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

Abstract

Evidence regarding the relationships between 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 of mathematical problems were examined before and after a monthlong intensive course on mathematical 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 statement). After the course the students perceived relatedness more like the experts—according to principles or methods relevant for solution. Thus, criteria for perception shift as a person's knowledge bases become more richly structured^ Theories of solving commonly hold that the mental representation of problems influences how people perceive problems. Moreover, as experience leads to better solving, the quality of representation is expected to improve with corresponding improvement in 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 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 perception is con