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Examination of the relationship between executive function and mathematical achievement

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Title:
Examination of the relationship between executive function and mathematical achievement
Creator:
Estes, James Thomas
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Denver, CO
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University of Colorado Denver
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English

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Degree:
Doctorate ( Doctor of psychology)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
School of Education and Human Development, CU Denver
Degree Disciplines:
School psychology
Committee Chair:
Crepeau-Hobson, Franci
Committee Members:
Harris, Bryn
Geissler, Lisa

Notes

Abstract:
This study examined the relationship between measures of executive function, working memory, behavioral inattention, and a child’s math calculation ability. Measures included the WISC-III Digit Span subtest (Digits Forward and Digits Backward trials), BRIEF scales, and CBCL Attention Problems subscale and Attention-Deficit/Hyperactivity Problems scale. Math ability was measured using the WJ-III Calculation subtest. Using previously collected data from the NIH Pediatric MRI Repository (PedsMRI), multiple regression analysis was utilized to examine relationships among study variables. Results of this study indicate 14% of the variance in a child’s math calculation ability can be predicted by three measures: WISC-III Digit Span subtest scaled score, BRIEF Working Memory scale score, and BRIEF Organization of Materials scale score. The predictive model established in this study has implications for assessment and intervention with students with mathematical learning disabilities (MLDs).
General Note:
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University of Colorado Denver
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Auraria Library
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Copyright James Thomas Estes. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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Full Text
EXAMINATION OF THE RELATIONSHIP BETWEEN EXECUTIVE FUNCTION AND
MATHEMATICAL ACHIEVEMENT by
JAMES THOMAS ESTES B.S., Regis University, 2012
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Psychology School Psychology Program
2018


©2018
JAMES THOMAS ESTES ALL RIGHTS RESERVED
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This thesis for the Doctor of Psychology degree by James Thomas Estes has been approved for the School Psychology Program by
Franci Crepeau-Hobson, Chair Bryn Harris Lisa Geissler
Date: May 12, 2018


