Cognitive Style, Operativity, and Mathematics Achievement

Cognitive Style, Operativity, and Mathematics AchievementAuthor(s): James J. Roberge and Barbara K. FlexerSource: Journal for Research in Mathematics Education, Vol. 14, No. 5 (Nov., 1983), pp. 344-353Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/748679 .

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Journal for Research in Mathematics Education 1983, Vol. 14, No. 4, 344-353

COGNITIVE STYLE, OPERATIVITY, AND MATHEMATICS ACHIEVEMENT

JAMES J. ROBERGE, Temple University BARBARA K. FLEXER, Temple University

Previous investigations of the effects of field dependence-independence or the level of operational development on the mathematics achievement of children in the lower elementary school grades have involved the administration of concrete operational tasks (e.g., classification, conservation, and seriation). The present study was designed to examine the influence of these factors on the mathematics achievement of sixth, seventh, and eighth graders by using formal operational tasks (i.e., combinations, propositional logic, and proportionality). Results were analyzed using total mathematics achievement test scores as well as scores on subtests of computation, concepts, and problem solving. Field-independent students scored significantly higher than field-dependent students on the total mathematics, concepts, and problem-solving tests. High-operational students scored significantly higher than their low-operational peers on all tests. Educational implications of the findings are discussed.

In the past decade, educational researchers and psychologists have shown considerable interest in the relationship between cognitive style and the mathematics achievement of elementary school children (Buriel, 1978; Kagan & Zahn, 1975; Kagan, Zahn, & Gealy, 1977; Robinson & Gray, 1974; Satterly, 1976; Vaidya & Chansky, 1980). The dimension of cognitive style that has received the most attention in this regard is field dependence-independence (Witkin, Dyk, Faterson, Goodenough, & Karp, 1962). Witkin and his associates described this dimension of cognitive style as a continuum ranging from an analytic to a global approach to perceptual and cognitive activities. Field-independent people are characterized by their ability to dis- tinguish and coordinate items extracted from a complex stimulus context that may be confusing for others. Field-dependent people, however, tend to preserve the holistic nature of the stimulus and conform to the prevailing field.

Interest in field dependence-independence as a potential factor in mathematics achievement is predicated on the assumption that the tests of this cognitive style have task demands similar to those of standardized mathematics achievement tests and that as children progress through the grades, achievement in quantitative areas demands increasing field independence (cf. Cohen, 1969). Recent studies have shown that elementary school children who are field independent obtain higher scores on standardized mathematics achievement tests than their field-dependent peers (Buriel, 1978; Kagan & Zahn, 1975; Kagan et al., 1977; Satterly, 1976; Stone, 1976; Vaidya & Chansky, 1980). However, in spite of the continuing controversy about the overlap between field dependence-independence and general intelligence (Roberge & Flexer, 1981; Satterly, 1976, 1979; Vernon, 1972; Weisz, O'Neill, & O'Neill, 1975; Zigler, 1963) and admonishments that IQ differences be statistically controlled when the relationships between field depend-



James J. Roberge and Barbara K. Flexer 345

ence-independence and cognitive abilities are being ascertained (Kagan & Kogan, 1970; Messick, 1976), the influence of IQ on the presumed effects of field dependence-independence on students' mathematics achievement was not considered in most of the studies cited. Moreover, none of these studies involved students at or beyond the junior high school level.

Piaget's theory regarding the development of logical thinking in children and adolescents (Inhelder & Piaget, 1958, 1964) has also been the catalyst for a substantial amount of empirical research on mathematics learning. In his

