Abstract
This chapter explores the visual and imagistic aspects of cybernetic manipulatives, how they may be designed to improve upon physical manipulatives, what their potentials and pitfalls may be, and how they fit into the larger evolution of technology use in support of mathematics learning. We will examine questions of ,physicality - of representations, dynamic connections between ,natural - and formal representations, and, especially, ways that new forms of records of actions may alter these connections and elevate levels of thinking involved in the doing of mathematics, from low-level computation to higher level planning, strategic and structural thinking. This paper extends earlier work [11, 12, 13], which also includes references to the wider literature relating to these topics, references not repeated here. It also relates closely to other papers on another aspect of the representational use of new technologies, dealing with dynamic linkages between formal mathematical notations and authentic human experience, particularly as instantiated in realistic simulations [14, 15, 16].
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
Block, N. (1981). Imagery. Cambridge, Massachusetts: MIT Press.
Bochner, S. (1966). The Role of Mathematics in the Rise of Science. Princeton, NJ: Princetown University Press.
Cajori, F. (1929) A history of mathematical notations, Vol. 2: Notations mainly in higher mathematics. La Salle, Illinois: Open Court.
Confrey, J. (1993). Function Probe (software). Santa Barbara, CA: Intellimation
Davis, R. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex.
Dennet, D. (1990) Thinking with a computer. In Images and understanding, H. Barlow, C. Blakemore, & M. Weston-Smith (eds.) Cambridge: Cambridge University Press, 297–309.
Dienes, Z. (1973). The six stages in the process of learning mathematics (trans P.L. Seaborne) New York: Humanities Press (orig. published in 1970).
Doerfler, W. (1991). Meaning: Image schemata and protocols. In F. Furingetti (ed.) Proceedings of the Fifteenth PME Conference, Assissi, Italy. Vol. 1, 17–32.
Grossberg, S. (1980). How does the brain build a cognitive code? Psychological Review, 87, 1–51.
Hall, R. (1990). Making mathematics on paper: Constructing representations of stories about related linear functions. Unpublished doctoral dissertation. University of California at Berkeley, Berkeley, CA.
Hancock, C, Kaput, J. & Goldsmith, L. (1992). Authentic inquiry with data: Critical barriers to classroom implementation. Educational Psychologist, 27(3), 337–364.
Kaput, J. (1989). Linking representations in the symbol system of algebra. In C. Kieran & S. Wagner (eds.) A research agenda for the teaching and learning of algebra. Reston, VA: National Council of Teachers of Mathematics; and Hillsdale, NJ: Erlbaum, 167–194.
Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glasersfeld (ed.) Constructivism and mathematics education,. Dordrect: Kluwar Academic Publishers, 53–74.
Kaput, J. (1992). Technology and mathematics education. In D. Grouws (ed.) Handbook on research in mathematics teaching and learning. New York: Macmillan, 515–55
Kaput, J. (1993). MathCars [Computer animation video available from the author] Department of Mathematics, University of Massachusetts at Dartmouth, No. Dartmouth, MA.
Kaput, J. (in press). Democratizing access to calculus: New routes using old roots. In A. Schoenfeld (ed.) Mathematicical thinking and problem solving. Hillsdale, NJ: Erlbaum.
Kaput, J. (1994) The representational roles of technology in connecting mathematics with authentic experience. In R. Bieler, R.W. Scholz, R. Strasser, & B. Winkelman (Eds.) Mathematics as a didactic discipline: The state of the art. Dordrect: Kluwar Academic Publishers, 379–397.
Kaput J., Upchurch, R., & Burke, M. (1994). Integrated tools for elementary mathematics [Software available from the first author]. Department of Mathematics, University of Massachusetts at Dartmouth, No. Dartmouth, MA.
Kosslyn, S. & Koenig, O. (1992). Wet mind: The new cognitive neuroscience. New York: The Free Press
Laborde, J-M. (1991). CABRI Geometry. New York, NY: Brooks-Cole Publishing Co.
Lakoff, G. (1987). Women, fire and dangerous things. Chicago: University of Chicago Press.
Latour, E. (1990). Drawing things together. In M. Lynch & S. Woolgar (Eds.) Representation in scientific practice. Cambridge, Mass.: MIT Press.
Mahoney, M. (1980). The beginnings of algebraic thought in the seventeenth century. In S. Gankroger (Ed.) Descartes: Philosophy, Mathematics and Physics. Sussex, England: Harvester Press.
Miera, L. (1991). Explorations of mathematical sense-making: An activity-oriented view of children’s use and design of material displays. Unpublished doctoral dissertation. University of California at Berkeley, Berkeley, CA.
Miller, J. (1990). Moving pictures, in Images and understanding, H. Barlow, C. Blakemore, & M. Weston-Smith (Eds.) Cambridge: Cambridge University Press, 180–194.
Salomon, G. (1979). Interaction of media, cognition, and learning. San Francisco: Jossey-Bass.
Saussure, F. de (1959). Course in general linguistics. Glasgow: Fontana (first published in French in 1916).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kaput, J.J. (1995). Overcoming Physicality and the Eternal Present: Cybernetic Manipulatives. In: Sutherland, R., Mason, J. (eds) Exploiting Mental Imagery with Computers in Mathematics Education. NATO ASI Series, vol 138. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57771-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-57771-0_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63350-8
Online ISBN: 978-3-642-57771-0
eBook Packages: Springer Book Archive