Estes, James Thomas (PsyD, School Psychology Program)
Examination of the Relationship Between Executive Function and Mathematical Achievement Thesis directed by Professor Franci Crepeau-Hobson
ABSTRACT
This study examined the relationship between measures of executive function, working memory, behavioral inattention, and a child’s math calculation ability. Measures included the WISC-III Digit Span subtest (Digits Forward and Digits Backward trials), BRIEF scales, and CBCL Attention Problems subscale and Attention-Deficit/Hyperactivity Problems scale. Math ability was measured using the WJ-III Calculation subtest. Using previously collected data from the NIH Pediatric MRI Repository (PedsMRI), multiple regression analysis was utilized to examine relationships among study variables. Results of this study indicate 14% of the variance in a child’s math calculation ability can be predicted by three measures: WISC-III Digit Span subtest scaled score, BRIEF Working Memory scale score, and BRIEF Organization of Materials scale score. The predictive model established in this study has implications for assessment and intervention with students with mathematical learning disabilities (MLDs).
The form and content of this abstract are approved. I recommend its publication.
Approved: Franci Crepeau-Hobson
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TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION..............................................1
II. LITERATURE REVIEW.........................................5
III. METHOD...................................................10
IV. RESULTS..................................................15
V. SUMMARY AND CONCLUSIONS..................................17
REFERENCES..................................................23
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CHAPTERI INTRODUCTION
Approximately 5% to 8% of children in the U.S. are diagnosed with mathematics learning disability (MLD; Lewis & Fisher, 2016). However, the definition of MLD differs by context (Mazzocco & Myers, 2003). In clinical settings, a math learning disability is currently defined and categorized by the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5; American Psychological Association, 2013). Within the DSM-5, math disabilities are diagnostically classified as a Specific Learning Disorder with impairment in mathematics. To qualify for this diagnosis, an individual must demonstrate difficulties mastering number sense, number facts, or calculation or difficulties with mathematical reasoning (American Psychiatric Association, 2013). Meanwhile, math learning disabilities are defined in educational settings by criteria from the Individuals with Disabilities Education Act (IDEA). Under IDEA (2004), children are eligible for special education services with the classification of a specific learning disability (SLD) in math if (a) they do not make sufficient progress towards grade-level standards with intervention, and (b) they demonstrate a pattern of strengths and weaknesses in performance related to their area of disability as measured by appropriate assessments. This vague classification system is further compounded and made more confusing by states having varying identification criteria. Regardless of the setting, practitioners involved in the assessment and diagnosis/classification of math disabilities are in need of a test battery conducive to correctly identifying children with math difficulties.
Assessment of Math Disabilities
Unlike reading disabilities, there is limited research that has examined MLDs and less uniformity in assessment of these disorders. Within the literature, math disabilities are referred
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to by different names including “mathematics learning disabilities,” “developmental dyscalculia,” and “dyscalculia,” with some researchers distinguishing each term as a separate diagnosis (Watson & Gable, 2012). In a literature review examining 164 studies on MLDs, it was found that qualification and inclusion for identification of MLD varied based on the skills which were evaluated within the assessment (Lewis & Fisher, 2016). The most commonly used assessments for identifying MLD were the Wide Range Achievement Test (WRAT; Wilkinson & Robertson, 2006) and the Woodcock-Johnson (WJ) Test of Achievement (Woodcock, Mather, & McGrew, 2001). However, at least eight additional measures were used in studies to identify students with an MLD. While 90% of the studies used a cutoff score, there was considerable variability in the cutoff score used ranging from the 2nd percentile to 46th percentile (Watson & Gable, 2012). Given the lack of consistency in assessment of MLD within research, there is no evidence to suggest practitioners have a good model for accurately identifying students with MLD.
Even if there was some agreement among researchers and practitioners regarding the type of measure and cutoff score appropriate for identifying MLD, this would be insufficient in to ensuring accurate identification of students with this disability. In one study, even while using more restrictive criteria, the percentage of students who met any single criteria for MLD ranged from 0 to 45 percent (Mazzocco & Myers, 2003). One issue is that current measures do not account for changes in cognitive abilities between childhood and adolescence and often fail to discriminate at the lower end of the spectrum. This can lead to more “false positives” (i.e., identifying MLD when it is not present) and “false negatives” (i.e., failing to identify MLD when it is present) than true positives (Watson & Gable, 2012).
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Recent research has shown how the use of a secondary screening process which targets specific skills can reduce the number of incorrectly identified students at risk for math difficulty (Fuchs et al., 2011). The use of multiple assessment measures reduces the number of falsely identified students with MLD and is aligned with IDEA, which does not allow the use of any single measure as the sole criterion for determination of a disability (IDEA, 2004). It is recommended that students with math difficulties be administered a comprehensive battery of tests, including assessment of basic cognitive processes associated with MLD (Watson & Gable, 2012).
Purpose of the Study
When evaluating a child for a math learning disability, either under DSM-V diagnostic classification or IDEA educational classification, there are two errors clinicians want to avoid: false positives and false negatives. False positives include diagnosing a child with an MLD when they do not have an MLD. In these cases, the child will often be subject to changes in their educational plans and receive services and resources which they do not need and could be used elsewhere. Meanwhile, in the case of false negatives, the clinician inaccurately indicates the child does not have an MLD. When this happens, the child often does not get the services they need in order to be successful and they can quickly fall further behind their peers in math achievement. Both false positives and false negatives can have detrimental impacts on children’s development and academic growth.
Given the inconsistency and diversity of methods for assessing MLDS, there is reason to suspect the rates of false positive and false negative results are high. While clinicians are encouraged to use multiple data points and assessments in the evaluation of an MLD, it is less clear what battery of tests may be most beneficial for increasing the percentage of accurate MLD
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diagnoses/classifications. The purpose of this study is to determine if measures of executive
function and behavioral measures of attention can predict math ability.
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CHAPTER II LITERATURE REVIEW
Math Difficulty and Cognitive Processes
In order to accurately identify students with MLD, it is necessary to recognize the relationship between math difficulties and underlying cognitive functions. Only 36% of MLD studies have focused on non-mathematical skills, such as memory, executive function, attention, and cognitive processes (Lewis & Fisher, 2016). One proposed theory of cognitive abilities examines cognitive processes based on planning, attention, simultaneous processing, and successive processing (PASS; Naglieri & Das, 1997 as cited in Cai, Li, & Deng, 2013).
Planning provides cognitive control and self-regulation in moving toward a goal. Attention provides regulated cognitive focus over a period of time. Simultaneous processing allows individuals to take parts and create a whole and successive processing puts stimuli in a serial order.
The cognitive abilities which comprise the PASS theory have been used to demonstrate a cognitive profile specific to students with MLD. Students with MLD have been shown to perform significantly worse on measures of each PASS cognitive ability (Kroesbergen, Van Luit, & Naglieri, 2003; Cai, Li, & Deng, 2013) in addition to demonstrating lower processing speed (Cai et al., 2013; Tolar, Fuchs, Fletcher, Fuchs, & Hamlett, 2016). Of the students with MLD, those with processing speed deficits and automaticity problems produced lower scores on measures of planning, attention, and successive processing (Kroesbergen et al., 2003). Simultaneous processing and planning were also found to be significant predictors of MLD (Cai et al., 2013).
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Math Difficulty and Executive Function
The term executive function is a construct which encompasses a group of interrelated functions which direct, guide, and manage cognitive, emotional, and behavioral action, particularly during active problem solving (Gioia, Isquith, Guy, & Kenworthy, 2015). Executive functions are skills that include abilities related to planning, organizing, and completing tasks. The professional literature is robust with studies examining the relationship between executive functions and math difficulty in children and adolescents (e.g., Lewis & Fisher, 2016; Mazzocco & Kover, 2007; Peng, Congying, Beilei, & Sha, 2012; Verdine, Irwin, Golinkoff, & Hirsh-Pasek, 2013). In particular, research examining working memory, an executive function involved in the holding and manipulation of information has been extensive, with studies demonstrating a significant impact on math achievement (Willcut et al., 2013; David, 2012; Swanson & Beebe-Frankenberger, 2004). The demonstrated impact of working memory was consistently found in studies across different countries, including Spain (Miranda, Colomer, Fernandez, & Presentacion, 2012); China (Cai et al., 2013; Chan & Ho, 2009); Canada (Mabbott & Bisanz, 2008); and the Netherlands (Toll, Van der Ven, Kroesbergen, Van Luit, 2011).
The multicomponent model of working memory proposed by Baddeley and Hitch (1974) has been useful in conceptualizing these difficulties in children with MLD. Within their model, working memory is comprised of four components: the central executive, the phonological loop, the visuospatial sketchpad, and the episodic buffer. Although research indicates that working memory has an impact on math achievement, there is evidence to suggest some components of working memory may have greater implications than others in mathematics. With regard to the phonological loop, children with math difficulties appear to have significantly lower scores on verbal working memory measures which are numerical in nature (Willcut et al., 2013; David,
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2012; Mabbott & Bisanz, 2008; Miranda et al., 2012; Chan & Ho, 2009; Peng et al., 2012). However, a study examining the relationship between working memory, math problem solving, and the phonological system showed that executive functions and working memory had an effect on math achievement regardless of performance on measures of phonological processing (Swanson & Beebe- Frankenberger, 2004). Meanwhile, there is evidence to suggest spatial skills, which involving the visuospatial sketchpad, are implicated in math difficulty (Miranda et al., 2012; David, 2012; Verdine et al., 2014), but these effects may be diminished with age (David,
