Educational Studies in Mathematics

, Volume 73, Issue 1, pp 21–53 | Cite as

Thinking aloud together: A teacher’s semiotic mediation of a whole-class conversation about percents



How does classroom interaction support students’ apprenticeship into the ways of speaking, writing, and diagramming that constitute the practice of mathematics? We address this problem through an interpretative analysis of a whole-group conversation about alternative ways of solving a problem involving percent discounts that occurred in a sixth grade classroom. This research study draws upon Dewey’s theory of inquiry, Vygotsky’s cultural–historical psychology, Freudenthal’s realistic mathematics education, and Halliday’s systemic functional linguistics (SFL). From Freudenthal, we borrow the notions of mathematizing and guided reinvention—the former notion offers a view of mathematics as an activity of structuring subject matter and the latter one provides insights into the processes whereby mathematizing is learned and taught in the classroom. We glean from Dewey his view of reflective thinking as inquiry and the role that conversations may serve therein. We rely upon Vygotsky’s notions of a verbal thinking plane and a social phase of learning in order to reconsider the function of whole-class interaction in apprenticing students into mathematizing. Finally, SFL provides us with tools for explaining the choices of grammar and vocabulary students and teachers make as they realize meanings in whole-group conversations. Treating the selected whole-class conversation as a text, we focus our analysis on how this text came to mean what it did. Our central questions are as follows: What meanings were realized in the whole-class conversation by teacher and students and how were these meanings realized? How did the teacher’s lexico-grammatical choices guide the students’ choices? In addressing these questions, we advance an interpretation of the conversation as paradigmatic of students and teacher thinking aloud together about percents.


Reflective thinking Mathematizing Semiotic mediation Whole-class interaction Functional grammar Percents Bar model 



We wish to thank Norma Presmeg as well as the three anonymous reviewers for their very helpful editorial comments and suggestions. We acknowledge Dor Abrahamson for his critical reading of several iterations of this paper. Thanks also to Mary Schleppegrell and Cecilia Colombi for contributing their functional grammar expertise. We also acknowledge Ana María Bressan and Fernanda Gallego, from the Grupo Patagónico de Didáctica de la Matemática (Argentina) as well as Silva Koethe for their invaluable help in coding and analyzing the text. Finally, we are especially grateful to Yeuk-Sze Leong, a teacher in whose classroom we first witnessed the kind of thinking aloud together conversations discussed in this paper. The research described herein has been partially supported by Brooklyn College (City University of New York) through a Leonard and Claire Tow Travel Fellowship.


