VISUAL-SPATIAL PROCESSING AND MATHEMATICS ACHIEVEMENT:
THE PREDICTIVE ABILITY OF THE VISUAL-SPATIAL MEASURES OF THE
STANFORD-BINET INTELLIGENCE SCALES, FIFTH EDITION AND THE
WECHSLER INTELLIGENCE SCALE FOR CHILDREN- FOURTH EDITION
By
Eldon Clifford
B.S.Ed. Black Hills State University, 1997
M.S. South Dakota State University, 2000
A Dissertation Submitted in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
Division of Counseling and Psychology in Education
School Psychology Program
In the Graduate School
The University of South Dakota
December 13, 2008
UMI Number: 3351188
Copyright 2008 by
Clifford, Eldon
All rights reserved.
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ELDON CLIFFORD
2008
All Rights Reserved
Members of the Committee appointed to examine
the dissertation of Eldon Clifford find it
satisfactory and recommend that it be accepted.
JL
ale Pietrzak, Ed.D.
Committee Chair
Bruce Proctor, Ph.D.
Co-Committee Chair
rbara Yutrzenka, Ph.D.
m
Eldon Clifford (Ph. D., The University of South Dakota, 2008)
Dissertation Directed By Dr. Dale Pietrzak
Visual-Spatial Processing and Mathematics Achievement:
The Predictive Ability of the Visual-Spatial Measures of Stanford-Binet Intelligence
Scales, Fifth Edition and the Wechsler Intelligence Scale for Children- Fourth Edition
In the law and the literature there has been a disconnect between the definition of a
learning disability and how it is operationalized. For the past 30 years, the primary
method of learning disability identification has been a severe discrepancy between an
individual's cognitive ability level and his/her academic achievement. The recent 2004
IDEA amendments have included language that allows for changes in identification
procedures. This language suggests a specific learning disability may be identified by a
student's failure to respond to a research based intervention (RTI). However, both
identification methods fail to identify a learning disability based on the IDEA 2004
definition, which defines a specific learning disability primarily as a disorder in
psychological processing. Research suggests that processing components play a critical
role in academic tasks such as reading, writing and mathematics. Furthermore, there has
been considerable research that suggests visual-spatial processing is related to
mathematics achievement. The two most well known IQ tests, the Stanford-Binet-Fifth
Edition (SB5) and the Wechsler Intelligence Scale for Children-Fourth Edition (WISCIV), were revised in 2003 to align more closely with the most current theory of
intelligence, the Cattell-Horn-Carroll theory of cognitive abilities (CHC). Research
supports both instruments have subtests that measure visual-spatial processing. The
purpose of the current study is to identify which visual-spatial processing measure (SB5
or WISC-IV) is the better predictor of poor mathematics achievement. The participants
were 112 6th- 8th grade middle school students. Of the 112 original participants, 109 were
included in the study. The comparison of the results of two separate sequential logistic
regressions found that both measures could significantly predict mathematics
achievement. However, given the relatively small amount of variance accounted for by
both the SB5 and WISC-IV visual-spatial processing measures, the results had
questionable practical significance.
This abstract of approximately 291 words is approved as to form and content and I
approve its publication.
B£u5atesPietrzak, Dissertation Committee Chair
IV
Acknowledgements
I would like to thank the members of my dissertation committee Dr. Dale
Pietrzak, Dr. Bruce Proctor, Dr. Amy Schweinle and Dr. Barbara Yutrzenka for their time
in this endeavor. I would specifically like to thank the committee chair Dr. Pietrzak for
his guidance and stepping in to take on that role when my previous chair left the
university. In addition, I would like to extend my appreciation to Dr. Schweinle for her
statistical expertise and taking the time to read a number of drafts and offer feedback
when she was under no obligation to do so. I would also like to express gratitude to
former University of South Dakota School Psychology professor Dr. Jordan Mulder for
helping me with the conceptualization of my dissertation and his direction and
constructive comments during the proposal stage. Finally, I would like to thank the
School Psychology Department at the University of South Dakota for providing me with
a career that has afforded me much, personally and professionally.
I would like to express my appreciation for my sister Dr. Jessteene Clifford-Kelly.
I am grateful to her for taking the time to read a number of early drafts and providing me
feedback. In addition, I would like to thank her for her encouragement and her
commiserating ear as she similarly went through this sometimes convoluted graduate
education process. I would like to thank my parents Dewayne and Kathy Clifford for
their gentle yet persistent encouragement. Without the strong foundation they built, I
would have not been able to complete this undertaking. I would also like to Ms. Jami
Johnson for her feedback on a number of drafts as well as her encouragement and
support.
Table of Contents
1.
2.
3.
4.
5.
6.
7.
Title Page
Copy Right Page
Signature Page
Abstract
Acknowledgments
_
List of Tables and Figures
_
Chapter 1/Introduction..__
a. Introduction
b. Significance of the Study.
c. Statement of the Problem
d. Definition of Terms
_
e. Limitations
_
f. Structure of the Proceeding Chapters
_.__
8. Chapter 2/ Literature Review
_
a. Literature Review
b. Learning Disabilities.....
i. Learning Disabilities Defined: Past and Present
ii. Learning Disabilities Classification and
Identification
iii. Models of Identification: IQ-Achievement
Discrepancy and Response to Intervention
iv. Summary
c. Psychological Processing and Learning Disabilities
i. Reading
ii. Writing
iii. Mathematics...
iv. Summary.
d. Mathematical Disabilities
i. Mathematical Disabilities: Definition and
Identification
ii. Specific Mathematical tasks and their
Cognitive Processes
iii. Subtypes of Mathematical Disabilities
iv. Summary
e. Visual Spatial Processing and Mathematics
i. Visual-Spatial Processing's relationship to
Mathematics
ii. Visual-Spatial Processing
iii. Summary.
f. Modern Intelligence Theory and Assessing VisualSpatial Processing
_____
i.
CHC Theory
vi
p. iii
p. vi
..p. v
...p. viii
p. 1
p. 1
p. 17
p. 19
p. 19
.....p. 20
p. 21
p. 22
p. 22
p. 22
p. 22
p. 26
p. 28
p. 31
p. 32
p. 32
p. 35
.p. 42
p. 45
p. 46
p. 47
p. 49
p. 61
p. 62
p. 63
p. 64
p. 67
p. 73
p. 73
p. 76
ii.
The Stanford-Binet Intelligence Test,
Fifth Edition
iii. The Wechsler Intelligence Scale for ChildrenFourth Edition
_____
_
iv.
Summary
g. Summary
_
__
p. 82
_p. 87
_p. 92
__p. 93
3. Chapter 3/ Methodology.
__p. 96
a. Methodology.
_
_
p. 96
b. Participants
____
p. 97
c. Instruments
_
p. 100
i. Intelligence Measure
_
p. 100
ii. Visual-Spatial Measures
...p. 101
iii. Measure of Mathematics Achievement
p. I l l
d. Procedures..
.p. 117
e. Data Analysis..
p. 119
f. Summary
__
____
____
_._p. 122
4. Chapter 4/ Results
...p. 123
a. Preliminary Analysis
p. 123
b. Correlation Analysis
p. 125
c. Multiple Regression Analysis
p. 126
d. Logistic Regression Analysis
___p. 129
i. SB5 Visual-Spatial Processing
__...p. 131
ii. WISC-IV Visual Spatial Processing
p. 132
e. Comparison of the SB5 and WISC-IV
p. 134
f. Summary. _
p. 136
5. Chapter 5/ Discussion
__
p. 138
a. Visual-Spatial Processing's Relationship to Mathematics p. 138
b. Predictive Ability of the SB5 and WISC-IV
p. 140
c. Comparison of the SB5 and WISC-IV.
p. 143
d. Further Implications
_
p. 146
e. Limitations.
____
___
p. 149
f. Future Research
p. 150
g. Importance of the Study
p. 151
6. Appendices
_
p. 152
a. Institutional Review Board Approval
p. 152
b. Approval Letters From Participating Schools
_
p. 154
c. Demographic Form
p. 165
6. References ___
p. 166
vn
List of Tables and Figures
1. Chapter 1
a. Tables:
i. Table 1.1: The 10 Cattell-Horn-Carroll(CHC) Broad Factors of
Intelligence and their Abbreviations
p. 8
ii. Table 1.2: The 12 CHC Visual Processing (GV) Narrow
Cognitive Abilities and their Abbreviations
p. 9
iii. Table 1.3: The Visual-Spatial Process Measures of the
WISC-IV
p. 16
b. Figures:
i. Figure 1.1: The Structure of the SB5
_ p. 11
ii. Figure 1.2: The Visual-Spatial Processing Measures of the
SB5
p. 14
iii. Figure 1.3: The Structure of the WISC-IV
p. 15
2. Chapter 2
a. Tables:
i. Table 2.1: Tasks Used to Measure Visual-Spatial
Processing in Current Literature
ii. Table 2.2: Subtests and Domain Construction
of the SB5 Full Scale IQ
iii. Table 2.3: Index and Subtests of the WISC-IV
that Combine to Form the Full Scale IQ
b. Figures:
i. Figure 2.1: CHC Broad and Narrow Cognitive
Abilities
__
_
3. Chapter 3
a. Tables:
i. Table 3.1: Participants' grade levels
ii. Table 3.2: Demographics
_
iii. Table 3.3: Language Spoken at Home
iv. Table 3.4: Level of Parental Education
__
4. Chapter 4
a. Tables:
i. Table 4.1: Descriptive Statistics
_
ii. Table 4.2: Correlation Analysis
iii. Table 4.3: R2 Change and Change Statistics
iv. Table 4.4: Coefficients and Significance Tests
for the Reduced and Full Model
v. Table 4.5: SB5 Model Statistics
vi. Table 4.6: SB5 Model Parameters
__
vii. Table 4.7: WISC-IV Model Statistics _ _
viii. Table 4.8: WISC-IV Model Parameters
vni
p . 74
.p. 82
p. 88
p. 75
_
_____
__
p. 97
p. 98
p. 98
p. 98
p. 125
__p. 126
p. 129
p.
__ p.
p.
p.
p.
132
134
134
136
136
ix. Table 4.9: SB5 & WISC-IV Comparison of Models p. 139
x. Table 4.10: Model Parameters of Both the SB5
and WISC-IV
p. 139
ix
CHAPTER I
INTRODUCTION
The definition of a specific learning disability (SLD) has changed little from
Samuel Kirk's conceptualization in 1962-1963. Kirk defined a SLD as an
underdeveloped processing disorder in the areas of speech, language, reading, spelling,
writing or mathematics (Hammill, 1990; Kirk & Kirk, 1983). Public Law 94-142,
adopted in 1975, also maintained that a SLD was based on a disorder in psychological
processing. Similarly, the subsequent revisions of the Individuals with Disabilities
Education Act in 1990 and 1997 defined a SLD as a disorder in one or more basic
psychological processes (Jacob & Hartshorne, 2003; Reschly, Hosp & Schmied, 2003).
The current Individuals with Disabilities Education Improvement Act (2004) continued
this trend:
(i) General. Specific learning disability means a disorder in one or more of
the basic psychological processes involved in understanding or in using
language spoken or written, that may manifest itself in the imperfect
ability to listen, think, speak, read, write, spell or to do mathematical
calculations including conditions such as perceptual disabilities, brain
injury, minimal brain dysfunction, dyslexia and developmental aphasia.
(U.S. Department of Education, 2006a, p. 46757)
In a previous review of all 50 state department of education rules, over 80% of
states have adopted this definition of a SLD (Reschly, Hosp & Schmied, 2003). In
addition, 96% of state education departments believe that a SLD is a processing disorder
(Reschly, et al. 2003). Furthermore, a recent unpublished review of how states currently
define a SLD, found that 49 of the 51 states (including the District of Columbia) use the
1
federal definition of a SLD or use the term "processing disorder" in their definition
(Clifford, 2008). However, the main disagreement in special education is not in the
definition, but in the identification of a SLD (Kavale, Holdnack & Mostert 2005).
There is a disparity in the law and the literature between the definition of a
learning disability and how it is operationalized. For the past 30 years, the primary
method of SLD identification has been a severe discrepancy between an individual's
ability level and their achievement (Hallahan & Mercer, 2002; Jacob & Hartshorne,
2003). However, the recent 2004 IDEA amendments have included language that allows
for changes in identification procedures to a procedure based on a student's failure to
respond to an intervention (RTI). In addition, a recent unpublished review of state special
education rules (adopted or in the processes of adoption) found that states are moving
away from using a discrepancy only identification procedure for a SLD (Clifford, 2008).
According to IDEA 2004 the identification of a SLD:
Must not require the use of a severe discrepancy between intellectual
ability and achievement for determining whether a child has a specific
learning disability, as defined in 34 C.F.R. 300.8(c)(10);
Must permit the use of a process based on the child's response to
scientific, research-based intervention; and
May permit the use of other alternative research-based procedures for
determining whether a child has a specific learning disability, as defined in
34 C.F.R. 300.8(c)(10) (U.S. Department of Education, 2006b).
Interestingly, both methods (discrepancy and response to an intervention) of
learning disability identification fail to address the definition, which states a SLD is "a
2
disorder in one or more of the basic psychological processes..." (U.S. Department of
Education, 2006a, p. 46757). If the definition of a SLD is based on the assumption it is a
psychological processing disorder, then it is appropriate that the identification of a SLD
include elements of a psychological processing disorder evaluation (Torgesen, 2002).
Understanding this idea requires a clear conceptualization of what is meant by
psychological processing.
Psychological processes are the cognitive abilities that allow the use of language,
attention, memory, complex problem solving, higher order thinking and perception in
academic and non-academic tasks (Gerring, & Zimbardo, 2002). The literature suggests
there are specific processing components in the three major academic tasks of reading,
writing and mathematics. Research maintains reading requires the psychological
processes of phonological processing, syntactic processing, working memory, semantic
processing, and orthographic processing (Badian, 2001; Gray & McCutchen, 2006;
Holsgrove & Garton, 2006; Hoskyn & Swanson, 2000; Nation & Snowing, 1998; Siegel,
2003). The literature supports that writing involves phonological processing,
orthographic processing, working memory, long-term memory, short-term memory, and
morphological processing (Berninger, Abbot, Thomson, & Raskind, 2001; Hauerwas &
Walker, 2003; Kellogg, 2001b, Swanson & Berninger, 1996). Studies have found
mathematical thinking incorporates working memory, phonological processing, attention,
long-term memory, and the PASS (planning; attention; successive; simultaneous)
cognitive processes (Fuchs et al, 2005; Fuchs et al., 2006; Kroesberger, Van Luit and
Naglieri, 2003; Swanson, 2004; Swanson & Beebe-Frankenberger, 2004; Swanson &
3
Jerman, 2006). Recent literature suggests of the three academic areas, mathematics has
the most need for additional research (Swanson & Jerman, 2006).
Failing to gain proficiency in mathematics while in elementary and middle school
will negatively influence a student's future, both academically and occupationally (Assel,
Landry, Swank, Smith & Steelman, 2003; Griffin, 2003). It is estimated that 4-8% of
public school students have a disability in the area of mathematics (Fleischner, &
Manhemier, 1997; Fuchs et al., 2005; Fuchs & Fuchs, 2003; Geary 2004; Geary & Hoard
2003; Swanson & Jerman, 2006). According to IDEA (2004) students who have a
mathematics disability (MD) have a psychological processing disorder in utilizing written
or spoken language that has resulted in a less than adequate ability to do mathematical
calculations (U.S. Department of Education, 2006a). Recent literature suggests that
understanding the cognitive aspects of mathematical thinking may increase the ability of
professionals to identify and treat students that struggle with mathematics (Fuchs et. al.,
2006). Furthermore, the literature supports there are specific psychological processes in
the areas of mathematical calculation, mathematical fluency, and mathematical word
problems.
Research suggests that attention, working memory, short-term memory, long-term
(semantic) memory, and phonological processing are involved in mathematical
calculation and fluency tasks (Floyd, Evans, McGrew, 2003; Fuchs et al., 2006; Fuchs et
al., 2005: Swanson, 2006; Swanson & Beebe-Frankenberger, 2004). Additionally, studies
have shown that mathematical word problems require the psychological processes of
attention, working memory, short-term memory, and phonological processing (Fuchs et
al., 2005; Swanson, 2006; Swanson & Beebe-Frankenberger, 2004; Swanson, Jerman, &
4
Zheng, 2008). The literature also supports, through understanding the processing
components of mathematical thinking, subtypes of MD can be identified (Cornoldi,
Venneri, Marconato, Molin & Montinari, 2003; Geary, 2004; Jordan 1995). David Geary
has contributed much to this area of research. Swanson & Jerman (2006) stated,
"Although not a quantitative analysis, one of the most comprehensive syntheses of the
cognitive literature on MD was conducted by Geary" (p. 249). Geary (1993; 1996; 2004)
suggests there are three separate subtypes of MD: 1) Procedural; 2) Semantic; 3) Visualspatial. Additional literature has also supported a visual-spatial processing deficit as a
subtype of MD (Jordan, 1995; Cornoldi et al., 2003; Swanson & Jerman, 2006).
Several studies suggest visual-spatial processing is indeed related to mathematical
thinking (Ansari et al, 2003; Assel et al., 2003; Busse, Berninger, Smith & Hildebrand,
2001; Cornoldi et al., 2003; Geary, 1993; Geary & Hoard, 2003; Hartje, 1987; Mazzocco,
2005; Reuhkala, 2001). A student who has a visual-processing disorder will have
difficulty conceptualizing mathematical problems that are spatially based (Geary, 2004).
Visual-spatial processing is involved in the mathematical skills of cardinality, estimation,
solving word problems and number alignment (Assel, et al, 2003; Augustyniak, Murphy,
& Phillip, 2005; Jordan, et al, 2003). Other, studies have also shown a relationship
between MD and deficits in visual-spatial processing (Busse et al., 2003; Harnadeck &
Rourke, 1994; McGlaughlin et al., 2005; Reuhkala, 2001). A recent meta-analysis of MD
research has confirmed this relationship (Swanson & Jerman, 2006). Fully understanding
this relationship requires an understanding of visual-spatial processing.
Visual-spatial processing is defined as "The ability to generate, retain, retrieve
and transform well-structured visual images" (Lohman, 1994, p. 1000). Perhaps, the most
5
comprehensive view of where visual-spatial material is processed may come from the
work of Alan Baddeley (Fisk & Sharp, 2003; Geary, 2004; Pickering & Gathercole,
2004; Reuhkala, 2001; Sholl & Fraone, 2004; Swanson, 2004; Swanson & BeebeFrankenberger, 2004). Visual-spatial processing is one aspect of working memory (WM).
WM is the ability to take-in information and mentally manipulate that information while
simultaneously retaining it (Geary, 2004). Baddeley's (1996) theory separates WM into
four parts: 1) Central executive; 2) Episodic buffer; 3) Phonological loop; 4) Visualspatial sketchpad. The central executive is viewed as the controller for the remaining
three elements (Baddeley, 1996; Pickering & Gathercole, 2004). The episodic buffer is
responsible for integrating WM and long-term memory (Pickering & Gathercole, 2004).
The phonological loop is the part of WM that holds information of a verbal nature
(Baddeley, 1996). The visual-spatial sketchpad is utilized in such tasks as anticipating
spatial transformations, mental rearrangement of items and visualizing the relationship of
parts to a whole (Sholl & Fraone, 2004). The visual-spatial sketchpad processes visualspatial information (Reuhkala, 2001; Pickering & Gathercole, 2004).
The visual-spatial sketchpad is responsible for processing information that is both
visual and spatial in nature (Pickering & Gathercole, 2004). The visual-spatial sketchpad
is of limited duration and serves as a storage and processing center (Baddeley, 1996).
Visual material and spatial material are processed separately; however, when visual and
spatial information is utilized it is done as a gestalt (Baddeley, 1996; Richardson &
Vecchi, 2002; Sholl & Fraone, 2004). Neuropsychologists believe the visual-spatial
material is mainly processed in the right hemisphere of the brain in the parietal cortex
(spatial) and the inferotemproal areas (visual) (Cornoldi, Venneri, Marconato, Molin &
6
Montinari 2003; Geary, 1993; Harnadeck & Rourke, 1994; Morris & Parslow, 2004;
Young & Ratcliff, 1983). Fully comprehending visual-spatial processing also requires an
understanding of how it is assessed.
McGrew (2005) posits tasks that are believed to measure visual-spatial possessing
involve figural or geometric structures that necessitate the visual perception and mental
manipulation of "visual shapes, forms, or images, and/or tasks that require or maintain
spatial orientation with regard to objects that may change or move through space"
(McGrew, 2005 p. 152). To understand how visual-spatial processing is assessed it is
important to conceptualize it in the context of the most current theory of intelligence. The
Cattell-Horn-Carroll theory of intelligence has had a significant impact on the
construction and interpretation of current measures of intelligence (Alfonso, Flanagan, &
Radwan, 2005). The CHC theory of intelligence has a three tiered structure that consists
of a general factor of intelligence or "g", 10 broad factors of intelligence, and
approximately 70 narrow factors of intelligence (Evans, Floyd, McGrew, & Leforgee
2002; McGrew, 2005; Sattler, 2001). The 10 broad factors include: 1) Fluid Intelligence
(Gf); 2) Crystallized Intelligence (Gc); 3) Short-Term Memory (Gsm); 4) Visual
Processing (Gv); 5) Auditory Processing (Ga); 6) Long-term Retrieval (Glr); 7)
Processing Speed (Gs); 8) Reading and Writing (Grw); 9) Quantitative Knowledge (Gq);
10) Decision/Reaction Time (Gt) (see table 1.1) (Evans et al, 2002; Keith, et al. 2006;
Roid, 2003a; Roid, 2003b; McGrew, 2005). The literature overwhelmingly views the
terms Visual-Spatial Processing and Visual Processing (Gv) as the same construct
(Alfanso et al., 2005; DiStefano & Dombrowski, 2006; Evans et al., 2002; Floyd, et al.
7
2003; McGrew, 2005; Osmon, Smerz, Braun, & Plambeck, 2006; Proctor et al., 2005;
Roid, 2003a).
Table 1.1
The 10 Cattell-Horn-Carroll Broad Factors of Intelligence and their Abbreviations
Factor
Abbreviation
1. Fluid Intelligence
(Gf)
2. Crystallized Intelligence
(Gc)
3. Short-Term Memory
(Gsm)
4. Visual Processing
(Gv)
5. Auditory Processing
(Ga)
6. Long-term Retrieval
(Glr)
7. Processing Speed
(Gs)
8. Reading and Writing
(Grw)
9. Quantitative Knowledge
(Gq)
10. Decision/Reaction Time
(Gf)
The Gv broad category of intelligence incorporates several processing tasks
including the production of visual images, mentally holding and manipulating visual
images and recalling visual images (McGrew, 2005). The Gv broad category of
intelligence includes the narrow cognitive abilities of: 1 ) Visualization (VZ); 2) Spatial
relations (SR); 3) Closure speed (CS); 4) Closure flexibility (CF); 5) Visual memory
(MV); 6) Spatial scanning (SS); 7) Serial perception integration (PI); 8) Length
estimation (LE); 9) Perceptual illusions (IL); 10) Perceptual alterations (PN); 11)
Imagery (IM); 12) Perceptual Speed (PS) (see table 1.2) (Carroll. 1993; Lohman, 1994;
McGrew, 2005; Sattler, 2001). Carroll's (1993) factor analytical work with cognitive
abilities may provide the best understanding of how Gv (i.e. visual-spatial processing) is
assessed.
8
Table 1.2
The 12 CHC Visual Processing (Gv) Narrow Cognitive Abilities and their Abbreviations
Narrow Ability
Abbreviation
1. Visualization
(VZ)
2. Spatial Relations
(SR)
3. Closure Speed
(CS)
4. Flexibility of Closure
(CF)
5. Visual Memory
(MV)
6. Spatial Scanning
(SS)
7. Serial Perception Integration
(PI)
8. Length Estimation
(LE)
9. Perceptual Illusions
(IL)
10. Perceptual Alterations
(PN)
11. Imagery
(IM)
12. Perceptual Speed
(PS)
Literature suggests specific tasks measure each of the 12 Gv narrow cognitive
abilities. The first and broadest narrow cognitive ability is Visualization (VZ). Measures
for the VZ factor include assembly type tasks, block counting tasks, block rotation tasks,
paper folding tasks, surface development tasks, and figural rotation tasks (Carroll, 1993;
Lohman, 1994). The Block Design and Object Assembly subtests of the Wechsler
intelligence assessment series and the Form Board and Form Patterns subtests of the
Stanford-Binet series also may measure VZ (Carroll, 1993; G. H. Roid, personal
communication, November, 7 2006; Lohman, 1994; Sattler, 2001; Sattler & Dumont,
2004). Tasks that are thought to measure spatial relations (SR) include irregular card
comparisons, cube comparison tasks and the Block Design subtest of the Wechsler
intelligence assessment series (Carroll, 1993; Lohman, 1994; Sattler, 2001; Sattler &
Dumont, 2004). Tasks that are suggested to measure closing speed (CS) are the Street
9
Gestalt Completion test, tasks that include concealed letters, numbers or figures, and the
Object Assembly task of the Wechsler Intelligence Test series (Carroll, 1993; Sattler,
2001; Sattler & Dumont, 2004). Measures of flexibility of closure (CF) include tests that
have hidden or embedded figures, designs, or patterns (Carroll, 1993). Measures of visual
memory (MV) include a brief exposure to, then recalling in part or whole maps, pictures,
designs or shapes (Carroll, 1993). The Memory for Objects subtest of the Stanford-Binet
Fourth Edition is considered a measure of MV (Sattler, 2001). Measures of spatial
scanning (SS) involve maze tracing or planning and following a route on a twodimensional map (Carroll, 1993). The Mazes subtest of the Wechsler series may be a
well-known measure of SS (Sattler, 2001).
There is limited research on measures of the serial perception integration (PI)
factor; however, Carroll (1993) suggests tasks that measure PI involve the rapid
recognition of patterns in ordered and segmented parts (Carroll, 1993). Tasks that are
suggested to measure the narrow ability of length estimation (LE) include length
discrimination, length estimation, and comparison or proximity analysis of lines and
points (Carroll, 1993). Tasks that measure perceptual illusions (IL) may include the
estimation, contrasting, shape identification or direction identification of illusions
(Carroll, 1993). Carroll (1993) suggests that perceptual alterations (PN) measurement
tasks involve mental alternations of stimuli under timed conditions. Measures of imagery
(IM) require the subject to visually manipulate an object and compare it to other similar
non-manipulated objects (Carroll, 1993). Tasks that are believed to measure perceptual
speed (PS) involve the efficiency of recognition and comparison of visual stimuli under
timed conditions (Carroll, 1993). Symbol Search and Cancellation of the Wechsler
10
Intelligence Scale for Children 4 Edition may be measures of PS (Sattler & Dumont,
2004). The most recent revision of the Stanford-Binet Intelligence series is purported to
be aligned more closely with current theory regarding the measure of visual-spatial
processing.
The Stanford-Binet Intelligence Scales, Fifth Edition (SB5) published in 2003,
was designed to adhere more directly to the modern CHC theory of intelligence. The SB5
was developed around five factor areas. The five factors (and their corresponding CHC
cognitive ability) are Fluid Reasoning (Gf), Knowledge (Gc), Quantitative Reasoning
(Gq), Working Memory (Gsm) and Visual-Spatial Processing (Gv) (see figure 1.1)
(DiStefano & Dombrowski, 2006; Roid, 2003a). Roid (2003a) used confirmatory factor
analysis to confirm the factor structure of the SB5. Research substantiating the five
factors however, has not been conclusive. However, DiStefano's & Dombrowski's
(2006) exploratory factor analyses confirmed the SB5 as an adequate measure of general
intelligence or "g", but did not confirm the five factors. Roid maintains the rigorous
research that he and the test development team conducted fully substantiates the factor
structure of the SB5 (G. Roid, personal communication, November 7, 2006). The SB5
has both verbal and non-verbal measures of visual-spatial processing.
11
Figure 1.1. The Structure of the SB5.
SB5
Full Scale
10
Nonverbal
Domain
Fluid
Reasoning
Knowledge
Verbal
Domain
Quantitative
Reasoning
Visual-Spatial
Processing
Working
Memory
The SB5 defines visual-spatial processing as "... the ability to see relationships
among figural objects, describe or recognize spatial orientation, identify the "whole"
among a diverse set of parts and generally see patterns in visual material" (Roid &
Pomplun, 2005 p. 328). The verbal and nonverbal visual-spatial subtests of the SB5 were
created through a review of previous visual-spatial assessments and consultation with
notable experts in the field of CHC (see table 1.2) (Dick Woodcock, John Horn & John
Carroll; G. Roid personal communication November 7, 2006). The verbal visual-spatial
measure of the SB5 is the Position and Direction subtest. Position and Direction requires
the subject to "identify common objects and pictures using common visual/spatial terms
such as "behind" and "farthest left," explain spatial directions for reaching a pictured
destination or indicate direction and position in relation to a reference point" (Roid,
2003b p. 139). This subtest was derived from previous Stanford-Binet scales (Roid,
2003a). In addition, the subtest is based on Lohman's (1994) conceptualization that
verbal visual-spatial tests that require a subject to create a mental image and answer
12
corresponding questions are representative of real-life usage of visual-spatial processing
(Roid, 2003a). It is unclear however, which narrow cognitive ability Position and
Direction measures. Neither the technical nor the administrative manual directly specifies
the narrow cognitive ability (Roid, 2003a; 2003b). The nonverbal visual-spatial measures
of the SB5 were also designed to align with CHC theory.
The nonverbal visual-spatial processing domain of the SB5 contains two different
measures. At the early levels (1 -2) the measure is the Form Board task. The Form Board
task has been used with previous versions of the Stanford-Binet (Roid, 2003a). The Form
Board task is believed to be a measure of Gv and the narrow cognitive ability of VZ
(Carroll, 1993; Roid, 2003b). In the remaining levels of the nonverbal visual-spatial
processing domain, the Form Patterns task is used. The Form Patterns subtest was
selected by the test developers based on the suggestions by John Carroll, for a hands on
assembly task (G. Roid, personal communication, November 7, 2006). The task requires
subjects to reconstruct visually presented stimuli with geometric shapes. Form Patterns is
a measure of the broad Gv and of the narrow cognitive ability of VZ (G. Roid, personal
communication, November 7, 2006; Roid, 2003a). Currently there is a lack of nonpublisher developed research using the SB5 as a visual-spatial measure. The Wechsler
Intelligence Scale for Children was also recently revised and has tasks that research
suggests measure visual-spatial processing.
13
Figure 1.2. Visual-Spatial Processing Measures of the SB5.
Visual-Spatial
Processing
Nonverbal
Verbal
Form Board /
Form Patterns
Position and Direction
The current revision of the Wechsler Intelligence Scale for Children (WISC-IV)
published in 2003 was undertaken to more accurately align the test with current
intelligence theory, elevate psychometric structure, broaden applicability, and enhance
evaluator usage of the instrument (Sattler & Dumont, 2004). The revision of the test
includes additional subtests to improve the measurement of Fluid Reasoning (Gf),
Working Memory (Gsm), and Processing Speed (Gs) (Wechsler, 2003a; Zhu & Weiss,
2005). The WISC-IV's four Index scores Verbal Compression, Perceptual Reasoning,
Working Memory, and Processing Speed combine to form the Full Scale IQ or measure
of "g" (see figure 1.3). Test developers utilized exploratory and confirmatory factor
analysis research to verify the four factors (Wechsler, 2003 a). However, recent research
on the WISC-IV has disputed the four factors as the most appropriate organization for the
assessment.
Keith et al. (2006) maintains the WISC-IV is better described using five factors of
the CHC Theory. Using factor analysis Keith et al. found a test framework structured on
the CHC factors of Crystallized Intelligence (Gc), Visual Processing (Gv), Fluid
Reasoning (Gf), Short-Term Memory (Gsm) and Processing Speed (Gs) provided the best
14
fit for the test (using the standardization data). Keith et al.'s work suggests that the
WISC-IV is an appropriate measure of visual-spatial processing or Gv.
