# Supporting students’ participation in authentic proof activities in computer supported collaborative learning (CSCL) environments

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## Abstract

In this paper, I review both mathematics education and CSCL literature and discuss how we can better take advantage of CSCL tools for developing mathematical proof skills. I introduce a model of proof in school mathematics that incorporates both empirical and deductive ways of knowing. I argue that two major forces have given rise to this conception of proving: a particular learning perspective promoted in reform documents and a genre of computer tools, namely dynamic geometry software, which affords this perspective of learning within the context of mathematical proof. Tracing the move from absolutism to fallibilism in the philosophy of mathematics, I highlight the vital role of community in the production of mathematical knowledge. This leads me to an examination of a certain CSCL tool whose design is guided by knowledge-building pedagogy. I argue that knowledge building is a suitable pedagogical approach for the proof model presented in this paper. Furthermore, I suggest software modifications that will better support learners’ participation in authentic proof tasks.

### Keywords

Mathematical proof NCTM reform Dynamic geometry software Knowledge forum## Notes

### Acknowledgments

The author would like to thank Murat Çakır for his valuable comments on an earlier version of this manuscript.

### References

- Allen, F. B. (1996). A program for raising the level of student achievement in secondary level mathematics. Retrieved January 16, 2008, from http://mathematicallycorrect.com/allen.htm.
- Bielaczyc, K. (2001). Designing social infrastructure: The challenge of building computer-supported learning communities. In P. Dillenbourg, A. Eurelings, & K. Hakkarainen (Eds.),
*The proceedings of the first European conference on computer-supported collaborative learning*(pp. 106–114). The Netherlands: University of Maastricht.Google Scholar - Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education.
*International Newsletter on the Teaching and Learning of Mathematical Proof*,*7*(8). Retrieved from http://www.lettredelapreuve.it/Newsletter/990708Theme/990708ThemeUK.html. - Chazan, D. (1993a). High school geometry students’ justification for their views of empirical evidence and mathematical proof.
*Educational Studies in Mathematics*,*24*, 359–387.CrossRefGoogle Scholar - Chazan, D. (1993b). 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?*. Hillsdale, NJ: Erlbaum.Google Scholar - Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students.
*British Educational Research Journal*,*20*(1), 41–53.CrossRefGoogle Scholar - Davis, P. J., & Hersh, R. (1981).
*The mathematical experience*. New York: Houghton Mifflin.Google Scholar - de Villiers, M. (1997). The role of proof in investigative, computer-based geometry: Some personal reflections. In R. J. King, & D. Schattschneider (Eds.),
*Geometry turned on: Dynamic software in learning, teaching, and research*. Washington, DC: The Mathematical Association of America.Google Scholar - de Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer, & D. Chazan (Eds.),
*Designing learning environments for developing understanding of geometry and space*. Mahwah, NJ: Erlbaum.Google Scholar - de Villiers, M. (2003).
*Rethinking proof with the Geometer’s Sketchpad*. Emeryville, CA: Key Curriculum.Google Scholar - di Sessa, A. (2000).
*Changing minds: Computers, learning, and literacy*. Cambridge, Massachusetts: MIT.Google Scholar - Edwards, L. D. (1997). Exploring the territory before proof: Students’ generalizations in a computer microworld for transformation geometry.
*International Journal of Computers for Mathematical Learning*,*2*, 187–215.CrossRefGoogle Scholar - Ernest, P. (1998).
*Social constructivism as a philosophy of mathematics*. Albany: State University of New York Press.Google Scholar - Furinghetti, F., Olivero, F., & Paola, D. (2001). Students approaching proof through conjectures: Snapshots in a classroom.