2012) . Since less than 20% of MLD studies examined middle school or high school students (Lewis & Fisher, 2016), it is unclear which aspects of working memory continue to impact math achievement through adolescence.
While there is an abundance of research on working memory and math achievement, less is known about the effect of other executive functions. Two executive functions examined in the literature to a limited degree include inhibition and shifting. Cognitive inhibition is the ability to stop automatic responses in favor of responses oriented towards a specific goal, while shifting includes the ability to change mindset in order to meet the demands of a new task (Gioia et al., 2015). The relationship between these executive functions and math ability is unclear. In one study, neither inhibition nor shifting were predictive of current or future math abilities (Toll et al., 2011). Yet, other studies demonstrated inhibition as a significant predictor of math ability (Verdine et al., 2014) and shifting as independently associated with math deficits (Willcut et al.,
2013) . Further research focused on executive functions outside of working memory will be helpful in determining those functions which impact math achievement, and thus should be examined as part of a diagnostic assessment battery for MLD.
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Math Difficulty and Inattention
In addition to the executive functions, there is a demonstrated relationship between attention processes and math difficulties. Children with Attention Deficit Hyperactivity Disorder (ADHD) were found to have deficits in math word problem solving and were typically slower and less accurate than children without ADHD at math calculation (Lucangeli & Cabrele, 2006). However, the difference and poor performance was suggested to be more predominantly associated with children with hyperactive and impulsive symptoms. These findings are supported by a separate study which demonstrated students with ADHD had significant difficulties with numerical knowledge (Miranda et al., 2012).
Behavioral Inattention is measured via observations of an individual’s ability to initiate and sustain attention on a task within a given setting. Ratings of behavioral inattention in students with MLD have been shown to have a negative effect on math problem solving achievement (Tolar et al., 2016). Regardless of disability status (having vs. not having MLD), children who were rated high on behavioral inattention demonstrated worse performance on overall computation accuracy, math fact errors, and procedural errors (Raghubar et al., 2009). Children who demonstrate external, or visible, symptoms of inattention tend to perform worse on arithmetic calculation problems.
The literature suggests that deficits in cognitive processes associated with executive function and attention are associated with lower math performance and MLD. Therefore, the primary purpose of this study is that measures used to assess executive function, working memory, and behavioral inattention can predict an individual’s math ability. Specifically, the following research questions was investigated:
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Is there a combination of executive function scores as measured by the WISC-III digit span subtests and the BRIEF clinical scales, and behavioral inattention scores as measured by the CBCL attention subscales that best predicts math computation skills as measured by the WJ-II Calculation subtest?
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CHAPTER III METHOD
Design
A quantitative non-experimental design using multiple regression analysis was utilized. Participants
Data for this project was obtained with permission from the National Institutes of Health - Pediatric MRI Study of Normal Brain Development (PedsMRI; National Institutes of Health, 2012). That study enrolled over 400 children, ranging from infancy to young adulthood. The NIH Pediatric MRI Study was organized around two objectives corresponding to two age ranges at the time of enrollment, each with its own protocol. The present study utilized data for children enrolled in objective 1 of the PedsMRI study; participants included children ages 4 years, 6 months through 18 years, 3 months. The NIH Pediatric MRI Data Repository does not include any participants’ identifiable information. Thus, none of the data used nor results obtained can be identified to any of the participants, guaranteeing anonymity.
Sample Strategy
The sample was originally recruited across the six Pediatric Study Centers using community based sampling to reflect the demographics of the United States in terms of income, race, and ethnicity. The subjects were studied with both imaging and clinical/behavioral measures at two-year intervals for three time points from November 2001 to August 2007 (National Institutes of Health, 2012). The present study utilized only clinical/behavioral measures aligned with study aims.
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Variables
The dependent variable examined in this study was math ability. For this study, math ability was measured using the Calculation subtest of the Woodcock-Johnson Tests of Achievement, Third Edition (WJ-III; Woodcock et al., 2001). Calculation assesses quantitative knowledge by measuring an individual’s ability to perform mathematical computations presented in a traditional problem format. Items include basic operations (addition, subtraction, multiplication, division) as well as some geometric, trigonometric, logarithmic, and calculus operations. This subtest is standardized to have a mean of 100 and a standard deviation of 15.
While there are a number of valid measures of executive function, working memory, and attention, the independent variables for this study were chosen based on the assessments used in the original PedsMRI Study and those which assess executive function and attentional processes associated with MLD. The measures analyzed in this study as potential predictors of math ability are as follows:
WISC-III Digit Span Forward and Backward trials
On the Wechsler Intelligence Scale for Children, Third Edition (WISC-III; Wechsler, 1991) the Digit Span subtest includes two parts: Digits Forward and Digits Backward. On Digits Forward, the child listened to and repeated a sequence of numbers spoken aloud by the examiner. On Digits Backward, the child is read a sequence of numbers and then asked to recall the numbers in reverse order (Wechsler, 2014). Digits Forward primarily taps short-term auditory memory. Meanwhile, Digits Backward involves working memory, transformation of information, mental manipulation, and may involve visuospatial imaging (Flanagan & Kaufman, 2009; Groth-Marnat, 2009; Reynolds, 1997; Sattler, 2008). Furthermore, poor performance on backward digit span tasks has been associated with MLD (Toll et al, 2011; Willcut et al, 2013;
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David, 2012; Mabbott & Bisanz; Chan & Ho, 2009) and will be examined as a separate predictor of math ability. On the WISC-III, Digit Span Forward and Backward are combined to create an overall Digit Span subtest score. Digit Span Forward and Backward were measured using separate raw scores from 0 to 14; while the Digit Span subtest used scaled scores which have a mean of 10 and a standard deviation of 3. The Digit Span scaled score, Digit Span Forward and Digit Span Backward raw scores were all examined as individual variables for predicting math ability.
Behavior Rating Inventory of Executive Function, First Edition (BRIEF)
The BRIEF is a rating scale completed by parents and teachers of school-age children (5 to 18 years) and by adolescents aged 11 to 18 years that assess everyday behaviors associated with executive functions in the home and school environments (Gioia, Isquith, Guy, & Kenworthy, 2000). The BRIEF Parent Form and Teacher Form each contain 86 items within eight validated clinical scales which measure domains of executive functioning: Inhibit, Shift, Emotional Control, Initiate, Working Memory, Plan/Organize, Monitor, and Organization of Materials. The Self-Report Form contains 80 items within eight validated clinical scales: Inhibit, Monitor, Shift, Emotional Control, Task Completion, Working Memory, Organization of Materials, and Plan/Organize. The BRIEF scales use T-scores which have a mean of 50 and a standard deviation of 10. Each of the clinical scales was analyzed as a potential predictor of math ability.
Child Behavior Checklist (CBCL)
The CBCL (Achenbach & Rescorla, 2001) is a questionnaire completed by parents or youth as a means of assessing a child’s competencies and/or emotional or behavioral problems. The NIH Pediatric MRI study used the following versions: CBCL 1:5 - 5 and 6-18parent
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report and the Young Adult Self-Report (YASR) self-report for subjects 18 years and older. In this study, the scores for the Attention Problems subscale and Attention-Deficit/Hyperactivity Problems scale were used as measures of behavioral inattention.
Analysis Methods
Data from the PedsMRI study were transferred to an SPSS spreadsheet. Multiple regression analyses were then conducted to examine the relationship between the independent variables of the WISC-III Digit Span subtest; WISC-III Digit Span Forward trials; WISC-III Digit Span Backward trials; eight executive function scales of the BRIEF2 (Inhibit, Shift, Emotional Control, Initiate, Working Memory, Plan/Organize, Monitor, and Organization of Materials); and the Attention Problems subscale and Attention-Deficit/Hyperactivity Problems scale of the CBCL and the dependent variable, math ability (as measured by the Calculation subtest of the WJ-III). This analysis was used to determine if there is a model using the independent variables that could predict math ability. An alpha value of .05 was used to determine significance.
The initial pool of data included a sample size of N = 854. However, there were participants who had missing data for at least one (and in some cases several) of the variables being examined. After removing participants with missing data, a sample size of N = 746 remained.
A step-wise multiple regression was run with all of the variables. Assumptions of independence, homoscedasticity, and normality were checked and met. The independent variables of WISC-III Digit Span Forward and WISC-III Digit Span Backward; BRIEF Emotional Control, Initiate, Inhibition, Monitor, Plan/Organize, and Shift scales; and CBCL Attention Problems subscale and Attention-Deficit/Hyperactivity Problems scale were removed
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because they were not found to be significant predictors. Following the final regression analysis, correlations were obtained for the remaining independent variables: WISC-III Digit Span subtest scaled score, BRIEF Working Memory scale, and BRIEF Organization of Materials scale.. These correlations did not suggest any issues with predictor multicollinearity.
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CHAPTER IV RESULTS
Descriptive Statistics
The sample consisted of 328 children, who ranged in age from 4 years, 6 months to 18 years, 3 months. Many of these children were tested multiple times across the two-year interval timepoints leading to the final sample size, N = 746.
In order to better understand math ability, data were collected for the WISC-III Digit Span subtest scaled scores, WISC-III Digit Span Forward trials raw scores, WISC-III Digit Span Backward trials raw scores, BRIEF Inhibit scale score, BRIEF Initiate scale score, BRIEF Emotional Control scale score, BRIEF Monitor scale score, BRIEF Plan/Organize scale score, BRIEF Shift scale score, BRIEF Working Memory scale score, BRIEF Organization of Materials scale score, CBCL Attention Problems subscale score, and CBCL Attention-Deficit/ Hyperactivity Problems scale score. Assumptions were checked and met. Table 1 provides descriptive statistics for the independent variables from the PedsMRI study.
