The Tacit-explicit Dimension of the Learning of Mathematics: An Investigation Report**

  • Cristina FradeEmail author
  • Oto Borges


This paper reports on study that investigated the tacit-explicit dimension of the learning of mathematics. The study was carried out in a secondary school and consisted of an episode analysis related to a class discussion about the difference between plane figures and spatial figures. The data analysis was based on integration between some aspects of Polanyi’s theory on tacit knowledge and Ernest’s model of mathematical knowledge, with reference to its mainly explicit and mainly tacit components. This integration has involved not only the types of knowledge – mainly explicit or mainly tacit – the students used in a psychological way to perform a mathematical task involving conversation, but also and particularly how much the projection of those types of knowledge on the task were manifest tacitly or formalized by the students. Among the results of the research, a strong finding was that the lack of correspondence between the students’ utterances and their original understandings is directly related to the manner in which the tacit co-operates with the explicit in the process of articulation.

Key Words

mathematical knowledge mathematical thought and speech mathematics learning tacit knowledge 


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  1. Boaler, J. (2002). Exploring the nature of mathematical activity: Using theory, research and ‘working hypotheses’ to broaden conceptions of mathematics knowing. Educational Studies in Mathematics, 51(1–2), 3–21.CrossRefGoogle Scholar
  2. Ernest P. (1998a). Social construtivism as a philosophy of mathematics. Albany: SUNY.Google Scholar
  3. Ernest, P. (1998b). Mathematical knowledge and context. In A. Watson (Ed.), Situated cognition and the learning of mathematics (pp. 13–29). Oxford: University of Oxford Department of Educational Studies.Google Scholar
  4. Fischbein, E. (1989). Tacit models and mathematical reasoning. For the Learning of Mathematics, 9(2), 9–14.Google Scholar
  5. Frade, C. (2003). Componentes Tácitos e Explícitos do Conhecimento Matemático de Áreas e Medidas. Tese de Doutorado. Faculdade de Educação-UFMG, 251 páginas.Google Scholar
  6. Frade, C. (2004). The tacit-explicit dynamic in learning processes. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 407414), Vol. 2. Bergen (Norway), 1418 July.Google Scholar
  7. Frade, C. & Borges, O. (2002). Tacit knowledge in curricular goals in mathematics. Proceedings of the 2nd International Conference on the Teaching of Mathematics (at the undergraduate level)ICTM2, Hersonissos, Greece, 1–6 July.Google Scholar
  8. Hiebert, J. & Lefevre, P. (1986). In J. Hiebert (Ed.), Conceptual and procedural knowledge in mathematics: An introductory analysis (pp. 1–27) Hillsdale (NJ), Chapter 1.Google Scholar
  9. Kitcher, P. (1984). The nature of mathematical knowledge. Oxford: Oxford University Press.Google Scholar
  10. Kuhn, T.S. (1998). A estrutura das revoluções científicas, 5 ed., São Paulo: Perspectiva.Google Scholar
  11. Lerman, S. (2001). Getting used to mathematics: Alternative ways of speaking about becoming mathematical. Ways of Knowing, 1(1), 47–52.Google Scholar
  12. Mellin-Olsen, S. (1981). Instrumentalism as an educational concept. Educational Studies in Mathematics, 12, 351–367.CrossRefGoogle Scholar
  13. Piaget, J. (1974). Réussir et comprendre. Paris: Presses Universitaires de France.Google Scholar
  14. Piaget, J. (1976). A equilibração das estruturas cognitivas. Rio de Janeiro: Zahar Editores.Google Scholar
  15. Polanyi, M. (1962). Personal knowledge. London: Routledge & Kegan Paul.Google Scholar
  16. Polanyi, M. (1969). In M. Grene (Ed.), Knowing and being. Chicago: Chicago University Press.Google Scholar
  17. Polanyi, M. (1983). The tacit dimension. Gloucester (Mass): Peter Smith.Google Scholar
  18. Ryle, G. (1949). The concept of mind. London: Hutchinson.Google Scholar
  19. Schoenfeld, A.H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning, Chapter 15. New York: MacMillan, pp. 334–370.Google Scholar
  20. Skemp, R.R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.Google Scholar
  21. Sternberg, R.J. (1995). Theory and measurement of tacit knowledge as a part of practical intelligence. Zeitschrift für Psychologie, 203, 319–334.Google Scholar
  22. Tirosh, D. (Ed.). (1994). Implicit and explicit knowledge: An educational approach. Norwood (NJ): Ablex Publishing Co.Google Scholar
  23. Vergnaud, G. (1993). Teoria dos Campos Conceituais. In L. Nasser (Ed.), Anais do 1° Seminário Internacional de Educação Matemática do Rio de Janeiro. Rio de Janeiro: Projeto Fundão do Instituto de matemática da UFRJ, 28–30 de julho.Google Scholar
  24. Vergnaud, G. (1998). A comprehensive theory of representation for mathematics education. Journal of Mathematical Behavior, 17(2), 167–181.CrossRefGoogle Scholar
  25. Vygotsky, L.S. (1986). In A. Kozulin (Ed.), Thought and language. The Massachusetts Institute of Technology.Google Scholar
  26. Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge, UK: Cambridge University Press.Google Scholar
  27. Wigner. E.P. & Hodgkin, R.A. (1997). Michael Polanyi. Biographical memoirs of Fellows of the royal society (413–438), Vol. 23. London: The Royal Society, Nov.Google Scholar
  28. Winbourne, P. (2002). Looking for learning in practice: How can this inform teaching? Ways of Knowing Journal, 2(2), 3–18.Google Scholar
  29. Wittgenstein, L. (1995). Tratado lógico-filosófico. Investigações filosóficas. Tradução de M.S. Loureiro. 2a edição revista. Lisboa: Fundação Calouste Gulbenkian.Google Scholar

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© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Universidade Federal de Minas Gerais-UFMGEscola Fundamental do Centro Pedagógico-CPCarlosBrazil

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