*International Journal of Mathematical Education in Science and Technology*,*32*(3), 319–335.CrossRefGoogle Scholar - Goldenberg, E. P., & Cuoco, A. A. (1998). What is dynamic geometry? In R. Lehrer, & D. Chazan (Eds.),
*Designing learning environments for developing understanding of geometry and space*. Mahwah, NJ: Erlbaum.Google Scholar - Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). He role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments.
*Educational Studies in Mathematics*,*44*, 127–150.CrossRefGoogle Scholar - Hanna, G. (2000). Proof, explanation and exploration: An overview.
*Educational Studies in Mathematics*,*44*, 5–23.CrossRefGoogle Scholar - Healy, L., & Hoyles, C. (2001). Software tools for geometric problem solving: Potentials and pitfalls.
*International Journal of Computers for Mathematical Learning*,*6*, 235–256.CrossRefGoogle Scholar - Holzl, R. (2001). Using dynamic geometry software to add contrast to geometric situations—A case study.
*International Journal of Computers for Mathematical Learning*,*6*, 63–86.CrossRefGoogle Scholar - Hoyles, C. (1997). The curricular shaping of students’ approaches to proof.
*For the Learning of Mathematics*,*17*(1), 7–16.Google Scholar - 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*. Dordrecht: Kluwer.Google Scholar - Jackiw, N. (1995).
*The Geometer’s Sketchpad (Version 3)*. Berkeley, CA: Key Curriculum.Google Scholar - Jackiw, N. (2001).
*The Geometer’s Sketchpad (Version 4)*. Berkeley, CA: Key Curriculum.Google Scholar - 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 - Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 515–556). New York: Macmillan.Google Scholar - Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving.
*Educational Studies in Mathematics*,*44*, 151–161.CrossRefGoogle Scholar - Laborde, C. (2001). Integration of technology in the design of geometry tasks with Cabri-geometry.
*International Journal of Computers for Mathematical Learning*,*6*, 283–317.CrossRefGoogle Scholar - Lakatos, I. (1978).
*Mathematics, science and epistemology*(vol. 2). Cambridge: University of Cambridge.Google Scholar - Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics.
*Educational Studies in Mathematics*,*46*, 151–161.CrossRefGoogle Scholar - Lipponen, L. (2002). Exploring foundations for computer-supported collaborative learning. In G. Stahl (Ed.),
*Computer support for collaborative learning: Foundations for a CSCL community*. Proceedings of CSCL 2002. Boulder, CO, USA. (pp. 72–81). Mahwah, NJ: LEA.Google Scholar - Livingston, E. (1999). Cultures of proving.
*Social Studies of Science*,*29*(6), 867–888.CrossRefGoogle Scholar - Livingston, E. (2006). The context of proving.
*Social Studies of Science*,*36*(1), 39–68.CrossRefGoogle Scholar - Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment.
*Educational Studies in Mathematics*,*44*, 25–53.CrossRefGoogle Scholar - Mariotti, M. A. (2001). Justifying and proving in the Cabri environment.
*International Journal of Computers for Mathematical Learning*,*6*, 257–281.CrossRefGoogle Scholar - 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 - Moss, J., & Beatty, R. (2006). Knowledge building in mathematics: Supporting collaborative learning in pattern problems.
*International Journal of Computer-Supported Collaborative Learning*,*1*(4), 441–465.CrossRefGoogle Scholar - Nason, R., & Woodruff, E. (2003). Fostering authentic, sustained, and progressive mathematical knowledge-building activity in computer supported collaborative learning (CSCL) communities.
*Journal of Computers in Mathematics and Science Teaching*,*22*(4), 345–363.Google Scholar - National Council of Teachers of Mathematics (1989).
*Curriculum and evaluation standards for school mathematics*. National Council of Teachers of Mathematics: Reston, VA.Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Norman, D. A. (1994).