Table 1
Descriptive Statistics
Mean Standard Deviation
WISC-III Digit Span - scaled score 10.67 2.80
WISC-III Digit Span - raw score 14.27 3.66
WISC-III Digit Span - digits forward raw score 8.78 2.14
WISC-III Digit Span - digits backward raw score 5.48 2.01
BRIEF - Initiate scale T-score 47.95 8.19
BRIEF - Inhibition scale T-score 47.13 7.62
BRIEF - Monitor scale T-score 46.76 9.28
BRIEF - Organization of Materials scale T-score 49.71 8.66
BRIEF - Plan/Organize scale T-score 16.94 4.20
BRIEF - Shift scale T-score 45.56 7.86
BRIEF - Working Memory scale T-score 48.06 8.78
CBCL - DSM: ADHD T-score 51.93 3.81
CBCL - YSR Attention Problems T-score 52.23 3.84
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Regression Analyses
The results of the stepwise regression models are reported in Table 2. Utilizing stepwise selection, the best combination of variables to predict math ability included WISC-III Digit Span subtest scaled scores, BRIEF Working Memory scale scores, and BRIEF Organization of Materials scale scores. The final model [F(3, 742) = 40.62, p < .001] accounted for approximately 14% of the variance (R =.141, adjusted R =. 138) in math ability (WJ-III Calculation subtest score). Thus, children with weaker working memory skills and those with deficits in the ability to organize their belongings and environment had lower math computation ability. None of the other independent variables significantly predicted WJ Calculation scores.
Table 2
Predictors of Math Calculation Ability
Coefficients Standard Error t Stat P-value
Intercept 104.28 3.17 32.92 <001
WISC-III Digit Span subtest scaled score 1.25 0.14 8.63 <001
BRIEF - Working Memory scale T-score -0.32 0.05 -5.94 <001
BRIEF - Organization of Materials T-score 0.17 0.05 3.11 .002
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CHAPTER V
SUMMARY AND CONCLUSIONS
The purpose of this study was to determine if there is a relationship between scores on measures of executive function and behavioral inattention and children’s math ability using a nationally representative sample. While only a few of the measures of executive functioning and none of the measures of behavioral inattention were significant predictors of math calculation scores, a model was created using measures of executive function which accounted for 14% of the variance in an individual’s math ability. Specifically, math ability, as measured by the WJ-III Calculation subtest can be explained to a limited degree by scores on measures of working memory (WISC-III Digit Span subtest scaled score, BRIEF Working Memory scale score) and a measure of the ability to organize one’s own materials and environment (BRIEF Organization of Materials scale score).
The results indicate that math computation ability is significantly associated with better performance on measures of working memory. This finding is consistent with previous research. Several studies have found working memory abilities to have a significant impact on math achievement (Willcut et al., 2013; David, 2012; Swanson & Beebe-Frankenberger, 2004). Baddely and Hitch’s (1974) multicomponent model of working memory encompasses the phonological loop, the visuo-spatial sketchpad, and episodic buffer. A number of studies have implicated the role of the phonological loop in math achievement (Willcut et al., 2013; David, 2012; Mabbott & Bisanz, 2008; Miranda et al., 2012; Chan & Ho, 2009; Peng et al., 2012). The present study’s finding that the WISC-III Digit Span subtest score is predictive of calculation abilities provides some additional evidence of the relationship between the phonological loop and math skills.
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Study findings also indicate that a child’s ability to keep materials and belongings well organized and readily available for projects or assignments is also predictive of calculation scores. Children who have significant difficulties in this area often do not function efficiently on academic tasks because they do not have ready access to what they need and must spend time getting organized rather than producing work.
Although prior research found a relationship between behavioral inattention and math achievement (e.g., Tolar et al., 2016; Raghubar et al., 2009), this association was not found within the current study. This may be due in part to the nature of the WJ-III Calculation task which is untimed and does not include any distractors. Behavioral inattention likely would have a greater impact on math achievement in the areas of math fluency, a timed task involving quick recognition of the sign and operation required, and in problem solving, in which word problems are presented orally and additional, irrelevant or distracting information is sometimes present. Further research examining these areas would help clarify the role and relationship of behavioral inattention and math achievement. Based on the current study, behavioral inattention does not appear to be a significant predictor of math calculation ability.
Limitations
While this study yielded significant results and a predictive model for math calculation ability, there are a number of limitations which affect the ability to generalize the results. First, the original NUT Pediatric MRI study was conducted from 2001 to 2007. Efforts were made during the NUT Pediatric MRI study to include regional samples which matched the census data at that time (National Institutes of Health, 2012). In the ten years since these data were collected, the demographics of the U.S. population have likely changed across all six regions included in the NIH Pediatric MRI study. Thus, the participants in the NIH Pediatric MRI study may not be
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reflective of the current population in terms of income, race, and ethnicity. Furthermore, some NIH Pediatric MRI study participants had missing data and were excluded from analyses. Although the sample in the present study was relatively large (N = 746), it is possible that the sample does not accurately reflect the demographics of the nation, thereby reducing generalizability.
Second, in the ten years since the Pediatric MRI study many of the measures used were updated with new editions and normative samples. For example, the Wechsler Intelligence Scale for Children is currently on its fifth edition (WISC-V), which was published in 2014. Compared to the WISC-III, the WISC-V Digit Span subtest has largely remained unchanged with one exception: in addition to the Forward and Backward trials, the most recent edition also has a Numerical Order set of trials. Since the WISC-III Digit Span subtest scaled score was a significant predictor of math ability, it will be important to see if the inclusion of a Numerical Order trial will enhance or detract from the Digit Span subtest scaled score’s predictive power.
Similarly, the BRIEF and Woodcock-Johnson Tests of Achievement have been updated with new normative samples and editions. The second edition of the BRIEF (BRIEF-2) was published in 2015 and the fourth edition of the Woodcock-Johnson Test of Achievement (WJ-IV) was published in 2014. Generally, the BRIEF-2 maintains the original BRIEF’s qualities and scales and has mostly identical measures of executive function. However, in this new edition, the Monitor scale from the BRIEF was separated into two scales: Task-Monitor and Self-Monitor. The WJ-IV still includes the Calculation subtest as one of three subtests within the Mathematics cluster and has remained generally unchanged.
The CBCL is one of three instruments included in the Achenbach System of Empirically Based Assessment (ASEBA). The ASEBA was last normed in 2001, prior to the original NIH
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Pediatric MRI study. At this point in time, the normative sample used for the ASEBA is no longer reflective of the demographics of the United States.
Finally, this study solely used the Calculation subtest of the WJ-III as the measure of a participant’s math ability. Although the Calculation subtest serves as a solid indicator of an individual’s foundational math skills and operations (addition, subtraction, multiplication, division), it does not reflect mathematical abilities examined within the WJ subtests of Applied Problems and Math Facts Fluency. In order to truly understand the significance of executive function and behavioral inattention on math ability, it would be important to examine their effect on these other areas of math ability in addition to their demonstrated impact on WJ-III Calculation subtest scores.
Future Directions and Implications
In our attempt to better understand MLDs, it is clear we are scratching the surface in terms of our knowledge of the underlying cognitive processes associated with math achievement. Results of the present study provide some directions in terms of next steps. First, to better clarify the relationship between working memory and math skills, future research should utilize both current and varied measures of working memory. In addition, it might be helpful to investigate if the association between working memory varies based on demographics such as age and race/ethnicity. Additionally, to better understand the relationship between executive functions and mathematical achievement, future studies may benefit from examining other areas of math ability beyond calculation (i.e., math fluency or word problem-solving). Lastly, just as dyslexia is not purely defined by the inability to read, dyscalculia or MLD should not be solely measured by a student’s difficulty to demonstrate math ability.
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The findings of the present study may be helpful in informing assessment practices with regard to MLDs. The use of multiple assessment measures reduces the number of falsely identified students with MLD. It has been recommended that students with math difficulties be administered a comprehensive battery of tests, including assessment of basic cognitive processes associated with MLD (Watson & Gable, 2012). Consistent with previous research, the present study indicated that working memory is involved in math calculations and thus measures of this cognitive process may be helpful in correctly identifying children with MLD.
Given the relationship between working memory and math calculation ability, recommended interventions for a child with difficulty in math calculations should likely include interventions and accommodations to address any working memory difficulties or deficits as well. By intervening and/or supporting the underlying cognitive processes involved in working memory, practitioners may be able to help children compensate for deficits and/or improve the cognitive skills needed to perform math calculations.
Meta-analyses have indicated that working memory training programs are associated with significant short-term improvements in working memory; however, there is no convincing evidence of generalization of working memory training to other abilities, such as mathematics (Melby-Lervag & Hulme, 2013). Furthermore, working memory capacity is highly stable from the ages 6 to 19 years (Flanagan, Ortiz, & Alfonso, 2013) and is unlikely to change during that time frame. Thus, it would seem to be a poor use of resources to attempt to improve a child’s working memory abilities. Instead, it may be more beneficial to provide the child with accommodations to support working memory deficits. For example, avoiding multi-step directions, reducing demands, and providing templates for step-by-step procedures are all
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accommodations which can ease a child’s need to rely on their working memory to solve math calculation problems.
Finally, the findings of the present study also suggest that children lacking organizational skills may have greater difficulty accurately completing math problems. As such, teaching a child to organize his or her belongings can be a useful, concrete tool for teaching greater task organization and may also be helpful in supporting improved math achievement.
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Full Text