writings on mathematics education, Piaget (1970, 1975) described the operational structures of a logico-mathematical nature that are assumed to develop spontaneously between the ages of 7 and 11 (concrete operations stage) and those that supposedly emerge between ages 11 and 15 (formal operations stage). In addition, he discussed the links between these operational structures and the abstract and general structures of modern mathematics. One line of recent research has focused on an examination of the level of operational development (i.e., concrete or formal) that is necessary for the success- ful solution of different kinds of mathematics tasks (Collis, 1971, 1973, 1976, 1978; Halford, 1978; Malpas & Brown, 1974). Another line of research has concentrated on the relationship between cognitive development and the mathematics achievement of elementary school children. In particular, it has repeatedly been found that students' level of operativity on Piaget- ian measures of concrete operational thought is significantly related to their performance on standardized mathematics achievement tests (Arlin, 1981; Dimitrovsky & Almy, 1975; Kaufman & Kaufman, 1972; Vaidya & Chansky, 1980). Surprisingly, however, there seems to be only one published investigation of the relationship between students' level of operativity on Piagetian measures of formal operational thought and their performance on standardized mathematics achievement tests. Specifically, Lawson (1982) found significant positive correlations between ninth graders' performance on two of the Inhelder and Piaget (1958) formal operations tasks (i.e., bending rods and balance beam) and their standardized mathematics achievement test scores. Despite the oft-reported finding of a considerable overlap between performance on Piagetian tasks and intelligence tests (Bart, 1971; Cloutier & Goldschmid, 1976; DeVries, 1974; Dudek, Lester, Goldberg, & Dyer, 1969; Keating, 1975), IQ differences have not been taken into consideration in most studies of the level of operativity and mathematics achievement.

The purpose of the present study was to examine the effects of field dependence-independence and the level of operativity on the mathematics achievement of students in the upper elementary school grades. In contrast to most previous studies that have analyzed the relationships among these factors for students in the lower elementary school grades, IQ differences were statistically controlled. Moreover, because of the age of the students, operativity was assessed through formal operational tasks rather than con-



346 Cognitive Style, Operativity, and Mathematics Achievement

crete operational tasks. We hypothesized that field-independent students would obtain significantly higher scores on standardized mathematics achievement tests than field-dependent students and that high-operational students would obtain significantly higher scores than low-operational students.

METHOD

Subjects The sample consisted of 450 students who spanned the ages (approxi-

mately 11 to 15 years) during which Piaget postulated that formal operational thinking would emerge (Inhelder & Piaget, 1958; Piaget & Inhelder, 1969). Specifically, 150 students (75 boys and 75 girls) were randomly selected at each of Grades 6, 7, and 8 in a suburban public school. The mean IQs on the Lorge-Thorndike Intelligence Tests of the sixth, seventh, and eighth graders were 115.50, 113.61, and 112.57; their mean ages were 11.33, 12.36, and 13.28. We expected that choosing subjects from a population with an above-average mean IQ would increase the likelihood of observing considerable growth in formal operational reasoning abilities during early adolescence (Keating, 1975; Keating & Schaefer, 1975; Roberge, 1976; Yudin, 1966).

Tests

Group Embedded Figures Test (GEFT). This group test developed by Oltman, Raskin, and Witkin (1971) was used as the measure of cognitive style. It consists of 7 practice items and 18 test items that require the subject to locate simple shapes embedded in complex figures. The subject's score is the number of simple shapes correctly outlined in a given period of time.

Witkin, Oltman, Raskin, and Karp (1971) described the GEFT as a valid and reliable alternative to individually administered measures of field dependence-independence. They reported correlations of .82 and .63 between scores on the GEFT and the individually administered Embedded Figures Test for male and female undergraduates. They also reported a split-half reliability coefficient of .82, for both males and females. Although Witkin and his associates presented no validity or reliability data for younger-than-college- aged samples, Flexer and Roberge (1983) reported test-retest reliability coefficients (1-year interval) of .78 and .79 for sixth and seventh graders.

Formal Operational Reasoning Test (FORT). This paper-and-pencil test constructed by Roberge and Flexer (1982) was used to evaluate subjects' level of reasoning for three essential components of formal operational thought: combinations, propositional logic, and proportionality (Greenbowe, Herron, Lucas, Nurrenbern, Staver, & Ward, 1981).

Roberge and Flexer illustrated the content validity of the FORT by describ- ing the relationship between each of the FORT subtests and the correspond-




ing Inhelder and Piaget (1958) formal operations scheme, and they presented factor analytic evidence of the FORT's construct validity. They reported test-retest reliability coefficients (2-week interval) of .81 and .80 for samples of seventh and eighth graders on the combinations subtest. They also reported internal consistency reliability coefficients (K-R Formula 20) for samples of seventh and eighth graders of .75 and .74 on the logic subtest and of .52 and .60 on the proportionality subtest.

Metropolitan Achievement Tests (MAT). The Mathematics Computation, Mathematics Concepts, and Mathematics Problem Solving tests of the MAT (Durost, Bixler, Wrightstone, Prescott, & Balow, 1970) were used as the measures of mathematics achievement. These tests were administered to all students as part of the school's annual testing program. Form F (Inter- mediate) was administered at Grade 6; Form G (Advanced) at Grades 7 and 8.

