The Relationship of Visuospatial Skills and Verbal Ability to Learning Disabilities in Mathematics Terry M. McLeod, EdD, and W. Donald Crump, PhD
Learning disabilities In mathematics have received relatively little attention despite the social relevance of basic arithmetic ability. Early theories of mathematics disability viewed visuospatial and visuomotor skills as critical. This study examines verbal skills in addition and finds them to be equally important. - G.M.S. This study examines the relationship among selected language and visuospatial measures and mathematics achievement in a group of children identified as having learning disabilities in mathematics. With a modification of the Myklebust learning quotient formula, 43 children were identified as having a learning disability in mathematics in a group of 385 children enrolled in public school learning disabilities programs. These children were administered a variety of measures. Canonical correlation analysis revealed a general canonical factor permeating both the predictor and criterion variables. The study indicates that verbal ability plays a stronger role in learning disabilities in mathematics than previously hypothesized in the literature.
disabilities in mathematics have been related to deficiencies in visuospatial organization (Strauss & Lehtinen 1947, Johnson & Myklebust 1967, Kaliski 1962) and visuomotor association (Strauss 1951, Kaliski 1962). While research in mathematics has been prolific, much of it has been concerned with content, student attitudes toward content, and
earning frequently
instructional techniques (Bums 1965). Studies providing information about basic cognitive processes involved in attaining mathematical concepts can be considered as dealing with five factors (Chalfant & Scheffelin 1969): (1) general
intelligence, (2) spatial ability, (3) neurophysiological factors, (4) verbal ability, and (5) approach to problem solving. The present study
was
designed with
two
purposes in mind: (1) To determine if certain factors among visuospatial skills, verbal ability, and learning
disabilities in mathematics account for a signifiof the variance among the set of variables, and (2) to investigate the extent of the relationships among visuospatial skills and verbal ability, and mathematics achievement in content, operations, and applications. The nature of these relationships can be clarified by the use of canonical correlation and correlation analysis. Canonical correlation starts with two sets of variables, the predictor set or left-hand variables and the criterion set or the right-hand variables. In this study the predictor cant amount
53 Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
consisted of the visuospatial and verbal ability variables while the criterion set consisted of mathematical ability variables. The primary hypothesis deriving from the major purpose of the study states that there is one statistically significant factor among the predictor and criterion variables. A second hypothesis predicts a statistically significant correlation between each of the possible pairings of the predictor variables, while the third hypothesis predicts a significant correlation between each of the possible pairings of criterion variables. set
’
SUBJECTS population for this study was restricted to first, second, third, fourth, and fifth graders identified as learning disabled in mathematics. Children (385) identified as learning disabled in public school learning disability programs were screened to determine the presence of a learning disability in mathematics. A learning quotient formula somewhat similar to that of Myklebust (1968) was employed. The variables in the formula are mental age (MA), chronological age (CA), grade age (GA), derived by adding 5.2 to the subjects’ present grade placement, and achievement age (AA), also derived by adding 5.2 to the grade equivalent score on the KeyMath Diagnostic Arithmetic Test. The
The first step in the formula is to determine an expectancy age (EA), accomplished by the following formula:
chronological age for this group was 114.58 months (s 13.81) with a range of 81 to 138 months while the mean grade age for this group was 108.63 months (s 11.87) with a range of 81 =
=
to 129 months. The mean achievement age as measured by the KeyMath Diagnostic Arithmetic Test was 98.98 months (s = 11.04) with a range of 80 to 131 months.
INSTRUMENTATION AND DATA ANALYSIS The instruments were selected with the purposes of the study and the variables to be measured in mind. These measures (see Table I) were administered to the 43 subjects who met criteria outlined previously. The three measures of visuospatial skills and three measures of verbal ability made up the predictor set while the three measures of mathematics skills formed the criterion set. Raw scores from these measures were converted to z-scores to standardize the variability among measures before correlations were computed. A correlation matrix and canonical variates were computed, and canonical roots were then extracted. A chi-square was then calculated to determine the probability these roots occurred by chance. The .05 level of significance was employed for the interpretation of canonical roots and intercorrelations. The CORR07 program from the Behavioral Sciences Statistics
Program Library (Barker 1973)
used in data
was
analysis.