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EXAMINATION OF THE RELATIONSHIP BETWEEN EXECUTIVE FUNCTION AND MATHEMATICAL ACHIEVEMENT by JAMES THOMAS ESTES B.S., Regis University, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Psychology School Psychology Program 2018

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ii © 2018 JAMES THOMAS ESTES ALL RIGHTS RESERVED

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iii This thesis for the Doctor of Psychology degree by James Thomas Estes has been approved for the School Psychology Program by Franci Crepeau Hobson, Chair Bryn Harris Lisa Geissler Date: May 12, 2018

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iv Estes, James Thomas (PsyD, School Psychology Program) Examination of the Relationship Between Executive Function and Mathematical Achievement Thesis directed by Professor Franci Crepeau Hobson ABSTRACT This study examined the relationship between measures of executive function, working me WISC III Digit Span subtest (Digits Forward and Digits Backward trials), BRIEF scales, and CBCL Attention Problems subscale and Attention Deficit/Hyperactivity Prob lems scale. Math ability was measured using the WJ III Calculation subtest. Using previously collected data from the NIH Pediatric MRI Repository (PedsMRI), multiple regression analysis was utilized to examine relationships among study variables. Results o f this study indicate 14% of the variance III Digit Span subtest scaled score, BRIEF Working Memory scale score, and BRIEF Organization of Materials scale score. The predictive model established in this study has implications for assessment and intervention with students with mathematical learning disabilities (MLDs). The form and content of this abstract are approved. I recommend its publication. Approved: Franci Crepeau Hobson

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v TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ ...................... 1 II. LITERATURE REVIEW ................................ ................................ ................................ ........... 5 III. METHOD ................................ ................................ ................................ ................................ . 10 IV. RESULTS ................................ ................................ ................................ ................................ . 15 V. SUMMARY AND CONCLUSIONS ................................ ................................ ....................... 17 REFERENCES ................................ ................................ ................................ ............................. 23

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1 CHAPTER I INTRODUCTION Approximately 5 % to 8 % of children in the U.S. are diagnosed with mathematics learning disability (MLD ; Lewis & F isher, 2016 ). However, the definition of MLD differs by context (Mazzocco & Myers, 2003) . In clinical settings, a math learning disability is currently defined and categorized by the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition ( DSM 5 ; American Psychological Association, 2013). Within the DSM 5, math disabilities are diagnostically classified as a Specific Learning Disorder with impairment in mathematics. To qualify for this diagnosis, an individual must demonstrate difficulties mastering number sense, number facts, or calculation or difficulties with mathematical reasoning (American Psychiatric Association, 2013). Meanwhile, math learning disabilities are defined in educational settings by criteria from the Individuals with Dis abilities Education Act (IDEA). Under IDEA (2004), children are eligible for special education services with the classification of a specific learning disability (SLD) in math if (a) they do not make sufficient progress towards grade level standards with intervention , and (b) they demonstrate a pattern of strengths and weaknesses in performance related to their area of disability as measured by appropriate assessments. This vague classification system is further compounded and made more confusing by state s having varying identification criteria. Regardless of the setting, practitioners involved in the assessment and diagnosis/classification of math disabilities are in need of a test battery conducive to correctly identifying children with math difficulties . Assessment of Math Disabilities Unlike reading disabilities, there is limited research that has examined MLDs and less uniformity in assessment of these disorders. Within the literature, math disabilities are referred

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2 diagnosis (Watson & Gable, 2012). In a literature review examining 164 studies on MLDs, it was found that quali fication and inclusion for identification of MLD varied based on the skills which were evaluated within the assessment (Lewis & Fisher, 2016). The most commonly used assessments for identifying MLD were the Wide Range Achievement Test (WRAT ; Wilkinson & R obertson, 2006 ) and the Woodcock Johnson (WJ) Test of Achievement (Woodcock, Mather, & McGrew, 2001) . However, at least eight additional measures were used in studies to identify students with an MLD. While 90% of the studies used a cutoff score, there w as considerable variability in the cutoff score used ranging from the 2 nd percentile to 46 th percentile (Watson & Gable, 2012 ) . Given the lack of consistency in assessment of MLD within research, there is no evidence to suggest practitioners have a good m odel for accurately identifying students with MLD. Even if there was some agreement among researchers and practitioners regarding the type of measure and cutoff score appropriate for identifying MLD, this would be insufficient in to ensur ing accurate iden tification of students with this disability . In one study, even while using more restrictive criteria, the percentage of students who met any single criteria for MLD ranged from 0 to 45 percent (Mazzocco & Myers, 2003). One issue is that current measures do not account for changes in cognitive abilities between childhood and adolescence and often fail to discriminate at the lower end of the spectrum . This can it is present) than true positives (Watson & Gable, 2012).

