Generalization of quadratic figural patterns: Shifts in student noticing

Quadratics, as the simplest non-linear function, are a foundational context for drawing together key algebraic concepts. Yet research has documented student difficulties in moving beyond rote procedures. This fine-grained study with Year 10 (15 or 16-year-old) students sought insights into how visualization and pair-work interactions might be harnessed for highlighting conceptual connections among quadratic concepts. Theorizations of symbolic generalization and student noticing were drawn on to investigate two pairs working on 19 quadratic generalization tasks. Different visual structures were found to elicit both productive and unproductive actions. Productive actions included building on each other’s visualizations, explaining disagreements verbally and with gestures, and verifying each other’s generalizations. Some unproductive actions were found that shifted focus away from symbolic generalization. An emergent framework to inform participative approaches to learning algebra is shared, along with suggested implications for including figural patterns when teaching quadratics and suggestions for further research.