【正文】 cal Problem Solving Mary Hegarty and Maria Kozhevnikov University of California, Santa Barbara Although visualspatial representations are used extensively in mathematics and spatial ability is highly correlated with success in mathematics education, research to date has not demonstrated a clear relationship between use of visualspatial representations and success in mathematical problem solving. The authors distinguished 2 types of visualspatial representations: schematic representations that encode the spatial relations described in a problem and pictorial representations that encode the visual appearance of the objects described in the problem. Participants solved mathematical problems and reported on their solution strategies. The authors were able to reliably classify their visualspatial representations as primarily schematic or primarily pictorial. Use of schematic spatial representations was associated with success in mathematical problem solving, whereas use of pictorial representations was negatively correlated with success. Use of schematic representations was also significantly correlated with one measure of spatial ability. The research therefore helps clarify the relationship between visual imagery, spatial ability, and mathematical problem solving. Visual imagery refers to the ability to form mental representations of the appearance of objects and to manipulate these representations in the mind (Kosslyn, 1995). Most researchers agree that such visual representations are important in mathematics education because they enhance an intuitive view and an understanding in many areas of mathematics (., Krutetskii, 1976。 Usiskin, 1987). There is a significant relationship between spatial ability and achievement in mathematics (., Battista, 1990). However, the wide use of visual images by students is not always effective in problem solving and can lead to erroneous solutions (., Lean amp。 Presmeg, 1992). In this study, we clarify the relationship between visual imagery, spatial ability, and mathematical problem solving by identifying two different types of visualspatial representations used in solving mathematical problems— schematic and pictorial representations— and by showing that they are differentially related to success in mathematical problem solving. VisualSpatial Representations in Mathematical Problem Solving There is extensive research in mathematics showing a correlation between spatial ability and mathematical performance (., Battista, 1990。 Sherman, 1979。 the second group, visualizers, involves those who prefer to use visual imagery。 Clements, 1981), and Presmeg (1986a, 1986b, 1992) recognized that individuals could be placed on a continuum with regard to their preference for using visual imagery while solving mathematical authors of these studies defined mathematical visuality as the extent to which a person prefers to use visual imagery or diagrams when attempting mathematical problems. Suwarsono developed an instrument to measure an individual39。 (b) pattern imagery (pure relationships depicted in a visualspatial scheme)。 (d) dynamic imagery, which involves dynamic transformations of geometric figures。s attention from the main elements in the original problem representation, whereas other kinds of imagery may play a more positive role. Presmeg ascribed the most essential role in mathematical problem solving to pattern imagery, in which concrete details are disregarded and pure relationships are depicted. This kind of imagery was also identified by other researchers (Johnson, 1987。 Calvanio, 1988。 Logie, 1995). Visual imagery refers to a representation of the visual appearance of an object, such as its shape, color, or brightness. Spatial imagery refers to a representation of the spatial relationships between parts of an object and the location of objects in space or their movement。s arm, whereas spatial imagery tasks are impaired by arm movements but not by viewing irrelevant pictures (Logie, 1995). Furthermore, cognitive neuroscience studies (., Farah et al., 1988) have demonstrated that following brain lesions, patients can be extremely impaired in tasks tapping visual aspects of imagery while showing normal performance in tests of spatial imagery. We argue that a dissociation between visual and spatial imagery also exists in individual differences in imagery— some individuals are especially good at pictorial imagery (., constructing vivid and detailed visual images), whereas others are good at schematic imagery (., representing the spatial relationships between objects and imagining spatial transformations). We consider spatial ability as a subset of imagery abilities, related to schematic imagery and not related to pictorial imagery (Poltrock amp。s attention from the main relationships in the problem statement. Second, we hypothesize that spatial ability is positively associated with use of schematic imagery but not with use of pictorial imagery. Finally, to test the alternative hypothesis that use of schematic imagery is related to general intelligence, rather than spatial ability specifically, we include measures of verbal and nonverbal general intelligence. Method Participants Thirtythree boys in sixth class (sixth grade) in an allboys primary school in Dublin, Ireland, took part in this study. The mean age of the participants was 12 years, 1 month (range = 11 years, 6 months13 years, 1 month). Materials The following measures were administered to the students: 1. The MPI consists of 15 problems, either taken from previous studies (Krutetskii, 1976。 Clements, 1981) or posed specifically for the study. A pilot study had determined that these problems were of appropriate difficulty level for the students and that students used a variety of strategies to solve the problems, including