Coordinating visual and algebraic reasoning with quadratic functions

Abstract

Quadratics provide a foundational context for making sense of many important algebraic concepts, such as variables and parameters, nonlinear rates of change, and views of function. Yet researchers have highlighted students’ difficulties in connecting such concepts. This in-depth qualitative study with two pairs of Year 10 (15 or 16-year-old) students investigated the potential of figural pattern generalisation—a context not traditionally used for teaching quadratics—to stimulate students’ coordination of visual and algebraic reasoning and attention to quadratic function concepts. Theorisations of embodied visualisation, algebraic thinking, and student noticing were drawn on to analyse the pairs responding to 19 quadratic figural pattern generalisation tasks interspersed throughout their class topic on quadratic equations. It was found that students became adept at connecting the generality of different types of structural aspects of figures (square, rectangular, linear, constant/invariant) to their symbolic expression in quadratic equations. Students’ construction of numeric instantiations of figural aspects was found to support pairs in moving towards symbolic generalisation. Task prompts to find different (but equivalent) algebraic equations for the same pattern evidenced pairs beginning to distinguish among general, factorised and standard forms of quadratic equations. One pair’s attention to first and second differences (between total quantities of figures in a sequence) highlighted both the difficulty of and potential for connecting quadratic rate-of-change concepts and parameters visually. Implications for including figural pattern generalisation when teaching quadratics and suggestions for further research are shared.