Children's Developing Mathematics in Collective Practices: A Framework for Analysis

This article presents a cultural–developmental framework for the analysis of chil-dren’s mathematics in collective practices and illustrates the heuristic value of the framework through the analysis of videotaped episodes drawn from a middle-school classroom. The framework is presented in 2 related parts. The first targets the chil-dren’s emerging mathematical goals in collective practices, with a particular focus on the complex role that artifacts play in children’s emerging goals. The second part fo-cuses on children’s developing mathematics that takes form in their goal-directed ac-tivities: (a) Microgenetic analyses concern the process whereby children structure cultural forms like artifacts to serve particular functions as they accomplish emerging mathematical goals; (b) sociogenetic analyses concern the spread or travel of mathe-matical forms and associated functions within a community of individuals; and (c) ontogenetic analyses concern the interplay between the forms that children use and the functions that they serve over the course of children’s development. The analyses of the classroom episodes points to the promise (and limitations) of the framework as a method for furthering our understanding of the interplay between social and devel-opmental processes in children’s mathematics. A growing body of classroom-based research in mathematics education is con-cerned with understanding the role of artifacts in processes of teaching and learn-ing. In her classroom-based research, Ball (1993), for example, pointed to the way in which the use of one kind of artifact—the area of geometrical shapes—provides a representational context to explore properties of fractions. Lampert (1986) pointed to the utility of currency as support for teaching and learning about