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ELEMENTARY STUDENTS’ TRANSFORMATIONAL GEOMETRY ABILITIES AND COGNITIVE STYLE Xenia Xistouri and Demetra Pitta-Pantazi Department of Education, University of Cyprus This study investigated 93 elementary students’ abilities in solving transformational geometry tasks and how they relate to cognitive style. A test was developed to assess students’ transformational geometry abilities, which included translation, reflection and rotation tasks. Students’ cognitive styles were assessed using the Object-Spatial Imagery and Verbal Questionnaire (OSIVQ) (Blazhenkova & Kozhevnikov, 2009). Results suggest: 1) that the elementary students had average performance in solving the transformational geometry tasks, with rotation tasks being the most difficult and 2) that although both spatial imagery and object imagery cognitive styles relate to performance in transformational geometry, highly spatial imagery students perform better, because their cognitive style gives them an advantage in the case of solving the most difficult tasks, those of rotations. Key-words: transformational geometry, cognitive style INTRODUCTION The growing emphasis on geometry teaching during the last few decades has modified its’ traditionally Euclidian-based content, by introducing new types of geometry such as transformational geometry (Jones, 2002). There are several suggestions that there is limited research on transformational geometry (Boulter & Kirby, 1994; Hollebrands, 2003), which is imputed to its’ underemphasis in mathematics curricula. However, it is considered important in supporting children’s development of geometric and spatial thinking (Hollebrands, 2003; Jones & Mooney, 2003) and it is related to a variety of activities in academic and every-day life, such as geometrical constructions, art, architecture, carpentry, electronics, mechanics, clothing design, geography, navigation and route following (Boulter & Kirby, 1994). Performance in geometric transformations has been previously connected to the holistic-analytic types of processing (Boulter & Kirby, 1994). However, despite its’ rather obvious relation to visual imagery and the fact that it has often been connected to spatial abilities in literature, there doesn’t seem to be a study that examines abilities in geometric transformations in relation to the visual-verbal cognitive style. This paper will study the relation between abilities in solving transformational geometry tasks and a new three-dimensional cognitive style model proposed by Blazhenkova and Kozhevnikov (2009) that distinguishes between Object imagery, Spatial Imagery and Verbal dimensions. Specifically, the aim of the paper was to investigate nine to eleven year old students’ abilities in transformational geometry tasks of translations, reflections and rotations, and to investigate the relationship between these abilities and the students’ cognitive style.
THEORETICAL FRAMEWORK Transformational Geometry in Mathematics Education. The inclusion of transformational geometry in mathematics curricula in the early 70’s raised an emphasis around the importance of teaching and understanding geometric transformations (Jones, 2002). Early studies focus on providing evidence for suggesting that teaching geometric transformations in elementary and high school education is feasible and may have positive effects on students’ learning of mathematics (Edwards, 1989; Williford, 1972). Later studies focus on more psychological aspects, such as students’ ability and misconceptions (Kidder, 1976; Moyer, 1978), strategies for solving transformational geometry problems (Boulter & Kirby, 1994) and configurations influencing students’ ability in transformational geometry (Schultz, 1983). During the early 90’s, research started to focus on investigating a hierarchy that describes students’ acquisition of transformational geometry (Molina, 1990; Soon, 1989). It seems that research in transformational geometry decreased substantially around the late 80’s, leaving unanswered questions on the cognitive development of transformations (Boulter & Kirby, 1994). For instance, Moyer (1978) raised questions on whether some geometric transformations are more difficult than others and emphasized the need to search for a successful sequence of learning activities in transformational geometry for children. There were also some issues raised concerning individual differences and different types of processing information in transformational geometry problem solving (Boulter & Kirby, 1994). It is thus important to understand the role that individual differences such as students’ cognitive style may have in their abilities to solve transformational geometry tasks. Such information would guide educators in providing further assistance to the less able students in transformational geometry to overcome their difficulties. The Object-Spatial-Verbal Cognitive Style Model. Cognitive styles refer to psychological dimensions representing consistencies in an individual’s manner of cognitive functioning, particularly