Mathematical Flexibility: Case Studies of Individual-Context Interactions

• In Event: Researching Self- and Coregulated Learning as Situated in Contexts: Implications for Theory, Research, and Practice

Abstract

Mathematics researchers and educators argue that mathematics education must equip students to use mathematics adaptively to solve authentic problems that arise in day-to-day life (Governments of Yukon, Alberta, British Columbia, Manitoba, Northwest Territories, Nunavut and Saskatchewan, 2008; Schoenfeld, 1992). In this research, we conceptualized this adaptive performance as mathematical flexibility. We aimed to inform understanding of how cognitive, metacognitive and contextual structures might be co-implicated in flexible performance, and to develop methodological approaches that might reveal individual-context transactions.

Research suggests that cognitive, metacognitive and contextual structures make important contributions to mathematical flexibility (Greeno & Sande, 2007; Dehaene, 1999, 2001). However, some researchers have foregrounded the role of cognitive structures in flexibility – like domain-specific structures for the whole number system (e.g., Case et al., 1996; Dehaene et al., 2004). Others have foregrounded the role of contextual structures that support flexibility – like availability of resources to thinking (e.g., Greeno, 1991, 1994). Still others have focused on the roles of self-regulation and metacognition (Farb et al., 2007; Schoenfeld, 1987). In this study, we integrated across research in neuroscience, psychology, and education/social psychology to consider how cognitive, metacognitive and contextual structures might be understood as functional components within an individual-contextual transaction (or network). We then pilot-tested a methodological approach designed to study the interaction between functional structures within transactions involving flexibility.

Specifically, we undertook case studies of 5 secondary-level students engaged in solving problems associated with linear equations. Following on an initial interview, each student was observed solving problems designed either to afford flexibility or to afford straightforward application of previously learned mathematical processes (termed competence in this study). During observations we gathered video-, audio- and behavioural-trace data of problem-solving processes in relation to context (e.g., how contextual resources were drawn on to support, shape or extend thinking). We also used brief interview probes to tap into students’ thinking directly after attempting to solve each problem. We engaged in an iterative process of results generation, where we drew on descriptions and traces of student activity to infer the activity of particular cognitive, metacognitive and contextual structures, and then related network activity to patterns in student flexibility.

Results suggested that the protocol used in this study afforded detection of important forms of cognitive, metacognitive and contextual activity that could be related meaningfully to flexibility, suggesting the usefulness of this general approach. However, important methodological limitations remain (e.g., challenges in probing structures for consciousness; or distinguishing important metacognitive structures). In addition to methodological contributions, results suggested qualities of mathematical flexibility associated with individual-context interactions. In particular, our results suggest that coordination between cognitive and contextual structures may be essential for flexibility, and may be mediated by particular metacognitive structures; that very similar flexible behavior can result from vastly different cognitive and contextual transactions; and that, for flexibility to occur, the student may need to recognize a need to adapt, and this recognition may be impaired by the use of competent (pre-learned) approaches to mathematics. Implications for research and practice are discussed.

Author

• Matt Giammarino, The University of British Columbia