The duality between ways of thinking and ways of understanding: Implications for learning trajectories in mathematics education

• We consider the various uses of learning trajectories in mathematics education.
• We consider the role of Harel's notion of duality, consisting of ways of thinking and ways of understanding, in the construction of these learning trajectories.
• We propose that duality provides a mechanism by which conceptual change in mathematical can be represented.
• We provide two case studies from calculus and combinatorics to substantiate the utility of duality and its use for learning trajectories.
• We consider how Simon's original conceptualization of a hypothetical learning trajectory might be extended by attending to ways of thinking and ways of understanding.

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 to characterize students’ thinking about mathematics in two case studies. We use these case studies 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.