This research investigates the development of central processes in mathematical problem solving. The project examines a central issue in theories of formal problem solving: how children and adults select an appropriate mathematical operation when they encounter a novel problem. Despite the fact that students learn a particular mathematical operation (e.g., division) in the context of relatively few problem domains (e.g., equally distributing candy), they can eventually apply the operation to solve problems in a tremendous variety of novel domains (i.e., domains in which the solution operation has not been identified for them). The proposed research investigates two central processes in mathematical problem solving: Domain Mapping and Operation Mapping. The Domain Mapping process is similar to analogical mapping and unfolds as follows. Students learn a formal operation (e.g., division) in the context of a particular base domain (e.g., equally distributing candy). As a consequence, the base domain and the formal operation become linked. When a novel problem domain is encountered the child attempts to analogically map the novel domain to the base domain. If the two domains can be mapped, the formal operation is applied. Mathematical operations can also be reached through Operation Mapping. Children induce the functional relationships that hold for a mathematical operation as they apply it to problems and practice computational procedures. Under Operation Mapping, these functional relationships (i.e., what the mathematical operation accomplishes) are directly mapped to the representation of the novel problem. In the first study, developmental changes in the use of the two mapping processes will be assessed using a correlational design with children from 4th through 8th grade. A second study will examine the effects of the introduction of rational numbers on students' representations of the mathematical operations and on their subsequent problem solving. The introduction of rational numbers effectively rewrites the functional relationships that hold for different operations. This should affect students' representations of mathematical operations and have subsequent detrimental effects on their ability to use the Operation Mapping process. The third study directly manipulates the Domain and Operation Mapping processes and test the causal relationships each predicts. The proposed project will substantially contribute to research on the development of problem solving, and to the study of problem solving in adults. This research will also provide tremendous educational opportunities for students, and involve collaboration with local teachers and administrators.
