Investigating secondary students’ generalization, graphing, and construction of figural patterns for making sense of quadratic functions

An important aim in school mathematics is to help students experience algebra's power to express generality. Researchers have been investigating figural growing pattern generalization as an early route to developing students’ understanding of functional relationships. More recently studies have been considering ‘open’ or ‘free’ tasks involving figural growing pattern construction. This study explored twelve Years 7–12 Australian students’ intuitions and connection of meanings for quadratic functions through growing pattern generalization, graphing, and construction activities during individual task-based interviews. The students who evidenced complementary correspondence and covariation views when generalizing appeared to handle pattern construction activities in productive ways, regardless of their prior formal study of quadratic functions. Areas of difficulty with graphing growing patterns and potential benefits of pattern construction tasks for exploring quadratic equations and for formative assessment are discussed.