Educational Studies in Mathematics

, Volume 46, Issue 1–3, pp 13–57 | Cite as

There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning

  • Anna Sfard


Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult questions, I propose to base research on a metaphor of thinking-as-communicating.This conceptualization entails viewing learning mathematics as an initiation to a certain well defined discourse. Mathematical discourse is made special by two main factors: first, by its exceptional reliance on symbolic artifacts as its communication-mediating tools, and second, by the particular meta-rules that regulate this type of communication. The meta-rules are the observer’s construct and they usually remain tacit for the participants of the discourse. In this paper I argue that by eliciting these special elements of mathematical communication, one has a better chance of accounting for at least some of the still puzzling phenomena. To show how it works, I revisit the episodes presented at the beginning of the paper, reformulate the ensuing questions in the language of thinking-as-communication, and re-address the old quandaries with the help of special analytic tools that help in combining analysis of mathematical content of classroom interaction with attention to meta-level concerns of the participants.


Mathematical Thinking Communicational Approach Mathematical Discourse Legitimate Peripheral Participation Sociomathematical Norm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Anderson, J.R., Reder, L.M. and Simon, H.A.: 1996, ‘Situated learning and education’, Educational Researcher 25(4), 5-11.CrossRefGoogle Scholar
  2. Bateson, G.: 1973, Steps to an Ecology of Mind, Frogmre, St. Albans: Paladin.Google Scholar
  3. Bauersfeld, H.: 1995, ‘Language games’ in mathematics classroom: Their function and their effects’, in P. Cobb and H. Bauersfeld (eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 271-292.Google Scholar
  4. Bourdieu, P.: 1999, ‘Structures, Habitus, practices’, in A. Elliot (ed.), The Blackwell Reader in Contemporary Social Theory, Blackwell, Oxford, UK, pp. 107-118.Google Scholar
  5. Bouveresse, J.: 1999, ‘Rules, dispositions, and the Habitus’, in R. Shusterman (ed.), Bourdieu: A Critical Reader, Blackwell, Oxford, UK, pp. 45-63.Google Scholar
  6. Brown, J.S, Collins, A. and Duguid, P.: 1989, ‘Situated cognition and the culture of learning’, Educational Researcher 18(1), 32-42.CrossRefGoogle Scholar
  7. Brownell, W.A.: 1935, ‘Psychological considerations in the learning and teaching of arithmetic’, in The Teaching of Arithmetic: Tenth Yearbook of the National Council of the Teachers of Mathematics, Columbia University Press, New York.Google Scholar
  8. Bruner, J.: 1983, ‘The acquisition of pragmatic commitments’, in R. Golinkoff (ed.), The Transition from Prelinguistic to Linguistic Communication, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 27-42.Google Scholar
  9. Bruner, J.: 1986, Actual Minds, Possible words, Harvard University Press, Cambridge, Massachusetts.Google Scholar
  10. Bruner, J.: 1990, Acts of Meaning, Harvard University Press, Cambridge, Mass.Google Scholar
  11. Davis, R.: 1988, ‘The interplay of algebra, geometry, and logic’, Journal of Mathematical Behavior 7, 9-28.Google Scholar
  12. Cazden, C.: 1988, Classroom Discourse, Heinemann, Portsmouth, NH.Google Scholar
  13. Cobb, P.: 1998, ‘Learning from distributed theories of intelligence’, Mind, Culture, and Activity 5(3), 187-204.CrossRefGoogle Scholar
  14. Cobb, P. and Bowers, J.: 1999, ‘Cognitive and situated perspectives in theory and practice’, Educational Researcher 28(2), 4-15.CrossRefGoogle Scholar