Figure 1.3. Structure of the WISC-IV
WISC-IV
Full Scale
IQ
Verbal
Comprehension
Perceptual
Reasoning
TnHex
TnHpv
1. Similarities
2. Vocabulary
3. Comprehension
4. Information
5. Word Reasoning
Working
Memory
InHe
I
1. Digit
Span
2. LetterNumber
Sequence
3. Arithmetic
1. Block
Design
2. Picture
Concepts
3. Matrix
Reasoning
4. Picture
Completion
Processing
Speed
TnHpx
I
1. Coding
2. Symbol
Search
3. Cancellation
The subtests in bold
typeface are the core
subtests of the WISC-IV
The subtests of the WISC-IV that purport to measure visual-spatial processing
(Gv) fall under the Perceptual Reasoning Index (see table 1.3). The Block Design subtest
of the WISC-IV may be the most complete measure of visual-spatial processing in the
Perceptual Reasoning Index. Block Design has been consistently utilized with the
Wechsler series. The literature supports Block Design as a measure of the broad cognitive
ability Gv and the narrow abilities of visualization (VZ) and spatial relations (SR)
(Carroll, 1993; Keith et al, 2006; Sattler & Dumont, 2004). In addition, studies often use
Block Design as a primary measure of visual-spatial processing (Carroll, 1993; Cornoldi
et al., 2003; Fuchs et al., 2005; Hegarty & Kozhevnikov, 1999; Lee et al, 2004). Sattler
(2001) cautions however, that children with visual or motor skill difficulties may not do
15
well on the task; suggesting that other abilities may influence students' performance. The
literature supports additional subtests of the WISC-IV as secondary measures of visualspatial processing.
For example, there is literature to support that Picture Completion (PCm) is a
measure of Gv. PCm involves visual responsiveness, visual perception, visual
discrimination and visual memory (Sattler & Dumont, 2004; Zhu & Weiss, 2005). In
addition, PCm is suggested to be a measure of the narrow cognitive ability, flexibility of
closure (CF) (Sattler & Dumont, 2004). Research also supports Matrix Reasoning (MR)
as a measure of visual-spatial processing (Keith et al., 2006). Sattler (2001) and Sattler
and Dumont (2004) maintain that because of MR's visual-perceptual and visual-spatial
processing elements it is a good measure of the broad Gv ability and the narrow VZ
cognitive ability. There is some disagreement with Symbol Search (SS) as a measure of
Gv. Keith et al.'s research with the WISC-IV found that SS loaded on the Gv cluster and
the Gs Cluster. Sattler and Dumont (2004) maintain that SS is more strictly a measure of
Processing Speed (Gs).
Table 1.3
Visual-Spatial Processing Measures of the WISC-IV
Subtest
CHC Cognitive Ability
Broad
Narrow
Block Design
Gv
Picture Completion
Gv
CF
Matrix Reasoning
Gv
VZ
Symbol Search*
Gv; Gs
VZ; SR
* Note: There is some disagreement in the literature regarding whether
Symbol Search is a measure of Visual Processing or
Processing Speed.
16
Significance of the Study
The current and past definition of a learning disability is grounded in the idea that
a SLD is a disorder in basic psychological processing. The most often used methods of
identifying a SLD involve the ability-achievement discrepancy paradigm and the more
recent response to intervention (RTI) process (Kavale et al., 2005; Reschly et al., 2003).
Both methods fail to diagnosis a SLD based on a disorder in processing (Torgesen, 2002).
It is logical if the definition of a SLD is stated as "a disorder one or more of the basic
psychological processes..." then an evaluation should include an assessment of
psychological processing (U.S. Department of Education, 2006a, p. 46757). There is
research to support that certain processing components play an important role in reading,
writing and mathematics achievement.
In comparison to reading and writing, mathematics achievement has had the least
amount of research in understanding the potential cognitive process involved (Swanson
& Jerman, 2006). Recent literature maintains improved understanding of the cognitive
components involved in mathematics achievement may increase the ability of
professionals to identify and treat disabilities in mathematics (Fuchs, et. al. 2006). There
are believed to be specific psychological processes involved in the basic mathematical
tasks of calculation, fluency and word problems. Of the psychological process involved
in the application and understanding of mathematics, working memory appears to
contribute to all areas of mathematical thinking (Swanson & Beebe-Frankenberger, 2004;
Swanson & Jerman, 2006). A significant sub-process of working memory is visualspatial processing (Baddeley, 1996; Pickering & Gathercole, 2004; Swanson & Jerman,
17
2006). Studies have shown that visual-spatial processing is related to mathematics
(Geary, 2004).
The recently revised Stanford-Binet Intelligence Scales, Fifth Edition (SB5) has
been designed to align closely with the most current theory of intelligence, the combined
Cattell-Horn-Carroll (CHC) theory of cognitive abilities (Roid, 2003a). The VisualSpatial factor of the SB5 is purported to be a measure of visual-spatial processing or Gv.
The Visual-Spatial factor of the SB5 includes verbal (Position and Direction) and
nonverbal (From Board; Form Patterns) measures of visual-spatial processing. There
currently is limited non-publisher developed research on the visual-spatial measures of
the SB5. In addition, the Wechsler Intelligence Scale for Children (WISC-IV) was also
recently updated to align more closely with the CHC theory of cognitive abilities (Sattler
& Dumont, 2004; Wechsler, 2003). Research has suggested that Bock Design, Picture
Completion, and Matrix Reasoning are measures of visual-spatial processing (Gv) (Keith
et al, 2006; Sattler & Dumont, 2006).
There are five reasons for the current study. First, if a SLD is defined as a disorder
in a basic psychological process it is important to show that processing deficits are related
to a SLD. Second, there is a literature supported need for increased research in
mathematics achievement. Third, there is a limited amount of research on the revised
visual-spatial measures (Position and Direction; Form Board; Form Pattern) of the SB5.
In addition, to date, there has been no research with visual-spatial measures of the SB5
and poor achievement in mathematics. Finally, to date there has been no research
investigating the relationship between the combined visual-spatial processing measures
18
of the WISC-IV (Block Design, Matrix Reasoning, and Picture Completion) and poor
mathematics achievement.
Statement of the Problem
The primary purpose of this study is to investigate the ability of the visual-spatial
measures of the Stanford-Binet-Fifth Edition (SB5) and the Wechsler Intelligence Scale
for Children- Fourth Edition (WISC-IV) to discriminate between students with and
without difficulties in mathematics achievement. It is suggested from a review of
literature, visual-spatial processing, as measured by the SB5 and the WISC-IV, will be
significantly different between students who have a potential disability in mathematics
and those who do not. In addition, the study will identify which visual-spatial measure or
index has the most potential as a discriminator between students who have poor
mathematics achievement and those who do not.
The following research questions will be used as a guide to the current study:
1. Is there a relationship between the psychological process of visual-spatial
processing (as measured by the SB5 and WISC IV) and mathematics
achievement (as measured by the Woodcock-Johnson III Tests of
Achievement-Normative Update (WJ-III-NU)?
2. Can the visual-spatial measures of the WISC-IV and the SB5 predict
mathematics achievement (as measured by the WJ-III-NU)?
3. What visual-spatial measure (SB5; WISC-IV) is the best
predictor of poor mathematics achievement (as measured
by the WJ-III-NU)?
19
Definition of Terms
The following definitions will be useful in understanding the preceding study.
Specific Learning Disability: ".. .Specific learning disability means a disorder in one or
more of the basic psychological processes involved in understanding or in using language
spoken or written, that may manifest itself in the imperfect ability to listen, think, speak,
read, write, spell or to do mathematical calculations including conditions such as
perceptual disabilities, brain injury, minimal brain dysfunction, dyslexia and
developmental aphasia" (U. S. Department of Education, p. 46757, 2006a).
Working Memory: The cognitive process that allows one to keep information at the
forefront of one's thoughts while mentally manipulating that information (Geary, 1996).
Visual-Spatial Processing: "The ability to generate, retain, retrieve and transform wellstructured visual images" (Lohman, 1994, p. 1000).
Limitations
One limitation of the current study may be some concerns regarding
generalizability. Using only middle schools students in grades 6A-Sth from specific
geographic locations in the West and Midwest may limit the application of the findings to
specific age groups and geographic locations. This limitation may prohibit the application
of the study's findings to students that are not in grades 6^-%^ and not from similar
geographic areas; making it difficult to generalize the study to students that are in
different age groups (younger or older) and/or come from larger or smaller communities.
Another factor that may cause some concerns regarding generalizability is, only students
from which parental or legal guardian consent is obtained will participate in the study
limiting the subject pool. This potentially limits the participants in the study to
20
individuals that are motivated enough to obtain parent consent. That in turn may exclude
those students that lack motivation to participate or may not be willing to participate do
to an aversion toward testing. An additional limitation may be that the measures of the
SB5 and the WISC-IV used in the study, purporting to measure visual-spatial processing,
may not accurately measure this construct. Due to the complexities of how the brain
analyzes and applies information, additional cognitive mechanisms may interfere with a
pure measure of relationship between visual-spatial processing and mathematical
achievement, confounding the results of the current study.
The Structure of the Proceeding Chapters
The literature review in Chapter 2 will provide a structural understanding of the
elements of the current study. It will identify the current literature regarding: 1) How
learning disabilities are defined operationally; 2) An understanding of mathematical
disabilities; 3) A conceptualization of visual-spatial processing and mathematics; 4) How
visual-spatial processing is assessed. Chapter 3 will provide the methodology for the
current study. The third chapter will address: 1) The participants used in the study; 2)
Instruments that were utilized; 3) The procedural aspects of the study; 4) How the data
were analyzed. Chapter 4 will present the results of the data analyses. Finally, Chapter 5
provides a summarization of the findings of the current study and a discussion of the
implications for this research.
21
CHAPTER II
LITERATURE REVIEW
Learning Disabilities
The assessment, identification and remediation of learning disabilities are a
significant focus of special education programs in today's public schools. According to
the most recent data from the United States Office of Special Education (2004) there are
over 2.8 million students identified as having a specific learning disability in the United
States. That number translates into approximately 47% of all students being served
through special educations services have a learning disability (Heward, 2006). There are
disagreements with both the definition and identification of a learning disability. This
first section will address the definition and identification of learning disabilities.
Learning Disabilities Defined: Past and Present
Defining a learning disability is complicated. In one article alone, the author
identified 11 separate definitions for a learning disability (Hammill, 1990). The
conceptualization of the term learning disability, in the United States, is credited to the
work of Samuel Kirk in 1962-1963 (Hallahan & Mercer, 2001; Hammill, 1990; Hammill,
Leigh, McNutt & Larsen, 1981; Heward, 2006; Kirk & Kirk, 1983; Reschly, Hosp, &
Schmied, 2003). In Kirk's original definition, he defines a learning disability as an
underdeveloped process disorder in the academic and non-academic areas of speech,
language, reading, spelling, writing or mathematics (Hammill, 1990: Kirk & Kirk, 1983).
The process disorder may originate from either a brain dysfunction, behavioral
dysfunction or emotional dysfunction (Hammill, 1990; Kirk & Kirk, 1983). Kirk's
definition excluded individuals with mental retardation, any type of sensory deficit, and
individuals whose abilities were negatively impacted by culture or instruction (Hammill,
22
1990; Kirk & Kirk, 1983). Kirk's learning disability definition is the framework for the
current definition.
The current learning disability definition used by special education professionals
has its roots in Kirk's original definition. One main reason is Samuel Kirk was the head
of the National Advisory Committee on Handicapped Children (NACHC) that formulated
and presented the original definition to congress and the U. S. Office of Education in
1969 (Hallahan & Mercer, 2001; Hammill, 1990; Kirk & Kirk, 1983; Reschly, Hosp, &
Schmied, 2003). The NACHC definition also identified a learning disability as a process
disorder. More specifically it stated a child with a specific learning disability has a
".. .disorder in one or more of the basic psychological processes involved in
understanding or using spoken language. These may be manifested in a disorder of
listening, thinking, talking, reading, writing, spelling or arithmetic" (NACHC, 1968, p.
34 as cited in Hammill, 1990, p. 75). That definition with minimal changes was adopted
into law in 1975 as part of Public Law 94-142.
The 1975 definition also identified a specific learning disability as a
psychological processing disorder. More specifically it states, "The term "specific
learning disability" means a disorder in one or more of the basic psychological processes
involved in understanding or in using language, spoken or written, which may manifest
itself in an imperfect ability to listen, speak, read, write, spell or to do mathematical
calculations" (U. S. Office of Education, 1977, p 65083 as cited in Hammill, 1990, p. 77).
Analyzing the current federal definition adopted by the U.S. Department of Special
Education reveals the definition of a specific learning disability (SLD) has remained
23
constant from the original definition in 1977. The Individuals with Disabilities
Improvement Act (2004) states a SLD is:
(i) General. Specific learning disability means a disorder in one or more of
the basic psychological processes involved in understanding or in using
language spoken or written, that may manifest itself in the imperfect
ability to listen, think, speak, read, write, spell or to do mathematical
calculations including conditions such as perceptual disabilities, brain
injury, minimal brain dysfunction, dyslexia and developmental aphasia,
(ii) Disorders not included. Specific learning disability does not include
learning problems that are primarily the result of visual, hearing, or motor
disabilities, of mental retardation, of emotional disturbance, or of
environmental, cultural or economic disadvantage (U.S. Department of
Education, 2006a, p. 46757).
Some have questioned the adequacy of the current definition (Reschly, Hosp, &
Schmied, 2003). The National Joint Committee on Learning Disabilities (NJCLD)
contends that there are limitations with the federal definition. The NJCLD believes the
federal definition: 1) Fails to include adults; 2) The use of the term "basic psychological
processes" is ambiguous; 3) Spelling as a disability category is redundant and can be
included under a written expression disability; 4) Terms such as dyslexia, minimal brain
dysfunction, perceptual impairments and developmental aphasia are outdated; 5) The
exclusionary clause in the second section is confusing by failing to clearly explain why
these areas are not included (NJCLD, 1991). Others have also suggested the federal
definition maybe inadequate. Kavale, Holdnack and Mostert (2005) suggest one of the
24
main problems with the category of SLD in special education is the definition not the
identification. They contend the federal definition lacks specificity and is fraught with
vagueness (Kavale, et al, 2005). Regardless of any dissatisfaction with the current
definition, little has changed regarding the federal definition of a SLD since its
acceptance in 1977. Analyzing the regulations used by state education departments
reveals wide spread adoption of the current federal definition of SLD.
The majority of state education departments have adopted the federal definition of
a SLD. Reschly, et al. (2003) investigated state education agencies (SEA) in all 50 states
and identified that over 80% of states have used the federal definition. Only nine states
diverted substantially from the federal definition (AL, CO, FL, MA, NV, VT, WV, NC,
WI) (Reschly, et al., 2003). In further analysis of Reschly, et al.'s study, the data reveals
of the 50 states, 48 states conceptualize a SLD as a possessing disorder. The only two
states that do not utilize a processing disorder as a main component of their state
definition of a SLD are West Virginia and Illinois (Reschly, et al, 2003). In addition, a
recent unpublished review of how states currently define a SLD, found that 49 of the 51
states use the federal definition of a SLD or use the term "processing disorder" in their
definition (Clifford, 2008).
To conclude this section, the definition of the term SLD was first conceptualized
in the early 1960's. The current definition of a SLD in the reauthorization of IDEA
(2004) has changed little from the original definition in 1977 as part of P. L. 94-142. The
idea that a processing disorder is a foundational element of a SLD has been held constant
throughout the revisions of the definition and the law. The majority of states utilize the
federal definition of a SLD. Finally, all but two of the SEAs explicitly state that a specific
25
learning disability is defined by a processing disorder. Where the majority of
disagreement occurs among SEAs and professionals in the field of learning disabilities is
how to best identify an individual with a SLD.
Learning Disability Classification and Identification
The current methods of identifying a SLD can be traced back to the U.S. Office
Education in 1976. The U.S. Office of Education stated that a SLD was identified by a
"severe" discrepancy between an individual's intellectual ability and academic
achievement (Hammill, 1990; Reschly, et al, 2003). Specifically, it operationalized a
severe discrepancy when achievement was at or below 50% of what could normally be
expected given the child's age and education (Hammill, 1990). The discrepancy criteria
of 50%, offered in 1976 received significant criticism by education professionals and
laypersons, and was not included in the final regulations adopted as P. L. 94-142 in 1977
(Hammill, 1990; Reschly, et al., 2003). In 1977 without further guidance, the majority of
states adopted the practice of classifying a SLD as a discrepancy between ability and
achievement (Reschly, et al., 2003). That practice has been consistently employed by
state departments of education over the past 30 years.
With the initial 1975 implementation of P. L. 94-142 and the subsequent
reauthorizations of the Individuals with Disabilities Education Act in 1990 and 1997 the
language continued to included identifying a SLD through a sever discrepancy between
ability and achievement (Hallahan & Mercer, 2002; Jacob & Hartshorne, 2003). The
regulations indicate the multidisciplinary team determines if an individual has a
significant discrepancy between their level of achievement and level of ability (U.S.
Department of Education, 2006a). The discrepancy can be in a single area or in any
26
combination of the areas of oral and written expression, listening and reading
comprehension, mathematics calculation and reasoning, and in basic reading skills (U.S.
Department of Education, 2006a). No precise criteria have been offered to quantify what
was meant by significant. Current regulations have offered SEAs more options. Recently
within the Individuals with Disabilities Improvement Act of 2004, there has been a shift
in the identification procedures involved with specific learning disabilities. No longer is
there an implied requirement to use only a severe ability-achievement discrepancy for
identification and classification purposes. The new regulations indicate that states may
use as an evaluation procedure based on whether or not the student responds to a
researched based intervention. In identifying a SLD SEAs:
Must not require the use of a severe discrepancy between intellectual
ability and achievement for determining whether a child has a specific
learning disability, as defined in 34 C.F.R. 300.8(c)(10);
Must permit the use of a process based on the child's response to
scientific, research-based intervention; and
May permit the use of other alternative research-based procedures for
determining whether a child has a specific learning disability, as defined in
34 C.F.R. 300.8(c)(10) (U.S. Department of Education, 2006b).
Some are in support of this change. Stanovich (2005) contends that the use of the
achievement-discrepancy paradigm for learning disability identification in some ways is
equitable to malpractice, and flies in the face of substantial research noting its
inadequacy. Others believe there are unknown questions and limitations with the use of
response to intervention that need to be explored before wholesale adoption (Kavale, et
27
al., 2005). To understand the complicated nature of SLD diagnosis it is relevant to
discuss both methods of identification.
Models of Identification: IQ-achievement Discrepancy and Response to Intervention
The three most commonly used discrepancy models are the grade level
discrepancy model, standard score/ standard deviation model, and the regression model
(Mercer, Jordan, Allsopp & Mercer, 1996; Proctor & Prevatt, 2003; Reschly, et al.,
2003). The grade level discrepancy model is the least frequently used and is often called
the deviation from grade level model (Mercer, et al., 1996; Proctor & Prevatt, 2003). In
this model, a SLD is identified by a difference between the child's actual grade level and
the child's achievement level (Mercer, et al, 1996; Proctor & Prevatt, 2003). The
difference is indicated by a grade equivalence score on an academic achievement test
(Mercer, et al., 1996; Proctor & Prevatt, 2003). In the model, the child is often required to
have a minimal IQ (often 80 or 85) to receive a diagnosis of SLD (Proctor & Prevatt,
2003). In addition, the difference required for SLD identification can vary from 1-2 grade
levels (Proctor & Prevatt, 2003). Concerns regarding this method include the potential for
over identification of slow learners, under identification of students with higher IQ sores
and the inaccuracy of grade level placements (Mercer, et al., 1996; Proctor & Prevatt,
2003).
The standard score/ standard deviation model, also called the simple discrepancy
model, is a frequently used model by state departments of education (Reschly, et al.
2003). This method identifies a SLD by a discrepancy between an intelligence
assessment score and an achievement test score. State criteria can vary for identifying a
severe discrepancy. Some states use standard deviation (SD) differences of between 1.0-
28
2.0 to indicate a severe discrepancy (Reschly, et al. 2003). Other states may use standard
score units with magnitude variations of between 15-20 standard score points (Reschly, et
al. 2003). The use of varying standard scores and SD levels produces inconsistencies in
SLD identification among state departments of education. Some contend that problems
with using this model lie in three areas: 1) Difference scores are unreliable; 2) The model
fails to identify poor readers; 3) The model does not account for regression to the mean
(Proctor & Prevatt, 2003).
The third model is the regression model. The regression model is also frequently
used by state departments of education (Mercer, et al. 1996; Reschly, et al. 2003). The
regression model improves on the simple discrepancy model by controlling for the
correlation between cognitive and achievement tests (Proctor & Prevatt, 2003). The
regression model for determining SLD is founded on two critical items: 1) The
discrepancy between the individuals' achievement score and the mean achievement score
of individuals with similar ability levels; 2) A discrepancy between the individual's level
of achievement and ability level (Proctor & Prevatt, 2003). Some suggest that issues with
this model center on a lack of consistency in implementation, and laypersons difficulty in
understanding the model (Mercer, et al. 1996; Proctor & Prevatt, 2003).
The most recent method of SLD identification, endorsed by federal legislation, is
centered on a student's failure to respond to a research based intervention. The failure to
respond method is often described in the literature as response to intervention (RTI). In
the reauthorization of IDEA, RTI is not specifically mentioned nor are any procedural
guidelines given (National Research Center on Learning Disabilities [NRCLD], 2005).
The lack of specific methodological requirements in the law leaves the process open to
29
interpretation by individual states. RTI bases the identification of a SLD on the failure of
a student to respond to rigorous implementation of empirically backed interventions
(Kavale, et al. 2005). Some experts in the field have defined RTI as an observable change
in academic performance or behavior precipitated by an intervention (Gresham, 2002).
The first step in identifying a SLD by RTI is to provide and implement well-researched
and proven instructional techniques in the classroom (Kavale et al, 2005; NRCLD,
2005). Second, each individual student's performance is monitored for changes (Kavale
et al, 2005; NRCLD, 2005). Third, students that fail to respond to research validated
instructional techniques receive additional intensive instruction (Kavale, et al., 2005;
NRCLD, 2005). Fourth, progress or lack of progress is again monitored (Kavale, et al,
2005; NRCLD, 2005). If a student does not adequately progress with intensive
instructional interventions, the student is identified with a SLD and qualifies for special
education services (Kavale, et al., 2005; NRCLD, 2005). Often in the RTI model,
students' progress is monitored by using curriculum-based measurements and graphing of
certain academic benchmarks (Gresham, 2002). There is some concern in the literature
regarding the use of this SLD identification model.
Some contend that RTI models focus heavily on reading disabilities and fail to
address other areas of academic weakness (Kavale, et al, 2005). In addition, an aspect
associated with RTI models is the need for validated screening of academic difficulties;
however, there is a lack of constancy regarding what type of screening method should be
used (Semrud-Clikeman, 2005). Another criticism of RTI is that previous research has
mainly been conducted with younger students (K-2) and there is a dearth of evidence of
appropriateness with older students (Semrud-Clikeman, 2005). Other areas of concern
30
regarding RTI include: 1) Identifying the best intervention for each individual student; 2)
Deciding how long and to what degree an intervention should be implemented; 3)
Uncertainty over who is responsible for implementing the intervention, monitoring the
intervention, and the rigor of implementation; 4) The associated costs of providing
intensive interventions to students (Gresham, 2002).
Summary
There are a substantial number of students in public schools identified as having a
learning disability. The definition of a SLD has changed little from its first acceptance in
1977 as part of P. L. 94-142 to the present IDEA improvement act of 2004. The
identification of a student with a SLD has in the past, primarily consisted of a
discrepancy between an individual's ability and their achievement level. Recently federal
regulations are allowing a student's failure to respond to a researched based intervention
as a classification method of SLD. It is apparent there is a disconnect between the current
definition of a SLD and how it is identified. The definition of SLD adopted by both the
federal government and the majority of states is centered on the concept that learning
disabilities are at their roots a processing disorder; however, processing disorders in the
identification of a SLD are often not considered. Of the previously noted 49 states that
define a SLD as processing disorder, only one utilizes processing in their classification
criteria (Clifford, 2008). If the definition of a SLD is based on the idea it is a processing
disorder, then it is prudent that SLD identification should include elements of a
processing disorder evaluation (Torgesen, 2002). If the classification of SLD does not
include the evaluation of processing components then the definition of a SLD may need
to be modified. Completely understanding the definition of a SLD requires understanding
31
what is meant by psychological processes. The next section will address the processing
components most often involved in academic abilities.
Psychological Processing and Learning Disabilities
Psychological processes are those processes that involve the effective use
of higher cognitive abilities such as the use of language, attention, utilization of memory,
thinking abstractly, solving problems, and perceptually based skills (Gerring, &
Zimbardo, 2002). Because the federal definition and the majority of state definitions of a
SLD emphasize a SLD as a processing disorder it is relevant to identify which
psychological process are involved in learning disabilities. The most common academic
learning disability diagnoses found in schools (excluding speech disorders) are learning
disabilities in reading, written language and mathematics (Heward, 2006). This next
section will address each learning disability area (reading, written language and
mathematics) identifying the most common psychological processes involved.
Reading
Reading difficulties are the most frequently diagnosed learning disability (Joseph,
2002). Some estimate almost 90% of students identified as learning disabled have a
reading disability (Heward, 2006). Others suggest that as many as 15% of all students
have reading difficulties (McCormick, 2003). The research suggests there are five main
cognitive processes involved in reading: 1) Phonological processing; 2) Syntactic
processing; 3) Working memory; 4) Semantic processing; 5) Orthographic processing
(Siegel, 2002).
Phonological processing is often considered the most important processing area in
reading development (Gray and McCutchen, 2006; Hoskyn & Swanson, 2000: Siegel,
32
2003). Phonological processing involves the association of sounds with single or
combined letters (Siegel, 2003). Specifically, it is the understanding of the relationship
between graphemes and phonemes in language (Siegel, 2003). Support for the
importance of phonological processing's role in reading comes for the work of Gray and
McCutchen (2006). Gray and McCutchen found a strong correlation between
phonological awareness (a significant component of phonological processing) and
reading tasks such as word reading and sentence comprehension. Gray and McCutchen
compared scores on the Test of Phonological Awareness Skills to timed word reading and
sentence comprehension tasks with students in kindergarten, first grade and second grade.
Gray's and McCutchen's results suggest children whose scores were high in phonological
awareness were more than twice as likely to score above the mean on word reading tasks
compared to those who scored low in phonological awareness (Gray and McCutchen,
2006). The results of the study suggest that aspects of phonological processing such as
phonological awareness may be important for early reading skills. Syntactic processing
also appears to be involved with reading skills.
The second significant processing component of reading is syntactic processing
(Siegel, 2003). Syntactic processing is the understanding of basic sentence structure or
the grammatical structure used in language (McCormick, 2003; Siegel, 2003). Support
for syntactic processing as a process of reading comes from the work of Holsgrove and
Garton (2006). The study involved assessing the reading comprehension of middle school
students. Holsgrove and Garton used measures of working memory, phonological
processing and syntactic processing. To measure syntactic processing Holsgrove and
Garton employed the aural moving-window technique that required students to analyze
33
syntactically ambiguous printed sentences. The authors found that syntactic processing
was a significant predictor of reading comprehension among the 13-year-old students.
Additionally, Holsgrove and Garton with regression analysis determined that syntactic
processing was a significant discriminator of students with and without reading
difficulties (Holsgrove & Garton, 2006). Working memory may also play a role in
student's ability to read.
In reading, working memory involves the ability to decode words while
simultaneously retaining what has been read (McCormick, 2003; Siegel, 2003). Swanson,
Howard and Saez (2006) found, with students varying in age from 7-to-17 years-of-age,
that working memory was a significant discriminator between students with and without
reading disabilities. Swanson, et al. (2006) used working memory measures such as digit
and sentence span tasks, a semantic association task, a listening span task and the
backward digit span of the Wechsler Intelligence Scale for Children-Ill to assess the
working memory of the subjects. Matching subjects for IQ and written math calculation
Swanson et al. found that students identified as reading disabled performed poorer on
working memory tasks when compared to non-reading disabled students. Swanson et
al.'s results suggest that working memory may be a contributing cognitive process in
reading ability. The literature suggests semantic processing may also be related to
students reading ability.
Semantic processing, understanding the meaning of sentences, is an important
cognitive process in reading (McCormick, 2003; Siegel, 2003). Evidence for this comes
from a study conducted by Nation and Snowling (1998). Nation's and Snowling's study
involved a comparison of average readers and students identified as having significant
34
difficulty with comprehension. Nation and Snowling matched students for decoding and
nonverbal ability. The authors used measures of both expressive and receptive language
to assess semantic processing differences between the two groups. Nation and Snowling
found that semantic processing significantly discriminated between readers with
comprehension difficulties and average readers. The results of the study suggest that
semantic processing may be an important component in children's ability to comprehend
written material. Some research also supports orthographic processing's relationship to
students' reading ability.
The final research identified significant cognitive process in reading is
orthographic processing. Orthographic processing is the knowledge or awareness of word
structure, specifically the knowledge of letters and spelling patterns (McCormick, 2003;
Siegel, 2003). Badian (2001) suggests a link between orthographic processing and
reading. Badian conducted a longitudinal study that followed the same group of children
from preschool to seventh grade. Badian used letter identification tasks as orthographic
processing measures. Badian found, among students with average to above average
intelligence, that orthographic processing skills at kindergarten were a significant
predictor of poor reading skills of those children in 7th grade. The results of the Badian
study suggest that deficits in orthographic processing may lead to poor reading ability in
later years. The next section will look at the cognitive processes involved in writing.
Writing
Prevalence rates of writing disabilities are difficult to estimate due to differences
in qualitative and quantitative distinctions (Hooper, Swartz, Wakely, de Kruif, &
Montgomery, 2002). As a measure of the number of students that struggle with writing,
35
14 % of all 4 graders, 15% of all 8 graders, and 26% of the 12 graders students who
took the National Assessment of Educational Progress in 2002 were below basic skill
levels in writing (National Center for Educational Statistics, 2002). The literature
regarding the cognitive processes involved in writing is less clear in comparison to
reading. The research suggests that there are six psychological processes involved in
writing: 1) Phonological processing; 2) Orthographic processing; 3) Working memory; 4)
Long-term memory; 5) Short term memory; 6) Morphological processing. The section
will look at the cognitive processes of writing in two ways. First, it will discuss the
processes in the holistic act of writing. Second, it will discuss spelling as a sub-skill
within writing.
Writing Processes
One of the more well know cognitive processing models of writing was developed
by Flower and Hayes in 1980 and latter expanded by Hayes (2000). The Hayes model
identifies the cognitive processes of writing as text interpretation, reflection and text
production (Hayes, 2000). Within those areas, Hayes states that working memory
(specifically phonological memory) is related to text interpretation because it
incorporates reading, listening and graphical scanning. Hayes theorizes that within
working memory the visual/spatial sketchpad is related to reflection skills. Hayes posits
that visual-spatial processing is involved when the individual utilizes internal
representations to prepare for the production of text. Hayes also believes text production
is related to long-term memory. Text production requires an individual to use previous
knowledge to construct text in a meaningful and coherent manner (Hayes, 2000). One
apparent criticism of Hayes model is, at best, it is a general model of cognitive process
36
and lacks specificity. In addition, Hayes offers little empirical research to support his
theory. In order to identify the specific processes involved one must look beyond his
model.
Research suggests that phonological processing is involved in writing (Berninger,
Abbot, Thomson, & Raskind, 2001; Johnson, 1993; McGrew & Knopik, 1993). McGrew
and Knopik (1993) found phonological processing to have a significant relationship to
writing achievement. McGrew and Knopik studied the cognitive clusters of the
Woodcock-Johnson Tests of Cognitive Ability-Revised (WJC-R) in comparison to
individual's Basic Writings Skills and Written Expression clusters scores on the
Woodcock-Johnson Psycho-Educational Battery-Revised (WJ-R). McGrew and Knopik,
using the WJ-R standardization sample, found that phonological processes (Auditory
Processing) were significantly related to basic writing skills and written expression skills.
Berninger et al. (2001) offers a more comprehensive study of the processing components
of writing.
Berninger, et al. (2001) found phonological processing to be a significant
predictor of writing skills. Berninger et al. discovered phonological measures contributed
unique variance to written composition abilities of students in first through sixth grades.
