MATHEMATICAL PROBLEM-SOLVING PERFORMANCE: A STUDY OF COGNITIVE AND NONCOGNITIVE FACTORS (GIFTED, MOTIVATION, BELIEF SYSTEMS)

The problem of this study was two-fold: to construct a model of problem-solving performance that incorporated both cognitive and noncognitive factors, and to determine if there were any observable qualitative differences in mathematical problem solving between gifted and average students. The subjects for this study were third- and fifth-grade students from a moderate-sized western city. Possible participants were identified on the basis of a mental age equivalent of 11 years (+OR-) 3 months on the Otis-Lennon Test of Mental Abilities, Elementary I Level, Form J (1967). Groups of three third-grade girls, third-grade boys, fifth-grade girls, and fifth-grade boys were randomly selected to participate in the problem-solving sessions. A modified teaching experiment (Steffe, von Glasersfeld, & Cobb, 1983) was conducted for eight weeks. Each group met twice a week for 30-45 minutes per session. Students were asked to solve a variety of mathematical problems. All sessions were videotaped. In addition to the information from these tapes, classroom teachers, parents, and the author as teacher/experimenter provided supplementary information about each student. To address the first problem of the study a global analysis of all available information was utilized to construct a Model of Mathematical Problem-Solving Performance. The model combined Skemp's (1978) cognitive theory of relational and instrumental mathematics learning, and Nicholls' (1983) achievement motivation theory of extrinsic motivation, ego-involvement, and task-involvement. Students were classified according to types within the model. Individual and group performances were used to illustrate the model. To answer the second problem of the study, four experts in gifted education judged the problem-solving performance of each group according to criteria established by these experts. They ranked the groups from highest to lowest quality; identified the gifted groups; indicated which groups were composed of girls and which of boys; and identified outstanding individuals within the groups. The fifth-grade boys' group was unanimously selected as demonstrating the highest quality problem solving. Rankings of the other groups were inconsistent. One fifth-grade boy was identified by all four judges as being an outstanding problem solver.