  15. Cobb, P., Wood, T. and Yackel, E.: 1993, ‘Discourse, mathematical thinking, and classroom practice’, in E. Forman, N. Minick and A. Stone (eds.), Contexts for Learning, Sociocultural Dynamics in Children's Development, Oxford University Press, New York, pp. 91-119.Google Scholar
  16. Cole, M.: 1988, ‘Cross-cultural research in the socio-historical tradition’, Human Development 31, 137-151.CrossRefGoogle Scholar
  17. Cole, M.: 1995, ‘Socio-cultural-historical psychology, some general reamrks and a proposal for a new kind of cultural-genetic methodology’, in J.V. Wertsch, P. del Rio and A. Alvarez (eds.), Sociocultural Studies of Mind, Cambridge University Press, Cambridge, Massachusetts, pp. 187-214.Google Scholar
  18. Cole, M.: 1996, Cultural Psychology, The Belknap Press of Harvard University Press, Cambridge, Massachusetts.Google Scholar
  19. Edwards, D.: 1993, ‘But what do children really think? Discourse analysis and conceptual content in children's talk’, Cognition and Instruction 11(3and4), 207-225.CrossRefGoogle Scholar
  20. Edwards, D.: 1997, Discourse and cognition, Sage, London.Google Scholar
  21. Engestrom, Y. and Middleton, D.: 1996, Cognition and Communication at Work, Cambridge University Press, Cambridge, Massachusetts.Google Scholar
  22. Fischbein, E.: 1989, ‘Tacit models and mathematical reasoning’, For the learning of mathematics 9(2), 9-14.Google Scholar
  23. Fischbein, E., Deri, M., Nello, M.S. and Marino, M.S.: 1985, ‘The role of implicit models in solving verbal problems in multiplication and division’, Journal for Research in Mathematics Education 16, 3-17.CrossRefGoogle Scholar
  24. Forman, E.: 1996, ‘Forms of participation in classroom practice, Implications for learning mathematics’, in P. Nesher, L. Steffe, P. Cobb, G. Goldin and B. Greer (eds.), Theories of Mathematical Learning, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 115-130.Google Scholar
  25. Forman, E. and Larreamendy-Joerns, J.: 1998, ‘Making the implicit explicit: Classroom explanations and conversational implicatures’, Mind, Culture, and Activity 5(2), 105-113.CrossRefGoogle Scholar
  26. Garfinkel, H.: 1967, Studies in Ethnomethodology, Polity Press, London.Google Scholar
  27. Goffman, E.: 1974, Frame Analysis, An Essay on Organization of Experience, Northeastern University Press, Boston, MA.Google Scholar
  28. Greeno, J.G.: 1997, ‘On claims that answer the wrong question’, Educational Researcher 26(1), 5-17.CrossRefGoogle Scholar
  29. Harre, R. and Gillett, G.: 1995, The discursive mind, Sage Publications, Thousand Oaks.Google Scholar
  30. Hiebert, J. and Carpenter, T.P.: 1992, ‘Learning and teaching with understanding’, in D.A. Grouws (ed.), The Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 65-100.Google Scholar
  31. Kieran, C. and Sfard, A.: 1999, ‘Seeing through symbols, The case of equivalent equations’, Focus on Learning Mathematics 21(1), 1-17.Google Scholar
  32. Kilpatrick, J.: 1992, ‘A history of research in mathematics education’, in D. Grouws, (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 3-38.Google Scholar
  33. Krummheuer, G.: 1995, ‘The ethnography of argumentation’, in P. Cobb and H. Bauersfeld (eds.), The Emergence of Mathematical Meaning, Interactions in Classroom Cultures, Erlbaum, Hillsdale, NJ, pp. 229-269.Google Scholar
  34. Lampert, M.: 1990, ‘When the problem is not the question and the solution is not the answer, Mathematical knowing and teaching’, American Educational Research Journal 27, 29-63.CrossRefGoogle Scholar