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3 Recent research has shown how the use of a secondary screening process which targets specific skills can reduce the number of incorrectly identified studen ts at risk for math difficulty (Fuchs et al., 2011). The use of multiple assessment measures reduces the number of falsely identified students with MLD and is aligned with IDEA, which does not allow the use of any single measure as the sole criterion for determin ation of a disability (IDEA, 2004). It is recommended that students with math difficulties be administered a comprehensive battery of tests, including assessment of basic cognitive processes associated with MLD (Watson & Gable, 2012). Purpose of t he Study When evaluating a child for a math learning disability, either under DSM V diagnostic classification or IDEA educational classification, there are two errors clinicians want to avoid: false positives and false negatives. False positives include d iagnosing a child with an MLD when they do not have an MLD. In these cases, the child will often be subject to changes in their educational plans and receive services and resources which they do not need and could be used elsewhere. Meanwhile, in the cas e of false negatives, the clinician inaccurately indicates the child does not have an MLD. When this happens, the child often does not get the services they need in order to be successful and they can quickly fall further behind their peers in math a chiev ement development and academic growth. Given the inconsistency and diversity of methods for assessing MLDS, there is reason to s uspect the rate s of false positive and false negative results are high. While clinicians are encouraged to use multiple data points and assessments in the evaluation of an MLD, it is less clear what battery of tests may be most beneficial for increasing t he percentage of accurate MLD

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4 diagnoses/classifications. The purpose of this study is to determine if measures of executive function and behavioral measures of attention can predict math ability.

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5 CHAPTER II LITERATURE REVIEW Math Difficulty and Cognitive Processes In order to accurately identify students with MLD, it is necessary to recognize the relationship between math difficulties and underlying cognitive functions. Only 36% of MLD studies have focused on non mathematical skills, such as me mory, executive function, attention, and cognitive processes (Lewis & Fisher, 2016). One proposed theory of cognitive abilities examines cognitive processes based on planning, attention, simultaneous processing, and successive processing (PASS ; Naglieri & Das, 1997 as cited in Cai, Li, & Deng, 2013). Planning provides cognitive control and self regulation in moving toward a goal. Attention provides regulated cognitive focus over a period of time. Simultaneous processing allows individuals to take parts and create a whole and successive processing puts stimuli in a serial order. The cognitive abilities which comprise the PASS theory have been used to demonstrate a cognitive profile specific to students with MLD. Students with MLD have been shown to pe rform significantly worse on measures of each PASS cognitive ability (Kroesbergen, Van Luit, & Naglieri, 2003; Cai, Li, & Deng, 2013) in addition to demonstrating lower processing speed (Cai et al., 2013; Tolar, Fuchs, Fletcher, Fuchs, & Hamlett, 2016). O f the students with MLD, those with processing speed deficits and automaticity problems produced lower scores on measures of planning, attention, and successive processing (Kroesbergen et al., 2003). Simultaneous processing and planning were also found to be significant predictors of MLD (Cai et al., 2013).

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6 Math Difficulty and Executive Function The term executive function is a construct which encompasses a group of interrelated functions which direct, guide, and manage cognitive, emotional, and behavio ral action, particularly during active problem solving (Gioia, Isquith, Guy, & Kenworthy, 2015). Executive functions are skills that include abilities related to planning, organizing, and completing tasks. The professional literature is robust with studi es examining the relationship between executive functions and math difficulty in children and adolescents ( e.g., Lewis & Fisher, 2016; Mazzocco & Kover, 2007; Peng, Congying, Beilei, & Sha, 2012; Verdine, Irwin, Golinkoff, & Hirsh Pasek, 2013). In particu lar, research examining working memory, an executive function involved in the holding and manipulation of information has been extensive, with studies demonstrating a significant impact on math achievement (Willcut et al., 2013; David, 2012; Swanson & Beeb e Frankenberger, 2004). The demonstrated impact of working memory was consistently found in studies across different countries, including Spain (Miranda, Colomer, Fernández , & Presentación , 2012) ; China (Cai et al., 2013; Chan & Ho, 2009) ; Canada (Mabbott & Bisanz, 2008) ; and the Netherlands (Toll, Van der Ven, Kroesbergen, Van Luit, 2011). The multicomponent model of working memory proposed by Baddeley and Hitch (1974) has been useful in conceptualizing these difficulties in children with MLD . Within their model, working memory is comprised of four components: the central executive, the phonological loop, the visuospatial sketchpad, and the episodic buffer. Although research indicates that working memory has an impact on math achievement, there is ev idence to suggest some components of working memory may have greater implications than others in mathematics. With regard to the phonological loop, children with math difficulties appear to have significantly lower scores on verbal working memory measures which are numerical in nature (Willcut et al., 2013; David,

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7 2012; Mabbott & Bisanz, 2008; Miranda et al., 2012; Chan & Ho, 2009; Peng et al., 2012). However, a study examining the relationship between working memory, math problem solving, and the phonolo gical system showed that executive functions and working memory had an effect on math achievement regardless of performance on measures of phonological processing (Swanson & Beebe Frankenberger, 2004). Meanwhile, there is evidence to suggest spatial skill s, which involving the visuospatial sketchpad, are implicated in math difficulty (Miranda et al., 2012; David, 2012; Verdine et al., 2014), but these effects may be diminished with age (David, 2012). Since less than 20% of MLD studies examined middle scho ol or high school students (Lewis & Fisher, 2016), it is unclear which aspects of working memory continue to impact math achievement through adolescence. While there is an abundance of research on working memory and math achievement, less is known about t he effect of other executive functions. Two executive functions examined in the literature to a limited degree include inhibition and shifting. Cognitive inhibition is the ability to stop automatic responses in favor of responses oriented towards a speci fic goal , while s hifting includes the ability to change mindset in order to meet the demands of a new task (Gioia et al., 2015) . The relationship between these executive functions and math ability is unclear. In one study, neither inhibition nor shifting were predictive of current or future math abilities (Toll et al., 2011). Yet, other studies demonstrated inhibition as a significant predictor of math ability (Verdine et al., 2014) and shifting as independently associated with math deficits (Willcut et al., 2013). Further research focused on executive functions outside of working memory will be helpful in determining those functions which impact math achievement, and thus should be examined as part of a diagnostic assessment battery for MLD.

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8 Math Difficulty and Inattention In addition to the executive functions, there is a demonstrated relationship between attention processes and math difficulties. Children with Attention Deficit Hyperactivity Disorder (ADHD) were found to have deficits in math w ord problem solving and were typically slower and less accurate than children without ADHD at math calculation (Lucangeli & Cabrele, 2006). However, the difference and poor performance was suggested to be more predominantly associated with children with h yperactive and impulsive symptoms. These findings are supported by a separate study which demonstrated students with ADHD had significant difficulties with numerical knowledge (Miranda et al., 2012). Behavioral Inattention is measured via observations and sustain attention on a task within a given setting. Ratings of behavioral inattention in students with MLD have been shown to have a negative effect on math problem solving achievement (Tolar et al., 2016). Rega rdless of disability status (having vs. not having MLD), children who were rated high on behavioral inattention demonstrated worse performance on overall computation accuracy, math fact errors, and procedural errors (Raghubar et al., 2009). Children who d emonstrate external, or visible, symptoms of inattention tend to perform worse on arithmetic calculation problems. The literature suggests that deficits in cognitive processes associated with executive function and attention are associated with lower math performance and MLD. Therefore, the primary purpose of this study is that measures use d to assess executive function, working Specifically, t he following research questions was investigated:

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9 Is there a combination of executive function scores as measured by the WISC III digit span subtest s and the BRIEF clinical scales, and behavioral inattention scores as measured by the CBCL attention subscales that best predicts math computa tion skills as measured by the WJ II Calculation subtest?

