Educational Studies in Mathematics

, Volume 61, Issue 1–2, pp 247–277 | Cite as

Mathematics and Mathematics Education as Triadic Communication? a Semiotic Framework Exemplified

  • Sigmund OngstadEmail author


The article investigates in the first part critically dyadic and essentialist understanding of signs and utterances in mathematics and mathematics education as opposed to a triadic view. However even Peircean semiotics, giving priority to triadic, dynamic sign may face challenges, such as explaining the sign as a pragmatic act and how signs are related to context. To meet these and other hurdles an explicit communicational, pragmatic and triadic view, found in parts of the works of Bühler, Bakhtin, Habermas, and Halliday, is developed. Two basic principles are combined and established in a theoretical framework. Firstly, whenever uttering, there will exist in any semiotic sign system, dynamic reciprocity and simultaneity between expressing through form, referring to content, and addressing as an act. Secondly, meaning will be created by the dynamics between given and new in utterances and between utterances and contextual genres. The latter principle explains how meaning merge in communication dynamically and create the basis for a discursive understanding of semiosis and hence even learning at large. The second part exemplifies each of the three main aspects and the dynamics of utterance and genre and given and new by excerpts from a textbook in mathematics education. The concept ‘positioning’, in use for operationalisation, is explained in relation to main principles of the framework. The article ends focusing crucial implications for validation when moving from a dyadic to a triadic understanding of mathematics and mathematics education.