Berninger et al. used structural equation modeling to compare the relationship between
phonological processing tasks such as phonemic deletion, segmentation, and nonword
memory and writing tasks including handwriting and written composition. Berninger et
al. found that phonological processing contributed unique variance to writing
composition beyond what could be accounted for by intellectual ability. In addition,
Berninger et al. found that orthographic processing appears to be important in written
37
composition tasks and handwriting. Berninger et al. compared letter cluster coding, an
orthographic measure that required no memory usage and an orthographic measure that
tapped long-term memory to students' story composition skills and handwriting. The
results suggested that a combined orthographic processing factor was a significant
contributor to story composition and handwriting ability. The role of working memory
and long-term memory in writing is less clear.
Some support for the role of working memory in writing comes from the work of
Kellogg (1994; 2001a), Hopper et al. (2002), and Swanson and Berninger (1996).
Kellogg (1994) maintains that in writing working memory is involved in temporarily
holding and manipulating ideas that are constructed into sentences. Kellogg (2001a)
supports this view through the study of text generation and response time analysis.
Kellogg's (2001a), study involved college students and writing ability. The study
compared the construction of narrative texts (in both longhand and word processing) in
combination with an interference task (a computer-generated tone that required students
to say their thoughts regarding their work at varying 10-15 second intervals). Kellogg
(2001a) suggests students' response times were an indication of working memory
capacity. Kellogg (2001a) contends because students' response times across the tasks of
planning, translating and reviewing were all consistent it provided evidence for the
utilization of working memory across all three areas. A caveat is warranted with this
study. First, Kellogg offered little researched support for the idea that response time and
reflection were an indication of working memory capacity. Second, Kellogg failed to use
any empirically validated measures of working memory in his study.
38
A study by Hooper et al. (2002) also offers inconclusive results regarding the role
of working memory in writing. Hopper et al.'s study involved the assessment of working
memory as a component of a larger assessment of central executive tasks, including
measures of inhibition, and attention. Hooper et al. compared a working memory task that
employed sentence construction from visually displayed pictures (while performing an
interference task) and student scores on a written narrative task. Hopper et al. maintains
the study's results suggested that working memory plays an important role in the
differentiation between good and poor writers of narrative material. As with Kellogg's
study, caution should be used in the interpretation of this study's results. First, Hopper et
al's measure of working memory capacity involved an interference task and not an
empirically validated measure of working memory. Second, Hopper et al's results, failed
to separate out the working memory assessment, leaving it as an element of a larger
domain that consisted of other nonworking memory related measures. Swanson's and
Berninger's (1996) study of fourth graders may offer a more concise explanation of
working memory's role in writing.
Swanson and Berninger (1996) used both verbal and visual-spatial working
memory measures to explore working memory in writing. The authors employed working
memory tasks that included sentence spans, rhyming, semantics (categorization and
association), phrase sequencing and story recall in conjunction with visual matrices,
mapping tasks and direction tasks (Swanson & Berninger, 1996). The authors compared
both verbal and visual-spatial measures to writing tasks that included expository and
narrative composition, handwriting and spelling. Swanson and Berninger, controlling for
the effects of age, found that overall there was a significant relationship between working
39
memory tasks and writing skills, particularly as it related to the executive system. An
important component of the Swanson and Berninger study is the authors found that shortterm memory contributed significantly to spelling and handwriting, but not text
construction. The results suggest a separation of roles of working memory and short-term
memory in writing. Long-term memory processing may also be involved in writing
ability.
Some suggest that long-term memory may play an important role in the
generation of text (Hayes, 2000; Kellogg, 1994; Kellogg 2001b). Limited support for the
role of long-term memory comes from a study conducted by Kellogg (2001b). Kellogg
conducted two experiments with college-aged students. In the first experiment, Kellogg
used domain knowledge as an indicator of long-term memory. Kellogg analyzed narrative
and persuasive text production in comparison to verbal ability (as measured by verbal
domain scores on a standardized test) and quality of production. Kellogg's results suggest
verbal ability did not affect text recall; rather text recall was affected by domain
knowledge. In the experiment, Kellogg used an interference task to measure response
time in combination with individual differences in verbal ability and domain knowledge.
The results of the study suggest that short-term memory (as measured by response time)
and verbal ability (as measured by verbal score on the standardized test) were not as
important as domain knowledge regarding text quality (Kellogg, 2001).
Caution should be used in unqualified acceptance of Kellogg's results. First, longterm memory as Kellogg conceptualizes is difficult to quantify. Second, Kellogg offers
little evidence regarding response time as a true measure of short-term memory.
Kellogg's study contributes confusion to psychological processing and writing. A better
40
understanding of the psychological processes involved in writing may come from an
analysis of processes involved in spelling.
Spelling Processes
Spelling is a key component of writing. As a subcomponent of writing, it appears
that phonological processing, orthographic processing, morphological processing, shortterm memory and working memory may play a role in spelling skills (Berninger &
Amtmann 2003). Cornwall (1992) studied phonological awareness and spelling skills in
elementary students. Controlling for age, IQ, SES, and behavior problems Cornwall,
identified that phonological awareness (measured by decoding, blending and phonemic
deletion) was a significant predictor of spelling ability. Hauerwas and Walker (2003)
investigated the phonological, orthographic and morphological processing in 11-13 yearold students with and without spelling deficits. The authors divided the students into two
groups (spelling deficit and non-spelling deficit) based on their scores on the spelling
subtest of the Wide Rage Achievement Test 3 and one group as an age-matched control.
The authors compared measures on phonemic deletion tasks, non-word cloze
tasks, non-word-choice tasks and inflection spelling tasks among the three groups.
Hauerwas and Walker (2003) found that orthographical and phonological awareness were
significant predictors of the spelling of base words, while morphological awareness was a
significant predictor of students' ability to spell inflected verbs. The results suggest
phonological, orthographic and morphological processing may play a role in the spelling
ability of students. Support for the role of short-term and working memory in spelling
comes from the previously mentioned Swanson and Berninger (1996) study. Swanson
and Berninger used the spelling subtest of the Wide Range Achievement Test-Revised to
41
identify the relationship between spelling, working memory and short-term memory. The
results indicated significant correlations between verbal short-term memory tasks, verbal
working memory tasks and executive processing (combined verbal and visual spatial
tasks) and spelling. The results suggest that working memory and short-term memory
both play a role in spelling. In addition, literature suggests psychological processes are
utilized in mathematical thinking.
Mathematics
Mathematical learning disabilities are not as prevalent as reading and writing
learning disabilities (Fleischner, & Manhemier, 1997; Fuchs & Fuchs, 2003; Geary 2004;
Geary & Hoard 2003; Jordan, & Montani 1997). In comparison to other disabilities (such
as reading) there has been substantially less research in the area of mathematics SLD
(Augustyniak, Murphy, & Phillips, 2004; Geary & Hoard, 2003; Robinson, Menchetti, &
Torgensen, 2002). There is research to suggest however, there are five significant
cognitive processes involved in mathematics; 1) Working memory; 2) Phonological
processing; 3) Attention; 4) Long term memory; 5) And the combined PASS cognitive
processes.
Working memory in mathematics involves holding mathematical concepts,
numbers and ideas at the forefront of thought for a short duration, while simultaneously
applying that information to mathematical processes (Fuchs, et al. 2005; Fuchs, et al.
2006; Swanson, 2004). Working memory appears to be a significant contributor to
children's mathematical thinking (Passolunghi & Siegel, 2004). Swanson and Sachse-Lee
(2001) employed working memory measures such as sentence span, auditory digit
sequencing, matrices, mapping and directions, with eight and eleven year-old elementary
42
students. Swanson and Sachse-Lee found the combined score on the working memory
measures contributed unique variance to solution accuracy in mathematical word
problems. The results of the study suggest a link between working memory processes and
mathematical problem solving. Research also supports phonological processing's role in
mathematical thinking.
Phonological processing also appears to be a notable cognitive process in
mathematical ability (Fuchs, 2005; Fuchs, et al. 2006; Swanson, 2004; Swanson &
Sachse-Lee 2001). Support for phonological processing's involvement in mathematical
thinking comes from the work of Swanson (2004). Swanson found a relationship between
phonological processing and the mathematical ability of eight and eleven year-old
elementary students. Swanson compared the phonological processing measures of
phonemic deletion, rapid digit naming and digit span with students' scores on the
Calculation subtest of the Woodcock Johnson Psychoeducational Battery. The calculation
abilities of both ages had a significant relationship with their phonological processing
abilities. The results of Swanson's study suggest that phonological processing may play
an important role in mathematics irrespective of the reading involved in word problem
solution. The psychological process of attention may also be involved with mathematical
thinking.
Attention also appears to be an important cognitive process across all areas of
mathematics (Fuchs, et al. 2005; Fuchs, et al., 2006; Swanson & Beebe-Frankenberger,
2004). Some support for this comes from the work of Fuchs, et al. (2005). In the Fuchs et
al. study, the authors examined the cognitive determinants of early mathematical
difficulties with students in first grade. Fuchs et al. utilized the short form of the Social
43
Skills Rating System to assess students' attention. The students were evaluated both at
beginning of the year and the end of the year. Students were assessed on an attentional
rating system, curriculum based measurements for mathematics and the Calculation and
Applied Problems subtests of the Woodcock Johnson-III. Fuchs et al. found that
attentional ratings of teachers were the most salient predictor of mathematical difficulty
or success across the areas of fact fluency, computation, story problems and mathematical
concepts/applications. Fuchs, et al.'s results confirmed previous research in the role of
attention and mathematics (Swanson & Beebe-Frankenberger, 2004). Research also
suggests that long-term memory may be an important cognitive process in mathematical
thinking.
The fourth cognitive process that appears to be influential in mathematics is longterm memory or semantic memory (Geary, 1993). Support for long-term memory's role
in mathematics comes from a meta-analysis of math-disabilities literature conducted by
Swanson and Jerman (2006). Swanson and Jerman identified 28 rigorously conducted
studies. All 28 studies included a comparison of average students, students with reading
disabilities, students with reading and math disabilities, and students with math
disabilities. Swanson and Jerman found that for each of almost 200 subjects, identified in
the separate studies, long-term memory was significantly correlated to their mathematical
abilities. The results suggest that long-term memory processes may have a relationship to
students' mathematical thinking. Researchers have also suggested alternative combined
processes may be involved in mathematical thinking.
Kroesberger, Van Luit and Naglieri (2003) posit a different theory of cognitive
processing may best identify how mathematical material is processed. Kroesberger, et al.
44
(2003) contends that the PASS cognitive processes of planning, attention, simultaneous
and successive (PASS) play an important role in mathematical thinking. The authors'
study was based on previous research that has linked all four of the PASS processes to
mathematical achievement (Kroesberger, et al. 2003). Kroesberger, et al. studied Dutch
elementary students identified with a math learning disability. The study compared low
scores on a standardized achievement test with student's scores on the Cognitive
Assessment System (CAS) (a measure of the PASS processes). Kroesberger et al. found
that when compared to a non learning-disabled reference sample, the students with
mathematical disabilities produced lower PASS scale scores across all four processes.
The authors' maintain, the results of the study suggest that the PASS cognitive processes
play an important role in students' mathematical thinking (Kroesberger et al., 2003).
Summary
The definition of SLD endorsed by the U.S. Education Department and the
majority of state education departments defines a SLD as a disorder in one or more of the
basic psychological process. Psychological processes include the areas of attention,
language, problem solving, memory, higher order thinking and perception. Three of the
most common SLD are learning disabilities in the areas of reading, writing and
mathematics. Research supports the idea that there may be certain psychological
processes involved in reading, writing and mathematics. The five main psychological
processing supported by research in reading include phonological processing, syntactic
processing, working memory, semantic processing and orthographic processing. The six
main research supported psychological process involved in writing include phonological
processing, orthographic processing, working memory, short-term memory, long-term
45
memory and morphological processing. The five main research identified psychological
processes involved in mathematics are working memory, phonological processing,
attention, long-term memory and the PASS cognitive processes.
It is apparent from a review of literature on SLD that in comparison to other
learning disabilities there has been considerably less research in mathematics
(Augustyniak, et al., 2004; Geary & Hoard, 2003; Robinson, Menchetti, & Torgensen,
2002; Swanson & Jerman, 2006). This suggests that further research in the area of
mathematical disabilities would be beneficial to the field of SLD. The following section
will look specifically at a SLD in the area of mathematics.
Mathematical Disabilities
The mathematical abilities children develop in school are important skills they
will need to be successful in their day-to-day functioning as adults (Assel, Landry,
Swank, Smith & Steelman, 2003; Griffin, 2003). The prevalence of mathematical
disabilities (MD) in school-aged children is notable (Jordan, & Montani 1997). It is
estimated that between 5-8% of students in public schools have MD (Fleischner, &
Manhemier, 1997; Fuchs & Fuchs, 2003; Geary 2004; Geary & Hoard 2003). Some
contented that those estimates lack accuracy and may be an actual overstatement of the
disability (Fuchs, et al., 2005; Geary, 2003). Fuchs, et al. (2005) maintains that there are
four reasons for the lack of accuracy: 1) Research has been neglectful in studying
complex forms of math difficulty; 2) The lack of proven prevention programs in primary
grades may contribute to over diagnosis; 3) Most studies fail to address the prevalence of
the disability as it is defined; 4) MD is often defined and identified differently across
studies. Regardless of the exact figure, the literature supports the existence of children
46
with mathematical difficulties. The following section will discuss the definition and
identification of MD. Next, it will address the cognitive processes involved in
mathematical calculation, mathematical fluency and mathematical word problems. The
section will conclude by exploring potential subtypes of disabilities in mathematics.
Mathematical Disabilities: Definition and Identification
The diagnosis and definition of MD are ambiguous. Exploring that ambiguity will
begin with a look at the definition of MD. According to the IDEA 2004, MD are a
disorder in a psychological process used in spoken or written language that has
manifested in a less than perfect ability to engage in mathematical calculations (U. S.
Department of Education, 2006). Others have suggested that MD are better understood by
unusual struggles in the areas of".. .the arithmetic module, conceptual knowledge base,
or problem-solving space of the domain-specific functional math system, given the
student's verbal quantitative, and/or visual-spatial reasoning ability" (Busse, Berninger,
Smith & Hildebrand, 2001, p. 238). The federal definition seems to lack specificity and
operationalzing MD based on Busse et al's (2001) definition may be more
comprehensive; however, the federal definition of learning disabilities is the accepted
definition by the majority of SEAs (Reschly, et al. 2003). For the purpose of this study,
MD will be defined as a disorder in psychological processing. The literature however,
does not agree on the most appropriate method to identify MD.
MD have been diagnosed and identified in the past with the IQ-achievement
paradigm employed with other learning disabilities. With the changes in the federal
regulations, some have suggested using RTI may be a more appropriate method of MD
identification. Fuchs et al. (2005) investigated preventative tutoring as SLD identification
47
technique with first grade students. Fuchs et al. was able to identify MD students at a
prevalence rate similar to the overall prevalence rate (4-7%) identified in other non-RTI
studies (Fuchs et al., 2005). The authors' suggest their results support RTI as an
alternative to IQ-achievement diagnosis (Fuchs et al., 2005). A review of literature on the
diagnosis and identification of MD reveals research most often employs standardized
assessments to identify MD, but criteria for inclusion is unclear.
The research is inconsistent regarding the criteria that should be used with
achievement and intelligence assessments to identify MD. Researchers have suggested
cut-off scores that range from the 10 percentile to the 45 percentile on mathematic
achievement tests with and without intelligence comparisons (Fuchs et al, 2005; Gersten,
Jordan & Flojo, 2005; Mazzocco, 2005: Mazzocco & Meyer, 2003; Murphy, Mazzocco,
Hanich & Early, 2007). Some experts suggest that scores lower than the 20th or 25th
percentile on a mathematical achievement test and in combination with a low average or
higher IQ score is the preferred method of identification (Fuchs & Fuchs 2003; Geary,
2004). Some literature suggests that an IQ-achievement discrepancy is not a necessary
factor in the determination of MD (Gersten, et al., 2005; Mazzocco & Meyer, 2003;
Murphy, et al, 2007). Support for this may come from a study conducted by Mazzocco
and Meyer (2003). Mazzocco and Meyer followed the same cohort of students from
kindergarten to 3 rd grade. The authors investigated students scoring in the 10th percentile
or less on the Test of Early Math Ability-Second Edition. Mazzocco and Meyer found the
majority of students scoring in 10th percentile across all three years of the study did not
have a consistently significant IQ-Achievement discrepancy. Mazzocco and Meyer posit
that the study's results suggest that using a mathematical achievement test alone may be a
48
more accurate predictor of MD. Fleischner and Manheimer (1997) diverge substantially
from other researchers in identifying MD. Fleischner and Manheimer, suggest that
achievement tests for MD lack the specificity needed for MD determination. Fleischner
and Manheimer advocate for error analysis of students' mathematical performance and
clinical interviews for diagnosis of MD. Fleischner and Manheimer suggest avoiding
standardized testing will provide more accurate strength and weakness information for
understanding and quantifying MD. In the literature, identification methods fail to
address the definition of MD as a processing disorder. In order to explore the use of
processing components in MD identification, it is important to further understand the
processes involved in mathematical thinking. The next section will look at the role
psychological processes play in specific mathematical tasks.
Specific Mathematical Tasks and Their Cognitive Processes
Literature has reported that little is known about the cognitive processes that are
involved in MD and mathematics achievement (Floyd, Evans, & McGrew, 2003).
Recently the literature suggests that increased understanding of the processing
components of mathematics may lead to improved identification and treatment (Fuchs, et
al., 2006). This suggests further understanding of the cognitive processes involved in
mathematics would benefit the field of SLD research. As mentioned previously, the
literature supports four basic cognitive processes in general mathematic ability including,
phonological processing, working memory, long-term memory, attention and the
collective PASS cognitive processes. This next section will explore specific cognitive
process involved in the specific areas of mathematical calculation, mathematical fluency
and mathematical word problems.
49
Calculation and Fluency
The literature often does not separate calculation tasks that involve pure
mathematical computation components from fluency tasks that are identified as
computational tasks under a time constraint. The lack of separation impedes clear
exploration of the cognitive processes involved in calculation and fluency tasks. The
literature does suggest there are five basic cognitive processes involved in both
calculation and fluency tasks. The psychological processes identified by research are: 1)
Attention; 2) Working memory; 3) Short-term memory; 4) Long-term (semantic)
memory; 4) Phonological processing. This subsection will provide support for these
processes in calculation and fluency tasks.
Attentional processes appear to play a significant role in the calculation abilities
of students (Fuchs et al., 2006; Fuchs et al, 2005). Fuchs et al. (2006) found that attention
was a significant predictor of the computational ability of third graders. Fuchs' et al.
(2006) measured third graders attention using a teacher rating scale and their
computational skills using an author derived double-digit addition and subtraction task.
Fuchs et al. (2006) identified that attention contributed unique variance beyond other
cognitive process (phonological, working memory, long-term memory) to solution
accuracy on the computational task. These results confirmed a previous study conducted
by Fuchs et al. (2005). In the Fuchs et al. (2005) study the authors explored the cognitive
factors associated with early mathematical development of first graders. The results
indicated the attentional rating by teachers was a significant predictor of first grade
students' year-end calculation skills on the Calculation subtest of the WoodcockJohnson-III Tests of Achievement (WJ-III) (Fuchs et al., 2005). The results of both
50
studies suggest there may be a relationship between calculation ability and attention, at
least in the early years of mathematical skill development. Aspects of memory processing
also appear to be influential in calculation and fluency skills.
Working memory appears to play a role in calculation and fluency tasks in
mathematics (Geary, 2004; McLean & Hitch, 1999; Swanson, 2004; Swanson, 2006;
Swanson & Beebe-Frankenberger, 2004). The literature is unclear however, on the exact
component of working memory (executive process; episodic buffer; phonological loop;
visual-spatial sketchpad) that is involved (these components will be explained later in the
chapter). Swanson (2006) investigated the role of working memory using tasks including
listening span, sentence/digit span, semantic association, visual-matrix tasks and
mapping-directions tasks. Swanson compared those working memory tasks to students'
calculations skills using the arithmetic computation subtests of the Wide Range
Achievement Tests-Third Edition, the Wechsler Individuals Achievement Test, and a
timed computational fluency task (creating a combined calculation cluster). Swanson
tested elementary students in grades first through third initially and one year later.
Swanson found that not only did working memory significantly correlate with the
calculation skills, but also discovered that working memory significantly predicted
second year calculation skills. Specifically, separate measures of working memory such
as tests of executive processes and tests of visual-spatial processes independently
predicted calculation performance (Swanson, 2006). The results suggest two components
of working memory (executive processing and visual-spatial processing) may play an
important role in students' calculations abilities. The results of this study confirmed
previous research conducted by Swanson (2004).
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Swanson's (2004) study examined age related differences between eight and
eleven-year-old children. Swanson (2004) compared students' scores on the Calculation
subtest of the WJ-III, to their combined working memory scores across digit and sentence
span tasks, listening spans tasks, visual-matrices, a mapping and a direction task.
Swanson (2004) found a significant relationship between calculation abilities and the
combined working memory tasks. The results suggest that working memory plays an
important role in the age related differences of calculation ability. Some caution is
warranted with both studies. Swanson (2006) did not separate out fluency computational
tasks and the calculation tasks. The lack of separation produces reduced clarity on
working memory's relationship to each separate mathematical task. Swanson (2004) did
not separate out the relationships between the separate working memory tasks. The lack
of clear distinction may provide evidence for a general working memory relationship, but
lack sufficient working memory substructure specificity. Floyd, et al. (2003) also found
significant relationships between working memory and calculation abilities.
Floyd et al. (2003) compared the cognitive clusters of the Woodcock-Johnson III
Test of Cognitive (WJ-III COG) abilities to Calculation subtests of the WoodcockJohnson III Tests of Achievement (WJ-III). Floyd et al., using the 6-19 year-old
standardization sample for the two assessments, found the working memory subtest of the
WJ-III COG had a consistently significant relationship to the calculation subtests
(Calculation and Fluency) of the WJ-III. Floyd et al.'s results disagree with Swanson
(2006) on the relationship of the visual-spatial processing component of working memory
to calculation skills. Using WJ-III COG measures of visual-spatial processing, Floyd et
al. did not find a significant relationship between calculation abilities of the WJ-III and
52
visual-spatial processing. The differences between the two studies may suggest further
clarification on visual-spatial processing's role in calculation abilities may be needed. In
addition, Floyd et al. reported that working memory's relationship to calculation abilities
was consistently stronger than the Working Memory and Memory Span combined
cluster. The results suggest the more pronounced role of working memory when
compared to passive short-term memory in calculation tasks. Floyd et al. however, failed
to separate subtests of mathematical fluency and mathematical calculation in the WJ-III
calculations cluster. The lack of specificity among these studies leads to ambiguity in the
role of working memory on each separate mathematical task. The literature also suggests
that short-term memory may be related to both mathematical calculation and fluency
skills.
The role of short-term memory in calculation and fluency is not clear. Short-term
memory processing tasks are passive tasks that do not require active mental manipulation
(Passolunghi & Siegel, 2004). Swanson and Beebe-Frankenberger (2004) combined
short-term memory measures including digit span forward, digit span backward, word
span and pseudo word span to form a short-term memory cluster. That cluster was then
compared to elementary students' combined scores on the arithmetic subtests of the Wide
Range Achievement Test, Wechsler Individual Achievement test and the Test of
Computational Fluency. Swanson and Beebe-Frankenberger found that short-term
memory had a significant correlation to the arithmetic calculation scores of first, second
and third grade students. Two cautions are apparent with the results of the Swanson and
Beebe-Frankenberger study. First, it could be argued that digit span backward is a
working memory task, because it involves active manipulation of numbers. Second, the
53
authors combined basic calculation subtests with a calculation fluency test for their
category of arithmetic calculation. The combination of these elements does not lend itsself to separate task and process identification. Floyd, et al. (2003) also assessed shortterm memory tasks and mathematical calculation. Floyd et al. found a significant
relationship between passive short-term memory tasks such as the Memory Span of the
WJ-III COG and the mathematical calculation composite on the WJ-III. Similar to the
Swanson and Beebe Frankenberger (2004) study Floyd et al. did not separate out tasks of
calculation and fluency. The lack of separation in both studies provides uncertainty of
short-term memory processes and their relationship to separate calculation and fluency
tasks. The cognitive process of long-term (semantic) memory and its relationship to
calculation and fluency tasks is also unclear.
Some researchers have asserted that the ability to retrieve mathematical facts used
in computational tasks is related to the processing component of semantic or long-term
memory (Geary, 1993; 2004). Research is mixed on the role of this construct; possibly do
to the difficulty in the measurement of long-term memory processes. Fuchs et al. (2005)
assessed third graders' long-term memory using the Retrieval Fluency subtest of the WJIII. Fuchs et al. did not find a significant relationship between long-term memory
processes and tasks of mathematical fluency and computation. The previously mentioned
Floyd et al. (2003) study however, did find a relationship between long-term retrieval
ability measured by the WJ-III COG and the mathematical calculation subtests of the WJIII. Cautions with the Floyd et al. study include, there was only a significant relationship
between measures at ages 6-years to 8-years old and no separation between the
Calculation and Fluency subtests of the WJ-III. Swanson and Beebe-Frankenberger
54
(2004) utilized a semantic fluency measure with mathematical computation and
mathematical fluency tasks on first, second and third grade students. Swanson and BeebeFrankenberger found a significant relationship between semantic processing and the nonseparated calculation and fluency tasks. Efficiently processing sound-letter combinations
may also play a role in calculation and fluency skills.
Phonological processing may play a role in mathematical calculation and fluency
skills. Support for the role of phonological processing and calculation comes from the
previously mentioned Swanson (2006) study. Swanson compared first, second and third
grader's scores on assessments of phonological processing (pseudo-words and
segmentation tasks), and measures of calculation and fluency. Swanson found students'
phonological processing was significantly related to their calculation skills. Swanson
however, did not endorse the role of phonological processing in calculation tasks. The
final regression model included measures of reading vocabulary, and phonological
processing. Phonological processing did not contribute unique variance in the final model
(Swanson did not differentiate between calculation and fluency tasks). Clearer support for
phonological processing in mathematical calculation tasks may come from Fuchs et al.
(2005). Fuchs et al.'s results suggest that phonological processing has an important
relationship with students' scores on the Calculation subtest of WJ-III. Fuchs et al. used
the rapid digit naming and sound matching subtests of the Comprehensive Test of
Phonological Processing to create a phonological processing index. The authors found the
phonological processing index was a significant predictor of first graders calculation
skills. While the majority of studies included did not separate calculation and fluency
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tasks, there is additional research that specifically explores mathematical fluency and
cognitive processes.
Fluency. There is some research that suggests there are specific cognitive
processes involved in mathematical fluency separate from calculation tasks.
McGlaughlin, Koop, and HoUiday (2005) maintain working memory is related to
mathematical fluency. McGlaughlin, et al. (2005) compared college students' scores on
the Wechsler Memory Scale-Third Edition (WMS-III) to students' scores on the
mathematical fluency test of WJ-III. The results indicated that with college students
previously identified as MD their scores were significantly lower on the working memory
measures of the WMS-III and the fluency subtest of the WJ-III. In addition, some of the
PASS cognitive process may also be related to mathematical fluency. Kroesberger et al.
(2003) used the PASS and a multiplication fluency task to assess the cognitive processes
of Dutch elementary students identified as MD and non-MD students. Kroesberger et al.
found that students with fluency difficulties had significantly lower scores in Planning,
Attention and Successive processing areas. Additionally, in the previously mentioned
Fuchs' et al. (2005) study the authors found phonological processing and attention were
unique predictors of elementary students' fluency on a timed one-minute addition fact
measure.
The literature suggests there may be specific cognitive processes involved in
mathematical calculation and fluency. Research indicates that, attention, working
memory, short-term memory, long-term or semantic memory, phonological processing,
and certain PASS processes (Planning; Attention; Successive) may all play a role in
mathematical calculation and fluency. The research however, often fails to separate out
56
the cognitive processes in specific calculation and fluency tasks. The following section
will look at the specific cognitive processes involved in mathematical word problems.
Word Problems
Some authors identify mathematical word problems as mathematical problems
set in a real world applicable form requiring significant cognitive demands (Garderen &
Montague, 2003). Solving word problems necessitates students to purposefully use
knowledge and skills, and apply them to mathematical situations (Fuchs & Fuchs, 2003).
To be skilled at solving word problems students need to be able to comprehend what they
read, convert what they have read into equations, and apply metacognition to obtain a
solution (Geary, 1996). These skills involve considerable cognitive demands and may be
more difficult for some children. The literature suggests that the psychological processes
involved in solving word problems are: 1) Attention; 2) Working memory; 3) Short-term
memory 4) Phonological processing. The following subsection will discuss the cognitive
processes used in mathematical word problems.
Attention appears to play a role in solving word problems. Fuchs et al. (2006)
studied the relationship between, simple arithmetic story problems and the attentional
ratings of third grade students. Fuchs et al. (2006) using path analysis identified that
attention was significantly correlated to third grade students' ability to solve arithmetic
word problems. The results of this study confirmed previous research by Fuchs' et al.
(2005). The Fuchs et al. (2005) study used a similar teacher attentional rating scale and
story problems with first grade students. This time using multiple regression analysis, the
authors' found that attention was a significant predictor of first graders end of school year
57
story problem accuracy. It also appears that working memory may play a role in the
solution of word problems.
Research suggests that working memory may be an important facet in children's
ability to solve word problems. Similar to working memory's relationship to calculation
and fluency tasks there is some disagreement regarding which particular aspect of
working memory (visual-spatial; phonological loop; executive; episodic buffer) plays the
most important role. Fuchs et al. (2005) compared the Listening-Recall subtest of the
Memory Test Battery for Children with story problems. In this study, working memory
was significantly related to the story problem solving ability of first grade students
(Fuchs et al., 2005).
Swanson and Beebe-Frankenberger (2004) also propose working memory may
play a role in the ability of students to solve mathematical word problems. In a study of
elementary students in grades first though third, Swanson and Beebe-Frankenberger
found that working memory contributed 30% of the variance in students' ability to solve
mathematical word problems. Swanson (2006) investigated working memory in those
same students one year later. Swanson used tasks the author suggest measure visualspatial working memory (matrix tasks; mapping tasks and direction tasks) and the
executive component of working memory (listening span, semantic association,
digit/sentence span and backward digit span). Swanson (2006) found executive function
tasks were the only tasks that predicted students' performance on word problems.
Swanson (2006) contradicts earlier work by Swanson (2004), and Swanson and SachseLee (2001). Using identical tests of working memory (both visual-spatial and executive)
both studies identified that the visual-spatial component of working memory was a
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unique predictor of 8 and 11 year-old students' solution accuracy on mathematical word
problems (Swanson, 2004; Swanson and Sachse-Lee, 2001). The role of short-term
memory in the solution of word problems is also unclear.
The literature is not clear on the relationship between mathematical word
problems and short-term memory. Swanson and Beebe-Frankenberger (2004) compared 8
and 11 year-old students' short-term working memory, using the Digit Span subtest of the
WISC-III, and a pseudo-word span task, to the students' ability to solve word problems.
Swanson and Beebe-Frankenberger found that short-term memory was a significant
predictor of solution accuracy on word problems. The use of the backward digit span
element of the WISC-III (as a measure of short-term memory) may be inappropriate
because it requires the subject to mentally reverse digits. Mental reversal is a task more
often associated with working memory. Passolunghi and Siegel (2001) also found a
relationship between the solution of word problems and short-term memory. Passolunghi
and Siegel investigated fourth graders differentiated as poor problem solvers and good
problem solvers. The authors used short-term memory tasks such as the Digit Span
subtest of the Wechsler Intelligences Scale for Children-Revised (WISC-R) and an author
constructed word span task. Passolunghi and Siegel found poorer problem solvers had
significantly lower scores on the combined forward and backward digit span subtest of
the WISC-R. As noted previously however, involving the Digit Span backward subtest of
the WISC-R subtest may reduce the clarity regarding the role of short-term memory.
Some research suggests that phonological processing may play a role in solving word
problems.
