Stephanie Uses Geometry To Explain Binomial Expansion

Researcher Carolyn Maher conducted interviews with math student Stephanie during her eighth grade year, exploring ideas about binomial expansion. Stephanie was also one of a group of children who participated in a longitudinal research study, doing math activities facilitated by the research team since first grade. Clips from these earlier years as well as the entire set of interviews are available for viewing on the Video Mosaic Collaborative (videomosaic.org). Researcher Maher met with Stephanie for seven separate interviews between November and April. The interviews provide us with a blueprint on how to integrate meaningful activities, questioning, and discovery into classroom lessons and provide specific illustrations of Common Core State Standards for Practice and Algebraic content. Stephanie constructs various representations and models of the algebraic ideas that she is exploring. She is asked to articulate her mathematical ideas precisely and to justify her conjectures. Researcher Maher asks Stephanie to restate her thinking throughout the interviews, allowing Stephanie time to think and communicate effectively. During the first four interviews, Stephanie is challenged to represent (a+b)^2 numerically, algebraically and as a geometric area model. She is able to work through her initial mistake of identifying (a+b)^2 = a^2 + b^2 and communicate effectively and quite profoundly her reasoning about the meaning of (a+b)^2. Finally, she is able to extend her thinking further to create a three dimensional model of (a+b)^3 that corresponds to the algebraic and numerical representations that she constructed earlier