LIMITATIONS OF THE STUDY
The learning quotient (LQ) is the ratio between achievement age (AA) and expectancy age (EA): ---
A
learning quotient
of 95%
or
below
was
considered the basis for classifying a child as learning disabled in mathematics. By this criterion, only 43 subjects were identified from the initial group of 385 children. The mean
This study was limited to 43 urban, suburban, and rural black and white students. The findings of the study were limited also by the extent to which the various visuospatial, verbal ability, and mathematics measures reflect the actual capacities of the children. It was assumed that these measures accurately reflect visuospatial skills, verbal ability, and mathematical ability. In canonical correlation, the number of significant roots which can be extracted is limited by
54 Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
TABLE I. Variables and
measures
used.
less than the number of variables in either the predictor set or criterion set, whichever is smaller. The number of significant roots in this study could not exceed two (criterion set 3). Additionally, the results of the study are limited by the fact that the number of subjects participating in the study does not meet the criterion of ten subjects per variable as suggested by Nunnally (1967). Thus, caution should be taken when generalizing the results of this study. one
=
RESULTS The intercorrelations between the variables and canonical variates are shown in Table II. Moderate to high correlations were found between the first canonical variate and all variables except visual closure and verbal expression. This suggests that there was a general factor permeating the predictor and criterion sets. This factor principally was identified with mathematical application, mathematical
content, and mathematical
operations, followed by verbal meaning, auditory association, spatial relations, and finally, right-left discrimination. The factor accounted for 61% of the variance among the variables and was significant at the .0001 level. There were no other factors significant at the .05 level. The second hypothesis was tested using an intercorrelation matrix consisting of the visuospatial and verbal ability measures previously designated as predictor variables. The correlations between spatial relations and right-left discrimination, spatial relations and verbal meaning, right-left discrimination and verbal meaning, and auditory association and verbal meaning were significant at the .05 level. No other correlations were significant. The third hypothesis was tested using an intercorrelation matrix consisting of the mathematical ability previously designated as criterion variables. All of the correlations were significant at the .05 level. 55
Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
TABLE II. Correlation matrix between variables and canonical variates.
DISCUSSION Canonical correlation analysis is a technique for determining maximal relationships among groups of variables. A major value of this technique is in exploratory research which provides a basis for subsequent inquiry. This observation, along with the limitations previously discussed, suggests that the results of the present study should be viewed tentatively with directions for additional research being a primary contribution. In viewing the findings, the significant relationship found among mathematics achievement, verbal ability, and visuospatial skills is noteworthy. It suggests that previous statements attributing learning disabilities in mathematics to disturbances primarily in visuospatial skills (Strauss & Lehtinen 1947, Strauss 1951, Johnson & Myklebust 1967, Kaliski 1962) may be an oversimplification of the issue. Considering the correlations between the separate variables and the general canonical root, the variables showing the highest correlation with the canonical root were the mathematics achievement variables, followed by two verbal ability variables, and
then two visuospatial skill variables. Although the degree of difference was small, this difference indicated that verbal ability variables were related more closely to mathematics achievement than visuospatial skills in the sample of youngsters with learning disabilities in mathematics. Since the substance of mathematics is symbols and language, verbal ability is integral to many facets of mathematics including concepts of quantity and classification, initial topics in many primary school mathematics programs. Hence, much of the foundation for mathematics achievement involves language concepts. In considering the findings regarding visuospatial skills, spatial relations and orientation were highly to moderately related to the canonical root. These results seem to support previous research regarding the relationship of visuospatial skills and mathematics achievement among school-aged children (Michael, Guilford, Fruchter, & Zimmerman 1957). These researchers generated three groups of spatial factors, spatial relations, orientation, and visualization. In the present study, visualization was not related
56 Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
closely to the general canonical root. Perhaps the
employed does not adequately assess the skills involved or does not emphasize convergence on a single correct response as do other measures which relate significantly to the measure
canonical
research could lead to improvement of mathematics instruction for children with learning disabilities. An unexpected, somewhat surprising finding
root.
