Abstract
In this paper, we aim to contribute to the discussion of the role of the human body and of the concrete artefacts and signs created by humankind in the constitution of meanings for mathematical practices. We argue that cognition is both embodied and situated in the activities through which it occurs and that mathematics learning involves the appropriation of practices associated with the sets of artefacts that have historically come to represent the body of knowledge we call mathematics. This process of appropriation involves a coordination of a variety of the semiotic resources—spoken and written languages, mathematical representation systems, drawings, gestures and the like—through which mathematical objects and relationships might be experienced and expressed. To highlight the connections between perceptual activities and cultural concepts in the meanings associated with this process, we concentrate on learners who do not have access to the visual field. More specifically, we present three examples of gesture use in the practices of blind mathematics students—all involving the exploration of geometrical objects and relationships. On the basis of our analysis of these examples, we argue that gestures are illustrative of imagined reenactions of previously experienced activities and that they emerge in instructional situations as embodied abstractions, serving a central role in the sense-making practices associated with the appropriation of mathematical meanings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Defectology is (a translation of) the term used by Vygotsky to denominate the scientific study of the processes of development of differently abled individuals.
Here, we are highlighting a difference in the sensory activities associated with “seeing” with one’s hands rather than with one’s eyes. We are not, at this point, claiming that this difference necessarily extends to cognitive mechanisms associated with interpreting this imagistic data—that is to what we call visualizing—and we have chosen, for the time being, not to enter the debate concerning the mental representations of images or whether mental imagery retains depictive properties of the images itself (Kosslyn, 2004) or has an essentially propositional nature (Pylyshyn, 2003).
We note the similarity between this description of the practice of abstracting and the process that Radford (2002) terms objectification or the becoming increasingly aware of a cultural object. If there is a difference, and we are not so sure that there is, perhaps it is in the emphasis on abstracting as a mathematical practice with objects, rather than on awareness of the objects themselves.
We are grateful to the funding we have received from FAPESP (Project no. 2004/15109-9) and CAPES (Project no. 23038.019444/2009-33) in the course of this programme of research.
We acknowledge that our use of the word “embodied” as a qualifier in the term “embodied abstraction”, might imply that we see some abstractions as embodied and other as not. This is not the case. In the same way that Noss and Hoyles (1996) have argued all abstractions are situated, we believe that all abstractions are embodied and that it is by looking at gestures alongside spoken language that the embodiments associated by different learners with different mathematical ideas become more evident.
Actually, this is not saying much as we collected no more video evidence with André concerning pyramids. Perhaps it is worth saying though, that during our numerous discussions of André’s activities, we seem to have appropriated this aspect of his mathematical practices and a tendency to reenact his gesture, without the presence of the pyramid, in a spontaneous manner.
References
Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–45.
Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 720–749). Mahwah: Lawrence Erlbaum Associates.
Cole, M., & Wertsch, J. V. (1996). Beyond the individual–social antinomy in discussions of Piaget and Vygotsky. Human Development, 39, 250–256.
Confrey, J. (1995). How compatible are radical constructivism, sociocultural approaches and social constructivism? In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 185–226). Hillsdale: Lawrence Erlbaum Associates.
Confrey, J. (1998). Voices and perspectives: Hearing epistemological innovation in students’ words. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), Constructivism and education (pp. 104–120). New York: Cambridge University Press.
Fernandes, S. H. A. A. & Healy, L. (2007). Ensaio sobre a inclusão na educação matemática. [Reflections on inclusion in mathematics education] Unión. Revista Iberoamericana de Educación Matemática. Federación Iberoamericana de Sociedades de Educación Matemática—FISEM, 10, 59–76.
Gallese, V., & Lakoff, G. (2005). The brain’s concepts: The role of the sensory–motor system in conceptual knowledge. Cognitive Neuropsychology, 22, 455–479.
Goldin-Meadow, S. (2003). Hearing gestures: How our hands help us think. Cambridge: Harvard University Press.
