had come to dominate the content of his mathematics courses. When language flowed from physical experience, Jim was quite ready to push very hard to coordinate and reconcile language with experience. As he said, "the thinking happens in geometry." Jim had a vision of what he expected of geometry, but that vision remained out of touch with school mathematics. Jim's vision was largely a 17th-century mechanical geometric vision, like that of Descartes and Pascal, which involved architecture, civil engineering, and mechanical devices. For example, Jim was disappointed that the geometry that he learned in high school never helped him even to begin to analyze the motion of the mechanical apparatus that reset pins in a bowling alley where he had worked.
Jim clearly benefited from his experience with these curve-drawing devices. His engagement with the curve-drawing devices was profound: They satisfied in him a longing for what he saw as the geometry of the world. We learned a great deal from watching and listening to Jim. Jim's phrase "these move in a fixed ratio," combined with his hand gestures, remain with us. They have already become part of our thinking about the learning and teaching of dynamic geometry.
If our curriculum is allowed to confront the uncertainties and ambiguities of how language interacts with the physical world--if mathematical language, symbols, and notations are allowed to grow directly from experiences and be shaped by them--then this fully circular feedback loop could evolve into a powerful epistemological model based on the coordination of multiple representations ( von Glasersfeld, 1978). The algebra of equations and functions would then be more than merely just what Jim despairingly referred to as "the rules." More students would then be able to say genuinely, as Jim did at the end of his derivation, "It makes me feel good to get that."
Artobolevskii, I. I. ( 1964). Mechanisms for the generation of plane curves. New York: Macmillan.
Confrey, J. ( 1993). "The role of technology in reconceptualizing functions and algebra". In J. R. Becker & B. J. Pence (Eds.), Proceedings of the 15th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 47-74). San Jose, CA: Center for Mathematics and Computer Science Education, San Jose State University.
Confrey, J., & Smith, E. ( 1991). "A framework for functions: Prototypes, multiple representations, and transformations". In R. Underhill & C. Brown (Eds.), Proceedings of the 13th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 57-63). Blacksburg, VA: Virginia Polytechnic Institute and Christianbury Printing Company.
Dennis, D. ( 1995). Historical perspectives for the reform of mathematics curriculum: Geometric curve-drawing devices and their role in the transition to an algebraic description of functions. Unpublished doctoral dissertation, Cornell University, Ithaca, NY.