Mediating artifacts and interaction in a computer environment – An exploratory study of the acquisition of geometry concepts

Jan Wyndhamn

doi:10.7146/nomad.v1i2.144996

Authors

Jan Wyndhamn

DOI:

https://doi.org/10.7146/nomad.v1i2.144996

Abstract

Nine dyads of twelve-year old students were engaged in collaborative small-group activity in order to find a way to determine the area of a parallelogram via a computer program. The program introduced two models or metaphors of a parallelogram, one changing the perimeter with a constant area ("the deck-of-cards model"), the other changing the area with a constant perimeter ("the frame model"). By changing the form and size of a displayed parallelogram the students had the opportunity to explore and obtain the characteristics of the parallelogram. The interaction between the students and their interaction with the computer were registered for analysis. After twenty minutes work with the computer, only one dyad failed to obtain the idea of the area determinants. The students' discussion, characterized by active inter- pretation and mutual adoption in combination with the deck-of-cards model as medi- ating artifact, established a situation in which the students acquired the conception that the area remained invariant through a specific kind of transformation. This know- ledge was then used to draw a rectangle with the same area as a given parallelogram.

References

Bruner, J. (1966). On cognitive growth II. In J.S. Bruner, R.R. Olver, & P.M. Greenfield (Eds.), Studies in cognitive growth (pp. 30-67). New York: Wiley.

Bruner, J. (1985). Vygotsky: A theoretical and conceptual perspective. In J.V. Wertsch (Ed.), Culture, communication and cognition: Vygotskianperspectives (pp. 21-34). Cambridge: Cambridge University Press.

Bruner, J., & Haste, H. (1987). Introduction. In J. Bruner, & H. Haste (Eds.), Making sense. The child's construction of the world (pp. 11-25). London: Methuen.

Cobb, P., Wood, T., Yackel. E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research Journal, 29(3), 573- 604. https://doi.org/10.3102/00028312029003573

Dreyfus, H.L. (1992). Being-in-the-world. A commentary on Heidegger's Being and Time, Division 1. Cambridge, MA: The MIT Press.

Feldman, C.F. (1987). Thought from language: the linguistic construction of cognitive representations. In J. Bruner, & H. Haste (Eds.), Making sense. The child's construction of the world (pp. 131-146). London and New York: Methuen.

Gellert, H., Kustner, H., Hellwich, M., & Kastner, H. (Eds.) (1977). The VNR concise encyclopedia of mathematics. New York: Van Nostrand Reinhold. https://doi.org/10.1007/978-1-4684-8237-9

Hart, K.M. (Ed.) (1981). Children's understanding of mathematics: 11-16. London: John Murray.

Hedrén, R. (1990). Logoprogrammering på mellanstadiet. En studie av fördelar och nack- delar med användning av Logo i matematikundervisningen under årskurserna 5 och 6 i grundskolan. [Programming in Logo at intermediate level of compulsory school. A study of advantages and disadvantages of the use of Logo in mathematics instruction in forms 5 and 6 of the Swedish compulsory school.] Linköping Studies in Education. Dissertations No. 28.

Hoyles, C. (1991). Developing mathematical knowledge through microworlds. In AJ. Bishop, S. Mellin-Olsen, & J. van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp.147-176). Dordrecht: Kluwer. https://doi.org/10.1007/978-94-017-2195-0_8

Hutchins, E. (1986). Mediation and automatization. The Quarterly Newsletter of the Laboratory of Comparative Human Cognition, 8(2), 47-58.

Kaput, J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 19-26). Hillsdale, NJ: Lawrence Erlbaum.

Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511609268

Nolder, R. (1991). Mixing metaphor and mathematics in the secondary classroom. In K. Durkin, & B. Shire (Eds.), Language in mathematical education. Research and practice (pp. 105-114). Milton Keynes, PA: Open University Press.

Olson, D.R. (1976). Culture, technology and intellect. In L.B. Resnick (Ed.), The nature of intelligence. Hillsdale, NJ: Lawrence Erlbaum.

Ong, W.J. (1982). Orality and literacy: The technologizing of the word. New York: Methuen. Pea, R.D. (1987). Cognitive technologies for mathematics education. In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 89-122). Hillsdale, NJ: Lawrence Erlbaum. https://doi.org/10.4324/9780203328064

Pufall, P.B. (1988). Function in Piaget's system: Some notes for constructors of microworlds. In G. Foreman, & P.B. Pufall (Eds.), Constructivism in the computer age (pp. 15-36). Hillsdale, NJ: Lawrence Erlbaum.

Säljö, R. (1992a). Human growth and the complex society. Notes on the monocultural bias of theories of learning. Cultural Dynamics, V (1), 43-56. https://doi.org/10.1177/092137409200500103

Säljö, R. (1992b). Kunskap genom samtal. [Knowledge through discourse.] Didaktisk tid- skrift (2-3), 16-29.

Säljö, R., & Wyndhamn, J. (1990). Problem-solving, academic performance and situated reasoning. A study of joint cognitive activity in the formal setting. British Journal of Educational Psychology, 60, 245-254. https://doi.org/10.1111/j.2044-8279.1990.tb00942.x

Sayeki, Y., Ueno, N., & Nagasaka, T. (1991). Mediation as a generative model for obtaining area. Learning and Instruction, 1(3), 229-242. https://doi.org/10.1016/0959-4752(91)90005-S

Schoenfeld, A. H. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 225-264). Hillsdale, NJ: Lawrence Erlbaum.

Scribner, S. (1984). Studying working intelligence. In B. Rogoff, & J. Lave (Eds.), Everyday cognition: Its development in social context (pp. 9-40). Cambridge, MA: Harvard University Press.

Sheingold, K. (1987). The microcomputer as a symbolic medium. In R.D. Pea, & K. Shein- gold (Eds.), Mirrors of minds: Patterns of experience in educational computing (pp. 198- 208). Norwood, NJ: Ablex.

Steinbring, H. (1989). Routine and meaning in the mathematics classroom. For the Learning of Mathematics, 8(1), 24-33.

van Hiele, P.M. (1986). Structure and insight. A theory of mathematics education. London: Academic Press.

Voigt, J. (1989). The social constitution of the mathematics province - A microethnographical study in classroom interaction. The Quarterly Newsletter of the Laboratory of Comparative Human Cognition, 11(1&2), 27-34.

Vygotsky, L.S. (1978). Mind in society. (M. Cole et al., Trans.) Cambridge, MA: Harvard University Press

Vygotsky, L.S. (1986). Thought and language. (A. Kozulin, Trans. and Ed.) Cambridge, MA: MIT Press.

Wertheimer, M. (1959). Productive thinking. London: Associated Book Publishers. Wertsch, J.V. (1991). A sociocultural approach to socially shared cognition. In L.B. Resnick, J.M. Levine, & S.D. Teasley (Eds.), Perspectives on socially shared cognition (pp. 85-100). Washington, DC: American Psychological Association. https://doi.org/10.1037/10096-004

Wyndhamn, J. (1992). Matematisk kompetens och sociokulturell aktivitet. En experimentell studie över hur elever löser problem med datumangivelser. [Mathematical competence and sociocultural activity. An experimental study of students solving problems concerning date information.] Arbetsrapport [Working paper] 1992:6. University of Linköping, Department of Communication Studies.