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1、外文文献资料Journal of Educational Psychology, 1999, 91, 4, 684-689.Types of Visual-Spatial Representations and Mathematical Problem SolvingMary Hegarty and Maria KozhevnikovUniversity of California, Santa BarbaraAlthough visual-spatial representations are used extensively in mathematics and spatial abili
2、ty 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: schematic represen
3、tations 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 th
4、eir 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 representations wa
5、s 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
6、 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, 1987). There is a s
7、ignificant 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). In this study, we cla
8、rify the relationship between visual imagery, spatial ability, and mathematical problem solving by identifying two different types of visual-spatial representations used in solving mathematical problemsschematic and pictorial representationsand by showing that they are differentially related to succ
9、ess in mathematical problem solving.Visual-Spatial Representations in Mathematical Problem SolvingThere is extensive research in mathematics showing a correlation between spatial ability and mathematical performance (e.g., Battista, 1990; McGee, 1979; Sherman, 1979; Smith, 1964). For example, Sherma
10、n (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 mental processes used in so
11、lving 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 (1976) concluded that indi
12、viduals 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 im
13、agery; 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 on a continuum with regard to t
14、heir 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 imagery or diagrams when attempting mathematical problems. Suwarsono developed an instrument to measure an in
15、dividuals level of visualitythe 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 mathema
16、tical 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) identi
17、fied 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 visual-spatial scheme); (c) kinesthetic imagery, which involves hand movement and other gestures; (d)
18、 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 reason
19、ing on irrelevant details that take the problem solvers 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 concret
20、e 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 different types of imagery and mathematical proble
21、m solving, nor have they examined the relationship between spatial ability and use of different types of imagery.In summary, although spatial ability correlates positively with mathematics achievement, preference to process information visually correlates with neither mathematical performance nor sp
22、atial ability tests. These results have cast doubt on the usefulness of classifying students as visualizers or verbalizers, and as a consequence, the number of educational studies related to the visualizer-verbalizer cognitive style has declined rapidly over the past decade.Types of Visual Imagery A
23、bilityThe current research differentiates between two different visual imagery abilities identified in cognitive psychology and neuroscience research. This research suggests that visual imagery is not general and undifferentiated but composed of different, relatively independent visual and spatial c
24、omponents (e.g., Farah, Hammond, Levine, & Calvanio, 1988; Kosslyn, 1995; 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 a
25、n object and the location of objects in space or their movement; further, spatial imagery is not limited to the visual modality (i.e., one could have an auditory or haptic spatial image). Cognitive studies have provided evidence for a dissociation between these two aspects of imagery. First, dual-ta
26、sk studies have shown that visual imagery tasks are impaired by concurrently viewing irrelevant pictures but not by moving ones arm, whereas spatial imagery tasks are impaired by arm movements but not by viewing irrelevant pictures (Logie, 1995). Furthermore, cognitive neuroscience studies (e.g., Fa
27、rah 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 diffe
28、rences in imagerysome individuals are especially good at pictorial imagery (i.e., constructing vivid and detailed visual images), whereas others are good at schematic imagery (i.e., representing the spatial relationships between objects and imagining spatial transformations). We consider spatial abi
29、lity as a subset of imagery abilities, related to schematic imagery and not related to pictorial imagery (Poltrock & 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 spat
30、ial 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 solvers attention from the main relationships in the problem statement. Second, we hypothesize th
31、at 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 n
32、onverbal general intelligence.MethodParticipantsThirty-three boys in sixth class (sixth grade) in an all-boys 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 months-13 years, 1 month).MaterialsThe following measu
33、res were administered to the students:1. The MPI consists of 15 problems, either taken from previous studies (Krutetskii, 1976; Lean & Clements, 1981) or composed specifically for the study. A pilot study had determined that these problems were of appropriate difficulty level for the students and th
34、at students used a variety of strategies to solve the problems, including use of diagrams and imagery and non-visual-spatial solutions. In the pilot study, the MPI gave internally consistent measures of problem solving success (Cronbachs a = .78) and solution strategy (i.e., tendency to use visual-s
35、patial representations Cronbachs 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 co
36、nsiderably 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 accompanying q
37、uestions are presented in Appendix B.2. Verbal reasoning ability was measured by the Drumcondra Verbal Reasoning Test (DVRT; 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 tes
38、ting 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
39、 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 dif
40、fer 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 ChildrenRevised (WISC-R; Wechsler, 1976) and the S
41、pace subtest of the Primary Mental Abilities Test (PMA Space; Thurstone & Thurstone, 1947), which are characteristic of two different spatial abilities factors. In the Block Design test, participants are presented with a set of blocks that are white on some sides, red on some sides, and half red-hal
42、f white on others. They are then presented with a picture of a two-dimensional red and white design. Their task is to arrange the blocks so that the design is shown. This test is characteristic of the spatial visualization factor (Carroll, 1993; Lohman, 1988). The PMA Space test is a mental rotation
43、 test. On each trial, participants are shown a standard figure on the left-hand side of the page and six comparison figures on the right-hand side of the page. Their task is to indicate whether each of six comparison figures is a planar rotation of the target figure (as opposed to its mirror image)
44、as quickly and accurately as possible. This test is a measure of the spatial relations factor, also referred to as speeded rotation (Carroll, 1993; Lohman, 1988).ProcedureThe measures were administered in two group sessions and one session in which students were tested individually. In the group ses
45、sions, 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
46、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 allowe
47、d 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 st
48、udent had answered the problem (or after 3 minutes, if the student did not complete 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 subtes
49、t of the WISC-R.Scoring of Mathematical Processing InstrumentFour 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 visual-spatial representations in solving the problems. Each student was given a score of 1 on each problem for which they reported use of a visual-spatial representation and 0 for each problem on which there was no evidence that they used such a representation. The th