Further research is needed to determine the nature of the relationship of both verbal ability and visuospatial skills to mathematics achievement. While the present study employed a
sample of children with learning disabilities in mathematics, subsequent studies might investigate the relationship of visuospatial skills and verbal ability to mathematics achievement among children with normal and high achievement in mathematics. Differing results might help clarify the assertion of deficits in visuospatial skills among youngsters with learning disabilities in mathematics. Such studies might eventually lead
addition, research should explore the relationship of visuospatial skills, verbal ability, and mathematics achievement at differing age levels. Such
to
improving mathematics
only 11% of the children enrolled in disabilities learning programs exhibited a learnwas
that
ing disability
quotient
in mathematics when
formula
was
a
learning
employed.
ABOUT THE AUTHORS
Terry McLeod received a doctorate from the University of Alabama. He is an assistant professor of special education at North Georgia College where he teaches courses in learning disabilities and the exceptional child. W. Donald Crump, trained at George Peabody College, is associate professor and chairperson of the Learning Disabilities Program at the University of Alabama. He was the recipient of the President’s Award for Outstanding Service from the Alabama ACLD in 1973 and 1978. Requests for reprints should be sent to Dr. Crump at the College of Education, University of Alabama, University, Ala. 35486.
instruction for these children.
addition, research investigating the effects of chronological age on the relationship of visuospatial skills and verbal ability to mathematics achievement is needed. Perhaps the relationship In
among these skills and mathematics achievement
may vary at different age levels, suggesting implications for remedial mathematics instruction at
varying grade levels.
Research needs to determine the reason for high correlation among the mathematical ability variables. The explanation could involve the fact that children with learning disabilities in mathematics are affected equally in the domains of mathematical content, operations, and applications. Another explanation could be that measures used to assess mathematical ability do not discriminate accurately among the different domains. In summary, a major implication from the present study is that mathematics achievement among children with learning disabilities appears to be related to verbal ability as well as to visuospatial skills. Research needs to explore the relationship of these skills among children with normal and high achievement in mathematics. In
the
ACKNOWLEDGEMENT This research was supported in part by Grant No. G00750355 f rom the Research Projects Branch, Division of Innovation and Development, Bureau of Education for the Handicapped, U.S. Office of Education. REFERENCES
Barker, H.: Behavioral Sciences Statistics Program Library. University, Ala.: University of Alabama, 1973. Burns, P.: Arithmetic research that has made a difference. Education Digest, 1965, 31, 16-18. Chalfant, J., and Scheffelin, M.: Central Processing Dysfunction in Children: A Review of Research. NINDS Monograph No. 9. Bethesda, Md.: U.S. Department of Health, Education, and Welfare, 1969. Johnson, D., and Myklebust, H.: Learning Disabilities: Educational Principles and Practices. New York: Grune and Stratton, 1967. Kaliski, L. : Arithmetic and the brain-injured child. Arithmetic Teacher, 1962, 9, 245-251. Michael, W., Guilford, J., Fruchter, B., and Zimmerman, W.: The description of spatial-visualization abilities. Educational and Psychological Measurement, 1957, 17, 185-199. Myklebust, H.: Learning disabilities: Definition and overview. In H.R. Myklebust (Ed.): Progress in Learning Disabilities, Vol. 1. New York: Grune and Stratton, 1968. Nunnally, J.: Psychometric Theory. New York: McGrawHill, 1967. Strauss, A.: The education of the brain injured child. American Journal of Mental Deficiency, 56, 1951, 712-718. Strauss, A., and Lehtinen, L.: Psyehopatholngy and education of the brain injured child. New York: Grune and Stratton, 1947. 57
Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
Learning disabilities In mathematics have received relatively little attention despite the social relevance of basic arithmetic ability. Early theories of mathematics disability viewed visuospatial and visuomotor skills as critical. This study examines verbal skills in addition and finds them to be equally important. - G.M.S. This study examines the relationship among selected language and visuospatial measures and mathematics achievement in a group of children identified as having learning disabilities in mathematics. With a modification of the Myklebust learning quotient formula, 43 children were identified as having a learning disability in mathematics in a group of 385 children enrolled in public school learning disabilities programs. These children were administered a variety of measures. Canonical correlation analysis revealed a general canonical factor permeating both the predictor and criterion variables. The study indicates that verbal ability plays a stronger role in learning disabilities in mathematics than previously hypothesized in the literature.
