Problem Solving, Modeling, and Local Conceptual Development

The research reported here describes similarities and differences between (a) modeling cycles that students typically go through during 60–90 min solutions to a class of problems thast we refer to as model-eliciting activities, and (b) stages of development that students typically go through during the “natural ” development of constructs (conceptual systems, cognitive structures) that cognitive psychologists consider to be relevant to these specific problems. Examples of relevant constructs include those that underlie children’s developing ways of thinking about fractions, ratios, rates, proportions, or other elementary, but deep mathematical ideas. Results show that, when problem solvers go through an iterative sequence of testing and revising cycles to develop productive models (or ways of thinking) about a given problem solving situation, and when the conceptual systems that are needed are similar to those that underlie important constructs in the school mathematics curriculum, then these modeling cycles often appear to be local or situated versions of the general stages of development that developmental psychologists and mathematics educators have observed over time periods of several years for the relevant mathematics constructs. Furthermore, the processes that contribute to local conceptual development in model-eliciting activities are similar in many respects to the processes that contribute to general cognitive development. Applying principles from developmental psychology to problem solving—and vice versa—is a relatively new phenomenon in mathematics education (Lesh