Evolving a three-world framework for solving algebraic equations in the light of what a student has met before

• Problematic aspects of linear and quadratic equations.
• A unified theoretical framework for solving algebraic equations.
• A three-world model combining embodiment, symbolism and reason.
• Blending many previous theoretical cognitive and affective theories.
• Re-interpretations of earlier research.

In this paper we consider data from a study in which students shift from linear to quadratic equations in ways that do not conform to established theoretical frameworks. In solving linear equations, the students did not exhibit the ‘didactic cut’ of Filloy and Rojano (1989) or the subtleties arising from conceiving an equation as a balance (Vlassis, 2002). Instead they used ‘procedural embodiments’, shifting terms around with added ‘rules’ to obtain the correct answer (Lima & Tall, 2008). Faced with quadratic equations, the students learn to apply the formula with little success. The interpretation of this data requires earlier theories to be seen within a more comprehensive framework that places them in an evolving context. We use the developing framework of three worlds of mathematics (Tall, 2004 and Tall, 2013), based fundamentally on human perceptions and actions and their consequences, at each stage taking into account the experiences that students have ‘met-before’ (Lima and Tall, 2008 and McGowen and Tall, 2010). These experiences may be supportive in new contexts, encouraging pleasurable generalization, or problematic, causing confusion and even mathematical anxiety. We consider how this framework explains and predicts the observed data, how it evolves from earlier theories, and how it gives insights that have both theoretical and practical consequences.