1、PDF外文：http:/ Journal of Educational Psychology, 1999, 91, 4, 684-689. Types of Visual-Spatial Representations and Mathematical Problem Solving Mary Hegarty and Maria Kozhevnikov University of California, Santa Barbara Although visual-spatial representations are used extensively in mathematics
2、and spatial ability is highly correlated with success in mathematics education, research to date has not demonstrated a clear relationship between use of visual-spatial representations and success in mathematical problem solving. The authors distinguished 2 types of visual-spatial representations: s
3、chematic 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 rel
4、iably classify their visual-spatial 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 r
5、epresentations 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
6、 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 (e.g., Krutetskii, 1976； Usiskin, 1
7、987). There is a significant relationship between spatial ability and achievement in mathematics (e.g., Battista, 1990). However, the wide use of visual images by students is not always effective in problem solving and can lead to erroneous solutions (e.g., Lean & Clements, 1981； Presmeg, 1992).
8、 In this study, we clarify the relationship between visual imagery, spatial ability, and mathematical problem solving by identifying two different types of visual-spatial representations used in solving mathematical problems schematic and pictorial representations and by showing that they are differ
9、entially related to success in mathematical problem solving. Visual-Spatial Representations in Mathematical Problem Solving There is extensive research in mathematics showing a correlation between spatial ability and mathematical performance (e.g., Battista, 1990； McGee, 1979； Sherman, 1979； Smith,
10、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 complexity of mathematical tasks (see Kaufmann, 1990, for a review). Other investigations have focused on the
11、mental processes used in solving mathematical problems, particularly the role of diagrams and visual-spatial 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
12、 (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
13、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 & Clements, 1981), and Presmeg (1986a, 1986b, 1992) recognized that individuals could be placed
14、 on a continuum with regard to their preference for using visual imagery while solving mathematical problems.The authors of these studies defined mathematical visuality as the extent to which a person prefers to use visual imageryor diagrams when attempting mathematical problems. Suwarsonodeveloped
15、an instrument to measure an individual'slevel of visuality the Mathematical Processing Instrument(MPI), which has been used extensively in further researchon this topic. A surprising result from this literature is thatthe wide use of visual images is not always effective and cansometimes lead to
16、 erroneous solutions of mathematicalproblems. Finding a negative correlation between mathematicalvisuality and both spatial ability and mathematicalperformance, Lean and Clements (1981) concluded thatverbalizers outperform visualizers on both mathematical andspatial ability tests. On this point, Pre
17、smeg (1986a, 1986b)identified five kinds of imagery used by high school studentsin solving mathematical problems: (a) concrete pictorialimagery (pictures in the mind)； (b) pattern imagery (purerelationships depicted in a visual-spatial scheme)； (c) kinestheticimagery, which involves hand movement an
18、d othergestures； (d) dynamic imagery, which involves dynamictransformations of geometric figures； and (e) memory offormulas, wherein visualizers typically imagine a formulawritten on a blackboard or in their notebooks. Presmeg (1986a, 1986b, 1992) argued that the use of concrete pictorial imagery ma
19、y focus the reasoning on irrelevant details that take the problem solver'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； Krutetskii, 1976). However, none of these researchers examined the quantitative relationships between use of