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The literature regarding the role of phonological processing and word problems is
nebulous. In the Swanson and Beebe-Frankenberger (2004) study, the authors found a
relationship between phonological processing and the solution accuracy of mathematical
word problems. Swanson and Beebe-Frankenberger compared scores across the
combined phonological measures of phonemic deletion, phonologic fluency, and pseudo
words to a set of mathematical story problems. The authors found that phonological
processing was significantly correlated to first, second and third grade elementary
students' accuracy in word problem solving. However, in the final hierarchical regression
model that included a reading factor, phonological processing was not a significant
predictor of the solution accuracy of word problems. The reading factor included tasks
such as word recognition, real-word fluency, rapid letter naming and reading
comprehension. Swanson's and Beebe-Frankenberger's results were confirmed in a more
recent Swanson (2006) study. Comparing the same group of students one year later,
Swanson found that phonological processing measured by the combined score on a
pseudo word task and a phonemic deletion task was significantly correlated to the
accuracy of word problems. When a reading factor that included measures of real word
fluency, word recognition, and reading comprehension were included in the final model,
phonological processing was not a significant predictor of accurate word problem
solving. The results are similar to the Fuchs et al. (2005) study. Fuchs et al. found that
phonological processes (measured by sound matching and rapid digit naming tasks of the
Comprehensive Test of Phonological Processing) were significantly correlated to first
graders ability to accurately solve mathematical word problems. When the data was
analyzed in a multiple regression, phonological processing was not a significant predictor
60
of accurate word problems solving. The literature suggests that while phonological
processing appears to be related to solving mathematical word problems some other
cognitive process may influence its ability to be a predictor.
The research while not always clear suggests there are specific cognitive
processes involved in the solution of mathematical word problems. The literature
maintains attention, working memory, short-term memory and phonological processing
may play a role in the solution of word problems. In relationship to the cognitive
processes involved in specific applications of mathematics, experts in the field of math
and learning disabilities suggest there may be different types of MD in students. The next
section will explore this paradigm.
Subtypes of Mathematical Disabilities
Children with MD are a heterogeneous group (Robinson et al., 2002; Kroesberger,
et al., 2003). Some have suggested there may be processing related differences between
students with MD (Cornoldi, Venneri, Marconato, Molin and Montinari, 2003; Geary,
2004; Jordan 1995). Perhaps one of the most useful approaches to identifying difference
among MD in children is the MD subtype categories proposed by David Geary (1993;
1996; 2004). Geary has conducted significant research in the area of MD, "Although not
quantitative analysis, one of the most comprehensive syntheses of the cognitive literature
on MD was conducted by Geary" (Swanson & Jerman, 2006, p. 249). Geary (1993; 1996;
2004) alone and with others (Geary and Hoard, 2003) proposed that MD could be divided
into three specific subtypes of disabilities: 1) Procedural; 2) Semantic Memory; 3)
Visual-spatial. The procedural MD subtype is identified in students who use less mature
mathematical procedures (i.e. finger counting in upper elementary grades), exhibit
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frequent procedural errors and have sequencing difficulties (Geary, 2004). Students with
the semantic memory MD subtype have difficulty with the retrieval of mathematical facts
from long-term memory (LTM) and produce a significant number of errors when facts
are retrieved from LTM (Geary, 2004). This subtype and its inability to retrieve facts
from LTM may bear marked similarity to individuals with reading disabilities, which also
often have LTM retrieval difficulties (Geary, 2004). The final MD subtype, identified by
Geary (2004), is the visual-spatial subtype. Students with a math disability in the visualspatial subtype may demonstrate difficulties with the application and comprehension of
information that is presented spatially (geometry problems and mathematical word
problems) (Geary, 2004).
Summary
The literature suggests there are a substantial number of students that struggle
with mathematics in public schools. The federal government and the majority of states
define a disability in mathematics as a disorder in a basic psychological process. There is
a disagreement in the literature and with SEAs on how to identify a MD among students.
Research supports that there are specific psychological processes involved in calculation
and fluency tasks (phonological processing, working memory, long-term memory,
attention and the PASS cognitive processes) and mathematical word problems (attention,
working memory, short-term memory and phonological processing). Research suggests
there may be three specific subtypes of MD (procedural, semantic memory, and visual
spatial).
The literature supports working memory is an important cognitive process used in
mathematical thinking (Geary, 2004; Floyd, et al. 2003; Fuchs et al., 2005; McLean &
62
Hitch, 1999; Swanson, 2006; Swanson, 2004; Swanson & Beebe-Frankenberger, 2004;
Swanson & Jerman, 2006). One sub-process of working memory involved with
mathematical calculation, mathematical fluency and mathematical word problems is
visual-spatial processing (Assel, et al., 2003; Augustyniak, Murphy, and Phillip, 2005;
Floyd, et al. 2003; Fuchs, et al., 2005; Geary, 2004; Hegarty and Kozhevnikov, 1999;
Jordan, et al, 2003; Swanson, 2004; Swanson and Sachse-Lee, 2001; Swanson & Jerman,
2006). Some have suggested that the role of visual-spatial processing in mathematical
thinking has not yet been fully explored and further research is needed (Floyd, et al.
2003; Forest, 2004; Fuchs, et al., 2005; Garderen & Montague, 2003; Geary, 1993;
Geary, 1996; Geary 2004; Mazzocco & Meyers, 2003; Reuhkala, 2001). Given that the
literature supports further investigation into the importance of visual-spatial processing
and mathematical thinking the next section will look specifically at visual-spatial
processing and mathematics.
Visual-Spatial Processing and Mathematics
Geary (1993; 1996; 2004), Jordan (1995) and Cornoldi et al. (2003) have
identified a subtype of student with MD that has a specific deficit in visual-spatial
processing. An individual with a visual-spatial processing MD subtype exhibits deficits in
representing both numerical and other forms of information that are mathematically
based (Geary, 2004). Additionally, individuals identified with this subtype often are
unable to correctly conceptualize mathematical concepts when exposed to information
that is spatially represented (Geary, 2004). Some specific skill deficits students in this
category demonstrate are discrimination difficulties between letters that look alike,
incorrectly transposing shapes and figures, improper number alignment in mathematical
63
problems, poor estimation skills and difficulty with word problems (Assel, et al, 2003;
Augustyniak, Murphy, and Phillip, 2005; Geary, 2004; Jordan, et al., 2003). The next
section will discuss the relationship between mathematics and visual-spatial processing.
Visual-Spatial Processing's Relationship to Mathematics
Visual-spatial processing has been identified as a foundational skill to the
understanding and development of basic mathematical skills, and a pathway to more
efficient problems solving (Assel, et al., 2003; Augustynaik, et al., 2005). Assel, et al.
(2003) tested children on visual-spatial processing and mathematical ability at 2, 3, 4, 6,
and 8 years of age. The authors compared the visual-spatial subtests of the Stanford
Binet- IV (SB-IV) (Pattern Analysis, Copying) to the Calculation subtest of the
Woodcock-Johnson Test of Academic Achievement- Revised (WJ-R) and the
Quantitative Reasoning subtest of the SB-IV. Assel et al.'s findings suggest a link
between the early visual-spatial processing at two and three years of age and the
mathematical ability of children at eight years old. Assel et al. also states the influential
nature of visual-spatial processing was not surprising given the study's math tasks
included word problems (Quantitative Reasoning). Finally, Assel et al. posits word
problems are highly related to visual-spatial skills in comparison to basic calculation
skills. Visual-spatial processing also appears to be an important factor in the development
of such basic skills as cardinality. Cardinality is the understanding the total number of
items in a set is equivalent to the last number in the count (Carr & Hettinger, 2003).
Ansari, et al. (2003) identified a relationship between visual-spatial processing
and cardinality in children's normal mathematical development. Ansari, et al. in their
study investigated children identified as having Williams Syndrome (WS), a disorder that
64
exhibits a profile of substantial impairment in non-verbal ability with intact verbal
abilities. The authors' found in comparison to the Williams Syndrome group, a control
group's visual-spatial processing rather than their language ability was a greater predictor
of their understanding of cardinality when controlling for age (Ansari, et al., 2003). The
results suggest a deficit in visual-spatial processing is related to a deficit in understanding
cardinality (Ansari et al, 2003). Research has also been conducted with older children
identifying a link between visual-spatial processing and mathematics. Reuhkala (2001)
used 15-16 year old high school freshman in her study of visual-spatial processing and
mathematics. Reuhkala investigated the relationship between three visual-spatial tasks
(static, dynamic and mental rotation) and students' scores on a mathematical achievement
test. Finally, Reuhkala found that visual-spatial processing correlated significantly with
the students' scores on a test of mathematical achievement. There also appears to be a
specific connection between poor visual-spatial processing and MD.
Research suggests there is a link between students with disabilities in
mathematics and less developed visual-spatial processing (Busse, et al., 2003). There is
however, a limited amount of research in this area and further research is suggested
(Fuchs, 2005; Geary. 1993; Mazzocco & Meyers, 2003; Reuhkala, 2001). Reuhkala
(2001) determined that visual-spatial processing was related to the mathematical skill
levels of 15-16 year old students. Using tasks such as a matrix, modified block tapping
and mental rotation, Reuhkala found a significant relationship between students that
demonstrate low ability in mathematics (MD) and poor visual-spatial processing. A link
between MD and less developed visual-spatial processing has also been found with
college level students. McGlaughlin, et al. (2005) compared college students identified as
65
MD and non-MD. College students identified with MD exhibited more pronounced
visual-spatial difficulties than non-MD students (McGlaughlin, et al, 2005). The results
of the study identified significantly lower Wechsler Adult Intelligence Scale-Third
Edition (WAIS-III) Performance IQ subtests scores in the MD group compared to the
non-MD group (McGlaughlin, et al., 2005). McGlaughlin, et al. contends that MD
students' lower Performance IQ scores indicated a deficient in visual-spatial processing.
There is also research suggesting visual-spatial skills relationship to mathematical
functioning is less clear.
Bull et al's (1999) investigation assessed visual-spatial skills using the Corsi
Blocks task. The authors did not find an association between mathematical ability and the
visual-spatial processing in a group of 7-year-old children. The authors however, admit
the potential weaknesses of their visual-spatial measure. Bull et al. posits that visualspatial tasks not involving a memory-span requirement (such as the Corsi Blocks) may
provide conclusive support for a linkage between visual-spatial processing and
mathematics. Other researchers have suggested a reason that Bull et al. did not find a
relationship may be due to the age of the participants in the study (Reuhkala, 2001).
Reuhkala (2001) suggests that because the working memory capacities of students 7
years of age or younger have yet to develop completely, it could affect the measurement
of the interaction between the use of working memory and mathematics. Additional
alternative evidence for a relationship between visual-spatial processing and mathematics
comes from Lee, Ng, Ng, & Lim, (2004). Lee et al. investigated central executive
functions (phonological loop and visual-spatial sketchpad) and mathematical
performance with 10-year-old students from Singapore. Lee et al. found that neither the
66
visual-spatial sketchpad nor the phonological loop contributed to mathematical
performance. Finally, Lee et al. implies overall executive functioning combining
elements of both the visual-spatial sketchpad and the phonological loop are the
contributing factors to mathematical performance, not one element in isolation (Lee et al,
2004)
The literature suggests there may be a relationship between the development of
mathematical skills and visual-spatial processing. Research also suggests a link between
poor visual-spatial processing and students with MD. Complete understanding of the
relationship between mathematics and visual-spatial processing requires clear
conceptualization of visual-spatial processing. The next section will explore visual-spatial
processing.
Visual-Spatial Processing
Visual-spatial processing has been defined as "The ability to generate, retain,
retrieve and transform well-structured visual images "(Lohman, 1994, p. 1000). The main
purpose of this sub-section is to coalesce what the literature reports about the term visualspatial and the concept of visual-spatial processing. The discussion will address the
visual-spatial sketchpad as a component of working memory. The section will continue
by looking at age and gender differences in regards to visual-spatial processing. It will
conclude with discussing the neuropsychological aspects of visual-spatial processing.
Constructs of visual-spatial working memory
Visual-spatial processing, is believed to be an element of the psychological
process of working memory. Working memory is the cognitive process that allows one to
keep information at the forefront of one's thoughts while mentally manipulating that
67
information (Geary, 1996). The most frequently identified theory of working memory in
the literature, is a theory conceptualized by Alan Baddeley (Fisk & Sharp, 2003; Geary,
2004; Pickering & Gathercole, 2004; Reuhkala, 2001; Sholl & Fraone, 2004; Swanson,
2004; Swanson & Beebe-Frankenberger, 2004) Baddeley's theory separates working
memory into four fractional parts central executive, phonological loop, visual-spatial
sketchpad and episodic buffer (Baddeley, 1996; Pickering & Gathercole, 2004).
Conceptualizing visual-spatial processing requires a discussion of Baddeley's model in
its totality.
The first component of Baddely's theory is the central executive system. The
central executive is a malleable system that maintains the responsibility for the regulation
of such processes as efficiency in multiple cognitive tasks, the vacillation among tasks or
strategies for retrieval, and the inhibition of and attention to incoming information
(Baddeley, 1996; Pickering & Gathercole, 2004). The central executive is aided by two
main assisting systems the phonological loop and the visual-spatial sketchpad (Baddeley,
1996; Pickering & Gathercole, 2004). More recently a third assisting system has been
identified by Baddeley, the episodic buffer (Pickering & Gathercole, 2004). The
phonological loop is a temporary storage area that holds auditorialy-based or speechbased information (Baddeley, 1996).The most recent component added by Baddeley in
2000, the episodic buffer, integrates information from the components of working
memory and long-term memory with multidimensional codes (Pickering & Gathercole,
2004). The visual-spatial sketchpad is responsible for processing visual-spatial
information (Reuhkala, 2001).
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Visual-Spatial Sketchpad
The visual-spatial sketchpad is a highly organized element of working memory
specializing in the maintaining and processing of information that demonstrates visual or
spatial characteristics (Pickering & Gathercole, 2004). According to Baddeley's model
the sketchpad is a limited duration storage center of mental representations (i.e. visualspatial properties of a physical stimulus) that has been formed from visual input or
retrieval from long term memory (Sholl, & Fraone, 2004). Most often visual-spatial
information that is temporarily stored in the visual-spatial sketchpad is used to solve
problems of a spatial nature such as anticipating spatial transformations, mental
rearrangement of items and visualizing the relationship of parts to a whole (Sholl &
Fraone, 2004).
Within the visual-spatial systems, both visual and spatial aspects are strongly
threaded together; however, it has been hypothesized that within the visual-spatial system
the visual and spatial aspects are processed differently (Baddeley, 1996; Richardson &
Vecchi, 2002; Sholl & Fraone, 2004). Sholl and Fraone (2004) and Baddeley (1996)
maintain that the evidence for this separate, but intertwined processing comes from
studies that use interference tasks to isolate each processing construct. Sholl and Fraone
(2004) posit that a task such as non-sighted tapping of the four corners of a square grid
during a visual-spatial assessment interferes with spatial ability, but does not interfere
with temporary visual storage. Conversely, tasks that utilize visual processes, such as
observing dot patterns or looking at abstract paintings, provide interference with visual
processing, but not with spatial processing (Sholl and Fraone, 2004). Baddeley (1996)
believes evidence for this separation can be found in neuropsychological studies that
69
show patients with a disruption of spatial imaging skills yet largely intact and unaffected
visual description ability.
The literature also suggests there are two types of visual-spatial processing,
passive and active (Richardson & Vecchi, 2002; Reuhkala, 2001; Vecchi & Cornoldi,
1999). Active processing involves taking in visual-spatial information while
transforming, manipulating or utilizing that information (Vecchi & Cornoldi 1999).
Passive visual-spatial processing requires retaining information that requires no
manipulation or modification (Vecchi & Cornoldi, 1999). Evidence for this
differentiation between passive and active visual-spatial memory comes from the
evaluation of visual-spatial processing differences in aging adults. The studies suggest
active visual-spatial processing deteriorates during the aging process, while passive
visual-spatial processing remains largely intact. Vecchi and Cornoldi (1999) investigated
different aged adults (averaged ages 22, 66 and 76) utilizing assessments that tapped both
passive visual spatial memory (tasks that only required short-term recall of visual-spatial
information) and active tasks (tasks that required mental rotation and/or visualization).
Vecchi and Cornoldi found there were distinct differences in passive and active
assessments among the age groups. Vecchi and Cornoldi results indicated there was a
significant decline in active visual-spatial processing in the older subjects, providing
support for the separation of passive and active visual-spatial processes. The decline in
active visual-spatial tasks was further confirmed by Richardson and Vecchi (2002).
Richard and Vecchi used a jigsaw-puzzle imagery task they suggest taps active visualspatial memory. Once again using three varying age groupings young adults (18-27), old
adults (60-75) and older adults (75-90) they found a significant decline in active visual-
70
spatial processing in the older population (60-90) in comparison to the younger
population (18-27).
Gender and Cultural Differences in Visual-Spatial Processing
In addition to differences in ages, there may be gender differences and possibly
cultural differences in visual-spatial processing. Both Geary (1996) and Richardson and
Vecchi (2002) in their analysis of literature on visual-spatial processing found evidence
to suggest that there are gender differences in visual-spatial tasks. The difference between
male and female visual-spatial processing may be a result of both biologically based and
environmentally based differences (Geary, 1996; Richardson & Vecchi, 2002). However,
the authors differ in regards to a specific superiority of one gender to the other. Geary
(1996) maintains that literature favors a male superiority in regards to visual-spatial
processing. Richardson and Vecchi (2002) maintain the literature supports that males
seem to be better at active visual-spatial tasks that require mental rotation or
transformation of images; however, there appears to be little difference in passive visualspatial tasks between males and females. In addition to gender differences, there may be
cultural differences in visual-spatial processing primarily as a function of educational
experience and type of measure (Rosselli & Ardila, 2003). Rosselli and Ardila (2003)
stress that caution should be used in utilizing nonverbal measures (such as visual-spatial
assessments) with different cultural groups. Differences in visual-spatial measures among
cultures may be due to a combination of factors, including the interaction of brain
organization, cultural experience and learning/education (Rosselli & Ardila, 2003).
Rosselli and Ardila suggest tests of visual-spatial processing may be lower or higher
among dissimilar cultures and that researchers need to be cognizant that those differences
71
will occur. The final subsection will identify relevant neuropsychological aspects of
visual-spatial processing.
Neuropsychology and Visual-Spatial Processing
It is important in understanding visual-spatial processing to have a
conceptualization of its relationship to neuropsychological functioning. Most researchers
agree that there is a clear link between the right hemisphere and visual-spatial processing
(Cornoldi, Venneri, Marconato, Molin & Montinari 2003; Geary, 1993; Harnadeck &
Rourke, 1994; Morris & Parslow, 2004; Young & Ratcliff, 1983). That is not to say that
visual-spatial functioning occurs in isolation in the right hemisphere, rather the right
hemisphere's contribution is more evident in complex visual-spatial tasks (Young &
Ratcliff, 1983). Less complex visual-spatial tasks are spread relatively moderately
between the two hemispheres (Young & Ratcliff, 1983). Support for the right
hemispheric localization of visual-spatial processing comes from studies that have used
patients with right and left hemispheric dysfunction or lesions. Consistently researchers
have found that individuals with right hemispheric dysfunction or lesions show severe
deficits in visual-spatial functions (Cornoldi, et al., 2003). Further research into the
neuro cognitive components of working memory, using computational modules, reveals
that the hippocampus is the main brain structure significantly involved with visual-spatial
functioning (Morris & Parslow, 2004). More specifically the parietal cortex is involved in
spatial aspects and the inferotemproal area is involved in the visual processing aspects
(Morris & Parslow, 2004).
72
Summary
The research indicates there may be a relationship between the development of
mathematical skills and visual-spatial processing. In addition, the literature suggests a
link between poor visual-spatial processing and students with MD. Visual-spatial
processing is a subpart of the larger cognitive process of working memory. Literature
suggests visual-spatial processing is controlled or regulated by the central executive
system. The literature also suggests visual and spatial information are processed
separately; however when information is recalled it is produced as a gestalt. Researchers
have identified that visual-spatial processing can be further broken down in to passive
and active processes. In addition, there are differences in visual-spatial processing in
regards to age, gender and culture. Finally, visual-spatial functioning is mainly a right
hemisphere activity that involves the hippocampus, the parietal cortex and inferotemproal
areas of the brain. A relatively recent theory of intelligence provides another mode
towards conceptualizing visual-spatial processing and how it is assessed.
Modern Intelligence Theory and Assessing Visual-Spatial Processing
There is a disagreement in the literature regarding how visual-spatial processing
is assessed (see table 2.1). The disagreement stems from which aspects of visual-spatial
processing are measured (Proctor, Floyd, & Shaver, 2005). According to McGrew (2005)
generally, tasks that are believed to measure visual-spatial possessing involve figural or
geometric structures that necessitate the visual recognition and manipulation of" visual
shapes, forms, or images, and/or tasks that require or maintain spatial orientation with
regard to objects that may change or move through space" (McGrew, 2005 p. 152).
Carroll (1993) in his work with the factor analysis of cognitive abilities may provide the
73
most comprehensive review of measures of visual-spatial processing. Understanding
Carroll's work with visual-spatial processing begins with understanding modern
intelligence theory.
Table 2.1
Tasks used to measure visual-spatial processing in the current literature
Task
Number of studies
Block Tapping Tasks
4
Block Design or Block Construction Task
5
Matrix Tasks
8
Map, Direction or Route Finding Tasks
8
Mental Rotation Tasks
3
Interview or Rating Scale
4
Jigsaw Puzzle Tasks
2
Pattern Construction/ Analysis
2
Copying Figures or Shapes
3
74
Fluid
Intelligenc
e(Qf)
Crystallize
d
Intelligenc
12.
1. General
Sequential
Reasoning
(RG)
2. Induction
0)
3. Quantitative
Reasoning
(QR)
4. Piagetian
Reasoning
(RP)
5. Speed of
Reasoning
(RE)
1. Language
Development
(LD)
2. Lexical
Knowledge
(VL)
3. Listening
Ability
(LS)
4. Verbal
Information
(KO)
5. Culture
Information
(K2)
6. Communication
Ability
(CM)
7. Oral
Production /
Fluency
(OP)
8. Grammatical
Sensitivity
(MY)
9. ForeignLanguage
Proficiency
(KL)
10. ForeignLanguage
Aptitude
(LA)
11. Science
Information
(Kl)
12. Geography
Achievement
(A5)
ShortTerm
Memory
±t
1. Memory
Span
(MS)
2. Working
Memory
(WM)
Visual
Processing
(Gv)
1. Visualization
(VZ)
2. Spatial
Relations
(SR)
3. Closure
Speed
(CS)
4. Flexibility
of Closure
(CF)
5. Visual
Memory
(MV)
6. Spatial
Scanning
(SS)
7. Serial
Perception
Integration
(pi)
8. Length
Estimation
(LE)
9. Perceptual
Illusions
$L)
10. Perceptual
Alterations
(PN)
11. Imagery
(IM>
12. Perceptual
Speed
fPSl
Long-term
Retrieval &
Storage
Auditory
Processing
(Ga)
(Glr)
Reading and
Writing
(Grw)
Processing
Speed
(Gs)
HE
±2.
1. Associative
Memory
(MA)
2. Meaningful
Memory
(MM)
3. Free-recall
Memory
(M6)
4. Ideational
Fluency
(FI)
5. Expressional
Fluency
(FE)
6. Naming
Facility
(NA)
7. Word
Fluency
(FW)
8. Figural
Fluency
(FF)
9. Figural
Flexibility
(FX)
10. Sensitivity
to Problems
(SP)
11. Originality/
Creativity
(FO)
12. Learning
Abilities
(Ll)
13. Associative
Fluency
(FA)
1. Phonetic
Coding
(Analysis and
Synthesis)
(PC)
2. Speech
Sound
Discrimination
(US)
3. Resistance
to Auditory
Distortion
(UR)
4. Memory
for Sound
Patterns
(UM)
5. General
Sound
Discrimination
(U3)
6. Temporal
Tracking
(UK)
7. Musical
Discrimination
and Judgment
(Ul, U9)
8. Maintaining
and Judging
Rhythm
(U8)
9. Sound
Intensity/
Duration
Discrimination
(U6)
10. Sound
Frequency
Discrimination
(U5)
11. Hearing
and Speech
Threshold
(UA, UT,
UU)
12. Absolute
Pitch
(UP)
13. Sound
Localization
(UL)
Figure 2.1. CHC Broad and Narrow Cognitive abilities were adapted
from Carroll (1993), McGrew (2005), and Alfanso, Flanagan, and
Radwan, (2005). Boldface type indicates Narrow Cognitive abilities
with some disagreement on placement among the three authors.
75
i£
1. Perceptual
Speed
(P)
1 a. Partem
recognition
(Ppr)
lh. Scanning
(Ps)
lc. Memory
(Pm)
Id. Complex
(Pc)
2. Rate of
Test
Taking
(R9)
3. Number
Facility
(N)
4. Speed of
Reasoning
(RE)
1. Reading
Decoding
(RD)
2. Reading
Comprdiension
(RC)
3. Verbal
Language
Comp reh en s ion
(V)
4. Cloze
Ability
(CZ)
5. Spelling
Ability
(CZ)
6. Writing
Ability
(WA)
7. English
Use
Knowledge
(EU)
8. Reading
Speed
(RS)
9. Writing
(WS)
Quantitative
Knowledge
(Gq)
Decision/Reaction
Time
(Gt)
i i
1. Mathematical
Knowledge
(KM)
2. Mathematical
Achievement
(MA)
1. Simple
Reaction
Time
(Rl)
2. Choice
Reaction
Time
(R2)
3. Semantic
Processing
Speed
(R4)
4. Mental
Comparison
Speed
(R7)
5. Inspection
Time
(IT)
6. Correct
Decision
Speed
(CDS)
* McGrew (2005) in his work on the CHC theory has added
the following Broad Cognitive abilities:
1. General knowledge (Gkn)
2. Psychomotor abilities (Gp)
3. Olfactory abilities (Go)
4. Tactile abilities (Gh)
5. Kinesthetic abilities (Gk)
Because these are less known cognitive abilities and are not
often cited in the literature they were not included in the
figure.
CHC Theory
The Cattell-Horn-Carroll (CHC) theory of intelligence combines the work of
Raymond Cattell, John Horn and John Carroll (Alfonso, Flanagan & Radwan, 2005;
McGrew, 2005). CHC theory has been influential with modern measurements of
cognitive abilities, "The CHC theory is the most comprehensive and empirically
supported psychometric theory of the structure of cognitive and academic abilities to
date" (Alfonzo et al., 2005 p. 185). Validation of the CHC theory has come from factor
analytical studies (Carroll, 1993; McGrew, 2005). The CHC theory of intelligence has a
three tiered structure that consists of a general factor of intelligence or "g", 10 broad
factors of intelligence, and approximately 70 narrow factors of intelligence (see Figure
2.1 for a diagram of the CHC Theory of Intelligence) ( Sattler, 2001; Evans, Floyd,
McGrew, & Leforgee 2002; McGrew, 2005). The following section will explore visualspatial processing and CHC theory.
Visual-Spatial Processing and CHC Theory
In the literature, visual-spatial processing and the Visual Processing (Gv) broad
cognitive ability of the CHC theory are consistently treated as the same construct
(Alfonzo et al., 2005; DiStefano & Dombrowski, 2006; Evans et al., 2002; Floyd, et al.
2003; McGrew, 2005; Osmon, Smerz, Braun, & Plambeck, 2006; Proctor et al., 2005). In
addition, tests that purport to measure Gv interchangeably use visual-spatial processing
and Gv to identify this cognitive ability (Roid, 2003a). Further support for equivalence of
the Gv and visual-spatial processing terms comes from an analysis of the narrow Gv
cognitive abilities identified by Carroll (1993). The Gv narrow cognitive abilities consist
of measures of both visual components and spatial components. The list of narrow
76
cognitive abilities suggests that terminology that includes visual and spatial elements is
equitable to Gv. The Gv area consists of a collection of processes that involve production,
mentally holding, recalling, and the manipulation of visual images (McGrew, 2005).
Under the Gv domain, there are approximately 12 narrow cognitive abilities including: 1)
Visualization (VZ); 2) Spatial relations (SR); 3) Closure speed (CS); 4) Closure
flexibility (CF); 5) Visual memory (MV); 6) Spatial scanning (SS); 7) Serial perception
integration (PI); 8) Length estimation (LE); 9) Perceptual illusions (IL); 10) Perceptual
alterations (PN); 11) Imagery (IM); 12) Perceptual Speed (PS) (Carroll. 1993; Lohman,
1994; McGrew, 2005; Sattler, 2001). The next section will explore each of these narrow
cognitive abilities and their measures.
Visualization (VZ). VZ is the ability to simultaneously view a spatial construct,
compare it to another spatial construct, often while mentally rotating the image in a two
or three-dimensional field (Carroll, 1993; McGrew, 2005). Tests that are believed to
measure this narrow cognitive ability are assembly tasks such as the Block Design and
Object Assembly tasks of the Wechsler series and the From Board tasks and Form
Patterns tasks of the Stanford-Binet series (Carroll, 1993; G. H. Roid, personal
communication, November, 7 2006; Sattler & Dumont, 2004). Other measures of this
factor include assembly type tasks, block counting tasks, block rotation tasks, paper
folding tasks, surface development tasks, and figural rotation tasks (Carroll, 1993;
Lohman, 1994).
Spatial relations (SR). SR is the rapid perception and manipulation of visual
stimuli and can also be the maintenance or orientation of objects in space (Carroll, 1993
McGrew, 2005). SR is differentiated from VZ by its emphasis on fluency (McGrew,
77
2005). Tasks that measure SR include irregular card comparisons, and cube comparison
tasks (Carroll, 1993; Lohman, 1994). Sattler and Dumont (2004) suggest that the Block
Design subtest of the Wechsler intelligence assessment series may also be a measure of
SR.
Closure speed (CS). CS is the rapid visual recognition of an incomplete object,
form or pattern without prior knowledge of the form, when it is presented in a masked
way (Carroll, 1993 McGrew, 2005). Carroll (1993) contends that there are four elements
of a CS measure: 1) The stimuli are obscured in some manner; 2) The stimuli are well
known; 3) The subject is asked to name the stimuli; 4) The subject's response is
evaluated for efficiency. Some tasks that are said to measure CS include the Street
Gestalt Completion test and tasks that included concealed letters, numbers or figures.
Sattler and Dumont (2004) state the Object Assembly task of the Wechsler intelligence
test series may be a measure of CS.
Closure flexibility (CF). CF is the efficient visual recognition of an imbedded and
obscured object or pattern with prior knowledge of the pattern or object (Carroll, 1993;
Lohman, 1994; McGrew, 2005). Four important elements of measures of CF are: 1) The
stimulus is "geometrically camouflaged" 2) The stimuli that is masked is a design based
on known shapes; 3) The subject has prior knowledge of the design; 4) The test is time
sensitive (Carroll, 1993). Tasks that measure CF include tests that have hidden or
embedded figures, designs or patterns (Carroll, 1993).
Visual memory (MV). MV is the ability to recognize or recall a visual stimulus
after a brief exposure (Carroll, 1993; McGrew, 2005). Some disagreement on MV as a
Gv narrow cognitive ability exists. Carroll (1993) believes that MV is more aptly placed
78
in the domain of Memory and Learning due to the memory components involved.
McGrew (2005) believes MV to be a Gv measure due to the visual elements involved.
Tasks that measure MV involve maps, pictures, designs or shapes (Carroll, 1993). Most
often subjects are briefly exposed to the stimuli and then must recall or redraw the
stimuli. Sattler (2001) implies that the Memory for Objects subtest of the Stanford-Binet
Fourth Edition may be a measure of MV.
Spatial scanning (SS). SS involves the ability to efficiently visually identify or
follow a path through a complicated visual field (Carroll, 1993; McGrew, 2005). Carroll
(1993) reports that little evidence for this factor exists in his analysis of data on Gv
measures. Some have suggested that tasks that involve maze tracing or planning and
following a route on a two dimensional map may measure this ability (Carroll, 1993).
Sattler, (2001) suggests that the Mazes subtest in the Wechsler intelligence assessment
series may measure this Gv component.
