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Educational Studies in Mathematics

, Volume 46, Issue 1–3, pp 187–228 | Cite as

The mathematical discourse of 13-year-old partnered problem solving and its relation to the mathematics that emerges

  • Carolyn Kieran
Article

Abstract

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving rational functions, was analyzed by means of recently developed methodological tools. The nature of the mathematics that emerged for each pair was found to be related to several factors that included the characteristics of the interpersonal object-level utterances both before and after the solution path had been generated, the degree of activity of the personal channels of the interlocutors, and the extent to which the thoughts of participants were made explicit in the public discourse. The analysis of the discursive interactions provided evidence that adolescents within novel problem situations can experience some difficulty in making their emergent thinking available to their partners in such a way that the interaction be highly mathematically productive for both of them.

communication discourse graphing of rational functions interaction mathematical discourse problem solving research in mathematics education student talk 

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REFERENCES

  1. Bartolini Bussi, M.G. and Mariotti, M.A.: 1999, ‘Semiotic mediation: from history to the mathematics classroom’, For the Learning of Mathematics19(2), 27–35.Google Scholar
  2. Cole, M. and Wertsch, J.V.: n.d., Beyond the individual-social antimony in discussions of Piaget and Vygotsky, Web site: http://www.massey.ac.nz/~ALock/virtual/colevyg.htm.Google Scholar
  3. Dembo, M.H. and McAuliffe, T.J.: 1987, ‘Effects of perceived ability and grade status on social interaction and influence in cooperative groups’,Journal of Educational Psychology79, 415–423.CrossRefGoogle Scholar
  4. Edwards, D.: 1993, ‘But what do children really think?: Discourse analysis and conceptual content in children's talk’, Cognition and Instruction11(3&4), 207–225.CrossRefGoogle Scholar
  5. Edwards, D. and Potter, J.: 1992, Discursive Psychology,Sage, London, UK.Google Scholar
  6. Fey, J. and Heid, M.K.: 1991, Computer-intensive algebra,The University of Maryland and The Pennsylvania State University.Google Scholar
  7. Forman, E.: 1989, ‘The role of peer interaction in the social construction of mathematical knowledge’, in N.M. Webb (ed.), Peer Interaction, Problem-Solving, and Cognition: Multidisciplinary Perspectives, Pergamon, Oxford, U.K., pp. 55–70.Google Scholar
  8. Forman, E.A. and Cazden, C.B.: 1985, ‘Exploring Vygotskian perspectives in education: The cognitive value of peer interaction’, in J.V. Wertsch (ed.), Culture, Communication, and Cognition: Vygotskian Perspectives,Cambridge University Press, New York, pp. 323–347.Google Scholar
  9. Fuson, K.C.: 1980, ‘An explication of three theoretical constructs from Vygotsky’, in T. Kieren (ed.), Recent Research in Number Learning, ERIC/SMEAC, Columbus, OH.Google Scholar
  10. Glachan, M. and Light, P.: 1982, ‘Peer interaction and learning: Can two wrongs make a right?’, in G. Butterworth and P. Light (eds.), Social Cognition: Studies in the Development of Understanding, University of Chicago Press, Chicago, pp. 238–262.Google Scholar
  11. Grice, H.P.: 1968, Logic and Conversation: The William James Lectures,Harvard University Press, Cambridge, MA.Google Scholar
  12. Harré R. and Gillett, G.: 1994, The Discursive Mind,Sage, Thousand Oaks, CA.Google Scholar
  13. Hatano, G. and Inagaki, K.: 1994, April, A Two-Level Analysis of Collective Comprehension Activity,Paper presented at the Symposium on Integrating the Cognitive and Social in the Construction of Mathematical and Scientific Knowledge, AERA meeting, New Orleans.Google Scholar
  14. Hershkowitz, R. and Schwarz, B.: 1997, ‘Unifying cognitive and sociocultural aspects in research on learning the function concept’, in E. Pehkonen (ed.), Proceedings of the 21st International Conference for the Psychology of Mathematics Education,PME Program Committee, Lahti, Finland, Vol. 1, pp.148–164.Google Scholar
  15. Kieran, C.: 1994, ‘A functional approach to the introduction of algebra: Some pros and cons’, in J.P. da Ponte and J.F. Matos (eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education,PME Program Committee, Lisbon, Portugal, Vol. 1, pp.157–175.Google Scholar
  16. Kieran, C. and Dreyfus, T.: 1998, ‘Collaborative versus individual problem solving: Entering another's universe of thought’, in A. Olivier and K. Newstead (eds.), Proceedings of the 22nd International Conference for the Psychology of Mathematics Education,PME Program Committee, Stellenbosch, South Africa, Vol. 3, pp. 112–119.Google Scholar
  17. Kieran, C. and Sfard, A.: 1999, ‘Seeing through symbols: The case of equivalent expressions’,Focus on Learning Mathematics21(1),1–17.Google Scholar
