Skip to main content

Epistemological change through peer apprenticeship learning: From rule-based to idea-based social constructivism

Abstract

The thrust of peer apprenticeship learning is in how an individual's personal beliefs, disposition, and epistemology can be transformed through an “apprenticeship-like” learning process. This paper describes the peer apprenticeship learning situation between two students, Dom (age: 13) and Ming (age: 14), in distributed computer-mediated co-construction of mathematical meanings. Ming was initially procedural or rule-based in his problem solving methods, encountering numerous difficulties with his approaches. Dom, however, was constantly engaged in playing with ideas through conjecturing and perceiving mathematical relationships. In the initial stages of co-construction efforts, Dom and Ming were solving the problems rather independently as each appropriated a radically different epistemology of mathematics – Ming was rule-based whereas Dom was idea-based. In the cause of increasing difficulties, Ming soon recognized that his methods were inadequate and, after a considerable struggle, positioned himself in an “apprentice-like” manner in order to follow Dom's conceptualizations. Through monitoring of Dom's conceptualizations and with personal experimentations to concretize his understanding, Ming was gradually able to assimilate Dom's mathematical meaning perspectives. We depicted such a learning situation as peer apprenticeship learning. As a result of assimilating the disposition towards playing with ideas, Dom and Ming were able to engage in meaningful idea-based social constructivism.

This is a preview of subscription content, access via your institution.

REFERENCES

  • Bakhtin, M.M. (1986). Speech Genres and Other Late Essays. Austin: University of Texas Press.

    Google Scholar 

  • Bartolini-Bussi, M. (1990). Mathematics knowledge as a collective enterprise. In Proceedings of the 4th SCTP Conference in Brakel, West Germany, IDM, Bielefeld.

    Google Scholar 

  • Barwise, J. and Perry, J. (1983). Situations and Attitudes. Cambridge, MA: MIT Press.

    Google Scholar 

  • Bateson, G. (1972). Steps to an Ecology of Mind. NY: Ballantine Books.

    Google Scholar 

  • Bateson, G. (1979). Mind and Nature: A Necessary Unity. NY: Bantam Books.

    Google Scholar 

  • Bickhard, M. H. (1992). How does the environment affect the person? In L. T. Wineger and J. Valsiner (Eds.), Children's Development in Social Context. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Bishop, A. (1988). Mathematical Enculturation: A Cultural Perspective on Mathematics Education. Dordrecht, Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Bredo, E. (1994). Reconstructing educational psychology: Situated cognition and Deweyan pragmatism. Educational Psychologist 29(1), 23-35.

    Google Scholar 

  • Brown, J., Collins, A. and Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher 18(1), 32-42.

    Google Scholar 

  • Butterworth, G. (1992). Context and cognition in models of cognitive growth. In P. Light and G. Butterworth (Eds.), Context and Cognition: Ways of Learning and Knowing. Harvester Wheatsheaf.

  • Carraher, T., Carraher, D. and Schliemann, A. (1983). Mathematics in the streets and in schools. British Journal of Developmental Psychology 3, 21-29.

    Google Scholar 

  • Chee, Y. S. (1996). Mind Bridges: A distributed, multimedia learning environment for collaborative knowledge building. International Journal of Educational Telecommunications 2(2/3), 137-153.

    Google Scholar 

  • Clancey, W. (1991). The frame of reference problem in the design of intelligent machines. In K. VanLehn (Ed.), Architectures for Intelligence. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Clancey, W. (1992). “Situated” Means Coordinating Without Deliberation.Santa Fe, NM: McDonnel Foundation Conference.

    Google Scholar 

  • Cobb, P., Wood, T. and Yackel, E. (1991). A constructivist approach to second grade mathematics. In E. von Glasersfeld (Ed.), Constructivism in Mathematics Education. Dordrecht: Kluwer Publications.

    Google Scholar 

  • Dewey, J. (1910/1981). The experimental theory of knowledge. In J. J. McDermott (Ed.), The Philosophy of John Dewey. Chicago: University of Chicago Press.

    Google Scholar 

  • Dewey, J. (1929/1988). The quest of certainty. In J. A. Boyston (Ed.), John Dewey: The Later Works, 1925 -1953, Vol. 4. Carbondale: Southern Illinois University Press.

    Google Scholar 

  • Dewey, J. (1933). How We Think. Boston: NY: D.C. Heath & Company.

    Google Scholar 

  • Dewey, J. and Bentley, A. (1949). Knowing and the known. Boston: Beacon Press.

    Google Scholar 

  • Gardner, H. (1983). Frames of Mind. NY: Basic Books.

    Google Scholar 

  • Garfinkel, H. (1967). Studies in Ethnomethodology.Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Greeno, J. (1988). For the study of mathematics epistemology. In R. Charles and E. Silver (Eds.), The Teaching and Assessing of Mathematical Problem Solving. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Geertz, C. (1983). Local Knowledge. NY: Basic Books.

    Google Scholar 

  • Goodwin, C. and Heritage, J. (1990). Conversation analysis. Annual Review of Anthropology 19, 283-307.

    Article  Google Scholar 

  • Hiebert, J. (Ed.) (1986). Conceptual and Procedural Knowledge. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Heritage, J. (1989). Current developments in conversation analysis. In D. Roger and P. Bull (Eds.), Conversation. NY: Multilingual Matters Ltd.