disabilities in mathematics have been related to deficiencies in visuospatial organization (Strauss & Lehtinen 1947, Johnson & Myklebust 1967, Kaliski 1962) and visuomotor association (Strauss 1951, Kaliski 1962). While research in mathematics has been prolific, much of it has been concerned with content, student attitudes toward content, and
earning frequently
instructional techniques (Bums 1965). Studies providing information about basic cognitive processes involved in attaining mathematical concepts can be considered as dealing with five factors (Chalfant & Scheffelin 1969): (1) general
intelligence, (2) spatial ability, (3) neurophysiological factors, (4) verbal ability, and (5) approach to problem solving. The present study
was
designed with
two
purposes in mind: (1) To determine if certain factors among visuospatial skills, verbal ability, and learning
disabilities in mathematics account for a signifiof the variance among the set of variables, and (2) to investigate the extent of the relationships among visuospatial skills and verbal ability, and mathematics achievement in content, operations, and applications. The nature of these relationships can be clarified by the use of canonical correlation and correlation analysis. Canonical correlation starts with two sets of variables, the predictor set or left-hand variables and the criterion set or the right-hand variables. In this study the predictor cant amount
53 Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
consisted of the visuospatial and verbal ability variables while the criterion set consisted of mathematical ability variables. The primary hypothesis deriving from the major purpose of the study states that there is one statistically significant factor among the predictor and criterion variables. A second hypothesis predicts a statistically significant correlation between each of the possible pairings of the predictor variables, while the third hypothesis predicts a significant correlation between each of the possible pairings of criterion variables. set
’
SUBJECTS population for this study was restricted to first, second, third, fourth, and fifth graders identified as learning disabled in mathematics. Children (385) identified as learning disabled in public school learning disability programs were screened to determine the presence of a learning disability in mathematics. A learning quotient formula somewhat similar to that of Myklebust (1968) was employed. The variables in the formula are mental age (MA), chronological age (CA), grade age (GA), derived by adding 5.2 to the subjects’ present grade placement, and achievement age (AA), also derived by adding 5.2 to the grade equivalent score on the KeyMath Diagnostic Arithmetic Test. The
The first step in the formula is to determine an expectancy age (EA), accomplished by the following formula:
chronological age for this group was 114.58 months (s 13.81) with a range of 81 to 138 months while the mean grade age for this group was 108.63 months (s 11.87) with a range of 81 =
=
to 129 months. The mean achievement age as measured by the KeyMath Diagnostic Arithmetic Test was 98.98 months (s = 11.04) with a range of 80 to 131 months.
INSTRUMENTATION AND DATA ANALYSIS The instruments were selected with the purposes of the study and the variables to be measured in mind. These measures (see Table I) were administered to the 43 subjects who met criteria outlined previously. The three measures of visuospatial skills and three measures of verbal ability made up the predictor set while the three measures of mathematics skills formed the criterion set. Raw scores from these measures were converted to z-scores to standardize the variability among measures before correlations were computed. A correlation matrix and canonical variates were computed, and canonical roots were then extracted. A chi-square was then calculated to determine the probability these roots occurred by chance. The .05 level of significance was employed for the interpretation of canonical roots and intercorrelations. The CORR07 program from the Behavioral Sciences Statistics
Program Library (Barker 1973)
used in data
was
analysis.
