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The duality principle and learning trajectories in mathematics education research
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The purpose of this paper is to argue that attention to students' ways of thinking should complement a focus on students' understanding of specific mathematical content, and that attention to these issues can be leveraged to model the development of mathematical knowledge over time using learning trajectories. To illustrate the importance of ways of thinking, we draw on Harel's (2008a, 2008b) description of mathematical knowledge as comprised of ways of thinking and ways of understanding. We use data to illustrate the explanatory and descriptive power that attention to the duality of ways of understanding and ways of thinking provides, and we propose suggestions for constructing learning trajectories in mathematics education research.
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