I had the great pleasure of being involved in a multiyear design research process in mathematics education. Specifically, I was involved the refinement and evaluation of an innovative mathematics curriculum for upper elementary school students. This curriculum, called Learning Mathematics Through Representations (LMR), uses the number line as a consistent representational context in order to support a rich, integrated understanding of integers and fractions. Unfortunately, integers and fractions are generally taught as two disconnected domains, reflecting a common critique of mathematics education in the United States — that it is “a mile wide and an inch deep.”
Among the many other distinctive features of this curriculum is that it supports a rich understanding of the structure of the number line through the use of mathematical definitions. These definitions are progressively introduced over the course of lessons, and constitute an evolving “common ground” for the classroom and a resource for students to explain their thinking and justify the reasoning (see figure at right for examples from the integers lessons).
The efficacy study I was a part of found strong learning gains in classrooms implementing the LMR curriculum, as compared to control classrooms. I also participated in the in-depth analysis of two case study classrooms, as a fruitful context to study processes of teaching and learning. Through this work, we advanced an analytic approach the treats classrooms as micro-cultures, in which various kinds of collective practices are reproduced and altered over time.
The figures below summarize the structure of the curriculum. For more information, see: http://culturecognition.com/lmr/
See my CV for my published articles related to this work.