A Models and Modeling Perspective on the Role of Small Group Learning Activities

The problem solving process in most twenty-first century contexts involves teams working on problem situations. It involves partitioning a complex situation into parts that can be addressed by specialists. It involves communicating information in forms that are meaningful for other specialists and suitable for their tools. It involves planning, monitoring and assessing intermediate results. In other words, the primary activity is team-based mathematical modeling. While the use of small groups in mathematics classes has been encouraged for over 20 years, research has focused on the achivement of individuals (as measured by standardized tests) and social benefits (such as self-esteem, social acceptance) (Davidson, 1990). These areas of emphasis and research have been confined to a frame of reference of school mathematics, whereas the needs of the current work place go beyond these interests. For example, schools are interested in the achievement of individual children, whereas business and industry are more interested in the products formed by teams of experts–with little interest in the contributions of the individual.\ud \ud The focus of this chapter reaches beyond the common mathematics classroom context to consider the cognitive development of groups as they work on well-crafted model-eliciting problems. The focus is on the kinds of tasks and processes valued in work place team problem solving. The chapter begins by teasing out the social aspects embedded in the six design principles for construct-eliciting (i.e., model-eliciting) problems described by Lesh, Hoover, Hole, Kelly and Post, (2000). The tasks require collaborative effort, leading to a discussion about how working in small groups amplifies the mathematical power of the individuals in the group. Since the products produced by the groups represent a collaborative effort, the cognitive development of the "group-as-a-unit" is important to consider. This discussion focuses on the role of perspective-taking and the internalization of once external group processes that take place in small group modeling episodes. Then individual cognitive development is examined with respect to "local" model development–i.e. the development of a construct within a model-eliciting activity. This is in contrast to traditional views, which are age-related, and unconnected to the local problem solving episode. Finally, selected classroom issues in management and implementation guidelines are given, based on the experiences of the authors when we have implemented collaborative problem solving with both children and adults