Standard scores furnished by the test publisher were used in the data analyses. These scores provide a continuous, equal-interval system for each test across all forms of the test. That is, the standard scores compose a scale that is continuous across grades.

Procedure

The students were tested in groups of 15 to 25 during regularly scheduled 45-minute classes. They completed the GEFT during the first testing session and the FORT during a second session 2 weeks later. The same experimenter administered all the tests. Mathematics achievement scores on the MAT were obtained from the permanent school records.

At each grade, the students were divided into cognitive style groups accord- ing to their performance on the GEFT. Because of the reported sex differences in performance on embedded figures tests (Witkin, Moore, Goodenough, & Cox, 1977), separate cognitive style groups were formed for each sex. More precisely, those students scoring in the top third of each grade level by sex group on the GEFT were classified as field independent, those in the bottom third of each group as field dependent, and the remaining students as intermediate. Descriptive data on the GEFT scores for the various groups are shown in Table 1.

Table 1 Means and Standard Deviations of Scores on the

Group Embedded Figures Test by Cognitive Style Group

Grade 6 Grade 7 Grade 8 Cognitive

style group Sex M SD M SD M SD

Field independent Boys 11.28 3.49 12.36 2.55 14.72 1.86 Girls 10.88 2.47 10.68 2.32 13.00 2.18

Intermediate Boys 4.80 1.12 6.72 1.59 8.68 1.41 Girls 5.64 1.50 6.88 0.78 7.96 1.24

Field dependent Boys 1.96 0.84 2.44 1.33 3.48 1.94 Girls 2.16 0.85 3.48 1.39 2.88 1.56

Note. n = 25 for each entry.




The students were also classified as high operational or low operational on the basis of performance on the FORT. To be classified as high operational, the students had to answer correctly at least 60% of the items on two (or more) of the FORT subtests. The students whose scores did not meet this criterion were classified as low operational.

Design A 3 x 3 x 2 (Grade Level x Cognitive Style x Operativity) analysis of

covariance (ANCOVA), with IQ as the covariate, was performed on the students' standard scores on the total mathematics test. In addition, a 3 x 3 x 2 (Grade Level x Cognitive Style x Operativity) multivariate analysis of covariance (MANCOVA), with IQ as the covariate, was performed on the students' standard scores on the three mathematics achievement tests: computation, concepts, and problem solving.

RESULTS

The ANCOVA of standard scores on the total mathematics test indicated significant main effects for grade level, F(2, 431) = 152.76, p < .001; cognitive style, F(2, 431) = 3.23, p < .05; and operativity, F(1, 431) =

15.11, p < .001. No significant interactions were found. The adjusted and unadjusted mean scores for these factors are presented in Table 2. Post hoc comparisons of the adjusted means (Tukey's HSD test) revealed that there was a significant (p < .01) increase in total mathematics achievement from grade to grade and that field-independent students scored significantly (p < .01) higher than field-dependent students. Moreover, an examination of the adjusted means indicated that high-operational students had significantly (p < .001) greater mean scores than low-operational students.

Table 2 Adjusted (and Unadjusted) Standard Score Means on the Total Mathematics Test

Grade 6 Grade 7 Grade 8 Cognitive Operativity

style group level n M n M n M

High 24 102.49 26 110.26 31 116.76 Field independent (109.33) (115.54) (122.29)

Low 26 96.04 24 107.61 19 112.54 (99.12) (109.63) (115.11)

High 18 98.02 19 111.33 20 112.90 Intermediate (99.83) (112.21) (113.40)

Low 32 96.74 31 105.73 30 113.70 (96.09) (104.74) (109.37)

High 15 99.38 14 105.28 18 113.89 Field dependent ( 98.67) (103.50) (112.56)

Low 35 97.57 36 105.46 32 109.75 (94.80) (100.67) (103.53)

The MANCOVA of standard scores on the mathematics achievement tests showed significant main effects for grade level, F(6, 858) = 49.01, p < .001;




cognitive style, F(6, 858) = 2.87, p < .01, and operativity, F(3, 429) = 4.09, p < .01. Again, no significant interactions were found. The adjusted and unadjusted mean scores for these conditions are presented in Table 3.