  1. Adler, J. (1999). The dilemma of transparency: Seeing and seeing through talk in the mathematics classroom. Journal for Research in Mathematics Education, 30(1), 47–64.CrossRefGoogle Scholar
  2. Atweh, B., Bleicher, R. E., & Cooper, T. J. (1998). The construction of the social context of mathematics classrooms: A sociolinguistic analysis. Journal for Research in Mathematics Education, 29(1), 63–82.CrossRefGoogle Scholar
  3. Bartolini Bussi, M. G. (1998). Verbal interaction in the mathematics classroom. In H. Steinbring, M. G. Bartolini Bussi & A. Sierpinska (Eds.), Language and communication in the mathematics classroom (pp. 65–84). Reston: NCTM.Google Scholar
  4. Bauersfeld, H. (1988). Interaction, construction and knowledge: Alternative perspectives for mathematics education. In D. A. Grows, T. J. Cooney & D. Jones (Eds.), Effective mathematics teaching (pp. 27–46). Reston: NCTM.Google Scholar
  5. Bernstein, B. (1990). The structuring of pedagogic discourse. London: Routledge.Google Scholar
  6. Bromme, R., & Steinbring, H. (1994). Interactive development of subject matter in the mathematics classroom. Educational Studies in Mathematics, 27, 217–248.CrossRefGoogle Scholar
  7. Christie, F. (2002). Classroom discourse analysis: A functional perspective. London: Continuum.Google Scholar
  8. Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical thinking and classroom practice. In E. Forman, N. Minick & C. Stone (Eds.), Contexts for learning: Socio-cultural dynamics in children’s development (pp. 91–119). New York: Oxford University Press.Google Scholar
  9. Delpit, L. (1995). Other people’s children. New York: New.Google Scholar
  10. Dewey, J. (1903). Studies in logical theory. Chicago: University of Chicago Press.Google Scholar
  11. Dewey, J. (1916a). Democracy and education: An introduction to the philosophy of education. New York: The Free.Google Scholar
  12. Dewey, J. (1916b). Essays in experimental logic. New York: Dover.Google Scholar
  13. Dewey, J. (1922/1930). Human nature and conduct: An introduction to social psychology. New York: Modern Library.Google Scholar
  14. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Massachusetts: Heath.Google Scholar
  15. Dewey, J. (1938). Logic: The theory of inquiry. New York: Holt.Google Scholar
  16. Eggins, S. (1994). An introduction to systemic functional linguistics. London: Continuum.Google Scholar
  17. Eggins, S., & Slade, D. (1997). Analyzing casual conversation. London: Cassell.Google Scholar
  18. Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. (1998). “You’re going to want to find out which and prove it”: Collective argumentation in mathematics classrooms. Learning and Instruction, 8(6), 527–548.CrossRefGoogle Scholar
  19. Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht: Kluwer.Google Scholar
  20. Gerot, L., & Wignell, P. (1994). Making sense of functional grammar. Sydney: Gerd Stabler Antipodean Educational Enterprises.Google Scholar
  21. Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25(5), 443–471.CrossRefGoogle Scholar
  22. Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling, and instructional design. In P. Cobb, E. Yackel & K. McClain (Eds.), Communicating and symbolizing in mathematics classrooms (pp. 225–273). Mahwah: Lawrence Erlbaum.Google Scholar
  23. Halliday, M. A. K. (1978). Language as social semiotic: The social interpretation of language and meaning. London: Arnold.Google Scholar
  24. Halliday, M. A. K. (1993). Towards a language-based theory of learning. Linguistics and Education, 5, 93–116.CrossRefGoogle Scholar
  25. Halliday, M. A. K. (1994). An introduction to functional grammar (3rd ed.). London: Arnold.Google Scholar
  26. Halliday, M. A. K. (1995). Language and the theory of codes. In A. Sadovnik (Ed.), Knowledge and pedagogy: The sociology of Basil Bernstein. Norwood: Ablex.Google Scholar
  27. Halliday, M. A. K., & Hasan, R. (1976). Cohesion in English. London: Longman.Google Scholar
  28. Halliday, M. A. K., & Martin, J. R. (eds). (1993). Writing science: Literacy and discursive power. London: Falmer.Google Scholar
  29. Halliday, M. A. K., & Matthiessen, C. M. I. M. (1999). Construing experience through meaning: A language-based approach to cognition. London: Cassell.Google Scholar
  30. Hasan, R. (1992). Speech genre, semiotic mediation, and the development of higher mental functions. Language Sciences, 14(4), 489–529.CrossRefGoogle Scholar