Key Words

communication language positioning semiotics theme-rheme triadic utterance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aas, P. A.: 2003, ‘Dannelse i krise’, doctoral dissertation, Oslo University, Oslo.Google Scholar
  2. Bakhtin, M.: 1986, Speech Genres and Other Late Essays, University of Texas Press, Austin.Google Scholar
  3. Bateson, G.: 1972, Steps to an Ecology of Mind, Ballantine Books, New York.Google Scholar
  4. Bauman, Z.: 1995, Postmodern Etik, Daidalos, Gothenburg.Google Scholar
  5. Bordum, A.: 2001, Diskursetik og den positive selvreference, Samfundslitteratur, Copenhagen.Google Scholar
  6. Brandist, C.: 2002, The Bakhtin Circle. Philosophy, Culture and Politics, Pluto Press, London.Google Scholar
  7. Braathe, H.J. and Ongstad, S.: 2001, ‘Egalitarianism meets ideologies of mathematical education – instances from Norwegian curricula and classrooms’, Zentralblatt für Didaktik der Mathematik 33(5), 147–157,
  8. Bühler, K.: 1934/1965, Sprachtheorie, Fischer, Stuttgart.Google Scholar
  9. Davies, B. and Harré, R.: 1990, ‘Positioning: The discursive production of selves’, Journal for the Theory of Social Behaviour 20(1), 43–63.CrossRefGoogle Scholar
  10. Dewey, J.: 1916, Democracy and Education, Macmillan, New York.Google Scholar
  11. Durkin, K. and Shire, B. (eds.): 1991, Language in Mathematics education – research and practice, Open University Press, Milton Keynes.Google Scholar
  12. Ernest, P.: 1997, ‘Mathematical activity and rhetoric: A semiotic analysis of an episode of mathematical activity’, Philosophy of Mathematics Education Journal, 10,
  13. Ernest, P.: 1998, Social Constructivism as a Philosophy of Mathematics, State University of New York Press, New York.Google Scholar
  14. Ernest, P.: This volume, A semiotic perspective of mathematical activity: the case of number.Google Scholar
  15. Evans, J. and Tsatsaroni, A.: 1994, Language and subjectivity in the mathematics classroom, in S. Lerman, (ed.), The culture of Mathematics Classroom, Kluwer, Dordrecht, pp. 169–190.Google Scholar
  16. Hernadi, P.: 1995, Cultural Transactions. Nature, Self, Society, Cornell, London.Google Scholar
  17. Habermas, J.: 1984, The Theory of Communicative Action, Vol. 1, Beacon, Boston.Google Scholar
  18. Habermas, J.: 1988, Kommunikativt handlande. Texter om språk rationalitet och samhälle, Daidalos, Gothenburg.Google Scholar
  19. Habermas, J.: 1998, Postmethaphysical Thinking: Philosophical Essays, Polity Press, London.Google Scholar
  20. Halliday, M.A.K.: 1978, Language as Social Semiotic, Arnold, London.Google Scholar
  21. Halliday, M.A.K.: 1994, An Introduction to Functional Grammar. 2nd edn., Arnold, London.Google Scholar
  22. Hansen, N.B.: 2000, Pædagogikkens Treklang. Et opgør med dualismen i dansk skoletænkning, Gyldendal Uddannelse, Copenhagen.Google Scholar
  23. Harré, R.: 1980, Social Being, Rowman and Littlefield, Totowa.Google Scholar
  24. Harré, R.: 1984, Personal Being, Harvard University Press, Cambridge.Google Scholar
  25. Harré, R.: 1991, Physical Being, Blackwell, Oxford.Google Scholar
  26. Harré, R. and van Langenhove, L.: 1991, Varieties of Positioning, Journal for the Theory of Social Behaviour 21(4), 393–407.CrossRefGoogle Scholar
  27. Hersh, R.: 1997, What is Mathematics, Really?, Jonathan Cape, London.Google Scholar
  28. Hiim, H. and Hippe, E.: 1998, Læring gjennom opplevelse, forståelse og handling, Universitetsforlaget, Oslo.Google Scholar
  29. Illeris, K.: 1999, Læring – aktuell læringsteori i spenningsfeltet mellem Piaget, Freud og Marx, Roskilde Universitetsforlag, Roskilde.Google Scholar
  30. Kant, I.: 1987, Critique of Jugdement, Hackett, Indianapolis.Google Scholar
  31. Kragh, H.: 1999, Det smukke og det sanne. Æstetiske Principper i de Eksagte Videnskaper, in J. Holmgaard (ed.) Æstetik og logik, Medusa, Ålborg, pp. 211–234.Google Scholar
  32. Khisty, C.J.: This volume, The significance of dialogue in problem-solving using Martin Buber's triad – “I-thou-we”.Google Scholar
  33. Künzli, R.: 1998, ‘The common frame and the places of didaktik’, in, B. Gundem and S. Hopmann, (eds.), Didaktik and/or Curriculum. An International Dialogue, Peter Lang, New York, pp. 29–46.Google Scholar
  34. Langenhove, L. van and Harré, R.: 1993, ‘Positioning and Autobiography: Telling your life’, in N. Coupland and J. Nussbaum (eds.), Discourse and Life-span Development, Sage, London, pp. 81–99.Google Scholar
  35. Maddy, P.: 1990, Realism in Mathematics, Oxford University Press, Oxford.Google Scholar
  36. Martin, J.: 1997, ‘Analysing genre: Functional parameters’, in Christie, F. and Martin, J. (eds.), Genre and Institutions, Cassell, London, pp. 3–39.Google Scholar
  37. Mellin-Olsen, S.: 1987, The Politics of Mathematics Education, Kluwer, Dordrecht.Google Scholar
