This project is a comparative investigation of changes in how students make and use representations, contrasting three innovative instructional settings where groups of elementary and middle school students work on "authentic" mathematical problems. In each setting, longitudinal video and documentary records of individual and group work are used to analyze changes in how students reason about quantity. Comparative case methods focus on how students in different instructional settings use the common media of talk, written notation, and bodily activity 1) to find mathematical problems, 2) to represent relevant quantities in these situations, 3) to make inferences about relations between quantities, 4) to organize calculation, and 5) to compare different approaches to solution. Cases are developed to serve as boundary objects between audiences with very different purposes: cognitive scientists interested in the development of quantitative reasoning, practitioners interested in new ways to support and evaluate their students' learning, and educational researchers interested in restructuring the teaching of mathematics.Rogers P. Hall is the principal investigator.