【正文】 外文文獻資料 Journal of Educational Psychology, 1999, 91, 4, 684689. Types of VisualSpatial Representations and Mathematical 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。 Clements, 1981。 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。 McGee, 1979。 Sherman, 1979。 Smith, 1964). For example, Sherman (1979) reported that the spatial ability factor was one of the main factors significantly affecting mathematical performance. This correlation increases with the plexity of mathematical tasks (see Kaufmann, 1990, for a review). Other investigations have focused on the mental processes used in solving mathematical problems, particularly the role of diagrams and visualspatial images in mathematical problem solving. In these studies, students reported their solution processes after solving problems or while solving problems. On the basis of such studies, Krutetskii (1976) concluded that individuals can be classified into three groups according to how they process mathematical information. The first group consists of verbalizers, who prefer verballogical rather than imagery modes when attempting to solve problems。 the second group, visualizers, involves those who prefer to use visual imagery。 and the third group, mixers, contains individuals who have no tendency one way or the other. Following the Krutetskii model, Moses (1980), Suwarsono (as cited in Lean amp。 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。s level of visuality— the Mathematical Processing Instrument (MPI), which has been used extensively in further research on this topic. A surprising result from this literature is that the wide use of visual images is not always effective and can sometimes lead to erroneous solutions of mathematical problems. Finding a negative correlation between mathematical visuality and both spatial ability and mathematical performance, Lean and Clements (1981) concluded that verbalizers outperform visualizers on both mathematical and spatial ability tests. On this point, Presmeg (1986a, 1986b) identified five kinds of imagery used by high school students in solving mathematical problems: (a) concrete pictorial imagery (pictures in the mind)。 (b) pattern imagery (pure relationships depicted in a visualspatial scheme)。 (c) kinesthetic imagery, which involves hand movement and other gestures。 (d) dynamic imagery, which involves dynamic transformations of geometric figures。 and (e) memory of formulas, wherein visualizers typically imagine a formula written on a blackboard or in their notebooks. Presmeg (1986a, 1986b, 1992) argued that the use of concrete pictorial imagery may focus the reasoning on irrelevant details that take the problem solver39。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 (Joh