Tracing the change in discourse in a collaborative dynamic geometry environment: From visual to more mathematical

  • Diler OnerEmail author


This case study investigated the development of group cognition by tracing the change in mathematical discourse of a team of three middle-school students as they worked on a construction problem within a virtual collaborative dynamic geometry environment. Sfard’s commognitive framework was employed to examine how the student team’s word choice, use of visual mediators, and adoption of geometric construction routines changed character during an hour-long collaborative problem-solving session. The findings indicated that the team gradually moved from a visual discourse toward a more formal discourse—one that is primarily characterized by a routine of constructing geometric dependencies. This significant shift in mathematical discourse was accomplished in a CSCL setting where tools to support peer collaboration and pedagogy are developed through cycles of design-based research. The analysis of how this discourse development took place at the group level has implications for the theory and practice of computer-supported collaborative mathematical learning. Discussion of which features of the specific setting proved effective and which were problematic suggests revisions in the design of the setting.


Mathematical discourse development Mathematical routines Group cognition Collaborative dynamic geometry Dependencies 



The author would like to thank the Fulbright Core Program for funding her sabbatical, and Gerry Stahl for welcoming her to his research team and sharing the VMT data analyzed in this study.


  1. Barron, B. (2000). Achieving coordination in collaborative problem-solving groups. The Journal of the Learning Sciences, 9(4), 403–436.CrossRefGoogle Scholar
  2. Berkowitz, M., & Gibbs, J. (1985). The process of moral conflict resolution and moral development. In M. Berkowitz (Ed.), Peer conflict and psychological growth (pp. 71–84). San Francisco: Jossey Bass.Google Scholar
  3. Chazan, D. (1993a). Instructional implications of students’ understanding of the differences between empirical verification and mathematical proof. In J. L. Schwartz, M. Yerushalmy, & B. Wilson (Eds.), The geometric supposer: What is it a case of? (pp. 107–116). Hillsdale, N.J: Lawrence Erlbaum Associates.Google Scholar
  4. Chazan, D. (1993b). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359–387.CrossRefGoogle Scholar
  5. Chazan, D., & Yerushalmy, M. (1998). Charting a course for secondary geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 67–90). Hillsdale, N.J.: Lawrence Erlbaum Associates.Google Scholar
  6. Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53.CrossRefGoogle Scholar
  7. Ellis, A. E., Lockwood, E., Williams, C. C. W., Dogan, M. F., & Knuth, E. (2012). Middle school students’ example use in conjecture exploration and justification. In L.R. Van Zoest, J.J. Lo, & J.L. Kratky (Eds.), Proceedings of the 34th Annual Meeting of the North American Chapter of the Psychology of Mathematics Education (Kalamazoo, MI). Google Scholar
  8. Gattegno, C. (1988). The awareness of mathematization. New York: Educational Solutions [also available as chapters 10–12 of Science of Education, part 2B].Google Scholar
  9. Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44, 127–150.CrossRefGoogle Scholar
  10. Harel, G., & Sowder, L. (1998). Students’ proof schemes. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on Collegiate Mathematics Education (Vol. III, pp. 234–283). Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
  11. Hölzl, R. (1995). Between drawing and figure. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 117–124). Berlin: Springer.CrossRefGoogle Scholar
  12. Hölzl, R. (1996). How does “dragging” affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1(2), 169–187.CrossRefGoogle Scholar
  13. Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 121–128). Dordrecht: Kluwer.Google Scholar
  14. Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44, 55–85.CrossRefGoogle Scholar
  15. Laborde, C. (2004). The hidden role of diagrams in students’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, & O. Skovsmose (Eds.), Meaning in mathematics education (pp. 1–21). Dordrecht: Kluwer Academic Publishers.Google Scholar
  16. Marrades, R., & Gutierrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44, 87–125.CrossRefGoogle Scholar
  17. Oner, D. (2013). Analyzing group coordination when solving geometry problems with dynamic geometry software. International Journal of Computer Supported Collaborative Learning, 8(1), 13–39.CrossRefGoogle Scholar
  18. Patton, M. (1990). Qualitative evaluation and research methods (2nd ed.). Newbury Park, CA: Sage.Google Scholar
  19. Roschelle, J., & Teasley, S. (1995). The construction of shared knowledge in collaborative problem solving. In C. O’Malley (Ed.), Computer-supported collaborative learning (pp. 69–197). Berlin: Springer Verlag.CrossRefGoogle Scholar
  20. Ryve, A., Nilsson, P., & Pettersson, K. (2013). Analyzing effective communication in mathematics group work: the role of visual mediators and technical terms. Educational Studies in Mathematics, 82(3), 497–514.Google Scholar
  21. Schoenfeld, A. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166. Google Scholar
  22. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
  23. Shaffer, D. W., & Kaput, J. J. (1999). Mathematics and virtual culture: an evolutionary perspective on technology and mathematics education. Educational Studies in Mathematics, 37, 97–119.Google Scholar
  24. Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: the development of the routine of shape identification in dynamic geometry environment. International Journal of Educational Research, 51–52, 28–44.Google Scholar
  25. Stahl, G. (2006). Group cognition: Computer support for building collaborative knowledge. Cambridge, MA: MIT Press.Google Scholar
  26. Stahl, G. (2009). Studying virtual math teams. New York, NY: Springer.Google Scholar
  27. Stahl, G. (2013a). Translating Euclid: Creating a human-centered mathematics. San Rafael, CA: Morgan & Claypool Publishers.Google Scholar
  28. Stahl, G. (2013b). Explore dynamic geometry together. Web: Accessed 27 Jan 2016.
  29. Stahl, G. (2016). Constructing dynamic triangles together: The development of mathematical group cognition. Cambridge, UK: Cambridge University Press.Google Scholar
  30. Stake, R. E. (1978). The case study method in social inquiry. Educational Researcher, 7(2), 5–8.CrossRefGoogle Scholar
  31. Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics instruction - the Wiskobas Project. Dordrecht, the Netherlands: Reidel Publishing Company.Google Scholar
  32. Van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic.Google Scholar
  33. Wheeler, D. (1982). Mathematization matters. For the Learning of Mathematics, 3(1), 45–47.Google Scholar

Copyright information

© International Society of the Learning Sciences, Inc. 2016

Authors and Affiliations

  1. 1.Department of Computer Education and Educational TechnologyBogazici UniversityBebekTurkey

Personalised recommendations