    Google Scholar 

  • Hopper, R. (1989). Conversation analysis and social psychology as descriptions of interpersonal communication. In D. Roger and P. Bull (Eds.), Conversation. Multilingual Matters Ltd.

  • Hung, D. (in press). Meanings, contexts, and mathematical thinking: The meaning-context model. Journal of Mathematical Behavior 16(3).

  • Hung, D. (1996). The Social Construction of Mathematical Meanings Through Computermediated Collaborative Problem Solving Environments. Unpublished doctoral dissertation, National University of Singapore.

  • Jefferson, G. (1979). A technique for inviting laughter and its subsequent acceptance declination. In G. Psathas (Ed.), Everyday Language. NY: Irvington Press.

    Google Scholar 

  • Jost, J. T. (1995). Toward a Wittgensteinian social psychology of human development. Theory & Psychology 5(1), 5-25.

    Google Scholar 

  • Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Princeton, NY: Cambridge University Press.

    Google Scholar 

  • Lave, J. (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. NY: Cambridge University Press.

    Google Scholar 

  • Lave, J. and Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, England: Cambridge University Press.

    Google Scholar 

  • Levinson, S. (1983). Pragmatics. Cambridge: Cambridge University Press.

    Google Scholar 

  • Maturana, H. and Verala, F. (1987). The Tree of Knowledge: The Biological Roots of Human Understanding. Boston: Shambhala.

    Google Scholar 

  • Mead, G. (1934). Mind, Self & Society. Chicago: University of Chicago Press.

    Google Scholar 

  • Newman, D., Griffin, P. and Cole, M. (1989). The Construction Zone: Working for Cognitive Change in Schools. Cambridge, England: Cambridge University Press.

    Google Scholar 

  • Pepper, S. C. (1942). World Hypothesis. Berkeley: University of California Press.

    Google Scholar 

  • Polanyi, M. (1964). Personal Knowledge: Towards a Post-critical Philosophy. NY: Harper & Row.

    Google Scholar 

  • Polya, G. (1954). How to Solve It (2nd. Ed.). NY: Anchor Books.

    Google Scholar 

  • Prawat, R. S. (1991). The value of ideas: The immersion approach to the development of thinking. Educational Researcher 20(2), 3-10.

    Google Scholar 

  • Prawat, R. S. (1996). Constructivism, Modern and Postmodern. Educational Psychologist 31(3/4), 215-225.

    Google Scholar 

  • Resnick, L. (1989). Problem solving as an everyday practice. In R. J. Charles and E. A. Silver (Eds.), The Teaching and Assessing of Mathematical Problem Solving. Hillsdale, NJ: Lawrence Erlbaum Association.

    Google Scholar 

  • Rogoff, B. and Lave, J. (Eds.) (1984). Everyday Cognition: Its Development in Social Contexts.Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Rollins, M. (1989). Mental Imagery: On the Limits of Cognitive Science. NewHaven: Yale University Press.

    Google Scholar 

  • Roschelle, J. (1989). The Construction of Shared Knowledge in Collaborative Problem Solving. Working Paper of the Institute for Research on Learning, University of California, Berkeley.

    Google Scholar 

  • Roschelle, J. (1992). Learning by collaborating: Convergent conceptual change. The Journal of the Learning Sciences 2(3), 235-276.

    Article  Google Scholar 

  • Rowe, H. A. H. (1991). Introduction: Paradigm and context. In H. A. H. Rowe (Ed.), Intelligence: Reconceptualization and Measurement (pp. 1-18). Hillsdale, NJ: Lawrence Erlbaum Association.

    Google Scholar 

  • Scardamalia, M. and Bereiter, C. (1992). Higher levels of agency for children in knowledge building: A challenge for the design of new knowledge media. The Journal of the Learning Sciences 1(1), 37-68.

    Google Scholar 

  • Schegloff, E. A. (1991). Conversation analysis and socially shared cognition. In L. B. Resnick, J. Levine and S. D. Behrend (Eds.), Socially Shared Cognition. Washington, DC: APA.

    Google Scholar 

  • Schegloff, E. and Sacks, H. (1973). Opening up closings. Semiotica 30, 289-327.

    Google Scholar 

  • Shell Center For Mathematical Education (1984). Problems with Patterns and Numbers. Manchester: The Joint Matriculation Board.

    Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. NY: Macmillan Publications.

    Google Scholar 

  • Suchman, L. A. (1987). Plans and Situated Actions: The Problem of Human-machine Communication. Cambridge, England: Cambridge University Press.

    Google Scholar 

  • Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge: Harvard University Press.

    Google Scholar 

  • Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch (Ed.), The Concept of Activity in Soviet Psychology. White Plains, NY: M. Sharpe.

    Google Scholar 

  • Walkerdine, V. (1988). The Mastery of Reason. London: Routledge.

    Google Scholar 

  • Wertsch, J. V. (Eds.) (1985). Vygotsky and the Social Formation of Mind. Cambridge: Harvard University Press.

    Google Scholar 

  • Wittgenstein, L. (1958). Philosophical Investigations. Cambridge: Basil Blackwell Ltd.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Loong, D.H.W. Epistemological change through peer apprenticeship learning: From rule-based to idea-based social constructivism. International Journal of Computers for Mathematical Learning 3, 45–80 (1998). https://doi.org/10.1023/A:1009786710753

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009786710753

  • meaning co-construction
  • epistemology
  • intersubjectivity
  • collaborative problem solving
  • assimilation
  • negotiation
  • internalization
  • appropriation