Serial perception integration (PI). PI is the ability to name a pattern (visual or
pictorial in nature) rapidly presented in ordered and segmented parts (Carroll, 1993;
McGrew, 2005). Carroll (1993) offers few examples of measures of PI. Carroll did find
that a gestalt completion task loaded on this factor. Further research regarding this factor
may now be possible due to technological advances (Carroll, 1993).
Length estimation (LE). LE is the unaided estimation or comparison of lengths or
distances (Carroll, 1993; McGrew, 2005). LE assessment can be based on liner segment
comparison or path length estimation (Carroll, 1993). Carroll (1993) states there is not a
preponderance of evidence from database analysis that LE is a identifiable factor in Gv.
79
Tasks that may measure LE include length discrimination, length estimation, and
comparison and proximity analysis of lines and points (Carroll, 1993).
Perceptual illusions (IL). IL involves the ability to inhibit the interference of the
inconsequential aspects of geometric shapes (Carroll, 1993; McGrew, 2005). Carroll
(1993) and McGrew (2005) disagree on the existence of IL. McGrew believes IL to be an
important narrow ability of Gv. Carroll (1993) notes that the data supporting IL is limited
and implies the idea that IL as a narrow cognitive ability is not conclusive. Researchers
have suggested tasks that measure IL may include the estimation, contrasting, shape
identification or direction identification of illusions (Carroll, 1993).
Perceptual alterations (PN). PN is the accuracy and efficiency in vacillation
between different visual stimuli (Carroll, 1993; McGrew, 2005). McGrew (2005)
suggests that PN is an important component of Gv. Carroll (1993) however, in his
analysis of data on Gv notes that PN did not correlate with any other measure of
perceptual abilities, suggesting separation from other Gv measures. Carroll suggests that
PN measurement tasks involve mental alternations of stimuli under timed conditions.
Imagery (IM). IM is the mental depiction or manipulation of a stimulus that is in a
spatial abstract figure (Carroll, 1993; McGrew, 2005). Researchers suggest that image
processing may play a significant role in solving spatially based tasks, and is separate
from SR and VZ factors (Carroll, 1993). Carroll (1993) cautions that existence of this
factor is equivocal because the datasets he explored did not consistently substantiate IM
as a factor. Tasks that may measure IM require the subject to visually manipulate an
object and compare it to other similar non-manipulated objects (Carroll, 1993).
80
Perceptual Speed (PS). PS is the efficacy in identifying an unmasked pattern in
isolation or comparing more than one unmasked pattern presented in a visual field
(Carroll, 1993). There is some disagreement where PS should fall as a narrow cognitive
ability. McGrew (2005) suggests that PS is best described as a narrow ability in the broad
cognitive ability of Cognitive Processing Speed (Gs). Carroll (1993) contends that PS is
better defined as a Gv narrow ability. Carroll maintains there are two types of tests that
may measure PS: 1) Tests that require efficiency in identifying visual stimuli in a display
without distracters; 2) Tests that require efficient comparison of stimuli in a broad or
narrow display. Sattler and Dumont (2004) suggest that the Cancellation and Symbol
Search subtests of the Wechsler Intelligence Scale for Children-Fourth Edition may be
measures of PS.
Recent developments in intelligence assessment have attempted to more
completely align IQ measures with the CHC Theory of intelligence. This alignment has
meant improved attempts to measure board and narrow CHC abilities such as Gv or
visual-spatial processing. In 2003, the two most well known measures of child
intelligence were revised, the Stanford Binet Intelligence Scales, Fifth Edition (SB5)
(Roid, 2003) and the Wechsler Intelligence Scale for Children- Fourth Edition (WISCIV) (Wechsler, 2003). It is implied that because the SB5 and the WISC-IV are more in
line with CHC theory that their visual-spatial measures will be grounded in one or more
of the 12 narrow Gv cognitive abilities The next section will discuss the SB5 and WISCIV as measures of visual-spatial processing.
81
Table 2.2
Subtests and Domain Construction of the SB5 Full Scale IQ
Domain
Factor
Subtest
Fluid
Object Series/Matrices
Reasoning
Procedural Knowledge
Nonverbal
Knowledge
Picture Absurdities
Quantitative
Quantitative Reasoning
Reasoning
Visual-Spatial
Form Board
Processing
Form Patterns
Working
Delayed Response
Memory
Block Span
Domain
Factor
Subtest
Early Reasoning
Fluid
Verbal Absurdities
Reasoning
Verbal Analogies
Verbal
Knowledge
Vocabulary
Quantitative
Quantitative Reasoning
Reasoning
Visual-Spatial
Position and Direction
Processing
Working
Memory for Sentences
Memory
Last Word
Note. Adapted from Stanford-Binet Intelligence Scales, Fifth Edition: Examiners Manual,
(p. 50) by G. H. Roid, 2003, Itasca, IL: Riverside Publishing.
Stanford-Binet: Fifth Edition
The ancestors of the modern Stanford-Binet intelligence scales can be consider
the starting point for all modern intelligence assessments. The Stanford-Binet scales have
undergone numerous revisions that began with the 1916 American adaptation of the
Binet-Simon scales by Lewis Terman (Becker, 2003). The Stanford-Binet scales have
continued to evolve from two parallel forms (Form L and From M), to a combined form
(Form L-M), followed by the Stanford-Binet Intelligence Scale: Fourth Edition in 1986
(Becker, 2003). The most recent revision is the Stanford-Binet Intelligence Scale: Fifth
Edition (SB5) authored by Gail H. Roid (Becker, 2003; Roid, 2003a; Roid, 2003b). The
82
SB5 is designed to align closely with modern CHC intelligence theory (Becker, 2003;
Melko & Burns, 2005). There is limited research regarding the SB5 factor structure.
Additionally, the literature that has been produced on the SB5 factor structure has been
mixed.
The SB5 is designed around five factors that fall across verbal and non-verbal
domains (DiStefano & Dombrowski, 2006; Roid, 2003a). The five factor areas (and their
corresponding CHC cognitive ability) are Fluid Reasoning (Gf), Knowledge (Gc),
Quantitative Reasoning (Gq), Working Memory (Gsm) and Visual-Spatial Processing
(Gv) (DiStefano & Dombrowski, 2006; Roid, 2003a). These factors are measured on both
the verbal and non-verbal domains and are combined to construct the Full Scale IQ or a
measure of "g" (see table 2.2). Roid (2003a) contends that the results of confirmatory
factor analysis support the legitimacy of the five-factor model. Roid's (2003a)
confirmatory factor analysis used the SB5 scores of 4,786 subjects across five age
groupings (2-5; 6-10; 11-16; 17-50; 51+). The results indicated the five-factor model
provided the best fit for the individuals' scores on the SB5 when compared to reduced
models (Roid, 2003a). Additionally, Roid maintains that additional support for the five
factors comes from cross-battery confirmatory factor analysis with the AnalysisSynthesis, Verbal Comprehension, Spatial Relations, Auditory Memory and the Applied
Problems subtests of the WJ-III batteries. Using the same 4,786 subjects across five age
groupings, the five-factor model provided the best fit for the individuals' scores on the
SB5 (Roid, 2003a). There is alternative factor analytical research that does not support
the five-factor model of the SB5. DiStefano and Dombrowski (2006), using both
exploratory and confirmatory analysis of the SB5 standardization sample, did not find
83
support for the five factors. DiStefano and Dombrowski suggest that a one-factor model
that identifies a general intelligence model or "g", best represents this assessment across
the five age groupings. DiStefano and Dombrowski further suggest that while the SB5
may be a good measure of general intelligence there is little support for the validity of the
five factors. Although Roid does caution using exploratory factor analysis (such as used
by DiStefano and Dombrowski) that does not extract the higher order "g" from the
correlations and using large age spans will not allow for the accurate analysis of lowerorder factors. Currently there is a limited amount of non-publisher generated research
with the SB5 as a measure of visual-spatial processing. The next area will address the
visual-spatial measures of the SB5.
Visual-Spatial Measures of the SB 5
The visual-spatial processing factor is identified in the SB5 as "... the ability to
see relationships among figural objects, describe or recognize spatial orientation, identify
the "whole" among a diverse set of parts and generally see patterns in visual material"
(Roid & Pomplun, 2005 p. 328). The visual-spatial measures of the SB5 were constructed
through consultation with Dick Woodcock, John Horn and John Carroll all experts in the
CHC theory (G. Roid personal communication November 7, 2006). The SB5 visualspatial processing factor consists of a verbal and non-verbal domain. The subtest for the
verbal visual-spatial processing domain is the Position and Direction Subtest (PD). The
PD subtest requires subjects to ".. .identify common objects and pictures using common
visual/spatial terms such as "behind" and "farthest left," explain spatial directions for
reaching a pictured destination or indicate direction and position in relation to a reference
point" (Roid, 2003b p. 139). The subtest begins at the earliest levels with the subject
84
orientating objects (ball and/or block) into certain positions such as "on", "inside" and
"outside" and progresses to more advanced orientation and direction items using terms
such as "left", "right" "east" and "west" (Roid, 2003b). The subtest has its roots in the
early L, and L-M editions of the Stanford-Binet series (Roid, 2003a). Additionally, Roid
(2003a) indicates that the work of Lohman (1994) and Carroll (1993) were influential in
the construction of this subtest. Lohman contends that verbal visual-spatial tests that
require a subject to create a mental image and answer corresponding questions are
representative of real-life usage of visual-spatial processing. However, caution with
application of Gv to the SB5 may be warranted. Roid (2003a) while suggesting that this
subtest was based on CHC theory and is a measure of Gv, does not suggest which narrow
cognitive abilities Position and Direction is said to measure. In addition, in reviewing the
work of McGrew (2005) and Carroll (1993) regarding narrow abilities within CHC
theory, it is uncertain as to exactly which narrow cognitive ability or abilities are
involved in this subtest. The visual-spatial factor of the SB5 also involves subtests in the
nonverbal domain.
The SB5 nonverbal visual-spatial domain involves different measures at different
ability levels. In the early levels of the SB5 (level 1 and level 2) the Form Board is
utilized as a measure of nonverbal visual spatial processing. The Form Board task was
used with previous editions of the Stanford-Binet scales (Roid, 2003b). The task utilizes a
plastic board with three geometric shapes (triangle, square and a circle) recessed into the
plastic. The subject uses whole or parts of geometric pieces to construct the circle, square
and triangle shapes on the Form Board. The SB5 Form Board task is said to incorporate
the broad Gv cognitive ability and the narrow visualization or VZ ability (Carroll, 1993;
85
Roid, 2003b). Upon successful completion of levels one and two, subjects progress to
level three. In level three, the nonverbal visual-spatial task changes to Form Patterns.
From Patterns is a new visual-spatial task designed by the SB5 test development team
(Roid, 2003b). The task involves the construction of recognizable shapes with ten
geomantic pieces (triangle, square, circle, rectangle, and parallelogram). Form Patterns
was developed after extensive review of previously used visual-spatial measures and
stringent field-testing (G. Roid personal communication, November 7, 2006). The task is
believed to measure the broad Gv and narrow cognitive ability of visualization or Vz
(Carroll, 1993; Roid, 2003a). There appears to be a dearth of supporting research
regarding the Gv measures of the SB5.
The Stanford-Binet scales have a long history as a measure of intelligence. The
Stanford-Binet scales have undergone numerous revisions. The current version the SB5
was designed to be more in line with the current CHC theory of intelligence. The SB5 is
composed of five factor scores, Fluid Reasoning, Knowledge, Quantitative Reasoning,
Visual-Spatial Processing, and Working Memory. The SB5 Visual-Spatial Processing
factor consists of verbal (Position and Direction) and non-verbal (Form Board; Form
Patterns) domains. Some subtests in the Visual-Spatial Factor have roots in earlier
Stanford-Binet scales (Form Board; Position and Direction), while Form Patterns is more
recently designed (Roid, 2003a). There appears to be a dearth of research validating the
use of the newly developed Visual-Spatial factor of the SB5 as a measure of visualspatial processing. The lack of research suggests that further research is warranted to
further understand this factor of the SB5. The next section will look at the recently
revised WISC-IV and the subtests said to measure visual-spatial ability.
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Wechsler Intelligence Scale for Children- Fourth Edition
The Wechsler Intelligence Scale for Children-Fourth Edition (WISC-IV) and its
predecessor the Wechsler Intelligence Scale for Children-Third Edition (WISC-III) may
be the most widely utilized and researched measures of children's intelligence (Zhu &
Weiss, 2005). The original Wechsler Intelligence Scale for Children was published in
1949 and was a downward adaptation of the Wechsler-Bellevue Intelligence Scale
(Wechsler, 2003b). Since its inception in 1949, the tests have undergone several revisions
in 1974 and again in 1991. The current revision, the WISC-IV published in 2003, was
undertaken to more accurately align the test with current intelligence theory, strengthen
psychometrics, increase overall applicability of the instrument, and ease evaluator usage
of the instrument (Sattler & Dumont, 2004). Some researchers contend however, that
there is a lack of understanding in what "current intelligence theory" the WISC-IV is
more closely aligned to (Keith et al, 2006).
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Table 2.3
Index and Subtests of the WISC-IV that combine to form the Full Scale IQ
Index
Subtests
Similarities
Verbal Comprehension
Vocabulary
Comprehension
Information (supplemental)
Word Reasoning (supplemental)
Block Design
Perceptual Reasoning
Picture Concepts
Matrix Reasoning
Picture Completion (supplemental)
Digit Span
Working Memory
Letter-Number Sequence
Picture Completion
Coding
Processing Speed
Symbol Search
Cancellation
The WISC-IV has four Index scores Verbal Comprehension, Perceptual
Reasoning, Working Memory, and Processing Speed that combine to form the Full Scale
IQ or measure of "g" (see table 2.3)(Wechsler, 2003a). The revision of the test includes
additional subtests redesigned to improve the measurement of the CHC cognitive abilities
of fluid reasoning, working memory, and processing speed (Wechsler, 2003a; Zhu &
Weiss, 2005). The test's authors maintain support for the four indexes comes from factor
analytical studies of the WISC-IV (Wechsler, 2003a; Zhu & Weiss, 2005). Test authors
used the four factors of the WISC-IV and a standardization sample of 1,525 both
collectively and across four ages groups (6-7; 8-10; 11-13; 14-16) in exploratory factor
analysis (Wechsler, 2003a). Wechsler (2003a) maintains the four factors provided the
best fitting model when compared to less developed models. Confirmatory factor analysis
with the same data confirmed the appropriateness of the four-factor model (Wechsler,
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2003a). Some researchers suggest however, the current revision is removed from any
clear adherence to a theoretical framework (Keith, et al, 2006).
Keith et al. (2006) in their factor analytical research on the WISC-IV suggest the
four index measures may not be the most appropriate structure for the assessment. Keith
et al. propose a five-factor model based on the CHC theory is a more appropriate
framework for the assessment. Keith et al. analyzed the WISC-IV scores of the agedifferentiated (6:0-16.0) 2,000 subject standardization sample identified in the WISC-IV
manual. Keith et al. found that a test framework structured on CHC factors of
Crystallized Intelligence (Gc), Visual Processing (Gv), Fluid Reasoning (Gf), Short-Term
Memory (Gsm) and Processing Speed (Gs) provided the best fit for the data. The results
of the study suggest two things: 1) The WISC-IV does measure intelligence as identified
by the widely accepted CHC theory of intelligence; 2) There may be certain subtests of
the WISC-IV that are specific measures of visual-spatial processing (Gv). The following
section will look at which subtests of the WISC-IV are said to measure visual-spatial
processing.
Visual-Spatial Measures of the WISC-IV
While the WISC-IV does not have tasks specifically labeled as visual-spatial
processing measures there is research to suggest that one or more subtests in the
Perceptual Reasoning Index may measure visual-spatial processing. In addition, some
researchers have suggested Perceptual Reasoning assesses two different measures of the
CHC theory, Fluid Intelligence (Gf) and Visual Processing (Gv) (Keith, et al., 2006). The
subtest of the WISC-IV that is most often referenced in the literature as a complete
measure of visual spatial processing is Block Design (BD).
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BD has been utilized extensively in both the child and adult instruments of the
Wechsler series. The BD task requires the subject to use red and white blocks to replicate
a model or picture under a time constraint (Wechsler, 2003b). Carroll (1993) identified
through factor analysis that BD falls under the broad cognitive ability of Gv and is
strictly a measure of the narrow cognitive ability of VZ. Sattler and Dumont (2004) also
suggest that it is a measure of spatial relations (SR) in addition to VZ. Further support for
BD as a visual-spatial measure comes from the Keith et al. (2006). Keith, et al. found in
their factor analysis of the WISC-IV that when BD was loaded on Gf and Gv factors that
the model did not improve from the original loading on the Gv factor. Keith, et al. posits
the results suggest that BD is more accurately a measure of Gv and not Gf. In addition, an
analysis of the literature regarding measures of visual-spatial processing suggests that BD
is often considered by researchers as a definitive measure of visual-spatial ability
(Carroll, 1993; Cornoldi et al., 2003; Fuchs et al., 2005; Hegarty & Kozhevnikov, 1999;
Lee, et al., 2004). Sattler (2001) cautions a child's performance on the BD may be
adversely affected by motor skills and vision. Sattler's caution suggests that other
abilities (besides visual-spatial processing) may affect an individual's score; calling into
question its purity as a measure of visual-spatial processing. Other Perceptual Reasoning
index subtests may also measure visual-spatial processing, but as a secondary not primary
cognitive ability (Keith, et al. 2006).
While not a pure measure of Gv, Picture Completion (PCm) may measure a
component of visual-spatial processing. PCm has been an element of the Wechsler series
since the original WISC was developed (Wechsler, 2003a). PCm involves the
identification of a missing part or parts of a familiar picture under a time constraint. PCm
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can be considered a measure of the Gv because the subtest requires visual responsiveness,
visual perception, visual discrimination and visual memory (Sattler & Dumont, 2004;
Zhu & Weiss, 2005). Sattler and Dumont (2004) more specifically state that PCm may
also be a measure of the narrow cognitive ability of flexibility of closure (CF). The
confirmatory factor analytical research of Keith et al. (2006) supports PCm as a measure
of Gv. Keith, et al. found that while PCm loaded on the Gc factor, it was primarily a
measure of Gv. Similarly, while not a pure measure, Matrix Reasoning (MR) may also
measure visual-spatial processing.
MR requires the subject to identify and select an item/picture from five relatively
similar items that will complete a matrix (Zhu & Weiss, 2005). Sattler (2001) states
visual-spatial ability may be a cognitive element tapped by MR with some individuals
who take the test. Sattler further explains that while MR may have a substantial verbal
mediation element, it also involves visual-perceptual and visual-spatial processing
elements. Sattler and Dumont (2004) suggest that MR is a measure of both general Gv
and the narrow cognitive ability of VZ. Keith et al. (2006) offers some support for MR as
a measure of visual-spatial processing as a secondary cognitive ability. Keith et al. found
that when Gv and Gf were loaded on MR there was a statistically significant
improvement of fit in comparison to the original model with only Gf. However, when
cross-validation was conducted the cross loading was not substantial. Keith et al.'s
research offers support for Sattler's claim that MR may have an undetermined role in
assessing visual-spatial processing. There is some disagreement regarding Symbol Search
as a measure of visual-spatial processing.
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Symbol Search (SS) was added the WISC scales in 1991 (Wechsler, 2003a). The
SS subtest requires the subject to visually scan a collection of symbols to identify if a
target symbol is present (Zhu & Weiss, 2005). Keith et al. (2006) maintains that SS
involves the attention to and discrimination of visual stimuli. In Keith et al's factor
analysis of the WISC-IV when Gv and Gs were both loaded on SS there was a significant
improvement in the fit of the model in comparison to loading Gs alone. Keith et al.
further explains that during cross-validation the cross-loading held constant. Keith et al.
suggests these results support SS as a measurement of visual-spatial processing.
However, Sattler and Dumont (2004) do not support this claim. Rather Sattler and
Dumont contend that SS is a measure of the broad cognitive ability of Gs and the narrow
cognitive abilities of perceptual speed (P) and rate of test taking (R9).
Summary
The WISC-IV and the SB5 are two recently revised tests of intelligence. The
WISC-IV and the SB5 were designed to align closely to the modern CHC theory of
intelligence. CHC theory has eight-ten broad categories of cognitive ability. One of those
categories is Visual Processing (Gv). The literature supports the equitability of the terms
visual-spatial processing and the CHC category of Gv (Evans et al, 2002; McGrew,
2005). The SB5 has both verbal and nonverbal measures of visual-spatial processing that
combine to form a Visual-Spatial Processing Factor. Two of the SB5 measure (Form
Board and Position and Direction) have roots in previous editions. Form Patterns is new
to the revised instrument. The author of the test suggests that Form Board and Form
Patterns are measures of Gv and the narrow cognitive ability of VZ. The author does not
specify which narrow cognitive ability is measured by the Position and Direction subtest.
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There appears in the literature to be a lack of research regarding the visual-spatial
measures of the SB5. The WISC-IV has four subtests that appear to measure visualspatial processing. BD is believed to be a primary measure of visual-spatial processing
according to CHC theory. BD is a measure of the broad category of Gv and the narrow
categories of VZ and SR. PCm is purported to also measure Gv and the narrow abilities
of CS and CF. PCm is not a pure measure of Gv because it is also believed to measure
Gc. Additionally, MR is believed to be a measure of the broad category of Gv and the
narrow category of VZ in addition to another broad cognitive ability (Cf). Finally, there is
limited support to suggest that SS is a measure of Gv.
Summary
Beginning with the initial special education mandate PL. 94-142 to the
present reauthorization of IDEA the definition of a SLD has not undergone significant
changes (Hammill, 1990; Kirk & Kirk, 1983; U.S. Department of Education, 2006). One
major aspect of the initial and current definition is the idea that a SLD is fundamentally a
psychological processing disorder. The majority of state education departments have
adopted the federal government's definition of a SLD as a processing disorder (Reschly,
et al, 2003). In the identification of students with a SLD, there has been a shift in the
literature and law from primarily an ability-achievement discrepancy model to
determination though failure to respond to interventions. Neither of the two models are
aligned with the definition of a SLD as a psychologically based processing disorder. To
increase continuity either the definition of a SLD may need be changed to exclude the
processing element or identification may need include a processing component. The
literature suggests that psychological processing is significantly involved with the three
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of the most commonly identified SLDs reading, writing and mathematics. (Badian, 2001;
Berninger & Amtmann 2003; Cornwall, 1992; Floyd, et al., 2003; Fuchs, et al. 2005;
Fuchs, et al. 2006; Hauerwas & Walker, 2003; Siegel, 2003; Swanson, et al, 2006). Of
the three most common SLD, the literature suggests that mathematics has the least
amount of research and further research is suggested (Augustyniak, et al., 2004; Geary &
Hoard, 2003; Robinson, Menchetti, & Torgensen, 2002; Swanson & Jerman, 2006).
The United States Office of Special Education (2006a) defines a mathematical
disability (MD) as a disorder in a basic psychological process. The literature suggests that
there are distinct psychological processes that are involved in the mathematical tasks of
calculation, fluency and word problems (Floyd, et al. 2003; Fuchs et al., 2006; Fuchs et
al, 2005; Geary, 2004; McLean & Hitch, 1999; Swanson, 2004; Swanson, 2006; Swanson
& Beebe-Frankenberger, 2004). Of those processes, it appears working memory plays a
significantly important role in mathematical calculation, fluency and word problems
(Geary, 2004; Floyd, et al. 2003; Fuchs et al, 2005; McLean & Hitch, 1999; Swanson,
2006; Swanson, 2004; Swanson & Beebe-Frankenberger, 2004; Swanson & Jerman,
2006). One aspect of working memory is visual-spatial processing (Baddeley, 1996;
Pickering & Gathercole, 2004). Visual-spatial processing may be related to all three
mathematical tasks (Assel, et al., 2003; Augustyniak, et al., 2005; Floyd, et al. 2003;
Fuchs, et al., 2005; Hegarty and Kozhevnikov, 1999; Jordan, et al., 2003; Swanson, 2004;
Swanson and Sachse-Lee, 2001; Swanson & Jerman, 2006). The literature suggests that
additional research is needed in the area of visual-spatial processing and mathematics
(Floyd, et al. 2003; Forest, 2004; Fuchs, et al., 2005; Garderen & Montague, 2003;
Geary, 1993; Geary, 1996; Geary 2004; Mazzocco & Meyers, 2003; Reuhkala, 2001).
94
In the literature there are a variety of tasks that are believed to measure visualspatial processing; however, there is some disagreement among researchers regarding
which task best measures this construct. The newly redesigned Stanford-Binet
Intelligence Scales, Fifth Edition (SB5) includes verbal and nonverbal measures of
visual-spatial processing (Becker, 2003; DiStefano & Dombrowski, 2006; Roid, 2003a;
Roid, 2003b). It has been suggested that Position and Direction, Form Board, and Form
Patterns are measures of visual-spatial processing. To date there has been limited
research with this redesigned instrument. The Wechsler Intelligence Scales for ChildrenFourth Edition (WISC-IV) has also been recently redesigned (Wechsler, 2003a). There
has been recent research that suggests certain subtests of the WISC-IV such as Block
Design, Picture Completion, Matrix Reasoning and Symbol Search may assess visualspatial processing (Gv) (Keith, et al, 2006). The next chapter (Chapter III) will describe
how a study will be constructed to further research in the areas of mathematical
achievement, visual-spatial processing and the SB5.
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Chapter III
METHODOLOGY
Statement of the Problem/ Research Questions
The primary purpose of this study was to investigate the ability of the
visual-spatial measures of the Stanford-Binet Intelligence Scales, Fifth Edition (SB5) and
the Wechsler Intelligence Scale for Children- Fourth Edition (WISC-IV) to discriminate
between students with and without difficulties in mathematics achievement. It is
suggested from a review of the literature that visual-spatial processing, as measured by
the SB5 and the WISC-IV, will be significantly different between students who have
lower ability in mathematics and those who do not. In addition, the study identified which
visual-spatial measure has the most potential as a discriminator between students who
have poor mathematics achievement and those who do not.
The following research questions were used as a guide for the current study:
1. Is there a relationship between the psychological process of visual-spatial
processing (as measured by the SB5 and WISC IV) and mathematics
achievement (as measured by the Woodcock-Johnson III Tests of
Achievement-Normative Update (WJ-III-NU)?
2. Can the visual-spatial measures of the WISC-IV and the SB5 predict
mathematics achievement (as measured by the WJ-III-NU)?
4. What visual-spatial measure (SB5; WISC-IV) is the best
predictor of poor mathematics achievement (as measured
by the WJ-III-NU)?
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Participants
A total of 112 students in grades 6-8 participated in the study. Recruitment
occurred during the Spring Semester of 2007 through the Spring Semester of 2008.
Participants were recruited from grades 6-8 in northeastern Nebraska (n = 50),
northeastern Wyoming (n = 42), north central South Dakota (n= 10), western Minnesota
(n= 6), southeastern South Dakota (n - 2), and north central Utah (n = 2). Data were
collected by either the researcher or research assistants. The research assistants all had
advanced degrees with Education Specialist Degree (Ed. S.) (n = 3) level training or
Doctorate (Ph.D.) (n - 1) level training in school psychology. All researchers were
required to meet Institutional Research Board (IRB) guidelines in-order to participate in
data collection.
The mean age of the participants was 12.8 (see table 3.1 for a break down by
grade). Participants in the study were relatively evenly distributed between males (49%)
and females (50%). Of the participants, 99% indicated English was the language they
were most comfortable with, while only 1% indicated they were more comfortable with a
language other than English (Spanish). See Tables 3.2-3.4 for additional demographic
information.
Table 3.1
Participants' Grade Levels
Grade
Number of Participants
6th
37
th
7
54
8^
21
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Table 3.2
Demographics
Ethnicity
Number of Participants
African American
3
Asian/Pacific Islander
2
Hispanic
29
Native American
6
White
69
Other*
3
*Note: The other category consists of participants who indicated
more than one category of ethnicity and no primary category.
Table 3.3
Language Spoken at Home
Language
English
Spanish
Both English and Spanish
Laotian
Togan/English
Number of Participants
81
21
7
1
1
Table 3.4
Level of Parental Education
Level of Education
Number of Participants
Less than High School
2
Some High School
8
High School
43
College*
53
Not Indicated
6
*Note: College includes technical college, 2-year college degree, a 4-year degree and/or an advanced
degree (M.S., Ph.D., J.D, etc.)
The sample for the study was composed of students in 6th, 7 th and 8 th grades
between the ages of 11-14. Students at this grade level were chosen for the current study
for several reasons. First, previous research conducted with students at this age level with
measures of visual-spatial processing and mathematics, has found a significant
relationship (Garden & Montague, 2003; Hegarty & Kozhevnikov, 1999; Reuhkala,
2001; Swanson, 2004; Swanson & Sachse-Lee, 2001). Second, by fifth grade, students
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are believed to have mastered higher-level mathematical skills including retrieval of basic
mathematical facts, decomposition, distribution and advanced word problem solving
(Carr & Hettinger, 2003). In addition, using 6th, 7th and 8th grade students will provide an
understanding of the relationship between visual-spatial processing and mature
mathematical thinking. Finally, the literature supports that visual-spatial material is
processed in working memory, which may not be fully developed in students in lower
grades (Baddeley, 1996; Reuhkala, 2001).
A power analysis was conducted to determine the appropriate number of subjects
for the logistic regression, using PASS statistical software developed by NCSS (2005).
The PASS power calculation software is based on the sample size calculation method for
logistic regression developed by Hsieh, Block, and Larson (1998) (NCSS, 2005).
Assuming a normal distribution and that the sample is similar to the population, it is
estimated that 25% (.25) of the participants will score below the 25th percentile in
mathematics using the WJ-III-NU. The use of 25th percentile as a cutoff to identify
students with poor mathematics achievement is consistent with previous research (Fuchs
et al. 2005; Murphy, Mazzocco, Hanich & Early, 2007). For an odds ratio of .53, 103
total subjects would be needed for power of .80 (NCSS, 2005). The current sample of 112
exceeds that criteria. In addition, it has been recommended, assuming a medium effect
size (a = .05; ft = .20), the sample size needed for adequate power in a multiple regression
is TV> 50 + 8m (Tabachnick & Fidell, 2007). In addition, for analyzing individual
predictors in a multiple regression Tabachnick and Fidell recommend JV> 104 + m (m =
the number of independent variables). Given three independent variables (N> 50 + 8(3)
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or iV> 74 and JV> 104 + (3) orN> 107), it appears the current sample size to meets
power requirements in both areas.
Participants scoring in the 25 percentile on the mathematics measure (WJ-III) for
the study were identified as having "poor mathematics achievement". Previous literature
has identified students that score in the 25th percentile or lower on standardized measures
of mathematics achievement have or may develop a mathematics disability that will
negatively influence their mathematical performance in the classroom (Geary, 1993;
Geary, 2004; Mazzocco & Meyer, 2003; Swanson & Beebe-Frankenberger, 2004;
Zeleke, 2004).
Instrumentation
The following identifies the instrumentation utilized in the study to assess IQ,
visual-spatial ability and mathematical ability.
Intelligence Measure
An estimated IQ or general ability index was obtained by using the Abbreviated
Battery of the Stanford-Binet Intelligence Scales-Fifth Edition (SB5). The Abbreviated
SB5 IQ assessment consists of the two routing subtests of the SB5 (Object
Series/Matrices and Vocabulary). The Object Series/Matrices subtest is a fluid reasoning
measure that requires subjects to not only conceptualize problems that have figural
elements, but also visually discriminate among pictured objects that have figural as well
as geometric properties (Roid, 2003a). The Vocabulary subtest is a measure of verbal
knowledge and requires the subjects to define vocabulary words (Roid, 2003a). Both
subtests are graduated in difficulty and have a mean score of 10 and a standard deviation
of3.
