This paper explores how models can support productive thinking. For us a model is a "thing", a tool to help make sense of something. We restrict attention to specific models for whole-number multiplication, hence the wording of the title. They support evolving thinking in large measure through the ways their users redesign them. They assume new…

Findings from prior research are drawn together to create a learning model for elementary school mathematics in the cognitive-constructivist tradition. A potential teaching/learning process consistent with the model was developed and applied in a longitudinal collaborative arrangement between university personnel and a local elementary school…