*Things that make us smart: Defending human attributes in the age of the machine*. Cambridge, MA: Perseus.Google Scholar - Papert, S. (1980).
*Mindstorms: Children, computers, and powerful ideas*. New York: Basic Books.Google Scholar - Pea, R. (1985). Beyond amplification: Using the computer to reorganize mental functioning.
*Educational Psychologist*,*20*(4), 167–182.CrossRefGoogle Scholar - Pea, R., Tinker, R., Linn, M., Means, B., Brandsford, J., Roschelle, J., et al. (1999). Toward a learning technologies knowledge network.
*Educational Technology Research and Development*,*47*, 19–38.CrossRefGoogle Scholar - Pólya, G. (1945).
*How to solve it: A new aspect of mathematical model*. London: Penguin Books.Google Scholar - Pólya, G. (1981).
*Mathematical discovery: On understanding, learning, and teaching problem solving*. New York: Wiley.Google Scholar - Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices.
*Philosophia Mathematica*,*3*(15), 291–320.CrossRefGoogle Scholar - Reiser, B. (2004). Scaffolding complex learning: The mechanisms of structuring and problematizing student work.
*Journal of the Learning Sciences*,*13*(3), 273–304.CrossRefGoogle Scholar - Reiser, B., Tabak, I., Sandoval, W. A., Smith, B. K., Steinmuller, F., & Leone, A. J. (2001). Bguile: Strategic and conceptual scaffolds for scientific inquiry in biology classrooms. In S. M. Carver, & D. Klahr (Eds.),
*Cognition and instruction: Twenty-five years of progress*(pp. 263–305). Mahwah, NJ: Erlbaum.Google Scholar - Reiss, K., & Renkl, A. (2002). Learning to prove: The idea of heuristic examples.
*Zentralblatt für Didaktik der Mathematik (ZDM), 34*(1), 29–35.CrossRefGoogle Scholar - Romberg, T. A. (1992). Problematic features of the school mathematics curriculum. In P. W. Jackson (Ed.),
*Handbook of research on curriculum: A project of the American Educational Research Association*. New York: MacMillan.Google Scholar - Scardamalia, M., & Bereiter, C. (2003). Knowledge building. In
*Encyclopedia of education*(2nd ed., pp. 1370–1373). New York: Macmillan Reference, USA.Google Scholar - Scardamalia, M., & Bereiter, C. (2004). Knowledge building: Theory, pedagogy, and technology. In K. Sawyer (Ed.),
*Cambridge handbook of the learning sciences*. New York: Cambridge University Press.Google Scholar - Senk, S. L. (1985). How well do students write geometry proofs?
*Mathematics Teacher*,*78*, 448–456.Google Scholar - Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one.
*Educational Researcher*,*27*(2), 4–13.Google Scholar - Silver, E., & Carpenter, T. (1989). Mathematical methods. In M. Lindquist (Ed.),
*Results from the fourth mathematics assessment of the national assessment of educational progress*(pp. 10–18). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Smith, B. K., & Reiser, B. (1998). National geographic unplugged: Designing interactive nature films for classrooms. In C. M. Karat, A. Lund, J. Coutaz, & J. Karat (Eds.),
*Proceedings of CHI98*(pp. 424–431). New York: ACM.Google Scholar - Stahl, G. (2006). Supporting group cognition in an online math community: A cognitive tool for small-group referencing in text chat.
*Journal of Educational Computing Research*,*35*, 103–122.CrossRefGoogle Scholar - Stegmann, K., Weinberger, A., & Fischer, F. (2007). Facilitating argumentative knowledge construction with computer-supported collaboration scripts.
*International Journal of Computer-Supported Collaborative Learning*,*2*(4), 421–447.CrossRefGoogle Scholar - Stylianides, G. J., & Silver, E. A. (2004). An analytic framework for investigating the opportunities offered to students. In D. E. McDougall, & J. A. Ross (Eds.),
*Proceedings of the 26th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(vol. 2, (pp. 611–619)). Toronto, ON, Canada: OISE/UT.Google Scholar - Usiskin, Z. (1987). Resolving the continuing dilemmas in school geometry. In M. Lindquist, & A. P. Shulte (Eds.),
*Learning and teaching geometry k-12, 1987 year-book*(pp. 17–31). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*. Reston, VA: NCTM.Google Scholar