic bilateral symmetry of mammalian bodies,etc.). In the “classical” cognitive science view, the senses function only as conduits for1  bringing “input” to the mind’s programs, and the motor system functions simply to carryout the results of the mind’s processing of information.During the past two decades, an alternative paradigm has emerged that challengesthe Cartesian assumptions of early cognitive science, and that instead acknowledges theembodied nature of cognition (Varela, Thompson & Rosch, 1991). This new paradigmsees the sensorimotor system as an essential aspect of cognition; as Varela states:“Embodied entails the following: (1) cognition dependent upon the kinds of experiencethat come from having a body with sensorimotor capacities; and (2) individualsensorimotor capacities that are themselves embedded in a more encompassing biologicaland cultural context...sensory and motor processes, perception and action, arefundamentally inseparable in lived cognition, and not merely contingently linked asinput/output pairs” (Varela, 1999, p. 12).Recently, research into the relationship between physical gesture and languagehas added a new dimension to the embodied cognition paradigm. According to work inthis area, not only is it the case that one must consider the connections between the body,the mind, and the world, but that human gestures form an integral part of language andthought. Indeed, there is one school of thought that holds that gesture preceded andscaffolded speech in human evolution, and evidence from neuroscience indicates that thesame areas of the brain are involved in the expressive use of gesture and oral language(Corballis, 1999).Foundational work on gesture and language has been carried out by psychologistand linguist David McNeill (1992, 2000). McNeill has proposed that spoken language2  and gesture form an integrated system of communication; that is, that “gesture and speecharise from a single process of utterance formation” (McNeill 1992, p. 29). In contrast totheories that see gesture as in some way epiphenomenal to speech, as embellishments, oreven as “translations” of previously conceptualized verbalizations, McNeill holds thatspeech and gesture are produced in concert in the mind, and act together to express thesame intended meaning. McNeill offers several kinds of evidence for this hypothesis,including the neurolinguistic findings described above, but also the fact that gesture andspeech are closely synchronized in time and meaning, that gesture occurs only duringspeech, and that gesture and spoken language develop together in children (ibid.).McNeill points out that spoken language and gesture have different yet complementarycharacteristics that allow them to synergistically support effective communication of meaning. Speech, on the one hand, is linearly segmented; it takes place one word at atime, with the order of the words playing an important, if not essential, role in themeaning of the utterance. Meaning is broken down into discrete segments, which areorganized hierarchically (a discourse, a sentence, a phrase, a word, a phoneme); viewedanother way, utterances are constructed by putting together these linearly organizedsegments, which combine to form more complex structures. In general, language isanalytic, with one meaning (or a cluster of related meanings) attached to a given word,and a fairly arbitrary mapping between the form of words and their meanings. In sum,oral language is linearly-segmented, hierarchical, and analytic .Gesture contrasts with oral language on all these dimensions. Rather thanpresenting meaning through a linearly segmented stream, gesture is what McNeill calls3  “global-synthetic”; the meaning comes from the motion as a whole, rather than beingbuilt up from simpler parts. In addition, one gesture can convey an entire complex of meanings, including aspects such as spatial location, physical shape, speed, and rhythmthat are difficult to convey through words. Gestures are also non-hierarchical and non-combinatoric; that is, more complex gestures are not built up from simpler ones; instead,“With gestures, each symbol is a complete expression of meaning unto itself” (ibid., p.21).McNeill points out that that gesture can play a mediating role between internal,subjective imagery, and shared, conventional speech. His central hypothesis is that“Speech and gesture are elements of a single integrated process of utterance formation inwhich there is a synthesis of opposite modes of thought — global-synthetic andinstantaneous imagery with linearly-segmented temporally extended verbalization.Utterances and thoughts realized in them are both imagery and language” (ibid., p. 35).Since mathematics itself has a dual identity comprising both an internal, individualexperience based at least in part in imagery, as well as a shared, conventional symbol-based language, gesture has the potential of illuminating its nature within the humanconceptual system.  Research on Gesture in Mathematics and Science In recent years, a body of empirical research into the role of gesture in doing, teachingand learning mathematics has emerged. These studies address topics ranging from theway young children use gesture in counting (Alibali & diRusso, 1999; Graham, 1999);4  teachers’ gestures when providing instruction in problem-solving (Goldin-Meadow, Kim& Singer, 1999); and how students working together make sense of various kinds of graphs (Moschkovich, 1996; Reynolds & Reeve, 2002). Similar studies have examinedgesture in science talk (Crowder, 1996; Roth & Welzel, 2001; see Roth, 2001 for a morethorough review). The results of these studies have suggested that when there is amismatch between gesture and speech, it can indicate readiness to learn a concept, thatstudents may be able to convey an understanding through gesture before they can do so inwords, and that appropriate (matched) gesture can enhance the effectiveness of spokeninstruction (summarized in Roth, 2001). Additional research has investigated bodilymovement as well as gesture in the context of technology-based tools for understandingphysics, mathematics, and the representation of both through graphing (Nemirovsky &Noble 1997; Nemirovsky, Tierney & Wright 1998).Given that the study of gesture in mathematics is still in its infancy, it seemsappropriate at this stage to undertake a descriptive analysis of kinds of gestures observedin use by mathematics learners and teachers. In the study described in this paper,prospective teachers (who are, in a sense, both teachers and learners) were interviewedabout fractions. The objective of the work is to describe the type and range of gesturesutilized by the students during these interviews, in a sense, to begin to develop a “naturalhistory” of gesture in this particular social and mathematical context. A further goal is toinvestigate a hypothesis put forward by McNeill in regards to mathematical gestures,specifically, that within a given mathematical domain, gestures may have a shared, semi-conventional structure (McNeill, 1992).5
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks