Sweeping Area across Physical and Virtual Environments

The concept of area measure is widely viewed as fundamental. It is both intrinsically important and also structurally significant, as a representative of the class of continuous multiplicative quantities. Operating flexibly in this context with meaning and conceptual understanding is thus a critical objective for elementary education. Meanwhile, the difficulties learners encounter with area are well documented in the literature and include challenges with visualizing structured 2D space and conceptualizing the referent-transforming action that converts two length measures into area measure. Responding to these challenges, we present an analysis of their activity where third-grade learners generated figures with area by sweeping one length (a ‘squeegee’) through another length. We describe a ‘duo’ of physical and virtual learning environments that we developed to enable this ‘sweeping’ approach to area – and we show how this duo supported a classroom group of students in engaging with the two challenges mentioned above. In our analysis, we draw upon Charles Goodwin’s framework of co-operative action, showing how, at both individual and group levels, learners began to build professional vision around area measure and how the shared gestures they developed pointed toward emerging collective understanding of area as a dynamic quantity.