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The Abbreviated Battery of the SB5 was standardized in conjunction with the
complete measure. SB5 was standardized on 4,800 subjects ranging in age from 2-80+
with demographics similar to the 2000 U.S. Census population (Roid, 2003b). The
standardization sample was constructed of 51% female, 49% male, 69% white, 12%
African American, 12% Hispanic, 4% Asian and 3% other (Roid, 2003a). The Full Scale
IQ and the Abbreviated IQ version of the SB5 have a mean of 100 and a standard
deviation of 15. The Abbreviated IQ battery has substantial internal consistency
reliability (r =. 91) (Roid, 2003b). The internal consistency reliability with children aged,
10 years (r = .88), 11 years (r = .87), 12 years (r = .90), 13 years (r = .85; and 14 (r =
.91) years is also moderate to high (Roid, 2003b). The Abbreviated SB5 IQ correlates
well with the SB5 Full scale IQ (r = .87) (Roid, 2003b).The Abbreviated SB5 correlates
moderately (r = .71) with the Composite Standards Age Score of the Stanford-Binet IV
(Roid, 2003b). In addition, the abbreviated battery correlates moderately with both the
Wechsler Intelligence Scale for Children-Third Edition (WISC-III) (r = .69) and the
Wechsler Adult Intelligence Scale-Third Edition (WAIS-III) (r = .81) (Roid, 2003b).
Visual-Spatial Measures
Nonverbal Visual-Spatial Measure of the SB5
Form Board. The Form Board task of the SB5 is a nonverbal measure of visualspatial processing and is used in the 1st and 2nd levels of the Nonverbal Visual-Spatial
Processing subtest of the SB5 (Roid, 2003a). The task involves the use of a form board
made of plastic with recessed shapes for a triangle, a square and a circle. The subject uses
pieces of geometric figures (circle, square and triangle) to form shapes on the form board.
The task was standardized in conjunction with the standardization of the full SB5
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instrument. The subtest has a mean standard score of 10 and a standard deviation of 3.
The Form Board task is combined with Form Patterns task to form the Nonverbal VisualSpatial Processing measures.
Form Patterns. The Form Patterns task of the Nonverbal Visual-Spatial
Processing subtest is a continuation of the Form Board task with increased complexity
and used on levels 3-6 of the subtest. The Form Patterns task measures visualization
ability through the analyzing and duplication of patterns based on a two-dimensional field
(Roid, 2003a). The Form Patterns utilizes geometric shapes and requires the subject to
duplicate figures or forms. The Form Patterns subtest has a mean standard score of 10
and a standard deviation of 3.
The Nonverbal Visual-Spatial Processing subtest is the combined Form Board and
Form Patterns tasks, each task was not separated out for reliability or validity. The
Nonverbal Visual-Spatial Processing subtest maintains adequate overall reliability (r =
.87) as well as moderate to adequate reliability at ages 10 (r = .76), 11 (r = .79), 12 (r =
.72), 13 (r =.79) and 14 (r = .83) (Roid, 2003a). The Nonverbal Visual-Spatial subtests
have moderate (r =.87) test-retest reliability with ages 6-20 (Roid, 2003a). The subtest
correlates moderately (Ages 6-10.11 years, r = .70; Ages 11-16.11 years r = .63) with the
Full Scale IQ of the SB5 (Roid, 2003b).
Verbal Visual-Spatial Measures of the SB 5
Position and Direction. Position and Direction is the Verbal Visual-Spatial
measure of the SB5. The task begins with the use of a small ball or green block and a cup
at the earliest levels and requires that subject to demonstration positional awareness
(inside, top, bottom, etc...). The more complex levels of the task involve the subject
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placing a green block on a stimulus page locating order and direction. The most complex
level of the task requires the subject to verbalize directions (left, right, north, south,
etc...) by visualizing a specific route from various printed pathways on a stimulus map
(Roid, 2003a). The subtest has 6 levels with a mean of 10 and a standard deviation of 3.
The Position and Direction subtest was standardized in conjunction with the
standardization of the full SB5 battery. The Verbal Visual-Spatial subtest correlates
moderately with the Full Scale IQ of the SB5 (Ages 6-10.11, r = .75; Ages 11-16.11, r =
.76) (Roid, 2003b). The Verbal Visual-Spatial subtest has adequate overall internal
consistency reliability (r = .86) as well as moderate internal consistency reliability at ages
10 (r = .89), 11 (r = .86), 12 (r = .84) 13 (r = .87) and 14 (r = .88) (Roid, 2003b).
Position and Direction has adequate tests-retest reliability (r = .79) with ages 6-20 (Roid,
2003a).
Specificity of the Visual-Spatial Measures of the SB5
Specificity is a measure of the unique variance of a subtest. Specific variance is
the part of the total variance that is unique to that specific subtest (Roid & Barram, 2004;
Sattler & Dumont, 2004). If a subtest's specific variance exceeds its error variance it is
said to have specificity (Sattler & Dumont, 2004). If a subtest in an intelligence measure
has ample (> 25%) or adequate (25-15%) specificity it is said to be measure something
distinctly different from the hypothesized "g" or general intelligence (Roid & Barram,
2004; Sattler & Dumont, 2004). The Nonverbal Visual-Spatial subtest of the SB5 (Form
Patterns) demonstrates ample specificity (specificity = .28; error variance = .13)
suggesting the subtest measures a construct unique to the hypothesized "g" or general
intelligence (Roid & Barram, 2004). This suggests the SB5's nonverbal measure of
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visual-spatial processing may be interpreted as a unique construct (Roid & Barram,
2004). The Verbal Visual-Spatial subtest (Position and Direction) of the SB5,
demonstrates adequate specificity (specificity = .17; error variance = .13); however, with
a specificity below .25 it suggests little unique variance is accounted for and caution
should be should be used in interpreting this subtests apart from the hypothesized "g" or
general intelligence (Roid & Barram, 2004; Sattler, 2008). This suggests that while the
verbal measure of visual-spatial processing demonstrates some unique variance in
comparison to the global factor of intelligence, some caution may be warranted in
detailed interpretation of the subtest (Roid & Barram, 2004).
Validity of the Visual-Spatial Measures of the SB5
Evidence of criterion-related validity of the visual-spatial measures of the SB5
comes from the SB5's relationship to subtests in previous Stanford-Binet editions that
measure visual-spatial processing and other tests that are believed to have measures of
visual-spatial processing. The SB5 Visual-Spatial Processing factor, which includes both
the Verbal and Nonverbal Visual-Spatial Processing subtests, correlates moderately (r =
.79) with the Abstract/Visual Reasoning scale of the previous Stanford-Binet, Fourth
Edition (SB IV) (Roid, 2003). Sattler (2001) suggests the subtests that combine to form
the Abstract/ Visual Reasoning scale (Pattern Analysis; Copying; Matrices; Paper
Folding and Cutting) of the SB IV are measures of visual-spatial processing. In addition,
the SB5 Visual-Spatial Processing factor correlates moderately (r =.70) with the
Performance IQ score of the Wechsler Preschool and Primary Scale of IntelligenceRevised (WPPSI-R). The WPPSI-R Performance IQ subtests (Object Assembly;
Geometric Design; Block Design; Mazes; Picture Completion; Animal Pegs) are believed
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to consist of measures of both visual and spatial constructs (Sattler, 2001). The Wechsler
Adult Intelligence Scale-Third Edition (WAIS-III) Performance IQ also correlates
moderately (r = .71) with the Visual-Spatial Processing factor of the SB5 (Roid, 2003a).
The Performance IQ subtests (Picture Completion; Block Design; Matrix Reasoning;
Digit-Symbol-Coding; Picture Arrangement; Symbol Search; Object Assembly) include
measures of both visual and spatial skills (Sattler, 2001). However, the SB5 VisualSpatial Processing factor does not correlate substantially (r = .42) with the Performance
IQ score of the Wechsler Intelligence Scale for Children-Third Edition (WISC-III). Roid
(2003a) attributes the lower correlation to the WISC-III's emphasis on time sensitive and
time-bonus elements. Finally, the SB5 Visual-Spatial Processing Factor correlates
adequately (r =.56) with the Spatial Relations subtest of the WJ-III Tests of Cognitive
Abilities which is designed to be a measure of visual-spatial processing (McGrew &
Woodcock, 2001; Roid, 2003a).
The SB5 visual-spatial processing measures also demonstrate predictive validity,
utility with English language learners and applicability with individuals from differing
SES levels. The Visual-Spatial Processing factor score of SB5 correlates moderately with
Broad Math (r = .61), and the Math Calculation Skills (r =.59) of the Woodcock Johnson
Test of Achievement-Third Edition (Roid, 2003a). The correlation (r = .69) between the
Visual-Spatial factor score of SB5 and the Mathematics Composite of the Wechsler
Individual Achievement Test-Second Edition is also moderate. The use of the SB5 visual
spatial processing measures with English language learners has been investigated. In a
study involving 65 students (49% identified as Hispanic English language learners), the
mean Visual-Spatial score was 93.8 (Roid, 2003a). The students' scores were
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approximately one-half of a standard deviation below the expected mean of 100 (Roid,
2003a). In the standardization of the SB5 battery, parent education level was used as a
measure of SES. In the standardization sample for ages 10-14, 54.5-52.5% of the subjects
had a parent with a 12th grade education or lower (Roid, 2003).
Visual-Spatial Measures of the WISC-IV
Block Design. Block Design is identified as a core Perceptual Reasoning subtest
of the Wechsler Intelligence Scale- Fourth Edition (WISC-IV) (Sattler & Dumont, 2004;
Wechsler, 2003a). The subtest has 14 items with a mean scaled score of 10 and a standard
deviation of 3. The subject is required to use red, white, and half-red/ half-white blocks to
replicate block pattern designs. The subtest is designed to measure spatial visualization,
visual perceptual abilities, and can be conceptualized as a task that involves relations of a
spatial nature (Sattler & Dumont, 2004; Wechsler, 2003a). Block Design was
standardized in conjunction with the complete WISC-IV battery on 2,200 children; with a
sample that was representative of the demographic characteristics of 2000 United States
census population (Sattler & Dumont, 2004; Wechsler, 2003 a). The standardization
sample included 63% Euro Americans, 16% African Americans, 15% Hispanic
Americans, 4% Asian Americans, and 1% other (Sattler & Dumont, 2004). Block Design
has adequate overall internal consistency reliability (r = .86) and at ages 10 (r = .84), 11
(r = .87), 12 (r = .88), 13 (r = .88) and 14 {r = .85) (Wechsler, 2003a; Sattler & Dumont,
2004). The subtest has moderate test-retest reliability with ages 10-11 (r = .86), 12-13 (r
= .82) and 14-16 (r = .86) (Wechsler, 2003a; Sattler & Dumont, 2004). The Block Design
subtest loads significantly on Perceptual Reasoning Index (.78) and correlates moderately
(r = .70) with the Full Scale IQ of the WISC-IV (Sattler & Dumont, 2004).
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Matrix Reasoning. Matrix Reasoning is a core subtest of the Perceptual Reasoning
Index. Matrix Reasoning requires the subject to view an incomplete matrix and visually
discriminate among five choices, then identify which figure will complete the matrix
(Wechsler, 2003a). Matrix Reasoning is suggested to be a measure of visual-spatial
processing (Keith et al., 2006; Sattler, 2001; Sattler & Dumont 2004). Matrix Reasoning
has 35 test items with a mean standard score of 10 and a standard deviation of 3. Matrix
Reasoning has moderate to high overall internal consistency reliability (r = .89) and
maintains moderate to high internal consistency reliability at ages 10 (r = .89), 11 (r =
.89), 12 (r = .92), 13 (r = .89) and 14 (r = .87) (Wechsler, 2003a; Sattler & Dumont
2004). The subtest has moderate to high test-retest reliability with ages 10-11 (r = .92),
12-13 (r = .80) and 14-16 (r = .78) (Sattler & Dumont, 2004; Wechsler, 2003a). Matrix
Reasoning has an adequate correlation with the Full Scale IQ (r = .72), and has a
substantial correlation with the Perceptual Reasoning Index (r = .84) (Sattler & Dumont,
2004).
Picture Completion. Picture Completion is a non-core subtest of Perceptual
Reasoning. The subtest requires the subject to visually identify a missing part of a picture
under a time constraint (Wechsler, 2003a). The Picture Completion subtest requires
visual responsiveness, visual perception, visual discrimination and visual memory
(Sattler & Dumont, 2004; Zhu & Weiss, 2005). The Picture Completion subtests is
believed to be a measure of visual-spatial processing (Keith et al. 2006; Sattler &
Dumont, 2004). The subtest has 38 items of increasing difficulty, with a mean standard
score of 10 and a standard deviation of 3. The subtest was standardized in conjunction
with the standardization of the entire WISC-IV assessment. Picture Completion has good
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overall internal consistency reliability (r = .84) and sufficient internal consistency
reliability at ages 10 (r = .85), 11 (r = .87), 12 (r = .84), 13 (r = .83) and 14 (r = .82)
(Sattler & Dumont, 2004; Wechsler, 2003a). The subtest has moderate to high test-retest
reliability at ages 10-11 (r= .85), 12-13 (r = .84) and 14-16 (r = .87) (Sattler & Dumont,
2004; Wechsler, 2003a). Picture Completion correlates adequately with the Full Scale IQ
(r = .60) and the Perceptual Reasoning Index (r = .57) (Sattler & Dumont, 2004).
Specificity of the WISC-IV Visual-Spatial Measures
Specificity is a measure of the unique variances of a subtest. Specific variance is
the part of the total variance that is unique to that specific subtest (Roid & Barram, 2004;
Sattler & Dumont, 2004). If a subtest's specific variance exceeds its error variance it is
said to have specificity (Sattler & Dumont, 2004). If a subtest in an intelligence measure
has ample (> 25%) or adequate (25-15%) specificity it is said to measure something
distinctly different from the hypothesized "g" or general intelligence (Roid & Barram,
2004; Sattler & Dumont, 2004). The Block Design (specificity = .36; error variance =
.14) Matrix Reasoning (specificity = .51; error variance = .11) and Picture Completion
(specificity = .40; error variance = .16) all demonstrate ample specificity (> 25%; error
variance < 25%) (Sattler, 2008; Sattler & Dumont, 2004). This suggests all three subtests
are distinctly different from "g".
Validity of the WISC-IV Visual-Spatial Measures
The evidence of criterion-validity for the WISC-IV subtest measures is limited
due to the lack of cross-battery studies conducted (Sattler & Dumont, 2004). The main
criterion validity studies involving the WISC-IV are with earlier versions of the test and
comparisons to Wechsler scales for ages outside of the age range of the WISC-IV. The
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Perceptual Reasoning Index of the WISC-IV that includes measures of visual-spatial
processing (Block Design; Picture Completion; Matrix Reasoning) correlate moderately
(r = .70) with the Perceptual Organization Index of the WISC-III that also includes
measures of visual-spatial processing (Block Design; Picture Arrangement; Picture
Completion; Object Assembly) (Sattler & Dumont, 2004; Wechsler, 2003a). In addition,
the visual-spatial measures of the WISC-IV correlate slightly higher (r = .73) with the
Performance IQ of the WISC-III that includes the additional measures of Coding, Symbol
Search, and Mazes (Sattler & Dumont, 2004; Wechsler, 2003a). The visual-spatial
measures of the WISC-IV also correlate similarly (r = .71) with both the Perceptual
Organization Index (Picture Completion; Block Design; Matrix Reasoning) and the
Performance IQ (Picture Completion; Block Design; Matrix Reasoning; Digit-SymbolCoding; Picture Arrangement; Symbol Search; Object Assembly) of the WAIS-III
(Sattler & Dumont, 2004; Wechsler, 2003a). Finally, the visual-spatial measures of the
WISC-IV correlate moderately (r = .74) with the Performance IQ of the Wechsler,
Preschool and Primary Scale of Intelligence-Ill (WPPSI-III) that includes measures of
visual-spatial processing (Block Design; Matrix Reasoning; Object Assembly; Picture
Completion) (Sattler & Dumont, 2004; Wechsler, 2003a).
The WISC-IV visual-spatial measures (Block Design, Picture Completion, and
Matrix Reasoning) also demonstrate predictive validity, utility with Hispanic American
students and applicability with individuals from differing SES levels. Overall the
Perceptual Reasoning index correlates moderately with the Mathematics Composite (r =
.67), Numerical Operations (r = .60), and Math Reasoning (r = .67) of the Wechsler
Individual Achievement Test-Second Edition (WIAT-II) (Wechsler, 2003a). Block
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Design correlates adequately with the Mathematics Composite (r = .57), Numerical
Operations (r = .50), and Math Reasoning (r= .58) of the WIAT-II (Wechsler, 2003a).
Picture Completion correlates adequately with the Mathematics Composite (r = .47),
Numerical Operations (r =.41), and Math Reasoning (r = .59) of the (WIAT-II)
(Wechsler, 2003a). Matrix Reasoning correlates moderately with Mathematics Composite
(r = .60), and adequately with Numerical Operations (r = .53), and Math Reasoning (r=
.59) of the WIAT-II (Wechsler, 2003a). In the standardization sample of the WISC-IV
researchers found the Full Scale IQ score obtained by Hispanic American children was
on average 10 points lower than Euro American children (Sattler & Dumont, 2004). In
addition, Hispanic American children scored 3-6 points higher on the Perceptual
Reasoning Index in comparison to their score on the Verbal Comprehension Index
(Sattler & Dumont, 2004). There is limited information available from the test publishers
or the literature on the use of the WISC-IV with English Language Learners. One reason
for this may be that the publisher has created a separate Spanish Version (with Spanish
language norms) for the WISC-IV. The WISC-IV also uses parent education level as an
indicator of SES. On the Perceptual Reasoning Index (that includes Block Design, Picture
Completion and Matrix Reasoning) parents of children with an 8 grade education or less
score approximately 15 points lower than children of parents with a college degree
(Sattler & Dumont, 2004). In addition, 42-42.5 % of the parents of the children in the
standardization sample (2,200) had a high school education or less. This suggests that
children of lower SES were represented in the standardization of the instrument.
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Measure of Mathematics Achievement
The administration of the standard mathematics battery of the Woodcock-Johnson
III Tests of Achievement-Normative Update (WJ-III-NU) was used to assess the
mathematical achievement of the participants. The Woodcock-Johnson III Tests of
Achievement (WJ-III) was renormed in 2007. The normative data was recalculated to
align with the 2005 U. S. Census statistics and is an update of the norm construction
procedures (McGrew, Schrank & Woodcock, 2007). The publishers of the test used the
same standardization sample that was used in the original norming of the WJ-III
(McGrew et al., 2007). The standardization sample of the WJ-III-NU included 4,740
students in grades K-12 (McGrew et al., 2007). The demographic makeup of the sample
was 78.4% White, 14.5% African American, 12% Hispanic, 5.1% Asian and Pacific
Islander, and 2% Native American (McGrew et al., 2007)
Information regarding the use of the WJ-III and the WJ-III-NU and English
language learners (specifically Hispanic/ Latino children) is limited. The main reason for
the lack of representative information may be the publisher provides an alternative
Spanish Language adaptation/translation called the Bateria III Woodcock-Munoz. This
measure is specifically designed for Spanish speaking populations. The Examiner's
Manual of the WJ-III suggests that it is important ensure the English language
proficiency of subjects prior to administration of the WJ-III (Mather &Woodcock, 2001).
The standardization sample excluded those individuals that had less than one year
experience in English speaking classes (McGrew &Woodcock, 2001). However, the
standardization sample included 570 students that identified their ethnicity as Hispanic
(McGrew &Woodcock, 2001; McGrew et al, 2007). In addition, as a measure of SES
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70% of the parents of the standardization sample had a high school diploma or less
(McGrew &Woodcock, 2001).
The WJ-III-NU Broad Math Cluster is the combined scores on the Calculation,
Fluency and Applied Problems subtest. In addition, the WJ-III also reports a Math
Calculation Skills Cluster that includes the Calculation and Fluency subtests. The
following provides reliability and validity information for the WJ-III-NU.
Broad Math Cluster
The overall Broad Math Cluster has high test-retest reliability at ages 8-10 (r .92), 11-13 (r= .92) and 14-17 (r = .91) (McGrew et al., 2007). The overall Broad Math
Cluster also has high internal consistency reliabilities at ages 10 {rcc= .94), 11 {rcc~ .95),
12 (rcc= .94), 13 (rcc= .95) and \4(rcc= .96) (McGrew et al., 2007). The validity of the
Broad Math Cluster comes mainly from a comparison with other measures of
achievement; this was not updated in the WJ-III-NU. The Broad Math Cluster correlates
moderately (r = .66) with the Mathematics Composite of the Kaufman Test of
Educational Achievement (KTEA) in grades 1-8 (McGrew & Woodcock, 2001). The
Broad Math Cluster also correlates moderately (r = .70) with the Mathematics Composite
of the Wechsler Individual Achievement Test (WIAT) in grades 1-8 (McGrew &
Woodcock, 2001). In addition, there is evidence of the relationship between the Broad
Math Cluster and measures of IQ. The Broad Math Cluster correlates moderately (r =.76)
with the Full Scale IQ of the SB5 (Roid, 2003a). The Broad Math Cluster also correlates
moderately at ages 9-13 (r = .67) and 14-19 (r = .67) with the General Intellectual Ability
cluster (standard battery) of the WJ-III Tests of Cognitive Abilities-Normative Update
(McGrew et al, 2007). Additionally, the Broad Math Cluster correlates moderately (r -
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.82) with the General Conceptual Ability composite of the Differential Ability ScalesSecond Edition with ages 8-13.11 (Elliott, 2007).
Math Calculation Skills Cluster
The Math Calculation Skills Cluster also has high test-retest reliability at ages 810 (r = .89), 11-13 (r = .86) and 14-17 (r = .82) (McGrew et al. 2007). The Math
Calculation Skills Cluster also has high internal consistency reliabilities at ages 10 (rcc =
.91), 11 (rcc= .92), 12 (rcc= .90), 13 (rcc= .93) and 14(rcc= .91) (McGrew et al, 2007).
Validity information for the Math Calculation Skills Cluster comes mainly from a
comparison with other measures of achievement and was not updated for the WJ-III-NU.
The Math Calculation Skills Cluster correlates moderately with the Mathematics
Composite (r = .60) and the Mathematics Computation (r = .67) scale of the KTEA in
grades 1-8 (McGrew & Woodcock, 2001). The Math Calculation Skills Cluster also
correlates moderately (r = .69) with the Mathematics Composite and adequately (r = .59)
with the Numerical Operations scale of the WIAT (McGrew & Woodcock, 2001). There
is evidence of the relationship between the Math Calculation Skills Cluster and measures
of cognitive ability. The Math Calculation Skills cluster correlates moderately (r = .74)
with the Full Scale IQ of the SB5 (Roid, 2003a). In addition, the Math Calculation Skills
Cluster correlates adequately at ages 9-13 (r = .54) and 14-19 (r =.73) with the General
Intellectual Ability Cluster (standard battery) of the WJ-III Tests of Cognitive Abilities
Normative Update (McGrew et al., 2007). Finally, the Math Calculation Skills Cluster
correlates moderately (r = .71) with the General Conceptual Ability composite of the
Differential Ability Scales-Second Edition with ages 8-13.11 (Elliott, 2007).
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Calculation
Calculation is a standard subtest of the WJ-III-NU. Calculation is a subtest in the
WJ-III-NU Broad Math Cluster and the Math Calculation Skills Cluster. The Calculation
subtest measures a student's ability to perform increasingly difficult mathematical
calculations (McGrew & Woodcock, 2001). The subtest requires the student to solve
subtraction, addition, multiplication, and division problems with fractions and whole
numbers. The subtest has 45 items with a mean standard score of 100 and a standard
deviation of 15. The calculation subtest was standardized in conjunction with the entire
WJ-III-NU. The Calculation subtest has moderate to high test-retest reliability at ages 8\Q(r= .83), 11-13 (r = .81) and 14-17 (r = .76) (McGrew et al, 2007). Calculation has
moderate internal consistency reliabilities at ages 10 (rn= .85), 11 (r,,= .87), 12 {r,,=
.84), 13 {r„= .86) and 14(ru= .83) (McGrew et al., 2007). Calculation also correlates
adequately (r = .56) with the Global Fluid-Crystallized Index of the Kaufman Assessment
Battery for Children-Second Edition in grades 6-10 (Kaufman & Kaufman, 2004).
Further information regarding the correlation of the Calculation subtest and cognitive
ability measures can be ascertained from the previously mentioned correlations
identifying the relationship of the Broad Math Cluster and the Math Calculation Skills
Cluster to the SB5, WJ-III Tests of Cognitive Ability and the Differential Ability ScalesSecond Edition.
Fluency
Fluency is a standard subtest of the WJ-III-NU. Fluency is a subtest in the WJ-IIINU Broad Math Cluster and the Math Calculation Skills Cluster. Fluency measures a
student's ability to rapidly and efficiently work with mathematical facts (Sattler, 2001).
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Fluency requires the subject to rapidly solve single digit addition, subtraction and
multiplication problems. The subtest has 160 items with a mean standard score of 100
and a standard deviation of 15 (McGrew & Woodcock, 2001). Fluency was standardized
in conjunction with the standardization of the entire WJ-III-NU. Fluency has moderate to
high test-retest reliability at ages 8-10 (r = .86), 11-13 (r = .89) and 14-17 (r = .92)
(McGrew, et al. 2007). Fluency has high internal consistency reliabilities at ages 10 (ru =
.96), 11 {r„= .97), 12 {r,,= .97), 13 (r 7/ = .98) and 14(r„= .98) (McGrew et al., 2007).
However, Fluency's correlation (r =.31) with the Global Fluid-Crystallized Index of the
Kaufman Assessment Battery for Children-Second Edition in grades 6-10 is not
substantial (Kaufman & Kaufman, 2004). Further information regarding the correlation
of the Fluency subtest and measures of cognitive ability can be ascertained from the
previously mentioned correlations identifying the relationship of the Broad Math Cluster
and the Math Calculation Skills Cluster to the SB5, WJ-III Tests of Cognitive Ability and
the Differential Ability Scales- Second Edition.
Applied Problems
Applied Problems is a standard subtest of the WJ-III-NU. Applied Problems is a
subtest in the Broad Math Cluster. The Applied Problems measures the ability to solve
problems of a mathematical nature that are organized around practical situations (Sattler,
2001). The WJ-III Applied Problems was standardized in conjunction with the
standardization of the entire WJ-III-NU. Applied Problems has 63 items, a mean standard
score of 100 and a standard deviation of 15. Applied Problems has moderate test-retest
reliability with ages 8-10 (r = .85), 11-13 (r = .88) and 14-17 (r = .89) (McGrew et al.,
2007). Applied Problems has high internal consistency reliabilities at ages 10 (r,,= .91),
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11 (r,,= .91), 12 (r,,= .92), 13 (r„= .92) and 14 (/•„= .95) (McGrew et al., 2007). In
addition, Applied Problems correlates moderately (r = .76) with the Global FluidCrystallized Index of the Kaufman Assessment Battery for Children-Second Edition in
grades 6-10 (Kaufman & Kaufman, 2004). Further information regarding the correlation
of the Applied Problems subtest to intelligence measures can be ascertained from the
previously mentioned correlations identifying the relationship of the Broad Math to the
SB5, WJ-III Tests of Cognitive Ability and the Differential Ability Scales- Second
Edition.
Procedures
Subjects were recruited from a small Midwestern school district in the Northeast
section of Nebraska, and a small Western school district in the Northeast section of
Wyoming. The school districts were informed of the nature, purpose, benefits and any
potential risks of the study. In addition, the researcher solicited assistance from practicing
school psychology graduates and a former professor from the School Psychology
Program at The University of South Dakota working for schools and educational
cooperatives. Each school district and/or cooperative was informed of the nature,
purpose, benefits and any potential risks of the study. Permission was obtained from each
school district or educational cooperative to allow the researcher or research assistants to
participate in the study (see appendix A). The researcher and research assistants met
Institutional Review Board (IRB) requirements prior to participation in the study. The
IRB received copies of all school district or educational cooperative permission letters.
Students in grades 6th, 7th and 8th at each of the participating school districts
and/or cooperatives were given a letter/permission form (English and Spanish translation
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where applicable) and asked to take the form home to their parents/guardian. The letter
identified the purpose, risks, benefits and method of the study. The researcher and
research assistants worked cooperatively with classroom teachers, and school
administrators to hand out and collect the permission forms. No individual was allowed
to participate in the study unless they had a signed parent/guardian consent form and had
signed the participant assent form.
After parent/guardian permission forms were returned, testing occurred during the
school day in the students' respective schools. The testing session lasted approximately
one (1) hour. Students were provided with a consent form, assent form and a form asking
for demographic information (see appendix C). Students were administered the
abbreviated IQ test of the SB5, Nonverbal and Verbal Visual-Spatial subtests of the SB5
the Block Design, Matrix Reasoning, Picture Completion subtests of the WISC-IV, and
the Calculation, Fluency and Applied problems subtest of the WJ-III. To account for
some practice effects, the order of administration of the visual-spatial measures of the
WISC-IV and SB5 was alternated, but not in any systematic fashion. In addition, in an
attempt to minimize fatigue effects the administration of the subtests was kept as fluid as
possible.
Data Analysis
To fully conceptualize the relationship of visual-spatial processing and
mathematical achievement the following research questions were used to guide the study.
Research Question 1
Is there a relationship between visual-spatial processing, (measured by the SB5
and the WISC-IV), and mathematics achievement (measured by the WJ-III-NU)?
117
Rationale and Statistical Technique
A correlation analysis was used to identify if a significant relationship existed
between the visual-spatial processing measures of the WISC-IV and the SB5 and the
mathematics achievement measures of the WJ-III-NU. If a significant relationship could
not be identified then it would suggest visual-spatial processing is possibly not related to
poor mathematics achievement. If a significant relationship was identified, then it would
be important to know if mathematical achievement can be predicted by visual-spatial
processing, and which measures (SB5; WISC-IV) of visual-spatial processing are the best
predictors.
The data were analyzed using Pearson product-moment correlation coefficient (r)
to understand the relationship among the variables across all subjects. All variables were
entered into the equation. This analysis technique was the most appropriate because it is
used to determine the strength and the direction of the relationship between independent
variables and a dependent variable (Glenberg, 1996).
Research Question 2
Can the visual-spatial measures of the WISC-IV and the SB5 predict mathematics
achievement (as measured by the WJ-III-NU) ?
Rationale and Statistical Technique
A multiple regression analysis was used to identify if the visual-spatial measures
of the WISC-IV and the SB5 could predict mathematics achievement. In addition,
participants' IQs were controlled for to identify the unique contribution of the visualspatial processing measures. If the visual-spatial measures can predict student
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mathematics achievement, it would then be important to identify between the two visualspatial measures, which is a better predictor of mathematics achievement.
A sequential multiple regression analysis was used to analyze the data. The
independent variables were: 1) The Combined WISC-IV Block Design, Matrix
Reasoning, and Picture Completion subtest scores; 2) SB5 Visual Spatial Processing
Factor (the Form Patterns and the Position and Direction subtests); 3) SB5 Abbreviated
IQ. The dependent variable was the participants' score on the Broad Math Cluster of the
WJ-III-NU. A sequential multiple regression was constructed with IQ in the first block
and the remaining independent variables in the second block. Comparing the R2, more
specifically the change in (A) R2, of the first model to the second model indicated the
unique predictive ability of the visual-spatial measures while holding IQ constant across
all participants. Due to the potential of multicollinearity and singularity among the
independent variables, (because they may be measures of the same construct) combined
factor/composite scores were used in the study. Combining variables is one way to reduce
potential multicollinearity and increase power (Mertler & Vannatta, 2005; Myers, &
Weld, 2003; Tabachnick & Fidel, 2007).
Research Question 3
Which visual-spatial measure (SB5; WISC-IV) is the better predictor of a student
with poor mathematics achievement (as measured by the WJ-III-NU)?
Rationale and Statistical Technique
A logistic regression allows for the prediction of a discrete outcome (poor
achievement/ adequate achievement). Logistic regression is often used in medical
research (disease/no-disease models). Logistic regression is often preferable to other
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techniques such as discriminant analysis because it has fewer restrictions and can be used
with continuous, discrete, and/ or dichotomous independent variables (Tabachnick &
Fidell, 2007). In addition, using a sequential logistic regression allowed for the control of
IQ, to identify the unique relationship of the visual-spatial measures. Using two separate
logistic regressions allowed for a comparison between the two tests (SB5 and WISC-IV).