with respect to acquiring and processing information (Witkin, Moore, Goodenough, & Cox, 1977). One of the most commonly acknowledged cognitive styles dimension is the Visual–Verbal (e.g. Paivio, 1971; Richardson, 1977), which describes consistencies and preferences in processing visual versus verbal information, and classifies individuals as either visualizers, who rely primarily on imagery when attempting to perform cognitive tasks, or verbalizers, who rely primarily on verbal-analytical strategies. However, neuropsychological data suggest the existence of two distinct imagery subsystems that encode and process visual information in different ways: an object imagery system that processes the visual appearance of objects and scenes in terms of their shape, colour information and texture and a spatial imagery system that processes object location, movement, spatial relationships and transformations and other spatial attributes of processing (Blazhenkova & Kozhevnikov, 2009). The
distinction between object and spatial imagery has been also found in individual differences in imagery (Kozhevnikov, Hegarty, & Mayer, 2002). Recent behavioural and neuroimaging studies have identified two distinct types of individuals, object visualizers, who use imagery to construct vivid high-resolution images of individual objects, and spatial visualizers, who use imagery to represent and transform spatial relations (Kozhevnikov, Kosslyn, & Shephard, 2005). Based on these distinctions, Blazhenkova and Kozhevnikov (2009) have developed a self-report instrument assessing the individual differences in object imagery, spatial imagery and verbal cognitive styles, the Object-Spatial Imagery and Verbal Questionnaire (OSIVQ). Mathematics education researchers have often linked the verbalizers/visualizers distinction to mathematical performance (Presmeg, 1986). Nevertheless, the results of the relationship between visualisation and mathematical performance are unclear. Some studies found that visual–spatial memory is an important factor which explains the mathematical performance of students (Battista & Clements, 1998; Tartre, 1990), while other studies showed that students classified as visualizers do not tend to be among the most successful performers in mathematics (Presmeg, 1986). In the case of transformational geometry, although it has not yet been linked to the verbalizers/visualizers distinction, it has been connected to spatial ability (Dixon, 1995; Kirby & Boulter, 1999), thus it is hypothesized in this study that the spatial imagery cognitive style will be related to abilities in transformational geometry tasks. METHODOLOGY The purpose of the study is to investigate elementary school students’ abilities in transformational geometry tasks and the relation of these abilities to the students’ cognitive style. Ninety three students were selected to participate in the study (34 fourth-graders and 59 fifth-graders), based on their teachers’ willingness to provide access to their classes during school-time. A transformational geometry ability test and a self-report cognitive style questionnaire were administered to all students at the same week, in groups of approximately 15 students. The transformational geometry ability test was used to measure students’ mathematical abilities in the concepts of translation, (axial) reflection and rotation. The test consisted of 33 tasks, of which seven were translations, fourteen were reflections and twelve were rotations. The test included multiple choice and drawing tasks which focused mainly in identifying the result of a given transformation, performing a specific transformation and finding the parameters of a given transformation (see for ex. Edwards, 1989; Dixon, 1995). It is noted that the students were taught 2-3 lessons on symmetry at every grade, and were informally introduced to the concept of transformations but not to the mathematical terms. They were given 40 minutes to solve the test during normal lesson time. Each correct response to an item in each of the tasks was assigned a positive point. Half point was assigned when a response was partially correct, for example when a requested transformation was performed correctly, but there was no accuracy in the shapes’ dimensions or
orientation. The points were summed up separately for translations, reflections and rotations, in order to give the students’ scores for each type of geometric transformation, and also in total to give an overall of each student’s performance. The students were then administered a modified version of the Object-Spatial Imagery and Verbal Questionnaire (OSIVQ) to assess the individual differences in spatial imagery, object imagery and verbal cognitive style. This is a self-report questionnaire, which includes 45 statements with a 5-point Likert scale for students to rate themselves on how much they agree with the content of the statement. Fifteen of the items measured object imagery preference and experiences, fifteen items measured spatial imagery preference and experiences and fifteen items