Problem perception and knowledge structure in expert and novice mathematical problem solvers.

Evidence regarding the relationships between problem perception and expertise has customarily been obtained indirectly, through contrasting-group studies such as expert-novice comparisons. Differences in perception have been attributed to differences in expertise, although the groups compared generally differ on a number of other major attributes (e.g., aptitude). This study explored the relationship between perception and proficiency directly. Students' perceptions of the structure of mathematical problems were examined before and after a monthlong intensive course on mathematical problem solving. These perceptions were compared with experts' perceptions. Subjects sorted problems on the basis of similarity. Hierarchical clustering analysis of the sorting data indicated that novices perceive problems on the basis of "surface structure" (i.e., words or objects described in the problem statement). After the course the students perceived problem relatedness more like the experts—according to principles or methods relevant for problem solution. Thus, criteria for problem perception shift as a person's knowledge bases become more richly structured^ Theories of problem solving commonly hold that the mental representation of problems influences how people perceive problems. Moreover, as experience leads to better problem solving, the quality of problem representation is expected to improve with corresponding improvement in problem perception (Chi, Feltovich, & Glaser, 1981; Hayes & Simon, 1974; Heller & Greeno, 1979; Newell & Simon, 1972). At one end of the spectrum, the correct perception of a problem may cue access to a "problem schema" that suggests a straightforwar d method of solution or a more or less automatic response (Chase & Simon, 1973; Kinsley, Hayes, & Simon, 1977). At the other end, an incorrect perception may send one off on a "wild goose chase." Since problem perception is con