The two logistic regression identified which visual-spatial measure was the better
predictor of poor achievement in mathematics while controlling for IQ. Identifying the
best predictor is important because it may indicate which visual-spatial measure (WISCIV; SB5) can more accurately be used to identify students with poor achievement in
mathematics. Additionally, this may in turn provide more accuracy in the assessment and
diagnosis of learning disabilities in the area of mathematics.
The dependent variable was the assignment to dichotomous groups depending on
mathematics performance. With those participants who scored in the 25 percentile or
lower on the Broad Math Cluster of the WJ-III-NU being placed in the "poor
mathematical performance" group. As mentioned previously, poor mathematics
achievement is identified by a score in the 25 percentile or less on the Broad Math
Cluster of the WJ-III-NU. There is a potential for multicollinearity with both logistic
regressions. However, the previous correlation and multiple regression analysis identified
any potential multicollinearity prior to the logistic regression. The researcher combined
or eliminated offending variables.
The data were analyzed with two separate sequential logistic regressions both
controlling for a participant's score on the SB5 Abbreviated IQ: 1) The first logistic
regression consisted of the WISC-IV Visual-Spatial Composite scores and a dichotomous
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poor achievement and adequate achievement dependent variable; 2) The second logistic
regression consisted of the SB5 Visual-Spatial Processing Factor scores and a
dichotomous poor achievement and adequate achievement dependent variable.
To identify the unique predictive ability of the visual-spatial measures (SB5;
W1SC-IV) while controlling for IQ, a sequential logistic regression was used in both
regressions. For both logistic regressions, IQ was entered in the first block and the
remaining independent variables were entered in the second block. A comparison of the
reported Nagelkerke R2, -2 Log likelihood, and Model Chi-Square indicated the
predictive ability of the visual-spatial measures over and above IQ. Specifically, the A
Chi-Square between the models suggested the predictive ability of the visual-spatial
measures while holding IQ constant across all participants.
To determine the better model a comparison of the -2 Log likelihoods and
Nagelkerke R of both sequential logistic regressions (WISC-IV; SB5) was conducted.
9
9
Nagelkerke R is an adjustment of the Cox & Snell R and is a variance interpretation
similar to R2 used in multiple linear regression (Tabachnick & Fidell, 2007). The more
accurate model showed the least amount of error (-2 Log likelihood) and accounted for
the largest amount of variance (Nagelkerke R ). In addition, to identify the best predictor
among the independent variables a comparison of the log odds/odds ratio was conducted.
An odds ratio can be interpreted as an effect size (Tabachnick & Fidell, 2007). Odds
ratios that are further away from 1 indicated a larger effect (Tabachnick & Fidell, 2007).
The best predictor was identified by the independent variable that most accurately
predicted poor mathematical achievement.
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Summary
The current study investigated the utility of the visual-spatial measures of the SB5
and the WISC-IV to identify students with low achievement in mathematics. The
independent variables for the study were: 1) WISC-IV Block Design subtest; 2) WISC-IV
Matrix Reasoning subtest; 3) WISC-IV Picture Completion WISC-IV; 4) The combined
WISC-IV Block Design subtest, Matrix Reasoning subtest, and Picture Completion
subtest composite score; 5) SB5 Position and Direction subtest; 6) SB5 Form Board
subtest; 7) SB5 Visual Spatial Processing Factor (Position and Direction and Form
Patterns); 8) SB5 Abbreviated IQ score. The dependent variable for the study was the
students' score on the Broad Math Cluster of the WJ-III-NU. The analysis of the data
included a correlation, a multiple regression and a logistic regression. The following
chapter (Chapter IV) reports the results of the data analysis. The final chapter (Chapter
V) discusses the implications of the results.
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CHAPTER IV
RESULTS
Data Analysis
This chapter presents the results of the data analysis. The data were initially
screened for assumption violations. After the initial data screening, each of the
subsequent research questions were addressed. As described in the methods section, a
correlation analysis, a multiple regression and a logistic regression were utilized to
address each respective research question.
Preliminary Analysis
Of the 112 participants, there were no missing or incomplete cases. The original
data set included nine variables: 1) Participants' scaled scores on the Block Design
subtest of the Wechsler Intelligence Scale for Children-Fourth Edition (WISC-IV); 2)
Participants' scaled scores on the Matrix Reasoning subtest of the WISC-IV; 3)
Participants' scaled scores on the Picture Completion of the WISC-IV; 4) Participants'
combined Block Design, Matrix Reasoning and Picture Completion scores; 5)
Participants' standard scores on the Abbreviated IQ Battery of the Stanford-Binet, Fifth
Edition (SB5); 6) Participants' scaled scores on the Form Patterns subtest of the SB5; 7)
Participants' scaled scores on the Position and Direction subtest of the SB5; 8)
Participants' standard scores on the Visual-Spatial Processing Factor Index of the SB5; 9)
Participants' standard scores on Broad Math Cluster of the WJ-III-NU.
A linear transformation was performed on the combined Block Design, Matrix
Reasoning and Picture Completion variable (variable indentified as #4 above), to
transform the variable into the scale used by the remaining composite scores. This
123
variable is the combination of the three visual-spatial subtests of the WISC-IV. The
combined Block Design, Matrix Reasoning and Picture completion score was first
transformed into a Z-score [z = (X- jx)la\ where X= the combined score, JU = mean of the
combined score, and a = the standard deviation of the combined score] and then
transformed into the standard score (ss) scale (ss =fi + z(a); where the ju = 100; o =15)
used by both the WJ-III-NU and the SB5 (Glenberg, 1996; Roid, 2003; Wechsler, 2003,
McGrew, Schrank, & Woodcock, 2007). A linear transformation does not change a
variable's relationship to other variables (Glenberg, 1996).
After the transformation, the data were analyzed for violations of normality and
the presences of univariate outliers. Only one variable demonstrated notable deviations
from normality. The Matrix Reasoning variable had notable positive skewness (z = 3.77)
and negative kurtosis (z = -5.16). This skewness may have been attributable to the
presences of two potential outliers as suggested in a histogram, stem and leaf plot and
box plot. Standardized residuals confirmed two subjects had standardized residuals
greater than z = 3.29 from the mean and were eliminated from the data analysis, as
recommended by Tabachnick and Fidel (2007). The removal of the cases reduced the
positive skewness (z = 0.14) and negative kurtosis (z = 0-.54) for the Matrix Reasoning
Variable to an acceptable level. None of the remaining variables indicated any univariate
outliers with standardized residuals greater than z = 3.29 from the mean.
The data were also examined for multivariate outliers. Mahalanobis distance
values were examined to identify multivariate outliers. Potential outliers were identified
as cases with Mahalanobis distances greater than^ 2 (9) = 27.88, p < .001 (Tabachnick &
Fidell, 2007). One subject had a Mahalanobis distance of/2 = 69.01 and was eliminated.
124
Further analysis identified this subject had notably lower scores on the visual-spatial
measures of the SB5 the WISC-IV and a comparatively higher score on the Broad Math
cluster of the WJ-III-NU. The data were also investigated for subjects with large
influence through the analysis of Cook's distance [3(k+l)/N; where k = number of
independent variables (9) and N = number of subjects (109)]. No subject was equal to or
exceeded a Cook's distance of 0.28. In addition, for each subject, leverage values were
examined and no comparatively large leverage values were found in the data. The
deletion of the univariate and multivariate outliers left a total of 109 cases (see table 4.1
for descriptive statistics).
Table 4.1
Descriptive Statistics
Variable
Block Design
Matrix Reasoning
Picture Completion
WISC-IV Combined
Visual-Spatial Measures
Abbreviated IQ (SB5)
Form Patterns
Position and Direction
Visual-Spatial Processing
Factor Index (SB5)
Broad Math (WJ-III-NU)
Standard
Deviation
2.55
2.26
2.60
Mean
9.26
8.49
9.78
99.65
91.82
10.05
9.63
13.65
10.92
2.35
2.53
98.69
98.43
11.62
12.60
Note. Composite/combined variables are indicated by bold typeface.
#=109
Correlation
A correlation analysis was employed, using the Pearson product-moment
correlation coefficient (r), to determine if there was a relationship between visual-spatial
processing and mathematics achievement. The results of the analysis suggested a
statistically significant relationship between participants' scores on the visual-spatial
measures of both the SB5 and the WISC-IV and their Broad Math score on the WJ-III125
NU (see table 4.2). In addition, using Cohen's (1998) conventions, the relationships
between the visual-spatial measures of the WISC-IV and SB5 and the Broad Math of the
WJ-III-NU demonstrated a medium effect size. The relatively large correlations (r >
0.70) between the Block Design subtest, the Matrix Reasoning subtest, the Picture
Completion subtest and the Combined WISC-IV Visual-Spatial Composite, as well as the
moderately large correlations between the Form Patterns subtest, the Position and
Direction subtest and the SB5 Visual-Spatial Factor suggested a notable amount of
overlap. The large correlations among these variables were not unexpected, as the
combined/factor scores are the result of combining the respective subtests. In addition,
using Cohen's conventions there was a large correlation [r (109) = .54] between the
visual-spatial measures of SB5 and WISC-IV.
Table 4.2
Correlations
Variables
1.
2.
3.
4.
1. Block Design (WISC-IV)
2. Matrix Reasoning (WISC-IV)
.46**
3. Picture Completion (WISC-IV) .30** .34**
4. WISC-IV Visual-Spatial Comp. .78** .77**
.73**
5. Form Patterns (SB5)
.52** .23*
.30** .47**
6. Position and Direction (SB5)
.35** .28**
.30** .42**
7. SB5 Visual-Spatial Factor (SB5) .53** .32**
.37** .54**
8. Broad Math (WJ-III-NU)
.39** .27**
.21** .38**
9. Abbreviated IQ (SB5)
.32** .44** .38** .50**
Note. The significance levels are based on uncorrected probabilities
5.
.31**
.79**
.34**
.24*
6.
7.
.83**
.42**
.30**
.47**
.33**
8.
9.
.50**
V < 0 . 0 5 ; **p< 0.001
Multiple Regression
A sequential multiple regression was used to determine if the visual-spatial
measures of the WISC-IV and the SB5 predicted mathematics achievement on the WJIII-NU (as measured by the Broad Math Cluster score). The previous correlation analysis
suggested a notable degree of overlap between the visual-spatial subtests and the
combined composite/factor scores of both the WISC-IV and SB5. While the larger
126
correlations were below .90, which would strongly suggest the potential for
multicollinearity, caution is warranted when including two or more variables with
bivariate correlations greater than .70 in a multiple regression (Mertler & Vannatta, 2005;
Tabachnick & Fidell, 2007). Given the potential for multicollinearity and singularity, as
well as the researcher's desire to look at the visual-spatial measures of the SB5 and
WISC-IV in totality, the researcher chose to use the composite/factor scores for the final
analysis (WISC-IV Visual-Spatial Composite; SB5 Visual-Spatial Factor; SB5
Abbreviated IQ; WJ-III-NU Broad Math).
As the data were previously screened for outliers and violations of normality, this
process was not repeated. An examination of residual scatter plots did not suggest any
violations of normality, linearity or homoscedasticity. An examination of the DurbinWatson statistic suggested independence of errors (Durbin-Watson = 1.70). After the
initial regression, the data were examined for outliers in the solution; no standardized
residuals greater than z = ±3.00 from the mean were found. No evidence of potential
multicollinearity was found in an examination of tolerance, condition index or
eigenvalues.
IQ significantly predicted mathematics achievement [R2= .24, adjusted R2 = .23;
F ( l , 107) = 33.59, MSE = 121.98,;? < .001]. In addition, participants' visual-spatial
composite/factor scores on WISC-IV and SB5 accounted for a statistically significant
amount of the variance (11%) in participants' Broad Math scores over and above IQ (see
table 4.3). Finally, the foil model, including IQ, SB5 Visual-Spatial Factor, and WISC-IV
Visual-Spatial Composite was significant [R2= .35, adjusted R2= .33; F (3,105) = 18.62,
MSE = 106.62,/» < .001]. This suggested that holding IQ constant across all participants,
127
visual-spatial processing accounted for a significant amount of the variance in
mathematical achievement. While the visual-spatial composite/factor scores did account
for a significant amount of the variance, there are some concerns regarding practical
significance. Based on Cohen's conventions for effect size [where R2=f2l (1 +f2)] in a
multiple regression (small = 0.02; medium = 0.13; large = 0.26), a change in (A) R of
0.11 demonstrated a medium effect size (Cohen, 1988). The regression coefficients and
significance tests for the full model (see table 4.4) indicated the slope of IQ-SB5 and the
SB5 Visual-Spatial Factor were significantly different from zero. Table 4.4 provides the
coefficients and significance tests for the reduced model. From an analysis of the
individual predictors, it appears IQ was the most important predictor, followed by the
SB5 Visual-Spatial Factor. Given its relatively small and nonsignificant /? weight, the
WISC-IV Visual-Spatial Composite appeared to be the least important predictor in the
model.
An additional sequential multiple regression, with the removal of the WISC-IV
Visual-Spatial Composite variable, was used to further evaluate the importance of the
SB5 Visual-Spatial Factor over and above a participants IQ. The full model that included
IQ and the SB5 Visual-Spatial Factor, was significant [R2 = .35, adjusted R2 = .33; F
(2,106) = 28.18, MSE = 105.62,/? < .001]. In addition, the change in R2 was significant
[AR2 = .11, AF(1, 106) = \7.57,p< .001]. This analysis further supported the lack of
importance of the WISC-IV Visual-Spatial Composite in the full model.
From these results, it can be concluded that visual-spatial processing does
significantly predict mathematics achievement over and above a participant's IQ. In
addition, only the visual-spatial measures of the SB5 were statistically significant in the
128
model. This suggested the WISC-IV was not a useful predictor of a participant's
mathematical achievement.
Table 4.3
R Change and Change Statistics
AR2
Model
.24
Reduced
(10)
Full
.11
(10, SB5, wise -IV)
dfl
1
df2
107
AF
33.59**
2
105
8.71**
Note. A= change; dependent variable = Broad Math score on the WJ-III-NU
**p<.001
Table 4.4
Coefficients and Significance Tests for the Reduced and Full Models
Model
b
Std. Error fi
Reduced (10)
Intercept
46.67**
9.00
Abbreviated 10 (AIO-SB5)
.56**
.10
.49
Full (10. SB5. WISC-IV)
Intercept
21.50*
10.37
Abbreviated IQ (AIQ-SB5)
.43**
.11
.37
WISC-IV Visual-Spatial Composite .01
.10
.01
SB5 Visual-Spatial Factor
.37**
.10
.34
Note. Dependent variable = Broad Math score on the WJ-III-NU
*p<.05, **p<-001
Logistic Regression
Two sequential logistic regressions were utilized to determine which measure of
visual-spatial processing (SB5; WISC-IV) was the better predictor of a participant with
poor mathematics achievement over and above IQ. The previous multiple regressions
identified which of the two visual-spatial measures was the better predictor of
mathematical achievement examined continuously. The use of the logistic regressions
allowed for a clearer link between analysis and practice. Dichotomizing the dependent
variable allowed for the visual-spatial processing measures to predict students with poor
mathematical performance. Identifying the best predictor was important because it
indicated which visual-spatial measure (WISC-IV; SB5) can more accurately be used to
129
identify students with poor achievement in mathematics. Additionally, this may in turn
provide enhanced accuracy in the assessment and diagnosis of learning disabilities in the
area of mathematics, above the understanding of the statistically significant predicative
relationship between visual-spatial processing and mathematics achievement found by
the previous multiple regression.
As the data were previously screened for univariate outliers, multivariate outliers
and normality, this process was not repeated for the logistic regression analysis. A
dichotomous variable was created based on a participant's standardized score on the WJIII-NU. Participants who scored above the 25th percentile rank were placed in the
adequate mathematical achievement category (coded as 0). Participants who scored at or
below the 25th percentile rank were placed in the poor mathematical achievement
category (coded as 1). In the current sample, 28 of the participants or 25.69% were placed
in the poor mathematical achievement category. Previous research suggests students who
score at the 25 percentile rank or below on mathematical achievement tests exhibited
notable difficulty in mathematical performance (Geary, 1993; Geary, 2004; Mazzocco &
Meyer, 2003; Murphy, Mazzocco, Hanich & Early, 2007; Swanson & BeebeFrankenberger, 2004; Zeleke, 2004). Both sequential logistic regressions were analyzed
for linearity in the logit using the Box-Tidwell approach prior to analysis; no significant
(p < .001) model interaction terms were found (Tabachnick & Fidell, 2007). Potential
multicollinearity was examined in the previous multiple regression and no evidence of
multicollinearity was found at that time.
130
SB5 Visual-Spatial Processing Factor
The first logistic regression analyzed the relationship between participants' scores
on the Visual-Spatial Factor Index of the SB5 and participants' mathematical
performance (as measured by participants' Broad Math score on the WJ-III-NU) while
controlling for IQ (see table 4.1 for descriptive statistics). After the initial regression
analysis, the data were screened for potential outliers in the solution. Two subjects were
identified as potential outliers in the solution, with studentized residuals greater than
+2.00 from the mean. However, after evaluating the results of the logistic regression with
the potential outliers removed, the potential outliers did not exert strong influence on the
results and were not eliminated.
IQ significantly predicted whether or not a participant would score below the 25th
percentile rank on the Broad Math Cluster of the WJ-III-NU (x2 (1) = 9.65, p < .05,
Nagelkerke R = .13). In addition, SB5 Visual-Spatial Processing score significantly
contributed to the prediction of mathematical performance (difference ^ (1) = 5.15,p <
.05,) suggesting a participant's SB5 visual-spatial score can predict mathematical
achievement over and above IQ (see table 4.6). Together IQ and the SB5 Visual-Spatial
Processing Factor score significantly predicted category, ^ 2 (2) = 14.81,/? < .05
Nagelkerke R2 - . 19. The Deviance Chi-Square for the full model was significant (-2LL =
109.40, p<. 05).
The full model (IQ and SB5 Visual-Spatial Processing Factor) correctly predicted
72.5% of all cases; which was an increase from the reduced model (IQ) that correctly
predicted 71.6% of all cases. In addition, the full model correctly predicted 14.3% of the
participants in the poor mathematical performance category and 92.6% in the adequate
131
mathematical performance category. In the reduced model, 7.1% were correctly predicted
in the poor mathematical performance category and 93.8% in the adequate mathematical
performance category.
The odds ratios for both IQ and SB5 Visual-Spatial were significant (see table
4.6). In addition, using the equation (rj2 = (f/cf+4; where d = ln(odds ratio)/l.81),
reported in Tabachnick and Fidell (2007), odds ratios can be converted into eta squared
values (ff) which can be interpreted as a measure of the proportion of the overall
variability attributed to an independent variable (Myers & Well, 2003). Using Cohen's
(1988) conventions, the eta squared values for the SB5 Visual-Spatial Processing Factor
odds ratio (77 2= .0002) demonstrates a small effect size. Both predictors appeared to be
relatively equal in strength, which suggested that as a participants' IQ or SB5 scores
increase the chances of being in the lower 25 percentile rank are .95 times the odds for
students with an IQ or SB5 score one point lower. The results indicated the SB5 VisualSpatial Processing Factor can predict mathematical achievement over and above IQ;
however, given the small amount of variance accounted for, practical significance may be
tenuous. This may mean the use of the SB5 Visual-Spatial Factor as a diagnostic tool for
poor mathematical performance is potentially not pragmatic, because it appears other
factors appear to account for a greater amount of variance.
Table 4.5
Model Statistics
Nagelkerke
Variable
B1
Abbreviated SB5 IQ
.13
SB5 Visual-Spatial Factor .19
*p<.05,
-ILL
114.57
109.81
**p<.001
132
Model
Chi-Square
9.65*
14.81**
A
Chi-Square
5.15*
Table 4.6
Model Parameters
Variable
Abbreviated SB5 IQ
SB5 Visual-Spatial Factor
Constant
Log Odds
Ratio
-0.05*
-0.05*
8.32*
S.E.
0.02
0.02
2.63
Wald
4.85
4.79
9.99
Odds Ratio
0.95
0.95
4110.45
*p < .05,
WISC-IV Visual-Spatial Composite
The second logistic regression analyzed the relationship between participants'
scores on the combined WISC-IV Visual-Spatial Composite (the combined and linear
transformed participant score on the Block Design, Matrix Reasoning, Picture
Completion subtests) and participants mathematical performance (as measured by
participants' Broad Math score on the WJ-III-NU) while controlling for participants' IQ
(see table 4.1 for descriptive statistics). Results of a preliminary logistic regression failed
to identify any potential outliers in the solution (indicated by studentized residuals greater
than ±2.00 from the mean).
A participant's WISC-IV Visual-Spatial Composite significantly contributed to
the prediction of mathematical performance over and above IQ (difference/2 (1) = 3.95,
p < .05,) suggesting that a participant's WISC-IV Visual-Spatial Composite can predict
mathematical achievement while controlling for IQ (see table 4.8). Together IQ and the
WISC-IV Visual-Spatial Processing Composite significantly predicted poor mathematics
performance/ (2) = 13.60,/? < .001, Nagelkerke R2=.\7. The Deviance Chi-Square for
the full model was significant (-2LL = 110.61,/) < .05).
The full model (IQ and WISC-IV Visual-Spatial Composite) correctly predicted
73.4 % of all cases, which was an increase from the reduced model (IQ). In addition, the
full model correctly predicted 17.9% of the participants in the poor mathematical
133
performance category and 92.6% in the adequate mathematical performance category.
The reduced model correctly predicted 7.1 % of the participants in the poor mathematical
performance category and 93.8% in the adequate mathematical performance category.
The odds ratio for WISC-IV Visual-Spatial Composite was significant (see table
4.8). ). In addition, using the equation (q2= d2/d2+4; where d = ln(odds ratio)/l.81),
reported in Tabachnick and Fidell (2007), odds ratios can be converted into eta squared
values (rj2) which can be interpreted as a measure of the proportion of the overall
variability attributed to an independent variable (Myers & Well, 2003). Using Cohen's
(1988) conventions, the eta squared values for the WISC-IV Visual-Spatial Composite
odds ratio (r/ 2= .0001) demonstrates a small effect size. Both predictors appear to be
relatively equal in strength, suggesting as a participants' IQ or WISC-IV score increases
the chances of being in the lower 25th percentile are .96 times the odds for students with
an IQ or WISC-IV score one point lower. The results of the sequential logistic regression
suggested the WISC-IV Visual-Spatial Composite can predict mathematics achievement
over and above IQ; however, given the small amount of variance accounted for by the
WISC-IV Visual-Spatial Composite, there may be practical significance concerns. This
may mean the use of the WISC-IV Visual-Spatial Composite measure as a diagnostic tool
for poor mathematical performance may not be useful in clinical or k-12 school settings,
because other factors appear to account for a greater amount of variance.
Table 4.7
Model Statistics
Nagelkerke
Variable
R2
-2LL
Abbreviated SB5 IQ
.13
114.56
WISC-IV Visual-Spatial Composite .17
110.61
*p < .05,**p<.001
134
Model
Chi-Square
9.65*
13.60**
A
Chi-Square
3.95*
Table 4.8
Model Parameters
Variable
Abbreviated SB5 IQ
WISC-IV Visual-Spatial
Composite
Constant
Log Odds
Ratio
-0.05
S.E.
0.03
Wald
3.31
-0.04*
6.84*
0.02
2.34
3.78
8.54
Odds Ratio
0.96
0.96
935.02
*p < .05
Comparison of the SB 5 and the WISC-IV
To identify which measure of visual-spatial processing (SB5; WISC-IV) was the
better predictor of mathematical achievement, the -2 Log likelihoods and variance
accounted for (Nagelkerke R2) in the final models were compared. In comparing the
overall models, that included IQ, both visual-spatial measures produced statistically
significant (p < .001) % statistics (see table 4.9). The SB5 Visual-Spatial Factor showed
the least amount of error and accounted for the largest amount of variance (SB5: -2LL =
109.40,/? < .001; Nagelkerke R2 = .19); however, the difference between the two final
models was small (WISC-IV: -2LL = 110.61,/? < .001; Nagelkerke R2 = .17). While the
comparison indicated the SB5 Visual-Spatial Factor was the better predictor, there was
little notable difference between the two measures ability to predict poor mathematical
performance.
In addition to a comparison of the -2 Log likelihoods and Nagelkerke R2, to more
clearly identify the better individual predictor of mathematics achievement, an
examination of the odds ratios of both logistic regressions was performed (see table
4.10). Both odds ratios were significant In addition, the eta squared values for the SB5
Visual-Spatial Processing Factor odds ratio (rj2= .0002) and for the WISC-IV VisualSpatial Composite odds ratio {n2 = .0001) both demonstrate a small effect size. The
results indicated .02% of the total variance is attributed to the SB5 Visual-Spatial Factor
135
and .01% of the total variance is attributed to the WISC-IV Visual-Spatial Composite in
their respective models. The eta squared values support, the SB5 Visual-Spatial
Processing Factor contributed slightly more variance; however, both appeared to lack
practical significance as diagnostic tools for poor mathematical achievement.
Table 4.9
Comparison of Models
Variables
SB5andIQ
WISC-IV and 10
Nagelkerke
it
0.19
0.17
-2LL
109.42
110.61
Model
Chi-Square
14.81**
13.60**
**p< .001
Table 4.10
Model Parameters of Both the SB5 and WISC-IV
Log Odds
Variable
Ratio
S.E.
Abbreviated SB5IQ
-0.05
0.03
WISC-IV Visual-Spatial Composite -0.04*
0.02
Constant
6.84*
2.34
Abbreviated SB5 IQ
-0.05*
0.02
SB5 Visual-Spatial Factor
-0.05*
0.02
Constant
8.32*
2.63
Wald
3.31
3.78
8.54
4.85
4.79
9.99
Odds
Ratio
0.96
0.96
935.02
0.95
0.95
4110.45
*p < .05
Summary
The data were subjected to an analysis for violations of normality, linearity and
outliers. The correlation analysis suggested a significant relationship between the visualspatial measures of the SB5 and WISC-IV and the Broad Math Cluster of the WJ-III-NU.
A multiple regression analysis identified visual-spatial processing could predict
mathematical achievement over and above IQ; however, given the small amount of
variance accounted for, there were practical significance concerns. Two sequential
logistic regressions were employed to determine which visual-spatial measure (SB5 or
WISC-IV) was the better predictor of poor mathematical achievement. The results of the
logistic regressions indicated both measures could significantly predict poor
136
mathematical performance; however, the difference between the two measures was
minimal. Finally, given the small amount of variance accounted for by both measures, the
results suggested questionable practical significance in the identification of students' with
poor mathematical performance with visual-spatial measures of both the SB5 and WISCIV. The proceeding chapter (Chapter V) will provide a discussion of the results found in
the study.
137
CHAPTERV
DISCUSSION
Discussion of Results
The final chapter will address the implications of the results found in Chapter IV.
In addition, the chapter will discuss potential areas of future research and the limitations
of the current study. Finally, the chapter will concluded by revisiting the significance of
the study.
The primary purpose of this study was to examine which measure of visual-spatial
processing, the Stanford-Binet Intelligence Scales, Fifth Edition (SB5) or the Wechsler
Intelligence Scale for Children-Fourth Edition (WISC-IV), was the better predictor of
poor mathematical performance as measured by the Broad Math Cluster of the
Woodcock-Johnson III Tests of Achievement-Normative Update (WJ-III-NU).
Participants' scores on the SB5 Abbreviated IQ measure were utilized to determine
visual-spatial processing's unique contribution to mathematical performance over and
above IQ. To address each research question, data were obtained from 109 students in
grades 6-8 at several Midwestern and Western locations in the United States. Participants
were administered: 1) The Block Design, Matrix Reasoning, and Picture completion
subtests of the WISC-IV; 2) The Form Patterns and Position and Direction subtests of the
SB5; 3) The Calculation, Fluency and Applied Problems subtests of the WJ-III-NU.
Visual-Spatial Processing's Relationship to Mathematical Achievement
The results of the correlation analysis indicated there was a medium positive
relationship between visual-spatial processing and mathematical achievement. The results
confirmed previous research that has found a statistically significant relationship between
visual-spatial processing and mathematics (Ansari et al., 2003; Assel, Landry, Swank,
138
Smith & Steelman, 2003; Busse, Berninger, Smith & Hildebrand, 2001; Cornoldi,
Venneri, Marconato, Molin & Montinari, 2003; Geary, 1993; Geary & Hoard, 2003;
Hartje, 1987; Mazzocco, 2005; Reuhkala, 2001; Swanson & Jerman, 2006). However, the
moderate effect sizes suggested, while there was indeed a statistically significant
relationship, that relationship was not as strong as the researcher would have hoped given
the previous research in this area. In examining all variables that measured visual-spatial
processing (WISC-IV; SB5), the SB5 Visual-Spatial Factor had the largest correlation to
the Broad Math Cluster of the WJ-III-NU; suggesting the visual-spatial processing
measures of the SB5 were the most related to mathematical achievement when compared
to the visual-spatial measures of the WISC-IV. Further, of the three WISC-IV subtests
purported to be measures of visual-spatial processing, the Block Design subtest had the
strongest correlation with the Broad Math Cluster of the WJ-III-NU.
The statistically significant correlation between the WISC-IV Block Design
subtest and the mathematics measures of the WJ-III-NU was in line with previous
research that has found a significant positive relationship between the Block Design
subtest of the WISC-IV and mathematical performance (Carroll, 1993; Cornoldi et al.,
2003; Fuchs et al, 2005; Hegarty & Kozhevnikov, 1999; Lee, Ng, Ng, & Lim, 2004). In
addition, there was a moderate to large correlation between the SB5 Visual-Spatial
Processing Factor and participants combined Block Design (BD), Matrix Reasoning
(MR), and Picture Completion (PC) score. The moderately large correlation suggested
that combined, the three subtests (BD, MR and PC) were related to the SB5 VisualSpatial Processing Factor. Additionally, given the ample specificity of the Block Design,
Matrix Reasoning, and Picture Completion subtests at all ages (> 25%) it suggests
139
individually they may measure a specific construct distinct from the hypothesized "g" or
general intelligence (Sattler & Dumont, 2004). Further, if the three combined subtests are
related to the Visual-Spatial Processing Factor of the SB5 (which visual-spatial subtests
demonstrate ample and adequate specificity), and their specificity indicates the subtests
measure distinct constructs from "g", there is support for a visual-spatial
factor/composite within WISC-IV as first suggested by Keith et al. (2006). Future
research may wish to further explore if these three combined subtests are a distinct
measure of visual-spatial processing.
The results of the correlation analysis found an additional notable correlation with
the Broad Math Cluster of the WJ-III-NU. Of all variables included in the analysis, the
SB5 Abbreviated IQ had the strongest positive correlation with the Broad Math Cluster
of the WJ-III-NU. The notable correlation indicated perhaps one, both or the
combination of the subtests used in the SB5 abbreviated measure of IQ (Object Series/
Matrices and Vocabulary) has a stronger relationship with a participant's mathematical
functioning then any of the SB5 or WISC-IV visual-spatial processing measures. This
finding may indicate either additional psychological processes or general intelligence (g)
is more related to mathematical achievement than visual-spatial processing. Future
research with the SB5 should explore which of the two Abbreviated IQ subtests is most
related to mathematical achievement.
The Predictive Ability of the Visual-Spatial Measures of the SB 5 and WISC-IV
The multiple regression demonstrated that visual-spatial processing can
significantly predict mathematical achievement over and above IQ. The SB5 VisualSpatial Processing Factor was a better predictor of a student's score on the Broad Math
140
Cluster of WJ-III-NU than the combination of the BD, MR, and PC subtests of the
WISC-IV. Of the three variables, the SB5 Abbreviated IQ was the best predictor of a
student's score on the Broad Math Cluster of the WJ-III-NU accounting for notably more
variance (24%) than the visual-spatial measures of both the SB5 and WISC-IV (11%)
combined (accounting for participants IQ).
It is important to note that given the less than adequate specificity of the Position
and Direction subtest of the SB5, there may have been some confounding among the
variables in the multiple regression. Because the Position and Direction subtest is not a
inherently distinct measure from "g", it may be that including both the Abbreviated IQ
and the Position and Direction subtest (as part of the SB5 Visual-Spatial
Factor/Composite) impacted the attenuation of R in final regression model that included
IQ, the SB5 Visual-Spatial Factor and the WISC-IV Visual-Spatial composite. Thereby
clouding the results of the multiple regression to some degree.