LIMITATIONS OF THE STUDY
The learning quotient (LQ) is the ratio between achievement age (AA) and expectancy age (EA): ---
A
learning quotient
of 95%
or
below
was
considered the basis for classifying a child as learning disabled in mathematics. By this criterion, only 43 subjects were identified from the initial group of 385 children. The mean
This study was limited to 43 urban, suburban, and rural black and white students. The findings of the study were limited also by the extent to which the various visuospatial, verbal ability, and mathematics measures reflect the actual capacities of the children. It was assumed that these measures accurately reflect visuospatial skills, verbal ability, and mathematical ability. In canonical correlation, the number of significant roots which can be extracted is limited by
54 Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
TABLE I. Variables and
measures
used.
less than the number of variables in either the predictor set or criterion set, whichever is smaller. The number of significant roots in this study could not exceed two (criterion set 3). Additionally, the results of the study are limited by the fact that the number of subjects participating in the study does not meet the criterion of ten subjects per variable as suggested by Nunnally (1967). Thus, caution should be taken when generalizing the results of this study. one
=
RESULTS The intercorrelations between the variables and canonical variates are shown in Table II. Moderate to high correlations were found between the first canonical variate and all variables except visual closure and verbal expression. This suggests that there was a general factor permeating the predictor and criterion sets. This factor principally was identified with mathematical application, mathematical
content, and mathematical
operations, followed by verbal meaning, auditory association, spatial relations, and finally, right-left discrimination. The factor accounted for 61% of the variance among the variables and was significant at the .0001 level. There were no other factors significant at the .05 level. The second hypothesis was tested using an intercorrelation matrix consisting of the visuospatial and verbal ability measures previously designated as predictor variables. The correlations between spatial relations and right-left discrimination, spatial relations and verbal meaning, right-left discrimination and verbal meaning, and auditory association and verbal meaning were significant at the .05 level. No other correlations were significant. The third hypothesis was tested using an intercorrelation matrix consisting of the mathematical ability previously designated as criterion variables. All of the correlations were significant at the .05 level. 55
Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
TABLE II. Correlation matrix between variables and canonical variates.
DISCUSSION Canonical correlation analysis is a technique for determining maximal relationships among groups of variables. A major value of this technique is in exploratory research which provides a basis for subsequent inquiry. This observation, along with the limitations previously discussed, suggests that the results of the present study should be viewed tentatively with directions for additional research being a primary contribution. In viewing the findings, the significant relationship found among mathematics achievement, verbal ability, and visuospatial skills is noteworthy. It suggests that previous statements attributing learning disabilities in mathematics to disturbances primarily in visuospatial skills (Strauss & Lehtinen 1947, Strauss 1951, Johnson & Myklebust 1967, Kaliski 1962) may be an oversimplification of the issue. Considering the correlations between the separate variables and the general canonical root, the variables showing the highest correlation with the canonical root were the mathematics achievement variables, followed by two verbal ability variables, and
then two visuospatial skill variables. Although the degree of difference was small, this difference indicated that verbal ability variables were related more closely to mathematics achievement than visuospatial skills in the sample of youngsters with learning disabilities in mathematics. Since the substance of mathematics is symbols and language, verbal ability is integral to many facets of mathematics including concepts of quantity and classification, initial topics in many primary school mathematics programs. Hence, much of the foundation for mathematics achievement involves language concepts. In considering the findings regarding visuospatial skills, spatial relations and orientation were highly to moderately related to the canonical root. These results seem to support previous research regarding the relationship of visuospatial skills and mathematics achievement among school-aged children (Michael, Guilford, Fruchter, & Zimmerman 1957). These researchers generated three groups of spatial factors, spatial relations, orientation, and visualization. In the present study, visualization was not related
56 Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
closely to the general canonical root. Perhaps the
employed does not adequately assess the skills involved or does not emphasize convergence on a single correct response as do other measures which relate significantly to the measure
canonical
research could lead to improvement of mathematics instruction for children with learning disabilities. An unexpected, somewhat surprising finding
root.
