Seeing quadratics in a new light: secondary mathematics pre-service teachers’ creation of figural growing patterns

Teachers whose mathematical meanings support understanding across different contexts are likely to convey them in productive ways for coherent student learning. This exploratory study sought to elicit 67 secondary mathematics pre-service teachers’ (PSTs) meanings for quadratics with growing pattern creation and multiple translation tasks. A framework for categorizing their patterns, in terms of structural correspondence to quadratic equation forms, is shared. PSTs’ translations and explanations were analyzed using an existing framework for representational fluency. Evidence was found for some PSTs applying prior productive meanings to an unfamiliar context and others discovering new meanings. It was found that some PSTs were not able to draw on their prior knowledge of quadratic functions successfully, and possible reasons for this are explored. Successful figural pattern creation was found to be associated with higher representational fluency. Seven PSTs were also interviewed to probe their cognitive and affective experiences with a “new” type of quadratics task; several described “aha” moments and used emotional languages like “horror” and “delight”. Implications for learning to teach quadratic functions, creative visualization tasks and future research are discussed.