【正文】 presentation. Cognitive Psychology, 20, 439462. Johnson, M. (1987). The body in the mind: The bodily basis of meaning, imagination, and reason. Chicago: University of Chicago Press. Kaufmann, G. (1990). Imagery effects on problem solving. In P. J. Hampson, D. E. Marks, amp。 Agnoli, F. (1986). Are spatial visualization ability and visual imagery ability equivalent? In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 3, pp. 255296). Hillsdale, NJ: Erlbaum. Presmeg, N. C. (1986a). Visualization in high school mathematics. For Learning of Mathematics, 63, 4246. Presmeg, N. C. (1986b). Visualization and mathematical giftedness. Educational Studies in Mathematics, 17, 297311. Presmeg, N. C. (1992). Prototypes, metaphors, metonymies, and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595610. Raven, J. (1958). Standard progressive matrices. London: H. K. Lewis. Sherman, J. A. (1979). Predicting mathematical performance in high school girls and boys.。 Peterson, 1985), which were most predictive of use of schematic representations in this study. Third, a recent study of physics problem solving carried out with male and female college students replicated this research, showing that high spatial ability students constructed more schematic spatial representations and low spatial ability students constructed more concrete pictorial images (Kozhevnikov, 1999). Therefore, we expect our results to generalize to other populations, although it is important to test this expectation empirically. Our research clearly shows that some visualspatial representations promote problemsolving success and others may present an obstacle to mathematical problem solving. Regardless of a student39。 hence, this problem was omitted from further analyses. We, serving as two independent raters, scored 20 of the participants39。 Educational Research Centre, 1968). This test was designed to measure general verbal intelligence and was developed as part of a large study of the effects of standardized testing in Ireland (Kellaghan, 1976). It is made up of sections on analogies, the identification of words opposite in meaning to a given stimulus, the identification of concepts as belonging to a single category, and inductive and deductive reasoning. The DVRT is standardized for children aged 10 years to 13 years, with reliability estimates ranging from .94 to .98 for different ages. 3. The Ravens Progressive Matrices Test (Raven, 1958) was used as a measure of nonverbal reasoning. Each item on this test shows a 3 X 3 matrix of figures with one missing cell. The figures in each row and column differ by some rule or set of rules. The task is to induce these rules and apply them to choose the missing figure from a set of eight choices. 4. Spatial ability was measured by two tests, the Block Design subtest of the Wechsler Intelligence Scale for Children— Revised (WISCR。 further, spatial imagery is not limited to the visual modality (., one could have an auditory or haptic spatial image). Cognitive studies have provided evidence for a dissociation between these two aspects of imagery. First, dualtask studies have shown that visual imagery tasks are impaired by concurrently viewing irrelevant pictures but not by moving one39。 (c) kinesthetic imagery, which involves hand movement and other gestures。 McGee, 1979。 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。 (b) pattern imagery (pure relationships depicted in a visualspatial scheme)。 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 a = .72]). The problems on the MPI were printed on cards. Each problem was followed by a set of questions, asked by the experimenter, about the strategy used to solve the problem. This method of questioning was adopted because we found that children of this age vary considerably in their ability to give concurrent verbal protocols while solving these types of problems. All students were asked all of the questions, unless they had already spontaneously provided the information asked in a question. The problems are presented in Appendix A, and sample acpanying questions are presented in Appendix B. 2. Verbal reasoning ability was measured by the Drumcondra Verbal Reasoning Test (DVRT。t see the lorry driver either. I just pictured 60 miles, it could have been 60 feet, 60 anything. Diagrams always showed the spatial relations between objects in a problem. Sample diagrams drawn for Problem 1 are shown in Figure 1. A visualspatial representation was scored as primarily pictorial if the student reported an image of the objects or persons referred to in a problem, rather than the relations between these objects. For example, the following imagery reported in Problems 1 and 11, respectively, was scored as primarily pictorial: I just saw the man going along planting trees. No, I just imagined him outside his house with his hand out, hitchhiking. If there was no evidence in the protocols that the relations between objects in a problem had been represented in an