  35. Lampert, M. and Blunk, M.L. (eds.): 1998, Talking Mathematics in School, Studies of Teaching and Learning, Cambridge University Press, Cambridge, UK.Google Scholar
  36. Lampert, M. and Cobb, P., in press, ‘White Paper on Communication and Language for Standards 2000 Writing Group’, in J. Kilpatrick,Martin, G. and Schifter, D. (eds.), A Research Companion for NCTMStandards, National Council for Teachers ofMathematics, Reston, VA.Google Scholar
  37. Lave, J.: 1988, Cognition in Practice, Cambridge University Press, Cambridge.Google Scholar
  38. Lave, J. and Wenger, E.: 1991, Situated Learning, Legitimate Peripheral Participation, Cambridge University Press, Cambridge.Google Scholar
  39. Leontiev, A.N.: 1930, ‘Studies in the cultural development of the child. II. The development of voluntary attention in the child’, Journal of Genetic Psychology 37, 52-81.Google Scholar
  40. Levinson, S.: 1983, Pragmatics, Cambridge University Press, Cambridge.Google Scholar
  41. Luria, A.R.: 1928, ‘The problem of the cultural development of the child’, Journal for Genetic Psychology 35, 493-506.Google Scholar
  42. Macnab, D.: 2000, ‘Raising standards in mathematics education, values, vision, and TIMSS’, Educational Studies in Mathematics 42(1), 61-80.CrossRefGoogle Scholar
  43. Mantovani, G.: 2000, Exploring Borders, Understanding Culture and Psychology, Routlege, London.Google Scholar
  44. Markovits, Z., Eylon, B. and Bruckheimer, M.: 1986, ‘Functions today and yesterday’, For the learning of mathematics 6(2), 18-24.Google Scholar
  45. Mayer, R.E.: 1983, Thinking, problem solving, cognition, W.H. Freeman and Company, New York.Google Scholar
  46. Morgan, C.: 1996, ‘The language of mathematics, Towards a critical analysis of mathematical text’, For the learning if mathematics 16(3), 2-10.Google Scholar
  47. NCTM (National Council of Teachers of Mathematics): 2000, Principles and standards for school mathematics, NCTM, Reston, VA.Google Scholar
  48. O'Connor, M.C.: 1998, ‘Language socialization in the mathematics classroom, Discourse practices and mathematical thinking’, in M. Lampert and M. Blunk (eds.), Talking Mathematics, Studies of Teaching and Learning in School, Cambridge University Press, Cambridge, NY, pp. 17-55.Google Scholar
  49. Schmidt, W.H., McKnight, C.C., Cogan, L.S., Jakwerth, P.M. and Houang, R.T.: 1999, Facing the Consequences-Using TIMSS for a Closer Look at U.S. Mathematics, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  50. Schutz, A.: 1967, ‘The problem of social reality’, in M. Natanson and H.L. van Breda (eds.), Collected Papers, Martinus Nijhoff, The Hague, vol. 1.Google Scholar
  51. Sfard, A.: 1997, ‘The many faces of mathematics, Do mathematicians and researchers in mathematics education speak about the same thing?’, in A. Sierpinska and J. Kilpatrick (eds.), Mathematics Education as a Research Domain, A Search for Identity, Kluwer Academic Publishers, Dordrecht, Vol. 2, pp. 491-512.Google Scholar
  52. Sfard, A.: 1998, ‘On two metaphors for learning and on the dangers of choosing just one’, Educational Researcher 27(2), 4-13.CrossRefGoogle Scholar
  53. Sfard, A.: 2000a, ‘Symbolizing mathematical reality into being, How mathematical discourse and mathematical objects create each other’, in P. Cobb, K.E. Yackel and K. McClain (eds.), Symbolizing and Communicating, Perspectives on Mathematical Discourse, Tools, and Instructional Design, Erlbaum, Mahwah, NJ, pp. 37-98.Google Scholar
  54. Sfard, A.: 2000b, ‘Steering (dis)course between metaphor and rigor: Using focal analysis to investigate the emergence of mathematical objects’, Journal for Research in Mathematics Education 31(3), 296-327.CrossRefGoogle Scholar
