Abstract
We conducted a design experiment to introduce children to the mathematics of position and direction by successively inscribing and symbolizing a large-scale space—their school’s playground. With the assistance of their teacher, the children (8 and 9 years of age) progressively ‘mathematized’ this familiar space. They initially produced drawings featuring playground equipment, but ultimately generated re-descriptions of the playground as sets of landmarks located within a space defined by polar coordinates. As they generated and revised their maps, children solved a series of mathematically productive problems, including measuring length and angle and developing correspondences between the worlds of paper and playground. This development relied upon the emergence of conceptions of scale, origin, and the appropriation of coordinates to describe position and direction. Children developed these forms of mathematical notation by modeling objects and their relations in the world. Measures administered individually six months later suggested that this learning was robust. The school experiences were then elaborated and extended in home-school partnership. Children and their parents created maps of spaces in their neighborhoods, often with the child assisting one or both parents. The design experiment resurfaced yet again in subsequent professional development. Teachers participated in many of the same forms of activity as they mapped the school’s wood lot to investigate its biological diversity. The chapter concludes by describing a skillful teacher’s orchestration of classroom talk and activity, and the implications of this orchestration for the identities of students as mathematical thinkers. These forms of activity also shaped teachers’ own identities as skilled professionals, attuned to mathematically fruitful qualities of student thinking.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Brown, A.L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions. Journal of Learning Sciences 2 pp. 137 178.
Brown, A.L. and Campione, J.C. (1996). Psychological theory and the design of innovative learning environments: On procedures, principles and systems. In L. Schauble and R. Glaser (Eds.), Innovations in learning: New environments for education. Mahwah, NJ: Lawrence Erlbaum Associates, Inc., pp. 289–325.
Clements, D.H. (1998). Development of geometric and measurement ideas. In Richard Lehrer and Daniel Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. Mahwah, NJ: Lawemce Erlbaum Associates, pp. 201–226.
Cobb, P. (in press). Supporting the improvement of learning and teaching in social and institutional context. In D. Klahr and S. Carver (Eds.), Cognition and Instruction 25 years of progress. Mahwah, NJ: Lawrence Erlbaum and Associates, Inc.
Dawkins, R. (1996). Climbing mount improbable. 1st American ed. New York: Norton.
DeLoache, J.S. (1987). Rapid Change in the Symbolic Functioning of Very Young Children. Science 238 pp. 1556–1557. diSessa, A. (2000). Changing Minds. Cambridge, MA: MIT Press.
Goldenberg, E.P., Cuoco, A.A. and Mark, J. (1998). A role for geometry in general education. In Richard Lehrer and Daniel Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. Mahwah, NJ: Lawemce Erlbaum Associates, 3–44.
Gravemeijer, K.P. (1998a). From a different perspective: Building on student’s informal knowledge. In Richard Lehrer and Daniel Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. Mahwah, NJ: Lawemce Erlbaum Associates, pp. 45–66.
Gravemeijer, K. (april 1998b). Fostering a Dialectic Relation between Theory and Practice. NCTM research presession Symposium “Perspectives on Classroom Research”, in Washington DC.
Hall, R. and Stevens, R. (1995). Making space: a comparison of mathematical work in school and professional design practices. In Susan Leigh Star (Ed.), The cultures of computing. Oxford, UK: Cambridge, MA, USA: Blackwell Publisher, pp. 118–145.
Latour, B. (1990). Drawing things together. In M. Lynch and S. Woolgar (Eds.), Representation in scientific practice. Cambridge, MA: MIT Press, pp. 19–68.
Lehrer, R. and Chazan, D. (Eds.) (1998). Designing learning environments for developing understanding of geometry and space. Edited by Alan H. Schoenfeld, Studies in Mathematical Thinking and Learning. Mahwah, NJ: Lawrence Erlbaum Associates.
Lehrer, R., Jacobson, C., Kemeny, V.and Strom, D. (1999). Building on children’s intuitions to develop mathematical understanding of space. In E. Fennema and T. Romberg (Eds.),Mathematics classrooms that promote understanding. Mahwah, NJ: Lawrence Erlbaum Associates, pp. 63–87.
Lehrer, R., Jacobson, C., Thoyre, G., Kemeny, V., Strom, D., Horvath, J., Gance, S. and Koehler, M. (1998). Developing understanding of space and geometry in the primary grades. In Richard Lehrer and Daniel Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. Mahwah, NJ: Lawernce Erlbaum Associates, pp. 169 200.
Lehrer, R., Jenkins, M. and Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In Richard Lehrer and Daniel Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. Mahwah, NJ: Lawrence Erlbaum Associates, pp. 137–167.
Lehrer, R. and Schauble, L. (in press). Modeling in mathematics and science. In R. Glaser (Ed.), Advances in instructional psychology. Mahwah, NJ: Lawrence Erlbaum Associates, pp. 101–105.
Liben, L.S. and Downs, R.M. (1993). Understanding person-space-map relations: cartographic and developmental perspectives. Developmental Psychology 29, no. 4, pp. 739–752.
Mead, G.H. (1934). Mind, self and society: From the standpoint of a social behaviorist. Edited by C. W. Morris. Chicago, IL: University of Chicago Press.
Mead, G.H. (1938). Philosophy of the act. Edited by C. W. Morris. Chicago, IL: University of Chicago Press.
Newell, A. and Simon, H.A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.
O’Connor, M.C. and Michaels, S. (1996). Shifting participant frameworks: Orchestrating thinking practices in group discussion. In Deborah Hicks (Ed.), Discourse, learning, and schooling. Cambridge; New York, NY: Cambridge University Press, pp. 63–103.
Olson, D.R. (1994). The world on paper: the conceptual and cognitive implications of writing and reading. Cambridge: Cambridge University Press.
Piaget, J., Inhelder, B. and Szeminska, A. (1960). The child’s conception of geometry. New York, NY: Basic Books.
Siegel, A.W. and White, S.H. (1975). The development of spatial representations of large-scale environments. In H.W. Reese (Ed.), Advances in Child Development. New York, NY: Academic Press, pp. 9–55.
Vygotsky, L.S. (Ed.) (1978). Mind in society: the development of higher psychological processes. Edited by Michael Cole, Vera John-Steiner, Sylvia Scribner and Ellen Souberman. Cambridge: Harvard University Press.
Watt, D.L. (1998). Mapping the classroom using a CAD program: Geometry as applied mathematics. In Richard Lehrer and Daniel Chazan (Eds.), Designing learning environments for developing understanding of geometry and space.
Mahwah, NJ: Lawemce Erlbaum Associates, pp. 419–438.
Wertsch, J.V. (1998). Mind as action. New York, NY: Oxford University Press.
Yackel, E and Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education 27, No. 4, pp. 458–477.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Lehrer, R., Pritchard, C. (2002). Symbolizing Space into Being. In: Gravemeijer, K., Lehrer, R., Van Oers, B., Verschaffel, L. (eds) Symbolizing, Modeling and Tool Use in Mathematics Education. Mathematics Education Library, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3194-2_5
Download citation
DOI: https://doi.org/10.1007/978-94-017-3194-2_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6180-5
Online ISBN: 978-94-017-3194-2
eBook Packages: Springer Book Archive