  18. Leikin, R. and Zaslavsky, O.: 1997, ‘Facilitating student interactions in mathematics in a cooperative learning setting’,Journal for Research in Mathematics Education28,331–354.CrossRefGoogle Scholar
  19. Leinhardt, G., Zaslavsky, O. and Stein, M.K.: 1990, ‘Functions, graphs, and graphing: Tasks, learning, and teaching’,Review of Educational Research60, 1–64.CrossRefGoogle Scholar
  20. Lerman, S.: 1998, ‘A moment in the zoom of a lens: Toward a discursive psychology of mathematics teaching and learning’, in A. Olivier and K. Newstead (eds.), Proceedings of the 22nd International Conference for the Psychology of Mathematics Education, PME Program Committee, Stellenbosch, South Africa, Vol. 1, pp.66–81.Google Scholar
  21. Linn, M.C. and Burbules, N.C.: 1993, ‘Construction of knowledge and group learning’, in K. Tobin (ed.), The Practice of Constructivism in Science Education, American Association for the Advancement of Science, Washington, DC, pp.91–119.Google Scholar
  22. Lotman, Y.M.: 1988, ‘Text within a text’,Soviet Psychology 26(3),32–51.Google Scholar
  23. Meira, L. and Carraher, D., (eds.): 1995,Proceedings of the 19th International Conference for the Psychology ofMathematics Education,PME Program Committee, Recife, Brazil.Google Scholar
  24. Noss, R. and Hoyles, C.:1996,Windows on Mathematical Meanings: Learning Cultures and Computers,Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  25. O'Connor, M.C.: 1996, ‘Managing the intermental: Classroom group discussion and the social context of learning’, in D.I. Slobin, J. Gerhardt, A. Kyratzis and J. Guo (eds.), Social Interaction, Social Context, and Language,Erlbaum, Mahwah, NJ, pp.495–509.Google Scholar
  26. Reddy, M.J.: 1979, ‘The conduit metaphor: A case of frame conflict in our language about language’, in A. Ortony (ed.), Metaphor and Thought,Cambridge University Press, Cambridge, pp.282–324.Google Scholar
  27. Rosenberg, J.: 1992, Math Connections: Algebra II[Computer software],WINGS for Learning, Inc., Scotts Valley, CA.Google Scholar
  28. Salomon, G. and Globerson, T.: 1989, ‘When teams do not function the way they ought to’, in N.M. Webb (ed.), Peer Interaction, Problem-Solving, and Cognition: Multidisciplinary Perspectives,Pergamon Press, Oxford, UK, pp. 89–99.Google Scholar
  29. Sfard, A.: 2000, ‘Steering (dis)course between metaphors and rigor: Using focal analysis to investigate an emergence of mathematical objects’,Journal for Research inMathematics Education31,296–327.CrossRefGoogle Scholar
  30. Sfard, A. and Kieran, C.: 2001, ‘Cognition as communication: Rethinking learning-bytalking through multi-faceted analysis of students' mathematical interactions’,Mind, Culture, and Activity8(1),42–76.CrossRefGoogle Scholar
  31. Shotter, J. and Billig, M.: 1998, ‘A Bakhtinian psychology: From out of the heads of individuals and into the dialogues between them’, in M.M. Bell and M. Gardiner (eds.), Bakhtin and the Human Sciences, Sage, Thousand Oaks, CA, pp. 13–29.Google Scholar
  32. Streefland, L., (ed.): 1985, Proceedings of the 9th International Conference for the Psychology of Mathematics Education,PME Program Committee, Noordwijkerhout, The Netherlands.Google Scholar
  33. Teasley, S.D.: 1995, ‘The role of talk in children's peer collaborations’,Developmental Psychology31(2),207–220.CrossRefGoogle Scholar
  34. Trognon, A.:1993, ‘How does the process of interaction work when two interlocutors try to resolve a logical problem?’,Cognition and Instruction11(3&4),325–345.Google Scholar
  35. Vion, R.: 1999, ‘Linguistique et communication verbale’, in M. Gilly, J.-P. Roux and A. Trognon (eds.), Apprendre dans l'Interaction: Analyse des Médiations Sémiotiques, Presses universitaires de Nancy, Nancy, FR, pp.41–67.Google Scholar
  36. Vygotsky, L.S.: 1981, ‘The instrumental method in psychology’, in J.V. Wertsch (ed.), The Concept of Activity in Soviet Psychology,M.E. Sharpe, Armonk, NY, pp. 134–143.Google Scholar
  37. Vygotsky, L.S.:1987, ‘Thinking and speech’, in R.W. Rieber and A.S. Carton (eds.), The Collected Works of L.S. Vygotsky: Vol. 1, Problems of General Psychology (trans. N. Minick), Plenum, NY, pp.39–285.Google Scholar
  38. Webb, N.M.: 1991, ‘Task-related verbal interaction and mathematics learning in small groups’,Journal for Research in Mathematics Education22, 366–389.CrossRefGoogle Scholar
  39. Wertsch, J.V., (ed.): 1985,Culture, Communication, and Cognition: Vygotskian Perspectives, Cambridge University Press, Cambridge, UK.Google Scholar
  40. Wertsch, J.V.: 1998,Mind as Action, Oxford University Press, New York.Google Scholar
  41. Wittgenstein, L.: 1953,Philosophical Investigations (trans. G.E.M. Anscombe), Blackwell, Oxford, UK.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Carolyn Kieran
    • 1
  1. 1.Département de MathématiquesUniversité du Québec à MontréalMontréalCanada

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