The cyclic nature of problem solving: An emergent multidimensional problem-solving framework

Abstract

This paper describes the problem-solving behaviors of 12 mathematicians as they completed four mathematical tasks. The emergent problem-solving framework draws on the large body of research, as grounded by and modified in response to our close observations of these mathematicians. The resulting Multidimensional Problem-Solving Framework has four phases: orientation, planning, executing, and checking. Embedded in the framework are two cycles, each of which includes at least three of the four phases. The framework also characterizes various problem-solving attributes (resources, affect, heuristics, and monitoring) and describes their roles and significance during each of the problem-solving phases. The framework's sub-cycle of conjecture, test, and evaluate (accept/reject) became evident to us as we observed the mathematicians and listened to their running verbal descriptions of how they were imagining a solution, playing out that solution in their minds, and evaluating the validity of the imagined approach. The effectiveness of the mathematicians in making intelligent decisions that led down productive paths appeared to stem from their ability to draw on a large reservoir of well-connected knowledge, heuristics, and facts, as well as their ability to manage their emotional responses. The mathematicians' well-connected conceptual knowledge, in particular, appeared to be an essential attribute for effective decision making and execution throughout the problem-solving process. © Springer 2005.