Greenwood, D., & Levin, M. (2000). Reconstructing the relationships between universities and society through action research. In N. K. Denzin & Y. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 85–106). Thousand Oaks: Sage Publications Inc.
Healy, L.(S.) (2002) Iterative design and comparison of learning systems for refection in two dimensions, Unpublished PhD thesis. University of London.
Healy, L. & Fernandes, S. H. A. A. (2008). The role of gestures in the mathematical practices of blind learners. In: Proceedings of the Joint Meeting of PME 32 and PME-NA XXX (Vol. 3, 137–144). Morelia, México: Cinvestav-UMSNH,
Healy, L. & Fernandes, S. H. A. A. (2009). Relationships between sensory activity, cultural artefacts and mathematical cognition. In: Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education. (Vol. 3, 137–144). Thessalonki, Greece.
Hoyles, C., & Noss, R. (Eds.). (1992). Learning mathematics and logo. Cambridge: MIT.
Hoyles, C., & Noss, R. (1993). Out of the Cul-De-Sac? In: Proceedings of the Fifteenth Annual Conference of North American Chapter of the International Group for the Psychology of Mathematics Education. (Vol. 1, 83–90). California: San Jose State University.
Iverson, J. M., & Goldin-Meadow, S. (1998). Why people gesture when they speak, vol. 396 (p. 228). London: Macmillan Publishers Ltd. Nature.
Kosslyn, S. T. (2004). Mental images and the brain. Cognitive Neuropsychology, 22(3), 333–347.
Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.
McNeill, D. (1992). Hand and mind: What gestures reveal about thought. Chicago: University of Chicago Press.
Noss, R., & Hoyles, C. (1996). Windows on mathematical meaning: Learning cultures and computers. Dordrecht: Kluwer Academic Press.
Ochaita, E. & Rosa, A. (1995). Percepção, ação e conhecimento nas crianças cegas. [Perception, action and knowledge of blind children]. In: C. Coll, J. Palacios, A. Marchesi (Eds.). Desenvolvimento psicológico e educação: necessidades educativas especiais e aprendizagem escolar (Vol. 3, 183–197). Tradução Marcos A. G. Domingues. Porto Alegre: Artes Médicas.
Presmeg, N. C. (1997). Generalization using imagery in mathematics. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 299–312). Mahwah: Lawrence Erlbaum Associates.
Pylyshyn, Z. (2003). Seeing and visualizing: It's not what you think. Cambridge: MIT.
Radford, L. (2002). The seen, the spoken and the written: A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23.
Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126.
Radford, L., Bardini, C., Sabena, C., Diallo, P., & Simbagoye, A. (2005). On embodiment, artifacts, and signs: A semiotic–cultural perspective on mathematical thinking. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, vol. 4 (pp. 113–120). Australia: University of Melbourne.
Radford, L., Edwards, L., & Arzarello, F. (Eds) (2009). Educational Studies in Mathematics, 70(2).
UNESCO. (1994). The Salamanca Statement and framework for action on special needs education. World Conference on Special Needs Education: Access and Quality. Salamanca: UNESCO.
Van Oers, B., & Poland, M. (2007). Schematising activities as a means for encouraging young children to think abstractly. In M. C. Mitchelmore & P. White (Eds.), Special issue: Abstraction in mathematics education. Mathematics Education Journal, 19(2), 10–22.
Vygotsky, L. S. (1978/1930). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press.
Vygotsky, L. S. (1981). The instrumental method in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 134–143). Armonk: M.E. Sharpe.
Vygotsky, L. (1997). Obras escogidas V—Fundamentos da defectología [The Fundamentals of defectology]. Traducción: Julio Guillermo Blank. Madrid: Visor.
Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 193–204). Norwood: Ablex.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Healy, L., Fernandes, S.H.A.A. The role of gestures in the mathematical practices of those who do not see with their eyes. Educ Stud Math 77, 157–174 (2011). https://doi.org/10.1007/s10649-010-9290-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-010-9290-1