【正文】 use of imagery in solving qualitative problems in kinematics. Unpublished doctoral dissertation, Israel Institute of Technology, Haifa, Israel. Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press. Lean, C, amp。 Clements, 1981。 Lohman, 1988). Procedure The measures were administered in two group sessions and one session in which students were tested individually. In the group sessions, all students were tested at once in their classroom. In the first group session, the DVRT was administered, and in the second group session, the Ravens Progressive Matrices and the PMA Space subtest were administered, according to the standard instructions for these instruments. Each session took approximately 1 hr. In the individual session, each student was first administered the MPI. The 15 problems were printed on cards and presented in different orders, such that no more than 6 students received the problems in any order. When each problem was presented, the student was first allowed up to 3 min to solve the problem, although students often gave an answer in less than this time. During this time the experimenter did not speak except to encourage a student to attempt a problem, but the experimenter did note any diagrams the student drew or gestures the student made. When the student had answered the problem (or after 3 minutes, if the student did not plete the problem), the student was asked the strategy questions about that problem (samples presented in Appendix B). Following the interviews, which were audiotaped, the students were administered the Block Design subtest of the WISCR. Scoring of Mathematical Processing Instrument Four different measures were scored from responses on the MPI. The first score was the number of problems solved correctly. The second score was a measure of the extent to which the student used visualspatial representations in solving the problems. Each student was given a score of 1 on each problem for which they reported use of a visualspatial representation and 0 for each problem on which there was no evidence that they used such a representation. The third and fourth scores measured the extent to which students39。 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 use of diagrams and imagery and nonvisualspatial solutions. In the pilot study, the MPI gave internally consistent measures of problem solving success (Cronbach39。 Calvanio, 1988。 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。 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。 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。 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。 Agnoli, 1986). The focus of this research is to identify how spatial and visual imagery abilities affect problem solving in mathematics. We first hypothesize that use of schematic spatial imagery in solving mathematical problems is associated with better performance, whereas use of pictorial imagery is associated with poorer performance in problem solving because it takes the problem solver39。 Thurstone amp。 in fact, this measure is significantly negatively associated with one of the spatial measures. As is to be expected, the psychometric tests were highly correlated with each other and with mathematical problem solving. We now consider the relationship between the more specific measures of use of schematic and pictorial representations, mathematical problem solving, and the psychometric measures. Use of schematic spatial representations was negatively correlated with use of pictorial representations (r = — .46, p .01). The correlations of these two measures with mathematical problem solving and the psychometric measures are presented in Table 3. As predicted, the pattern of correlations is very different for the two strategy measures. First, use of schematic spatial representations is positively correlated with mathematical problem solving。 Calvanio, R. (1988). Visual and spatial memory: Dissociable systems of re