In recent years, considerable progress has been made in understanding the nature of complex human thought, with particular insights into the mechanisms of skilled problem solving. Much of the early research on human problem solving concentrated on general problem-solving skills and used puzzle problems and highly structured tasks with a dearth of semantic content, such as the "Towers of Hanoi" or "Missionaries and Cannibals" problems. More recent research, however, has focused on problem-solving behavior in semantically rich knowledge domains with direct or indirect relevance to mathematics. The participants in this research enterprise have come from many fields, but primarily from mathematics education, cognitive psychology, science education, and artificial intelligence.
Corresponding to the intense level of research attention to the mechanisms of mathematical problem solving has been a growing interest in the topic among educational practitioners. Since the publication by the National Council of Teachers of Mathematics of the Agenda for Action, which asserted that the acquisition of problem-solving skills should be one of the goals of school mathematics instruction in the 1980s, problem solving has been a dominant topic at virtually all professional meetings of mathematics teachers and supervisors. Rarely in the history of education has a topic simultaneously captured so much of the attention of both researchers and practitioners. Usually, the research community is busy investigating a topic long after it ceases to be of real interest to practitioners.
Another interesting trend--a focus on the processes of learning--has made the time ripe for fruitful contributions of research on human cognition to classroom practice. Although some early workers in artificial intelligence were apparently quite interested in learning, it is only in recent years that much attention has been given to creating systems that learn. Similarly, cognitive psychologists who had heretofore focused their attention on the issue of performance, perhaps as a negative reaction to the obvious interest of behaviorist psychologists in learning (e.g., operant conditioning, maze learning), have re-discovered learning as an issue of import and interest. In recent years, a great deal of enthusiasm for tackling the mechanisms of learning has appeared in the cognitive science community. Thus, the moment seems opportune for the mathematics education and cognitive science communities to benefit from one another.
Despite the fact that a considerable amount of research on the topic of mathematical problem solving has been amassed, and despite the widespread interest on the . . .