Abstract
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.
Öner, D. (2016). Tracing the change in discourse in a collaborative dynamic-geometry environment: From visual to more mathematical. International Journal of Computer-Supported Collaborative Learning, 11(1), 59–88. https://doi.org/10.1007/s11412-017-9251-0
Received: 21 July 2015/Accepted: 17 January 2016/Published online: 17 February 2016 © International Society of the Learning Sciences, Inc. 2016.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The instructions specified that, “point H is an arbitrary point on line FG.” In Euclidean geometry, that would mean that even though H can be any point on line FG, it is not something that moves. Thus, although one looks for a solution that would work for any point H, any treatment of H would be static. In dynamic geometry, however, an arbitrary point H is a free point that can be dragged along line FG. Thus, there is some legitimacy to the students’ solution. Ultimately, however, the solution fails the drag test of dynamic geometry. If one properly constructs the perpendicular through point H, then one should be able to drag point H along line FG and have the perpendicular to FG move with it so that it always passes through H and remains perpendicular to FG. Cheerios, however, had only dragged their final construction by moving point G.
- 2.
In a similar analysis of all eight sessions of the Cereal Team, Stahl (2016) conceptualizes the development of the group’s mathematical cognition in terms of the successive adoption of group practices, rather than routines, in order to emphasize that they are being theorized as group-level rather than individual phenomena. As illustrated in the six episodes here, the Cereal Team questions, negotiates, and adopts new practices through their discourse (including shared GeoGebra actions). This meaning-making process creates a shared understanding within the team. Once the team agrees to use a routine, it may become a group practice, which can be used in the future without further discussion.
References
Barron, B. (2000). Achieving coordination in collaborative problem-solving groups. The Journal of the Learning Sciences, 9(4), 403–436.
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, FL: Jossey Bass.
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, NJ: Lawrence Erlbaum Associates.
Chazan, D. (1993b). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359–387.
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, NJ: Lawrence Erlbaum Associates.
Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53.
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. KalamazooMI: Western Michigan University.
Gattegno, C. (1988). The awareness of mathematization. New York, NY: Educational Solutions. [also available as chapters 10-12 of Science of Education, part 2B].
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.
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.
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, Germany: Springer.
Hölzl, R. (1996). How does “dragging” affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1(2), 169–187.
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, The Netherlands: Kluwer.
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.
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, The Netherlands: Kluwer Academic Publishers.
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.
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.
Patton, M. (1990). Qualitative evaluation and research methods (2nd ed.). Newbury Park, CA: Sage.
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.
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.
Schoenfeld, A. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, MA: MIT Press.
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.
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.
Stahl, G. (2006). Group cognition: Computer support for building collaborative knowledge. Cambridge, MA: MIT Press.
Stahl, G. (2009). Studying virtual math teams. New York, NY: Springer.
Stahl, G. (2013a). Translating Euclid: Creating a human-centered mathematics. San Rafael, CA: Morgan & Claypool Publishers.
Stahl, G. (2013b). Explore dynamic geometry together. Web: http://GerryStahl.net/elibrary/topics/explore.pdf. Accessed 27 Jan 2016.
Stahl, G. (2016). Constructing dynamic triangles together: The development of mathematical group cognition. Cambridge, UK: Cambridge University Press.
Stake, R. E. (1978). The case study method in social inquiry. Educational Researcher, 7(2), 5–8.
Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics instruction – The Wiskobas project. Dordrecht, The Netherlands: Reidel Publishing Company.
Van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic.
Wheeler, D. (1982). Mathematization matters. For the Learning of Mathematics, 3(1), 45–47.
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Öner, D. (2021). Investigation 9. Tracing the Change in Discourse in a Collaborative Dynamic-Geometry Environment: From Visual to More Mathematical. In: Stahl, G. (eds) Theoretical Investigations. Computer-Supported Collaborative Learning Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-49157-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-49157-4_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-49156-7
Online ISBN: 978-3-030-49157-4
eBook Packages: EducationEducation (R0)