  55. Sfard, A.: 2000c, ‘On reform movement and the limits of mathematical discourse’, Mathematical Thinking and Learning 2(3), 157-189.CrossRefGoogle Scholar
  56. Sfard, A. and Kieran, C.: 2001a, ‘Preparing teachers for handling students' mathematical communication, Gathering knowledge and building tools’, To appear in F.L. Lin and T. Cooney (eds.), Making Sense of Mathematics Teacher Education, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  57. Sfard, A. and Kieran, C.: 2001b, ‘Cognition as communication, Rethinking learning-bytalking through multi-faceted analysis of students' mathematical interactions’, Mind, Culture, and Activity 8(1), 42-76.CrossRefGoogle Scholar
  58. Smith, J.P., diSessa, A.A. and Rochelle, J.: 1993, Misconceptions reconceived, A constructivist analysis of knowledge in transition', The Journal of the Learning Sciences 3(2), 115-163.CrossRefGoogle Scholar
  59. Steinbring, H., Bartolini-Bussi, M.G. and Sierpinska, A. (eds.): 1998, Language and communication in mathematics classroom, The National Council of Teachers of Mathematics, Reston, VA.Google Scholar
  60. Stigler, J. and Hiebert, J.: 1999, The Teaching Gap, Best Ideas from the World's Teachers for Improving Education in the Classroom, The Free Press, New York.Google Scholar
  61. Tomasello, M.: 1999, The Cultural Origins of Human Cognition, Harvard University Press, Cambridge, Massachusetts.Google Scholar
  62. Tall, D. and Schwartzenberger, R.: 1978, ‘Conflicts in the learning of real numbers and limits, Mathematics Teaching 82, 44-49.Google Scholar
  63. Tall, D. and Vinner, S.: 1981, ‘Concept image and concept definition in mathematics with particular reference to limits and continuity’, Educational Studies in Mathematics 12, 151-169.CrossRefGoogle Scholar
  64. Vinner, S. and Dreyfus, T.: 1989, ‘Images and definitions for the concept of function’, Journal for Research in Mathamtics Education 20(4), 356-366.CrossRefGoogle Scholar
  65. Voigt, J.: 1985, ‘Patterns and routines in classroom interaction’, Recherches en Didactique des Mathématiques 6, 69-118.Google Scholar
  66. Voigt, J.: 1996, ‘Negotiation of mathematical meaning in classroom processes, Social interaction and learning mathematics’, in L.P. Steffe, P. Nesher, P. Cobb, G.A. Goldin and B. Greer (eds.), Theories of Mathematical Learning, Kluwer Academic Publishers, Mahwah, NJ, pp. 21-50.Google Scholar
  67. Vygotsky, L.S.: 1931/1981, ‘The genesis of higher mental functions’, in J. Wertsch (ed.), The Concept of Activity in Soviet Psychology, Sharpe, New York.Google Scholar
  68. Vygotsky, L.S.: 1978, Mind in Society. The Development of Higher Psychological Processes, Harvard University Press, Cambridge, MA.Google Scholar
  69. Vygotsky, L.S.: 1987, ‘Thinking and speech’, in R.W. Rieber and A.C. Carton (eds.), The Collected Works of L.S. Vygotsky, Plenum Press, New York, Vol. 1, pp. 39-285.Google Scholar
  70. Wenger, E.: 1998, ‘Practice’, in E. Wenger, Communities of Practice: Learning, Meaning, and Community, Cambridge University Press, New York, pp. 43-102.Google Scholar
  71. Wittgenstein, L.: 1953, Philosophical Investigations, G.E.M. Anscombe, Trans. Oxford, UK, Blackwell (Original work published 1953).Google Scholar
  72. Yackel, E. and Cobb, P.: 1996, ‘Sociomathematical norms, argumentation, and autonomy in mathematics’, Journal for Research in Mathematics Education 27, 458-477.CrossRefGoogle Scholar

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Anna Sfard
    • 1
  1. 1.Faculty of EducationThe University of Haifa, Mount CarmelHaifaIsrael

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