The journey metaphor as a lens to interpret students' problem solving with rates and ratios

Specific acts of problem solving with rates and ratios were interpreted using a journey metaphor derived from Skemp’s (1979) construct of director systems. To successfully undertake a problem solving journey a learner must recognise their starting place (present state), have a sense of destination (goal state), and co-ordinate sub-routes along the way (navigate the chosen schema). The metaphor was evaluated against four of Niss’s (2007) six criteria for productive theory; as a structured lens for interpretation of phenomena, as a way to describe the phenomena, as a predictive model, and as providing guidance for action. The data revealed instances of successful and unsuccessful journeys affected by learners’ interpretation of the relationship between parts and wholes, the schema initially called up, available knowledge and calculation strategies, and connections to other situations. Establishing appropriate present state presented as a necessary but not sufficient condition for successful journeys. The results supported the usefulness of the journey metaphor in terms of predicting learners’ success at problem solving and for suggesting productive scaffolding strategies for learners by an expert other.