Table 3 Adjusted (and Unadjusted) Standard Score Means on the

Mathematics Achievement Tests

Grade level Cognitive Operativity style group level 6 7 8

Mathematics computation

High 95.13 103.17 111.76 Field independent (101.54) (108.12) (116.94)

Low 89.54 100.03 107.65 ( 92.42) (101.92) (110.05)

High 92.86 105.13 106.99 Intermediate ( 94.56) (105.95) (107.45)

Low 92.17 99.76 108.85 ( 91.56) ( 98.84) (104.80)

High 93.80 100.17 107.92 Field dependent ( 93.13) ( 98.50) (106.67)

Low 92.51 100.10 104.75 ( 89.91) ( 95.61) ( 98.94)

Mathematics concepts


Low 92.29 103.94 103.74 (95.50) (106.04) (106.42)


Low 90.36 101.10 105.49 (89.69) (100.07) (100.97)


Low 92.32 100.01 104.15 (89.43) ( 95.00) ( 97.66)

Mathematics problem solving


Low 92.06 104.15 108.01 (95.08) (106.13) (110.53)


Low 93.98 101.97 108.08 (93.34) (101.00) (103.83)


Low 94.26 101.60 102.41 (91.54) ( 96.89) ( 96.31)

Note. Cell frequencies are given in Table 2.

Univariate ANCOVAs showed quite similar results for the three dependent variables: computation, concepts, and problem solving. For the computation test scores, there were significant main effects for grade level, F(2, 431) = 144.18, p < .001, and operativity, F(1, 431) = 9.96, p < .01. For the




concepts test scores, all the main effects were significant; grande level, F(2, 431) = 103.02, p < .001; cognitive style F(2, 431) = 5.05, p < .01; and operativity, F(1, 431) = 8.53, p < .01. Lastly, for the problem-solving test scores, the main effects were significant for grade level, F(2, 431) = 79.49, p < .001; cognitive style, F(2, 431) = 4.96, p < .01; and operativity, F(1, 431) = 7.03, p < .01. Thus, the only divergent finding was the nonsignificant cognitive style effect for the computation test scores.

Pairwise comparisons of the adjusted means (Tukey's HSD test) indicated significant (p < .01) developmental gains across grade levels for the three mathematics achievement tests. Furthermore, field-independent students scored significantly (p < .01) higher than field-dependent students on both the concepts and problem-solving tests. Finally, the high-operational students outperformed their low-operational peers on all three measures of mathematics achievement.

DISCUSSION

As predicted, both cognitive style and the level of operational development had a significant effect on the mathematics achievement of sixth, seventh, and

eighth graders. In particular, the analytic abilities displayed by field-

independent students and the logical-thinking abilities manifested by high- operational students had a pronounced influence on their mathematics achievement. These findings confirm and extend the results of previous investigations of the impact of such factors on the mathematics achievement of students in the lower elementary school grades (Arlin, 1981; Buriel, 1978; Dimitrovsky & Almy, 1975; Kagan & Zahn, 1975; Kagan et al., 1977; Kaufman & Kaufman, 1972; Satterly, 1976; Stone, 1976; Vaidya &

Chansky, 1980). These results also have important educational implications. Specifically,

the significant influence of field dependence-independence on students' mathematics achievement at all three grade levels suggests the need for future

investigations that examine the feasibility of using instructional strategies and mathematics materials that are optimally suited to the cognitive styles of individual learners. Most importantly, further attention should be given to

examining variables that might impinge on the mathematics achievement of field-dependent students (e.g., problem structure, explicitness of instructions, task-related experience).

Likewise, these results indicate that the knowledge of a student's level of

operativity is an essential component of our understanding of the student's mathematics achievement. In light of such evidence, a critical problem con-

fronting educational psychologists and curriculum planners is to determine the cognitive demands of various school mathematics tasks so that they can be used with students at an appropriate level of operational development. Recent attempts to ascertain the constraints on mathematics achievement imposed by students' level of operativity have provided valuable information




about the cognitive capacities required to assimilate specific mathematics concepts successfully (Collis, 1971, 1973, 1976, 1978; Halford, 1978; Mal- pas & Brown, 1974).

In sum, the findings of the present study highlight the need for investigations of different approaches to instructional design that are geared to the developmental capacities and cognitive styles of the individual learner (cf. Case, 1975). A vital element that should be incorporated into any such approach is the analysis of school mathematics tasks from the learner's point of view.

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Cognitive Style, Operativity, and Mathematics Achievement