Area Models to Image Integer and Binomial Multiplication

Middle-grades mathematics prospective teachers (PSTs) participated in three iterations of a design experiment that explored their current conceptualizations of integer and binomial multiplication; this study analyzed whether that knowledge changed after engaging in a series of scaffolded tasks while using area models as concrete manipulatives. Understandings were measured by participants composing word problems to model the multiplication of two integers, with at least one negative, and the multiplication of two binomials, first as a pre-assessment then as a post-assessment. Changes in participants' word problems were examined. Findings from the three iterations of the design experiment exhibited shifts in participants' understandings of multiplication across all three. The inclusion of "Algeblocks" and quadrant mat as area models facilitated conversations for PSTs to make meaning of integer and binomial multiplication. From the findings of this study, the researchers argue that future iterations of the design experiment should include an additional measure of PST pedagogical content knowledge related specifically to algebraic reasoning as well as opportunities for more exploration structural comparisons of integers to binomials so as to facilitate conversations around the concept of what is a binomial in a real-world context.