The nature of control in the problem-solving process: A study of Ph.D. mathematicians

Date of Completion

January 1999


During the past three decades several national organizations have recommended that problem solving be the focus of the K–16 mathematics curriculum (MAA, 1983; NCTM, 1980; 1989; NRC, 1989). In response, researchers have sought to develop instructional strategies to attain this goal by studying the mechanisms underlying successful problem solving behavior. An outcome of this research is a model of human problem-solving behavior that attributes problem-solving expertise to a range of factors including domain knowledge, problem-solving skills, beliefs, aesthetics, and how a problem solver manages and regulates these factors, that is, control. The purpose of this research was to answer the question: To what extent and in what manner does domain knowledge, problem-solving skills, beliefs and aesthetics impact on issues of control during problem solving endeavors? ^ The participants were Ph.D. mathematicians employed at research institutions in the Northeast. Data was collected during two phases. In each phase subjects were asked to “think aloud” while solving four complex mathematics problems—two problems that shared a deep structure and two problems that shared a surface structure. In addition, subjects were asked to respond to questions immediately after reading the problem and after completing the solution to the problem. Statistical analyses were used to determine whether there was a relationship between (a) problem representation and problem-solving success and (b) problem representation and control behavior. Subjects’ protocols were analyzed for emerging themes regarding those factors that influenced control. ^ The results of the study indicate: (a) in approximately half of the problem-solving protocols, subjects failed to identify the correct deep structure of the problems, (b) the relationship between correct deep structure characterization and problem-solving success was not significant, (c) the relationship between correct deep structure characterization and efficient control was not significant, and (d) none of the subjects identified the shared deep structure between problems and therefore did not employ similar techniques in their solutions. Additional analyses examined those factors impacting control behavior exhibited by subjects during the solution process. Several themes were identified with respect to domain knowledge, problem-solving skills, beliefs, aesthetics and the interaction of these factors with respect to control. ^