measured verbal preference and experiences. Examples of the statements are: “If I were asked to choose among engineering professions or visual arts I would choose visual arts” (measuring Object Imagery dimension), “My images are more schematic than colourful and pictorial” (measuring Spatial Imagery dimension), and “I usually do not try to visualize or sketch diagrams when reading a textbook” (measuring Verbal dimension). The test was translated in Greek and was modified to be comprehensive to elementary students. The students were given 30 minutes to complete the questionnaire. For each student, the fifteen item ratings for each factor were averaged to create object imagery, spatial imagery and verbal scale scores. RESULTS The main purpose of the study was to investigate elementary school students’ abilities in transformational geometry tasks of translations, reflections and rotations, and how these are related to the students’ cognitive style. Students’ means of performance were calculated to describe their abilities in transformational geometry. The object-spatial-verbal cognitive style dimension was used as predictor variable for students’ performance in transformational geometry tasks. Specifically, through multiple regression analyses with criterion (dependent) variable the students’ performance in translation tasks, performance in reflection tasks, performance in rotation tasks and overall performance, and predictors (independent) the spatial imagery, object imagery and verbal cognitive style scores. Χ
SD
Translations
0.59
0.26
Reflections
0.54
0.20
Rotations
0.38
0.21
Overall
0.49
0.18
Table 1: Means and Standard Deviations for each type of tasks in the transformational geometry abilities test
Table 1 presents the means and standard deviations for each type of transformational geometry task, as well as for overall performance. It appears that students performed slightly better in the translation tasks ( Χ=0.59, SD=0.26), rather than the other categories, indicating that these might be the easiest type of tasks for elementary school students. Mean performance in reflections ( Χ=0.54, SD=0.20) is next, but very close to performance in translation tasks. In order to test these observations, students’ means in translation tasks and in reflection tasks were compared in a paired sample t-test analysis, which showed that this mean difference is not significant (t=1.842, p=0.069). This finding is in accord with Moyer (1978), who found that translations are as easy as reflections. The most difficult tasks for the students seem to be the rotation tasks, where this group of students had a much lower mean performance ( Χ=0.38, SD=0.21). A paired sample t-test for comparing students’ mean performance in reflection tasks and in rotation tasks revealed this mean difference is statistically significant (t=7.266, p=0.000). Students’ overall performance mean in transformational geometry tasks is 0.49 (SD=0.18), which is near average, considering zero as minimum value and one as maximum. Translations
Reflections
Rotations
Overall
Spatial Imagery Cognitive Style
.162
.231*
.236*
.266*
Object Imagery Cognitive Style
.174
.231*
.199
.254*
Verbal Cognitive Style
.031
.169
-.097
.050
* Correlation is significant at the 0.05 level (2-tailed). Table 2: Correlations among performance scores and spatial imagery, object imagery and verbal cognitive styles.
The correlations among students’ cognitive styles and their performance in translations, reflections, rotations and overall are presented in Table 2. As can be seen from Table 2, the spatial imagery cognitive style is significantly correlated with students’ transformational geometry abilities in reflections, rotations and overall performance, while object imagery cognitive style is significantly correlated only with reflections and overall performance. The verbal cognitive style dimension did not correlate with any of the students’ abilities in transformational geometry. We further examined the nature of these correlations and non correlations between cognitive styles and students’ abilities in translation, reflection and rotation tasks, as well as their overall performance in the test. Table 3 presents the results of the multiple regressions, using the stepwise method. It should be noted that the regression analysis for students’ abilities in the translation tasks did not enter any variables in the equation, which means that none of the cognitive style variables - object, spatial or verbal - can significantly predict
performance in translation tasks. This was expected, as none of these variables were significantly correlated to the translation tasks, as seen in Table 2. Therefore, Table 3 presents the coefficients and levels of significance for predicting performance in reflections, rotations and overall. Reflections
Rotations
Overall
b
p
b
p
B
p
Spatial Imagery Cognitive Style
.154
.209
.327
.005**
.266
.013*
Object Imagery Cognitive Style
.231
.031*
.222
.083
.161
.183
Verbal Cognitive Style
.075
.536
-.228
.047*
-.068
.554
* Indicate statistical significant p