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10 CHAPTER III METHOD Design A quantitative non experimental design using multiple regression analysis was utilized. Participants Data for this project was obtained with permission from the National Institutes of Health Pediatric MRI Study of Normal Brain Development ( PedsMRI ; National Institutes of Health, 2012 ). That study enrolled over 400 children, ranging from infancy to young adulthood. The NIH Pediatric MRI Study was organized around two objectives corresponding to two age ranges at the time of enrollment, each with its own protocol. Th e present study utilized data for children enrolled in objective 1 of the PedsMRI study; p articipants i ncluded children ages 4 years, 6 months through 18 years , 3 months . The NIH Pediatric MRI Data Repository does not include any p Thus, n one of the data used nor results obtained can be identified to any of the participants , guaranteeing anonymity . Sample Strategy The sample was originally recruited across the six Pediatric Study Centers using community based sampling to reflect the demographics of the United States in terms of income, race, and ethnicity. The subjects were studied with both imaging and clinical/be havioral measures at two year intervals for three time points from November 2001 to August 2007 ( National Institutes of Health, 2012 ). The present study utilized only clinical/behavioral measures aligned with study aims.

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11 Variables The dependent variable examined in this study was math ability. For this study, math ability w as measured using the Calculation subtest of the Woodcock Johnson Tests of Achievement, Third Edition (WJ I II ; Woodcock et al., 2001 ). Calculation assesses quantitative knowledge by m in a traditional problem format. Items include basic operations (addition, subtraction, multiplication, division) as well as some geometric, trigonometric, logarithmic, and ca lculus operations. This subtest is standard ized to have a mean of 100 and a standard deviation of 15. While there are a number of valid measures of executive function, working memory, and attention, the independent variables for this study were chosen based on the assessments used in the original PedsMRI Study and those which assess executive function and attentional processes associated with MLD. The measures analyzed in this study as potential predictors of math ability are as follows: WISC III Digit Span Forward and Backward t rials On the W echsler I ntelligence Scale for Children, Third Edition (WIS C III ; Wechsler, 1991 ) the Digit Span subtest includes two parts: Digits Forward and Digits Backward. O n Digits Forward, the child listened to and re peated a sequence of numbers spoken aloud by the examiner. O n Digits Backward , the child is read a sequence of numbers and then asked to recall the numbers in reverse order (Wechsler, 2014). Digits Forward primarily taps short term auditory memory. Meanwhi le, Digit s Backward involves working memory, transformation of information, mental manipulation, and may involve visuospatial imaging (Flanagan & Kaufman, 2009; Groth Marnat, 2009; Reynolds, 1997; Sattler, 2008). Furthermore, poor performance on backward digit span tasks has been associated with MLD (Toll et al, 2011; Willcut et al, 2013;

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12 David, 2012; Mabbott & Bisanz; Chan & Ho, 2009) and will be examined as a separate predictor of math ability. On the WISC III , Digit Span Forward and Backward are combin ed to create an overall Digit Span subtest score. Digit Span Forward and Backward were measured using separate raw scores from 0 to 1 4 ; while the Digit Span subtest used scaled scores which have a mean of 10 and a standard deviation of 3 . The Digit Span sc aled score, Digit Span Forward and Digit Span Backward raw scores were all examined as individual variables for predicting math ability. Behavior Rating Inventory of Executive Function, First Edition (BRIEF) The BRIEF is a rating scale completed by parent s and teachers of school age children (5 to 18 years) and by adolescents aged 11 to 18 years that assess everyday behaviors associated with executive functions in the home and school environments ( Gioia, Isquith, Guy, & Kenworthy, 20 00 ). The BRIEF Parent Form and Teacher Form each contain 86 items within eight validated clinical scales which measure domains of executive functioning: Inhibit, Shift, Emotional Control, Initiate, Work ing Memory, Plan/Organize, Monitor, and Organization of Materials. The Self Report Form contains 80 items within eight validated clinical scales: Inhibit, Monitor, Shift, Emotional Control, Task Completion, Working Memory, Organization of Materials, and Plan/Organ iz e. The BRIEF s cales use T scores which have a mean of 50 and a s tandard deviation of 10. Each of the clinical scales w as analyzed as a potential predictor of math ability . Child Behavior Checklist (CBCL) The CBCL (Achenbach & Rescorla , 20 01 ) is a questionnaire completed by parents or youth as a means of assessing a The NIH Pediatric MRI study used the following versions: CBCL 1:5 5 and 6 18 parent

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13 report and the Young Adult Self Report (YASR) self report for subjects 18 years and older. In this study, the scores for the Attention Problems subscale and Attention Deficit/Hyperactivity Problems scale were used as measures of behavioral inattention. Analysis Methods Data from the PedsMRI study were transferred to an SPSS spreadsheet. Multiple regression a nalyses were then conducted to examine the relationship between the independent variables of the WISC III Digit Span subtest; WISC III Digit Span Forward trials ; WISC III Digit Span Backward trials ; eight executive function scales of the BRIEF2 (Inhibit, S hift, Emotional Control, Initiate, Working Memory, Plan/Organize, Monitor, and Organization of Materials) ; and the Attention Problems subscale and Attention Deficit/Hyperactivity Problems scale of the CBCL and the dependent variable, math ability (as measu red by the Calculation subtest o f the WJ I II ) . This analysis was used to determine if there i s a model using the independent variables that could predict math ability. An alpha value of .05 was used to determine significance. The initial pool of data included a sample size of N = 854. However, there were participants who had missing data for at least one (and in some cases several) of the variables being examined. After removing par ticipants with missing data, a sample size of N = 746 remained . A step wise multiple regression was run with all of the variables. Assumptions of independence, homoscedasticity, and normality were checked and met. The independent variables of WISC III Digit Span Forward and WISC III Digit Span Backward; BRIEF Emotional Contr ol, Initiate, Inhibition, Monitor, Plan/Organize, and Shift scales; and CBCL Attention Problems subscale and Attention Deficit/Hyperactivity Problems scale were removed

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14 because they were not found to be significant predictors. Following the final regressio n analysis , correlations were obtained for the remaining independent variables: WISC III Digit Span subtest scaled score, BRIEF Working Memory scale, and BRIEF Organization of Materials scale . . The se correlations did not suggest any issues with predictor m ulticollinearity.