  38. Miller, R.C.: 1998, ‘Three versions of objectivity: Aesthetic, moral and scientific’, in Levinson, J. (ed.) Aesthetics and Ethics. Essays at the Intersection, Cambridge University Press, Cambridge, U.K., pp. 26–58.Google Scholar
  39. Morgan, C.: This volume, What Does Social Semiotics Have to Offer Mathematics Education Research? Google Scholar
  40. Morris, C.W.: 1938, Foundation of the theory of signs, Chicago University Press, Chicago.Google Scholar
  41. Morris, C.W.: 1946, Writings on the General Theory of Sign, Mouton, The Hague.Google Scholar
  42. Nygaard, O., Hundeland, P.S. and Pettersen, P.: 1999, Aha. Matematikk og matematikkdidaktikk, Høyskoleforlaget, Kristiansand.Google Scholar
  43. Nöth, W.: 1990, Handbook of Semiotics, Indiana University Press, Bloomington.Google Scholar
  44. Ogden, C. and Richards, I.A.: 1923, The Meaning of Meaning, Routledge and Keagan Paul, London.Google Scholar
  45. Ongstad, S.: 1995, The concept of ‘positioning’: bridging a ‘false dichotomy’ between structuralisms and their ‘posts’? Paper at AARE' sconference, Hobart, Australia, November,
  46. Ongstad, S.: 1996, ‘Literacies and Mother Tongue Education. The challenge of task ideologies in a semiotic-didactic perspective’, in F. Christie and, J. Foley (eds.), Some Contemporary Themes in Literacy Research, Waxmann, New York and Berlin, pp. 266–296.Google Scholar
  47. Ongstad, S.: 1997, Sjanger, posisjonering og oppgaveideologier, Doctoral Dissertation, University of Trondheim/NTNU, Trondheim.Google Scholar
  48. Ongstad, S.: 1999a, Sources of ‘didaktization’. On defining disciplines and their ‘(fag-) didaktik’ across borders illustrated with examples from Mother Tongue Education (MTE), in B. Hudson, F. Buchberger, P. Kansanen and H. Seel (eds.), Didaktik/Fachdidaktik as Science(s) of the Teaching Profession? TNTEE Publications, Vol. 2, no.1, Umeå University, Umeå, pp. 173–186.Google Scholar
  49. Ongstad, S.: 1999b, ‘Self-positioning(s) and students’ task reflexivity – a semiotic macro concept exemplified’, Journal of Structural Learning & Intelligent Systems 14(2), 125–152.Google Scholar
  50. Ongstad, S.: 1999c, ‘Vad är positioneringsanalys? “Självpositionering” i en (post)modern skola som exempel’, in C. A. Säfström and L. Östman (eds.), Textanalys. En introduktion til syftesrelaterade analyser, Studentlitteratur, Lund, pp. 148–176.Google Scholar
  51. Ongstad, S.: 2002a, ‘Genres – from static, closed, extrinsic, verbal dyads to dynamic, open, intrinsic semiotic triads’, In R. Coe, L. Lingard, and T. Teslenko (eds.), The Rhetoric and Ideology of Genre: Strategies for Stability and Change. Hampton Press, Cresskill, New Jersey, pp. 297–320.Google Scholar
  52. Ongstad, S.: 2002b, ‘Positioning early Norwegian research on writing’, Written Communication, 19(3), 345–381.CrossRefGoogle Scholar
  53. Ongstad, S.: 2003, Teacher Education between Aesthetics, Epistemology and Ethics – focusing Mother Tongue Education. Reviewed paper at NZARE/AAREs Conference in Auckland, NZ, November/December,
  54. Ongstad, S.: 2004a, Språk, kommunikasjon og didaktikk, Fagbokforlaget and LNU, Bergen.Google Scholar
  55. Ongstad, S.: 2004b, ‘Bakhtin's triadic epistemology and ideologies of dialogism’, in F. Bostad, C. Brandist, L.S. Evensen, and H. Faber. (eds.), Bakhtinian perspectives on language and culture: Meaning in language, art and new media, Palgrave Macmillan, London.Google Scholar
  56. Ongstad, S.: forthcoming a, ‘Context’, in P. Strazny (ed.), Encyclopedia of Linguistics, Routledge, New York.Google Scholar
  57. Ongstad, S.: forthcoming b, ‘Genre’, in P. Strazny (ed.), Encyclopedia of Linguistics, Routledge, New York.Google Scholar
  58. Poulsen, A.: 1994, Børns udvikling, Gyldendal, Copenhagen.Google Scholar
  59. Ricoeur, P.: 1981, Hermeneutics and the Human Sciences, Cambridge University Press, Cambridge.Google Scholar
  60. Rommetveit, R.: 1972, Språk, tanke og kommunikasjon, Universitetsforlaget, Oslo.Google Scholar
  61. Sáenz-Ludlow, A.: 2003, “Classroom mathematics discourse as an evolving interpreting game,” in M. Anderson, A. Sáenz-Ludlow, and V. Cifarelli (eds.), Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing, Legas Press, Ottawa, pp. 253–284.Google Scholar
  62. Sáenz-Ludlow, A.: This volume, Classroom interpreting games as mediators in the construction of meaning.Google Scholar
  63. Saussure, F. de: 1916, Course in General Linguistics, Fontana, London.Google Scholar
  64. Westbury, I.: 1998, ‘Didaktik and curriculum studies’, in B. Gundem, and B. Hopmann (eds.) Didaktik and/or Curriculum. An International Dialogue, Peter Lang, New York, pp. 47–78.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Faculty of EducationOslo University CollegeOslo

Personalised recommendations