  31. Herbst, P. (2002). Engaging students in proving. Journal for Research in Mathematics Education, 33(3), 176–203.CrossRefGoogle Scholar
  32. Inagaki, K., Hatano, G., & Moritas, E. (1998). Construction of mathematical knowledge through whole class discussion. Learning and Instruction, 8, 503–526.CrossRefGoogle Scholar
  33. Lampert, M. (1990). When the problem is not the question and the solution is not the answer. American Education Research Journal, 27(1), 29–63.Google Scholar
  34. Lampert, M., & Blunk, M. (1998). Talking mathematics in school. Cambridge: Cambridge University Press.Google Scholar
  35. Leikin, R., & Dinur, S. (2007). Teacher flexibility in mathematical discussion. Journal of Mathematical Behavior, 26, 328–347.CrossRefGoogle Scholar
  36. Lemke, J. (1990). Talking science: Language, learning, and values. Norwood: Ablex.Google Scholar
  37. Lobato, J., Clarke, D., & Burns Ellis, A. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101–136.CrossRefGoogle Scholar
  38. Lotman, Y. (1988). Text within a text. Soviet Psychology, 24, 32–51.Google Scholar
  39. Lubienski, S. T. (2000). A clash of class cultures?: Students’ experiences in a discussion-intensive seventh-grade mathematics classroom. Elementary School Journal, 100, 377–403.CrossRefGoogle Scholar
  40. Morgan, C. (1998). Writing mathematically: The discourse of investigation. London: Falmer.Google Scholar
  41. Morgan, C. (2006). What does social semiotics have to offer mathematics education research? Educational Studies in Mathematics, 61, 219–245.CrossRefGoogle Scholar
  42. Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19, 11–19.Google Scholar
  43. Nathan, M., & Knuth, E. (2003). A study of whole-class mathematical discourse and teacher change. Cognition and Instruction, 21(2), 175–207.CrossRefGoogle Scholar
  44. O’Halloran, K. L. (2003). Educational implications of mathematics as a multisemiotic discourse. In M. Anderson, A. Saenz-Ludlow, S. Zellweger & V. Cifarelli (Eds.), Educational perspectives on mathematics as semiosis (pp. 185–214). Ottawa: Legas.Google Scholar
  45. O’Connor, M. C. (2001). Can any fraction be turned into a decimal?: A case study of a mathematical group discussion. Educational Studies in Mathematics, 46, 143–185.CrossRefGoogle Scholar
  46. Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421–481.Google Scholar
  47. Sherin, M. G. (2002). A balancing act: Developing a discourse community in a mathematics classroom. Educational Studies in Mathematics, 5, 205–233.Google Scholar
  48. Staples, M. (2007). Supporting whole-class collaborative inquiry in a secondary mathematics classroom. Cognition and Instruction, 25(5), 161–217.Google Scholar
  49. Steinbring, H. (2007). Epistemology of mathematical knowledge and teacher–learner interaction. ZDM, 39, 95–106.CrossRefGoogle Scholar
  50. Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm for developmental research. Dordrecht: Kluwer.Google Scholar
  51. van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9–35.CrossRefGoogle Scholar
  52. Veel, R. (1997). Learning how to mean—scientifically speaking: Apprenticeship into scientific discourse in the secondary school. In F. Christie & J. R. Martin (Eds.), Genre and institutions: Social processes in the workplace and school (pp. 161–195). London: Cassell.Google Scholar
  53. Veel, R. (1999). Language, knowledge, and authority in school mathematics. In F. Christie (Ed.), Pedagogy and the shaping of consciousness. London: Cassell.Google Scholar
  54. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological functions. Cambridge: Harvard University Press.Google Scholar
  55. Vygotsky, L. S. (1986). Thought and language. Cambridge: MIT.Google Scholar
  56. Wells, G. (1999). Dialogic inquiry: Towards a socio-cultural practice and theory of education. Cambridge: Cambridge University Press.Google Scholar
  57. Zack, V., & Graves, B. (2001). Making mathematical meaning through dialogue: Once you think of it, the z minus three seems pretty weird. Educational Studies in Mathematics, 46, 229–271.CrossRefGoogle Scholar
  58. Zolkower, B., & Shreyar, S. (2007). A teacher’s mediation of a thinking aloud discussion in a 6th grade mathematics classroom. Educational Studies in Mathematics, 65, 177–202.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Brooklyn CollegeCity University of New YorkNew YorkUSA
  2. 2.Teachers CollegeColumbia UniversityNew YorkUSA
  3. 3.Grupo Patagónico de Didáctica de la MatemáticaSan Carlos de BarilocheRío NegroArgentina

Personalised recommendations