Further research is needed to determine the nature of the relationship of both verbal ability and visuospatial skills to mathematics achievement. While the present study employed a
sample of children with learning disabilities in mathematics, subsequent studies might investigate the relationship of visuospatial skills and verbal ability to mathematics achievement among children with normal and high achievement in mathematics. Differing results might help clarify the assertion of deficits in visuospatial skills among youngsters with learning disabilities in mathematics. Such studies might eventually lead
addition, research should explore the relationship of visuospatial skills, verbal ability, and mathematics achievement at differing age levels. Such
to
improving mathematics
only 11% of the children enrolled in disabilities learning programs exhibited a learnwas
that
ing disability
quotient
in mathematics when
formula
was
a
learning
employed.
ABOUT THE AUTHORS
Terry McLeod received a doctorate from the University of Alabama. He is an assistant professor of special education at North Georgia College where he teaches courses in learning disabilities and the exceptional child. W. Donald Crump, trained at George Peabody College, is associate professor and chairperson of the Learning Disabilities Program at the University of Alabama. He was the recipient of the President’s Award for Outstanding Service from the Alabama ACLD in 1973 and 1978. Requests for reprints should be sent to Dr. Crump at the College of Education, University of Alabama, University, Ala. 35486.
instruction for these children.
addition, research investigating the effects of chronological age on the relationship of visuospatial skills and verbal ability to mathematics achievement is needed. Perhaps the relationship In
among these skills and mathematics achievement
may vary at different age levels, suggesting implications for remedial mathematics instruction at
varying grade levels.
Research needs to determine the reason for high correlation among the mathematical ability variables. The explanation could involve the fact that children with learning disabilities in mathematics are affected equally in the domains of mathematical content, operations, and applications. Another explanation could be that measures used to assess mathematical ability do not discriminate accurately among the different domains. In summary, a major implication from the present study is that mathematics achievement among children with learning disabilities appears to be related to verbal ability as well as to visuospatial skills. Research needs to explore the relationship of these skills among children with normal and high achievement in mathematics. In
the
ACKNOWLEDGEMENT This research was supported in part by Grant No. G00750355 f rom the Research Projects Branch, Division of Innovation and Development, Bureau of Education for the Handicapped, U.S. Office of Education. REFERENCES
Barker, H.: Behavioral Sciences Statistics Program Library. University, Ala.: University of Alabama, 1973. Burns, P.: Arithmetic research that has made a difference. Education Digest, 1965, 31, 16-18. Chalfant, J., and Scheffelin, M.: Central Processing Dysfunction in Children: A Review of Research. NINDS Monograph No. 9. Bethesda, Md.: U.S. Department of Health, Education, and Welfare, 1969. Johnson, D., and Myklebust, H.: Learning Disabilities: Educational Principles and Practices. New York: Grune and Stratton, 1967. Kaliski, L. : Arithmetic and the brain-injured child. Arithmetic Teacher, 1962, 9, 245-251. Michael, W., Guilford, J., Fruchter, B., and Zimmerman, W.: The description of spatial-visualization abilities. Educational and Psychological Measurement, 1957, 17, 185-199. Myklebust, H.: Learning disabilities: Definition and overview. In H.R. Myklebust (Ed.): Progress in Learning Disabilities, Vol. 1. New York: Grune and Stratton, 1968. Nunnally, J.: Psychometric Theory. New York: McGrawHill, 1967. Strauss, A.: The education of the brain injured child. American Journal of Mental Deficiency, 56, 1951, 712-718. Strauss, A., and Lehtinen, L.: Psyehopatholngy and education of the brain injured child. New York: Grune and Stratton, 1947. 57
Downloaded from ldx.sagepub.com at CARLETON UNIV on May 9, 2015
