Abstract
In this paper, I review both mathematics education and CSCL literature and discuss how we can better take advantage of CSCL tools for develo** 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.
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Notes
Along with verification, these functions include explanation (providing insight into why a mathematical statement is true), discovery (the discovery or invention of new results), and communication (the negotiation of meaning), intellectual challenge (the self-realization/fulfillment derived from constructing a proof), and systematization (the organization of various results into a deductive system of axioms, concepts and theorems) (de Villiers 2003).
The first generation was Computer Supported Intentional Learning Environments (CSILE).
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.
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.
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.
Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53.
Davis, P. J., & Hersh, R. (1981). The mathematical experience. New York: Houghton Mifflin.
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.
de Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer, & D. Chazan (Eds.), Designing learning environments for develo** understanding of geometry and space. Mahwah, NJ: Erlbaum.
de Villiers, M. (2003). Rethinking proof with the Geometer’s Sketchpad. Emeryville, CA: Key Curriculum.
di Sessa, A. (2000). Changing minds: Computers, learning, and literacy. Cambridge, Massachusetts: MIT.
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.
Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany: State University of New York Press.
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.
Goldenberg, E. P., & Cuoco, A. A. (1998). What is dynamic geometry? In R. Lehrer, & D. Chazan (Eds.), Designing learning environments for develo** understanding of geometry and space. Mahwah, NJ: Erlbaum.
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.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23.
Healy, L., & Hoyles, C. (2001). Software tools for geometric problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6, 235–256.
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.
Hoyles, C. (1997). The curricular sha** of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7–16.
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.
Jackiw, N. (1995). The Geometer’s Sketchpad (Version 3). Berkeley, CA: Key Curriculum.
Jackiw, N. (2001). The Geometer’s Sketchpad (Version 4). Berkeley, CA: Key Curriculum.
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.
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.
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.
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.
Lakatos, I. (1978). Mathematics, science and epistemology (vol. 2). Cambridge: University of Cambridge.
Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics. Educational Studies in Mathematics, 46, 151–161.
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.
Livingston, E. (1999). Cultures of proving. Social Studies of Science, 29(6), 867–888.
Livingston, E. (2006). The context of proving. Social Studies of Science, 36(1), 39–68.
Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44, 25–53.
Mariotti, M. A. (2001). Justifying and proving in the Cabri environment. International Journal of Computers for Mathematical Learning, 6, 257–281.
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.
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.
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.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. National Council of Teachers of Mathematics: Reston, VA.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Norman, D. A. (1994). Things that make us smart: Defending human attributes in the age of the machine. Cambridge, MA: Perseus.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.
Pea, R. (1985). Beyond amplification: Using the computer to reorganize mental functioning. Educational Psychologist, 20(4), 167–182.
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.
Pólya, G. (1945). How to solve it: A new aspect of mathematical model. London: Penguin Books.
Pólya, G. (1981). Mathematical discovery: On understanding, learning, and teaching problem solving. New York: Wiley.
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.
Reiser, B. (2004). Scaffolding complex learning: The mechanisms of structuring and problematizing student work. Journal of the Learning Sciences, 13(3), 273–304.
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.
Reiss, K., & Renkl, A. (2002). Learning to prove: The idea of heuristic examples. Zentralblatt für Didaktik der Mathematik (ZDM), 34(1), 29–35.
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.
Scardamalia, M., & Bereiter, C. (2003). Knowledge building. In Encyclopedia of education (2nd ed., pp. 1370–1373). New York: Macmillan Reference, USA.
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.
Senk, S. L. (1985). How well do students write geometry proofs? Mathematics Teacher, 78, 448–456.
Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13.
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.
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.
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.
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.
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.
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.
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.
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The author would like to thank Murat Çakır for his valuable comments on an earlier version of this manuscript.
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Öner, D. Supporting students’ participation in authentic proof activities in computer supported collaborative learning (CSCL) environments. Computer Supported Learning 3, 343–359 (2008). https://doi.org/10.1007/s11412-008-9043-7
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DOI: https://doi.org/10.1007/s11412-008-9043-7