In a recent book titled Beyond Constructivism: A Models & Modeling Perspective on Mathematics Problem Solving, Learning & Teaching (Lesh & Doerr, 2003a), the concluding chapter describes a number of specific ways that a models and modeling perspective moves significantly beyond the implications that can be drawn from constructivist theories in the context of issues that are priorities to address for teachers, curriculum developers, or program designers. In that chapter (Lesh & Doerr, 2003b), the following topics were treated as cross-cutting themes: (a) the nature of reality, (b) the nature of mathematical knowledge, (c) the nature of the development of children’s knowledge, (d) the mechanisms that drive that development, (e) the relationship of context and generalizability, (f) problem solving, and (g) teachers’ knowledge and the kinds of teaching and learning situations that contribute to the development of children’s knowledge. In this article, we organize our comments directly around the preceding topics and describe how a models and modeling perspective provides alternative ways of thinking about mathematics teaching and learning that enable teachers, researchers and others to produce useful and sharable conceptual tools that have powerful implications in the context of decision-making issues that are of priority to practitioners.
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