can do much to facilitate negotiation and thus provide rich learning opportunities. Actually, the facilitation itself is a negotiatory process. By encouraging the negotiation of social norms whereby students form the intention to make sense and feel comfortable to do so is an important component in establishing a productive learning environment. In this process there will be times when the focus of the discussion will be the social norms. The negotiation of social norms is necessary for the negotiation of mathematical meaning to occur.
In the episode discussed, the value of a task orientation, rather than an ego orientation, becomes clear. Brett is guided by his intention to make sense. Once this intention is formed and Brett is in an environment where others will negotiate, the potential for him to become a powerful mathematics student exists. As long as Sam maintains his ego orientation, his learning opportunities are severely limited. By being in a classroom where social norms are negotiated rather than rules being imposed by the teacher, a student like Sam could shift from his ego orientation to a more productive stance.
A thesis of this chapter is that mathematics learning (all learning?) is greatly facilitated by negotiation. Analysis of a discussion between two students allows us to see the complexity and power of negotiation. By striving to establish the conditions for negotiation, teachers can increase the probability that meaningful learning will occur.
Brown D., and G. Wheatley. Spatial Visualization and Arithmetic Knowledge. Proceedings of the Eleventh Annual Meeting of the North American Chapter-International Group for the Psychology of Mathematics Education. New Brunswick, NJ, 1989.
Confrey L. "What Constructivism Implies for Teaching." Journal for Research in Mathematics. Monograph No. 4 ( 1990): 107-122.
Csikszentmihalyi M. Flow: The Psychology of Optimal Experience. New York: Harper and Row, 1990.
Goodman N. Of Mind and Other Matters. Cambridge, MA: Harvard University Press, 1984.
Hallet D.H. "Visualization and Calculus Reform." In W. Zimmerman, and S. Cunningham (Eds.), Visualization in Teaching and Learning Mathematics. Washington, DC: Mathematics Association of America, 1990.
Lo J., G. Wheatley and A. Smith. "Learning to Talk Mathematics." Paper, annual meeting of the American Educational Research Association, Chicago, 1991a.
Lo J., G. Wheatley and A. Smith. "Negotiation of Social Norms in Mathematics Learning." Paper, thirteenth meeting of the Psychology of Mathematics Education--North American Chapter, Blacksburg, VA, 1991b.
Nicholls J. "Conceptions of Ability and Achievement Motivation: A Theory and its Implications for Education." In S. G. Paris, G. M. Olson, and H. W. Stevenson (Eds.), Learning and Motivation in the Classroom. Hillsdale, NJ: Erlbaum, 1983.
Presmeg N.C. The Role of Visually Mediated Processes in High School Mathematics: A Classroom Investigation. Unpublished doctoral dissertation, University of Cambridge, Wolfson College, 1985.
Richards J. "Mathematical Discussions." In E. von Glasersfeld (Ed.), Constructivism in Mathematics Education. Holland: Klewer, 1991.
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