One implication of the results is related to the identification of which
psychological process is the most highly predictive of students' mathematical
achievement. Visual-spatial processing was related to students' mathematical
achievement, suggesting a student with more developed visual-spatial processing abilities
performs better in the area of mathematics regardless of their IQ; however, given the
relatively small amount of variance accounted for by visual-spatial processing, the
practical utility of both measures as diagnostic tools for poor mathematical performance
is questionable. The amount of variance that was left unexplained by the combined
measures of visual-spatial processing is sizeable (65%). That would indicate some other
processing area or cognitive ability may be a better predictor of mathematical
141
achievement. Furthermore, perhaps it is a student's overall cognitive ability that is the
best predictor of how well a student will do mathematically and visual-spatial processing
is too narrow of a processing area to be useful in predicting a student's mathematical
abilities.
Similar to the correlation analysis, the SB5 Abbreviated IQ appeared to be a
more important predictor of mathematical achievement than the visual-spatial measures
used in the current study. The SB5 Abbreviated IQ with its two subtests (Object
Series/Matrices and Vocabulary) explained more variance than both measures of visualspatial processing combined. This suggested that an abbreviated measure of intelligence
may be a better predictor of mathematical achievement than measures of visual-spatial
processing. The results of this study were consistent with previous research that has
found measures of fluid reasoning and comprehension-knowledge/crystallized reasoning
were moderate predictors of mathematical performance on the mathematical achievement
tests of WJ-III (Floyd, Evans, McGrew, 2003). This suggests that while previous research
supports the importance of visual-spatial processing in mathematical achievement,
general intelligence may be considerably more important. In addition, the literature
identifies additional areas of cognitive processing that are related to mathematical
achievement and maybe significant predictors of mathematical performance.
Prior research has found additional cognitive processes such as attention, working
memory, short-term memory, long-term (semantic) memory, speed of processing and
phonological processing were related to mathematical achievement (Floyd, et al., 2003;
Fuchs et al., 2006; Fuchs et al, 2005; Geary, Hoard, Byrd-Craven, Nugent & Numtee,
2007; Murphy, Mazzocco, Hanich & Early, 2007; Swanson, 2006; Swanson & Beebe-
142
Frankenberger, 2004; Swanson, Jerman, and Zheng, 2008). The results of the current
study suggested, that while visual-spatial processing did appear to be a statistically
significant predictor of mathematical achievement, there are other cognitive areas
(processing, general intelligence) that may prove more important. Further research into
additional processing areas (working memory, speed of processing, etc.), general
intelligence and mathematical achievement is needed to fully articulate which area of
cognitive functioning is the best predictor of mathematical performance.
The results of the regression analysis may demonstrate some limited practical
significance concerns regarding the utility of both measures as predictors of mathematical
performance in a clinical application; however, the multiple regression did add to the
theoretical understanding of the relationship between visual-spatial processing and
mathematical achievement. The visual-spatial measures of the WISC-IV and SB5
accounted for 11% of the variance over and above a participant's IQ. In other words,
irrespective of a participants IQ, the visual-spatial measures of the WISC-IV and SB5
were able to account for a significant amount of the variance in mathematical
performance. This finding may add theoretical support to the importance of visual-spatial
processing in mathematical performance.
Comparison of the SB 5 and WISC-IV Visual-Spatial Measures
In the literature, students are often defined as having poor mathematical
achievement if they score at or below the 25th percentile on mathematics achievement
tests (Geary, 1993; Geary, 2004; Mazzocco & Meyer, 2003; Murphy, Mazzocco, Hanich
& Early, 2007; Swanson & Beebe-Frankenberger, 2004; Zeleke, 2004). To tie the current
study with previous research, participants were separated into two dichotomous groups to
143
compare the predictive ability of the visual-spatial measures of the SB5 and WISC-IV.
Participants who scored at or below the 25th percentile rank on the Broad Math Cluster of
the WJ-III-NU were placed in the poor mathematics achievement category and students
who scored greater than the 25 percentile rank were placed in the adequate
mathematical achievement category.
Two separate sequential logistic regressions were employed and compared to
determine which measure of visual-spatial processing was the better predictor of poor
mathematics achievement accounting for the effects of IQ. The results suggested both the
SB5 Visual-Spatial Processing Factor and the WISC-IV Visual-Spatial Composite
significantly predicted mathematical performance over and above IQ. Meaning regardless
of a student's IQ, both measures of visual-spatial processing were able to predict whether
or not a student demonstrated poor mathematical skills. However, further examination
suggested that while the measures were both statistically significant, the amount of
variance accounted for by both measures was small. The eta squared values indicated a
small effect size for both the WISC-IV and SB5 in their respective models. In
comparison of the final models, that included IQ and visual-spatial processing, the SB5
demonstrated the least amount of error and accounted for the most variance. The odds
ratios and eta squared values for the visual-spatial measures of the SB5 and WISC-IV in
their respective models were not notably different, suggesting little practical difference.
This suggests the two measures were not notably different in their ability to predict
mathematical performance. The comparison of the results of both logistic regressions
answered the main research question of this study, determining that neither of the visual-
144
spatial measures of the SB5 or the WISC-IV were decisively better in identifying poor
mathematical performance.
One implication for these results is that neither measure of visual-spatial
processing appeared to be a practical tool for use in the prediction of poor mathematical
performance. The results suggested, due to the relatively small amount of variance
accounted for, by both measures, the practical utility of the two visual-spatial measures as
diagnostic tools for poor mathematical performance was minimal. In addition, the
relatively minimal difference between the two visual-spatial measures, as predictors of
poor mathematical performance, suggested neither measure was notably better than the
other. The lack of a clear difference between the two instruments and small amount of
variance accounted for may have implications for how specific learning disabilities
(SLD) are defined and assessed in the practice of school psychology.
Again it should be emphasized that while, the study suggests limited practical
difference between the visual-spatial measures of the WISC-IV and SB5 in the prediction
of mathematical performance, both were able to statistically predicted the category of
mathematical performance over and above a student's IQ. Indicating regardless of a
student's overall cognitive ability (i.e. IQ) both visual-spatial measures can predict if a
student is struggling in mathematics. This finding may add to the theoretical
understanding of the relationship between visual-spatial processing and students who
struggle in mathematics; identifying the importance of visual-spatial processing in
mathematical achievement over and above a student's level of intelligence.
145
Further Implications for the Current Study
The results of the current study found visual-spatial processing was a statistically
significant predictor of mathematical achievement over and above a student's level of
intelligence. However, the results of the analysis suggested both measures of visualspatial processing may have limited practical utility in identifying poor mathematical
performance. Given the small amount of variance accounted for by visual-spatial
processing in mathematical achievement, it may be that defining a SLD, particularly a
math disability, as a visual-spatial processing disorder may not be important.
Furthermore, if state departments of education do not assess for deficits in processing,
then defining a SLD as a processing disorder may not be inherently useful.
As identified in the previous literature review, little has changed in regards to the
definition of a SLD since its conceptualization by Samuel Kirk in the 1960's and
introduction into special education law in the 1970's. In an unpublished recent review of
how states currently define a SLD, this researcher found that 49 of the 51 states
(including the District of Columbia) use the federal definition of a SLD or use the term
"processing disorder" in their definition (Clifford, 2008). In this review, the only two
states that did not use the term "processing disorder" were Kentucky and Louisiana.
Although the wording used by both states, " acquisition, organization, or expression"
(Kentucky) and "to acquire, comprehend, or apply" (Louisiana) suggested similar
terminology to the definition of psychological processing (i.e. the cognitive abilities that
allow the use of language, attention, memory, complex problem solving, higher order
thinking and perception in academic and non-academic tasks) (Gerring, & Zimbardo,
146
2002; Kentucky Administrative Regulations, p. 8,2007; Louisiana Administrative Code:
Chapter 28, 2008, p. 59).
States adherence to the federal definition has two possible implications. First, if
the federal definition is the accepted definition, then more research should be conducted
regarding psychological processing and learning disabilities (particularly mathematics).
Second, if psychological processing is not considered a valid part of a SLD then an
evolution of how a SLD is defined is needed. Support for a change in how a SLD is
defined may come from the substantial degree of difference between the federal and
states' SLD definition and how it is operationalized for identification purposes.
Currently, how a SLD is identified has undergone a substantial change and state
departments of education appear to be moving away from the IQ-achievement
discrepancy model. The federal regulations allow a student to be diagnosed with a SLD
utilizing a response to intervention framework (RTI) that is inclusive of a comprehensive
evaluation (Federal Register, 2006). The federal regulations do not explicitly state as part
of a comprehensive evaluation, for a SLD, that a student must be administered
instruments that assess cognitive processing. Rather, the regulations note a student should
be assessed in all areas related to the suspected disability, which could include measures
of general intelligence, academic performance, communication, motor functioning,
health, vision, hearing, and social and emotional functioning (Federal Register, 2006).
Some proponents of the link between neuropsychological functioning and SLD
identification maintain best practices for SLD identification should include measures of
psychological processing as part of a comprehensive evaluation (Hale, Flanagan &
Naglieri, 2008). However, as the current research suggests, full conceptualization of the
147
relationship between psychological processing and SLDs may not be complete
particularly in the area of mathematics, and further exploration may be needed in-order to
consider it a valid component in SLD determination. In a recent unpublished review of
how states operationalize SLDs, only one state (Maine) noted the use of processing in
how a learning disability was identified (Clifford, 2008). The remaining states required a
RTI model, a discrepancy model with or without an RTI model, a discrepancy model in
combination to a RTI model, or a pattern of strengths and weakness with or without an
RTI model (Clifford, 2008). This indicates that few states adhere to the definition of a
SLD and specifically look for processing deficits in the identification of learning
disabilities. With few states looking for processing components in SLD identification
defining a SLD as a processing disorder does not appear to be pragmatic.
An additional implication for this study may be concerned with where the
majority of school psychologists' time should be spent. Should school psychologists
spend a significant amount of time using instruments that have shown, at times, a
somewhat tenuous link between psychological processing and mathematics achievement
to determine a SLD; or should school psychologists focus the majority of their efforts in
trying to help students prior to a point where they begin to have notable struggles in
mathematics? The results of the current study suggested, perhaps, school psychologists'
time might be better spent through incorporating an RTI-mathematics-SLD model based
on early screening and intervention, rather than investigating learning difficulties through
processing disorders. There has been substantial research in the area of RTI, early
intervention and reading. Only recently, has there been an increase in the research with
RTI as a component of early mathematics SLD identification and intervention.
148
There is research that has found a link between early screening and the prediction
of a SLD in mathematics in later grades. Mazzocco and Thompson (2005) in their
longitudinal study followed 226 kindergartners through third grade. The researchers
found they were able to predict with 80-83% accuracy students who had a potential SLD
in mathematics from their results on early screening curriculum based measurements
(CBM). Fuchs et al. (2007) in their longitudinal study followed 225 students from the
beginning of first grade through the end of second grade. Fuchs et al. (2007) found by
using CBM measures of mathematics they were able to predict those students who had a
potential disability in the area of mathematics. In addition, one recent study on the
efficacy of early screening and intervention in the area of mathematics was conducted by
Bryant, Bryant, Gersten, Scammacca, and Chavez (2008). The study consisted of first
and second grade students placed in a RTI model tier two intervention that involved 4560 minutes of small group tutoring in mathematics. The results of the study were mixed.
The authors found when study participants were compared to a control group there was a
significant increase in second grade students' scores on a test of early mathematics at the
end of the intervention cycle, but did not find significant increase in first grade students
scores (Bryant, et al. 2008). The results of the noted studies indicate research in the area
of mathematics RTI, early intervention, and SLD has been conducted, but substantially
more is needed.
Limitations
There are three principle limitations for the current study that should be
addressed. The first limitation is in regard to the sample used in the study. The size of the
sample in the current study was relatively small (N = 109). While the sample size was
149
adequate to meet predetermined power requirements, a larger sample size may have
provided increased power. In addition, the sample used in the study may have limited
generalizability. The sampling procedure was not random and only students who were
motivated enough to engage in the required tasks were participants. Furthermore,
participants were from Midwest and Western communities and in a specific grade range
(6-8) limiting generalizability to other areas of the country and age groups.
A second limitation may be fatigue affects. While the evaluator attempted to keep
the pace of administration at a moderate tempo, to keep participants engaged, it is
possible participants became fatigued towards the end of the testing session. This may
have contributed to participants performing better at the beginning of testing session than
at the end. In addition, while the administration order of the WISC-IV and SB5 varied,
the variation was not random or specifically accounted for. The lack of random variation
may have impacted the study's results as fatigue set in with each participant.
The final limitation of the current study relates to the administration of the visualspatial measures of both the WISC-IV and SB5. While both instruments utilize relatively
different measures of visual-spatial processing, it may be that results were clouded by
practice effects. So closely administering the subtests of both instruments may have
allowed the participants to have some practice in performing tasks that involve visualspatial skills thereby possibly clouding some of the results.
Future Research
While the current study found students' mathematical functioning could be
predicted by their visual-spatial processing abilities, the findings did seem to have
questionable practical utility. Future research should explore which cognitive areas are
150
most related to mathematical achievement (fluid reasoning; crystallized knowledge;
general intelligence etc.). Only through further examination of the link between cognitive
processing and academic abilities can the utility of measures of processing in K-12
school settings be realized. In addition, future research in the area of early interventionRTI and mathematics SLD is needed. Fully conceptualizing how to best identify and
intervene with students who have mathematic SLDs is critical to their future academic
success.
Importance of the Study
The current study was concerned with the utility of the visual-spatial processing
measures of the SB5 and WISC-IV as predictors of mathematical performance. The
fundamental question the study addressed was the predicative relationship between the
psychological processes of visual-spatial processing and poor mathematics performance.
If the current definition of a SLD is centered on the idea a SLD is a disorder in a
psychological process, then it is important research is conducted to identify which
psychological processes are the most related to which areas of learning difficulties. If a
clear link cannot be found between psychological processes and SLDs and states do not
identify SLDs based on processing deficits, then a definition based on the
conceptualization that a SLD is a disorder in a psychological process may be antiquated.
151
Appendix A
IRB Approval
152
Office of Human Subjects Protection
(605) 677-6184
(605) 677-3134 Fax
The University of South Dakota.
March 23, 2007
Jordan Mulder
The University of South Dakota
Counseling aid Psychology in Education
Project Title:
PI:
Level of Review;
Date Approved:
100-07-049-Visual-spatial Processing and Mathematics Achievement: The
Predictive Ability of the Stanford-Binet, Fifth Edition and the Wechsler
Intelligence Scale for Children Fourth Edition
Jordan Mulder
Student PI:
Eldon Clifford
Exempt 1
Risk:
No More than Minimal
3/22/2007
The proposal referenced above has received an Exempt review and approval via the procedures
of the University of South Dakota Institutional Review Board 01.
Annual Continuing Review is not required for the above Exempt study. However, when this
study is completed you must submit a Closure Form to the IRB. You may close your study when
you no longer have contact with the subject.
Prior to initiation, promptly report to the IRB, any proposed changes or additions (e.g., protocol
amendments/revised informed consents/ site changes, etc.) in previously approved human subject
research activities.
The forms to assist you in filing your: project closure, continuation, adverse/unanticipated event,
project updates /amendments, etc. can be accessed at
http://www.usd.edu/oorsch/compliance/applicationforms.cfm.
If you have any questions, please contact me: lkorcusk(5),usd.edu or (605) 677-6184.
Sincerely,
Lisa Korcuska
Director-Office of Human Subjects Protection
University of South Dakota
Institutional Review Boards
The University of South Dakota IRBs operate in compliance with federal regulations and applicable laws and are registered
with the Office for Human Subject Protections (OHRP) under FWA # 00002421.
414
Easl
Clark
Street
•
Vermillion,
SD 5 7 0 6 9 - 2 3 9 0
•
1 - 8 7 7 - C O YO T E S
•
Fax:
605-677-6323
•
www.usd.edu
Appendix B
Approval LettersfromParticipating Schools
154
Page 1 of2
Clifford, Eldon S
m
From:
Tracy Heiiman [tracy.heilman@ssccardinals.org]
Sent:
Wednesday, March 14, 2007 3:57 PM
To:
Clifford, Eldon S
Cc:
Rozy Warder; John Laughhunn
Subject;
FW: USD student dissertation research at Middle
Follow Up Flag: Follow up
Flag Status:
Red
Hi EldonYou have the green light. Please read Rozy's email below.
Success! Now the hard work begins!
Tracy
Tracy Heiiman Kennedy, Ph.D.
Project Director
Safe Schools/Healthy Students
South Sioux City Community Schools
(402) 412-2883
(712) 259-0808
' fracy.heilman@ssccardinals.org
From: Rozy Warder
Sent: Wednesday, March 14,2007 10:02 AM
To: Tracy Heiiman
Cc: John Laughhunn
Subject: RE: USD student dissertation research at Middle
As long as the student is not missing class time he can go ahead and start. I am sorry that it took so long, but
generally the district doesn't allow this type of work to occur. This is the exception. Eldon and go ahead and get
started. Please let Eldon know it is okay to begin.
Rozanne Warder
Student Services Director
Special Education, Talented and Gifted, At-risk Programs,
Health Services, Safe Schools Healthy Students
South Sioux City Community Schools
&outh Sioux City, NE
From: Tracy Heiiman
Sent: Tuesday, March 13, 2007 9:09 PM
To: Rozy Warder; John Laughhunn
Subject: USD student dissertation research at Middle
Importance: High
Eldon Clifford has visited with Mr. Laughhunn and emailed me to say John was agreeable to Eldon coming in to
his school during study halls to work with kids. You had previously told me to let you know when John was
informed and if he was agreeable, to let you know to get the final approval. Eldon has his IRB approval and is
eager to get to work. He is extremely grateful for the chance to collect data in SSC.
3/19/2007
Clifford, Eldon S
From:
Seh't;
To:
Subject:
Klinette Brandon [brandonk@weston1.k12.wy.us]
Tuesday, February 27, 2007 12:07 PM
Clifford, Eldon S
Re: Hello and a request
Hi Eldon,
I am sorry that it has taken so long to respond - I visited with Brad and Scott - they are
ok with your proposal - we can talk about the details. I can buy some of the assessments
if they are ones that you will perfer to use in our programs, I would like to visit with
you on the phone - it has been so busy! I will be travelling to Aberdeen today and back
tomorrow - I had a student at the school for the visually impaired that had a blow out and
got suspended so I have been to Aberdeen and have been working on this student's program it has been all consuming - anyway I will catch up with you!
I hope that things are going well - take good care.
Klinette
On Feb 20, 2007 08:24 AM, "Clifford, Eldon S" wrote:
>Klinette,
>
>
>
>Good morning, I hope your weekend was good.
>
>
> .
>I was wondering if you and the district could help me out? I am
experiencing some challenges finding a district around here that will
>allow me to gather data for my dissertation, and was hopeful that I
>could talk you and Brad into letting me gather a goodly chunk of it in
>Newcastle.
>
>
>
>What it involves is giving 2 subtests of the SB5 (Position and
>Direction; Form Board/ Form Patterns), three measures of the WISC-IV
>(Block Design; Matrix Reasoning; Picture completion) and 3 measures of
>tfc* WJ-III (Calculation; Fluency; Applied Problems). In total, it takes
>a little under 1 hour.
>
>
>
>I would be giving the battery to middle school students in 6th-8th
>grades. I need about 106 students total; however, I understand that
>Newcastle only has about 160-200 students (combined) in those grades.
>It
>would be unrealistic for me to think that I could obtain all of what I
>need from Newcastle. Although the more I can get the less difficult it
>would be for me in the long run.
>
>
>
>If you were to agree, I would send out permission forms/ consent forms
>in the early fall of 07. I would of course emphasize that students and
>parents are under no obligation to participate. In addition, I would
>gather data at times that would not interfere with my school psych
responsibilities or with students' core classes (say testing kids,
>duying study hall, non-core classes, before/ after school, etc...). In
>ad^ition, with your permission and parental consent I could also add
>those measures when I test a student for a learning disability.
1
>
>
>I would of course purchase all test protocols myself. In addition, I
>think I remember someone saying that the district does not have the
>WJ-III so I would of course purchase that test kit myself (and the SB5
>if need b e ) .
>
>
>
>While I know the district does not get much out of this I could provide
>tOv each parent a summary of the results for their child and could
>p'$bvide the district with a summary of my results, if desired. In
>addition, if you were to help me out I would be willing to commit in
>writing to at least three years with Newcastle if you decide you would
>like me to stay,
>
>
>
>Please let me know what you think.
>
>
>
>Have great week!
>
>
>
>Eldon
>
2
Page 1 of2
Clifford, Eldon S
From:
Lentz, Jim [Jim.Lentz@pas.k12.mn.us]
Sent:
Tuesday, March 13, 2007 7:27 AM
10 To:
Clifford, Eldon S
Subject: RE: permission
If permissions are obtained, that would be fine by me.
Jim Lentz
From: Clifford, Eldon S [mailto:Eldon,Clifford@usd.edu]
Sent: Monday, March 12,2007 2:37 PM
To: Lentz, Jim
Subject: permission
Mr. Lentz,
My name is Eldon Clifford, I a doctoral student at the University of South Dakota. Mr. Lammers referred me to
you. Your school psychologist, Renae, has agreed to assist me with some data collection for my dissertation. I am
seeking your permission to allow her to do that The study looks at ability of the visual-spatial measures of the
? Stanford-Binet, Fifth Edition ( Position and Direction; Form Patters/ Form Board) and the WISC-IV ( Block Design,
Picture Complete and Matrix Reasoning) to identify middle school students with a potential disability in
mathematics. To collect the data Renae would add on the subtest concurrently with her evaluation, and/ or ask a
few students to see if the would be willing to participate (with parental permission). A total of 8-12 students would
be evaluated over this and the next academic year (06-07; 07-08). The addition of the subtests to her current
evaluation time would amount to an extra 15-20 min. After parental permission is received (by giving their
signature on an informed consent form), the students would go through the following process:
• Renae will explain the study to the child and have the child sign a form indicating she/he
gives assent to participate.
• Renae will ask the child to complete a short demographic form that asks for age and
ethnicity.
• First, the child will be asked to complete the abbreviated cognitive ability test of the StanfordBinet, Fifth Edition.
• Next, the child will be asked to complete the 2 visual spatial processing tests of the Stanford
Binet Fifth, Edition (Position and Direction; Form Patterns).
• Then, the child will be asked to complete 3 tests of the Wechsler Intelligence Scale for Children
(Block Design; Matrix Reasoning; Picture completion).
• Finally, the child will be asked to complete 3 tests of the Woodcock-Johnson Tests of Achievement
(Calculation; Fluency; Applied Problems).
• The child may refuse to participate at any time with no penalties.
• The child will only meet with the Renae 1 time and for less than 1 hour or the subtests would be
added concurrently with an evaluation.
I am wondering if you would be willing to let Renae participate in this study at your middle school? If
you have any questions I would be happy to further explain the study.
Thank you for your time.
Eldon Clifford M.S. NCC, NCSC
3/19/2007
Page 1 of 1
Clifford, Eldon S
From:
Dave.Lammers [Dave.Lammers@swsc.org]
Sent:
Thursday, March 08, 2007 8:33 AM
To:
Clifford, Eldon S
Cc:
Jim Lentz; Renae Christensen
Subject: RE: classmate of Renae's
Hi Eldon - To have Renae assist you in collecting data is fine with me, but because the
data will be coming from Pipestone Area Schools, I think that formal permission should
come from Mr. Jim Lentz, Superintendent of Schools at PAS. His e-mail address is
jim.lentz<a>pas.k12.mn.us.
Good luck to you on your data collection project.
Director of Special Education
SW/WC Service Cooperatives
dave. lamme rs@swsc. org
Phone: 507-825-5858
FAX: 507-825-4035
From: Clifford, Eldon S [mailto:Eldon.Clifford@usd.edu]
Sent: Monday, March 05, 2007 11:03 AM
To: Dave.Lammers
Subject: classmate of Renae's
Mr. Lammers,
My name is Eldon Clifford, I contacted one of your school psychologists (Renae) and she stated you would be
willing to let her aide me in collection of some data for my dissertation. Would it be possible to for you to send me
an email saying that you are giving Renae permission to help me out? The institutional review board for USD
requires some type of documentation for permission. If you have any questions regarding the study please
contact me and I will be happy to answer them.
Thank you for you assistance.
Eldon Clifford M.S. NCC, NCSC
Doctoral Student
School Psychology
University of South Dakota
3/19/2007
Clifford, Eldon S
From:
Sent:
Tci! •
Cc:
Subject:
Joe.Lenz@k12.sd.us
Friday, March 16, 2007 8:55 AM
Clifford, Eldon S
Diana.Holzer@k12.sd.us
RE: A classmate of Diana
Eldon,
Sorry for not responding a little earlier in the week. I am fine with Diana helping you
out providing 1) she is comfortable with the extra time and work 2) it doesn't take time
away from our schools and 3) she receives parental consent on all Students. It looks like
you have addressed these below and I have recently discussed this with Diana where it
doesn't appears that it would be a problem. Good Luck!
Joe Lenz
Director
NWAS Educational Cooperative
PO Box 35 Isabel, SD 57 633
joe.lenz@kl2.sd.us
605-466-2206 - Isabel
605-845-5880 - Mobridge
From: Clifford, Eldon S [mailto:Eldon.Clifford8usd.edu]
Sent: Mon 3/12/2007 2:45 PM
To: Lenz, Joe
Cc: Diana Holzer
Subject: A classmate of Diana
Mr. Lenz,
My name is Eldon Clifford, I a school psychology doctoral student at the University of
South Dakota. Your school psychologist, Diana, has agreed to assist me with some data
collection for my dissertation. I am seeking your permission to allow her to do that. The
study looks at ability of the visual-spatial measures of the Stanford-Binet, Fifth Edition
( Position and Direction; Form Patters/ Form Board) and the WISC-IV ( Block Design,
Picture Complete and Matrix Reasoning) to identify middle school students with a potential
disability in mathematics. To collect the data Dina would add on the subtests concurrently
with her evaluation, and/ or ask a few students to see if the would be willing to
participate (with parental permission). A total of 8-15 students would be evaluated. The
addition of the subtests to her current evaluation time would amount to an extra 15-20
min. After parental permission is received (by giving their signature on an informed
consent form), the students would go through the following process:
* Diana will explain the study to the child and have the child sign a form indicating
she/he
gives assent to participate.
* Diana will ask the child to complete a short demographic form that asks for age, birth
date, gender, and
ethnicity.
* First, the child will be asked to complete the abbreviated cognitive ability test of the
StanfordBinet, Fifth Edition (2 subtests).
* Next, the child will be asked to complete the 2 visual spatial processing tests of the
Stanford
Binet Fifth, Edition (Position and Direction; Form Board/ Form Patterns).
* Then, the child will be asked to complete 3 tests of the Wechsler Intelligence Scale for
Children
•(Block Design; Matrix Reasoning; Picture Completion).
* Finally, the child will be asked to complete 3 tests of the Woodcock-Johnson Tests of
Achievement
(Calculation; Fluency; Applied Problems).
* The child may refuse to participate at any time with no penalties.
* The child will only meet with the Diana 1 time and for less than 1 hour, or the subtests
would be added concurrently with an evaluation.
I am wondering if you would be willing to let Dina participate in this study? If you have
any questions I would be happy to further explain the study.
Thank you for your time.
ElSon Clifford M.S. NCC, NCSC
2
Langstraat, Deb L
From:
Sent:
To:
Subject:
Clifford, Eldon S
Sunday, October 21, 2007 4:25 PM
Langstraat, Deb L
FW: Dr. Jordan Mulder
From: Dave Forbush [mailto:david.forbush@cache.kl2.ut.us]
Sent: Fri 10/5/2007 7:04 PM
To: Jordan.mulder@cache.kl2.ut.us
Cc: Clifford, Eldon S
Subject: FW: Dr. Jordan Mulder
Dr. Mulder,
Please see the message below from one of your prior students. It appears that his request for your time would connect
with much of your day to day work, and where it doesn't it includes a small number of students. With regard to this
request, if you feel that you can assist Clifford without substantial impact to your day to day services, please proceed.
Mentoring others is good stimulation. Enjoy!
Dave
David Forbush Ph.D.
Director of Special Education Services
Cache County School District
(435) 792-7631
david.forbush(S)cache.k12.ut.us
From: Clifford, Eldon S [mailto:Eldon.Clifford@usd.edu]
Sent: Tuesday, September 25, 2007 1:30 PM
To: david.forbush@cache.kl2.ut.us
Subject: Dr. Jordan Mulder
Mr. Forbush,
My name is Eldon Clifford. I am a doctoral student at the University of South Dakota and a school psychology intern in
Newcastle, WY. A school psychologist with your district Dr. Jordan Mulder referred me to you.
I am a former student of Dr. Mulder and he has graciously agreed to assist me with some data collection for my
dissertation. I am seeking your permission to allow him to do that. The study looks at ability of the visual-spatial measures
of the Stanford-Binet, Fifth Edition (Position and Direction; Form Patters/ Form Board) and the WISC-IV (Block Design,
Picture Complete and Matrix Reasoning) to identify middle school students with a potential disability in mathematics. To
collect the data Dr. Mulder would add on the subtest concurrently with his evaluation, and/ or ask a few students to see if
the would be willing to participate (with parental permission). A total of 8-12 students would be evaluated over this
academic year (07-08). The addition of the subtests to his current evaluation time would amount to an extra 15-20 min.
l
The study has received Institutional Review Board and dissertation committee approval. After parental permission is
received (by giving their signature on an informed consent form), the students would go through the following
process:
• Dr. Mulder will explain the study to the child and ask the child sign a form indicating she/he gives assent to
participate.
• Dr. Mulder will ask the child to complete a short demographic form that asks for age grade, gender and
ethnicity.
• First, the child will be asked to complete the abbreviated cognitive ability test of the Stanford- Binet, Fifth
Edition.
• Next, the child will be asked to complete the 2 visual spatial processing tests of the Stanford Binet Fifth,
Edition (Position and Direction; Form Patterns).
• Then, the child will be asked to complete 3 tests of the Wechsler Intelligence Scale for Children (Block
Design; Matrix Reasoning; Picture completion).
• Finally, the child will be asked to complete 3 tests of the Woodcock-Johnson Tests of Achievement
(Calculation; Fluency; Applied Problems).
• The child may refuse to participate at any time with no penalties.
• The child will only meet with the Dr. Mulder 1 time and for approximately 1 hour or the subtests would be
added concurrently with an evaluation.
I am wondering if you would be willing to let Dr. Mulder participate in this study at your middle school? If you
have any questions I would be happy to further explain the study.
Thank you for your time.
Eldon Clifford M.S. NCC, NCSC
School Psychology Intern
Weston County SD#1
Newcastle, WY
ecliffor@usd.edu
307-746-3107 (h)
307-629-0349 (w)
2
Langstraat, Deb L
From:
Sent:
To:
Subject:
Attachments:
Clifford, Eldon S
Monday, November 05, 2007 6:40 PM
Langstraat, Deb L
FW:
4-FORM--ProjectUpdateAmendmentForm.doc
Deb
Please, see attachement for project update/ addendum.
Eldon Clifford
From: Frank.Seiler@kl2.sd.us [mailto:Frank.Seiler@kl2.sd.us]
Sent: Mon 10/29/2007 9:41 AM
To: Clifford, Eldon S
Subject:
I grant permission to Diana Holzer, school psychologist for our
district, to assist Eldon Clifford, USD graduate student, with his
dissertation by conducting some testing of middle school students'
visual-spatial ability and math achievement. Sincerely, Frank Seiler
Superintendent Timber Lake School District 20-3
1
Appendix C
Demographic Information
Testing Date
Grade
Age
Birth Date
Gender:
Male
Female
Language you are Most Comfortable With
Language Spoken at Home
Ethnicity (circle all that may apply): White Non-Hispanic
African American
Native American/ Alaskan Native
Hispanic
Asian/ Pacific Islander
I do not wish to disclose my ethnicity
Highest Grade Completed by Parent or Guardian: Less than High School
Some High School
High School
College
I wish not to disclose the highest grade completed by my parent or guardian
165
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