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15 CHAPTER IV RESULTS Descriptive Statistics The sample consiste d of 328 children , wh o r a n ged in age from 4 years, 6 months to 18 years, 3 months . Many of these children were tested multiple times across the two year interval timepoints leading to the final sample size, N = 746 . In order to better understand math ability, data were collected for the WISC III Digit Span subtest scaled scores, WISC III Digit Span Forward trials raw scores, WISC III Digit Span Backward trials raw scores, BR IEF Inhibit scale score, BRIEF Initiate scale score, BRIEF Emotional Control scale score, BRIEF Monitor scale score, BRIEF Plan/Organize scale score, BRIEF Shift scale score, BRIEF Working Memory scale score, BRIEF Organization of Materials scale score, CB CL Attention Problems subscale score, and CBCL Attention Deficit/ Hyperactivity Problems scale score. Assumptions were checked and met . Table 1 provides d escriptive statistics for the independent variables f rom the PedsMRI study. Table 1 Descriptive Statis tics Mean Standard Deviation WISC III Digit Span scaled score 10.67 2.80 WISC III Digit Span raw score 14.27 3.66 WISC III Digit Span digits forward raw score 8.78 2.14 WISC III Digit Span digits backward raw score 5.48 2.01 BRIEF Initiate scale T score 47.95 8.19 BRIEF Inhibition scale T score 47.13 7.62 BRIEF Monitor scale T score 46.76 9.28 BRIEF Organization of Materials scale T score 49.71 8.66 BRIEF Plan/Organize scale T score 16.94 4.20 BRIEF Shift scale T score 45.56 7.86 BRIEF Working Memory scale T score 48.06 8.78 CBCL DSM: ADHD T score 51.93 3.81 CBCL YSR Attention Problems T score 52.23 3.84

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16 Regression Analyses The results of the stepwise regression models are reported in Table 2. Utilizing stepwise selection, the best combination of variables to predict math ability included WISC III Digit Span subtest scaled scores, BRIEF Working Memory scale scores, and BRIEF Organization of Materials scale scores . The final model [F(3, 742) = 40.62, p < .001] acco unted for approximately 14% of the variance ( R 2 =.141, adjusted R 2 = . 138) in math ability (WJ III Calculation subtest score) . Thus, children with weaker working memory skills and those with deficits in the ability to organize their belongings and environment had lower math computation ability. None of the other independent variables significantly predicted WJ Calculation scores. Table 2 Predictors of Math Calculation Ability Coefficients Standard Error t Stat P value Intercept 104.28 3.17 32.92 <.001 WISC III Digit Span subtest scaled score 1.25 0.14 8.63 <.001 BRIEF Working Memory scale T score 0.32 0.05 5.94 <.001 BRIEF Organization of Materials T score 0.17 0.05 3.11 .002

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17 CHAPTER V SUMMARY AND CONCLUSIONS The purpose of this study was to determine if there i s a relationship between scores on measures of executive function and behavioral inattention and children using a nationally representative sample . While only a few of the measures of executive function ing and none of the measures of behavioral inattention were significant predictors of math calculation scores , a model was created using measures of executive function which account ed for 14% of the variance in an indi Specifically, math ability , as measured by the WJ III Cal cul ation subtest can be explained to a limited degree by scores on measures of working memory ( WISC III Digit Span subtest scaled score, BRIEF Working Memory scale score ) and a measure of the ability to organize one s own materials and environment ( BRIEF Organization of Materials scale score ) . The results indicate that math computation ability is significantly associated with better performance on measures of working memory. Th is finding is consistent with previous research. Several studies have found working memory abilities to have a significant impact on math achievement (Willcut et al., 2013; David, 2012; Swanson & Beebe Frankenberger, 2004). component model of working memory encompasses the phonological loop, the visuo spatial sketchpad, and episodic buffer. A number of studies have implicated the role of the phonological loop in math achievement (Willcut et al., 2013; David, 2012; Mabbott & B isanz, 2008; Miranda et al., 2012; Chan & Ho, 2009; Peng et al., 2012). The that the WISC III Digit Span subtest score is predict ive of calculation abilities provides some additional evidence of the relationship between the phonolog ical loop and math skills.

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18 organized and readily available for projects or assignments is also predictive of calculation scores. Children who have significant diffic ulties in this area often do not function efficiently on academic tasks because they do not have ready access to what they need and must spend time getting organized rather than producing work. Although prior research found a relationship between behavior al inattention and math achievement ( e.g., Tolar et al., 2016; Raghubar et al., 2009), this association was not found within the current study. This may be due in part to the nature of the WJ III Calculation task which is untimed and does not include any d istractors . Behavioral inattention likely would have a greater impact on math achievement in the areas of math fluency, a timed task involving quick recognition of the sign and operation required, and in problem solving, in which word problems are present ed orally and additional, irrelevant or distracting information is sometimes present. Further research examining these areas would help clarify the role and relationship of behavioral inattention and math achievement. Based on the current study, behaviora l inattention does not appear to be a significant predictor of math calculation ability. Limitations While this study yielded significant results and a predictive model for math calculation ability, there are a number of limitations which affect the ability to generalize the results. First, the original NIH Pediatric MRI study was conducted from 2001 to 2007. Efforts were made during the NIH Pediatric MRI study to include regional samples which matched the census data at that time (National Institutes of Health, 2012) . In the ten years since the se data were collect ed , the demographics of the U.S. population have likely changed across all six regions included in the NIH Pediatric MRI study. Th us, the participants in the NIH Pediatric MRI study may no t be

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19 reflective of the current population in terms of income, race, and ethnicity. Furthermore, some NIH Pediatric MRI study participants had missing data and were excluded from analys e s. Although the sample in the present study was relatively large (N = 74 6), it is possible that the sample does not accurately reflect the demographics of the nation, thereby reducing generalizability. Second, in the ten years since the Pediatric MRI study many of the measures used were updated with new editions and normative samples. For example, t he Wechsler Intelligence Scale for Children is currently on its fifth edition (WISC V) , which was published in 2014. Compared to the WISC III, the WISC V Digit Span subtest has largely remained unchanged with one exception : in addition to the Forward and Backward trials, the most re c ent edition also has a Numerical Order set of trials. Since the WISC III Digit Span subtest scaled score was a significant predictor of math ability, it will be important to see if the inclusion of a Numerical Similarly, the BRIEF and Wo odcock Johnson Test s of Achievement have been updated with new normative samples and editions. The second edition of the BRIEF (BRIEF 2) was published in 2015 and the fourth edition of the Woodcock Johnson Test of Achievement (WJ IV) was published in 2014. Generally, the BRIEF and scales and has mostly identical measures of executive function. However, in this new edition , the Monitor scale from the BRIEF was separated into two scales: Task Monitor and Self Monitor . The WJ IV still includes the Calculation subtest as one of three subtests within the Mathematics cluster and has remained generally unchanged . T he CBCL is one of three instruments included in the Achenbach System of Empirically Based Assessment (ASEBA) . The ASEBA was last normed in 2001, prior to the original NIH

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20 Pediatric MRI study. At this point in time, the normative sample used for the ASEBA is no longer reflective of the demographics of the United States. Finally, this study solely used the Calcul ation subtest of the WJ III as the measure of a ions (addition, subtraction, multiplication , division ), it does n ot reflect mathematical abilities examined within the WJ subtests of Applied Problems and Math Facts Fluency. In order to truly understand the significance of executive function and behavioral inattention on math ability, it would be important to examine t heir effect on these other areas of math ability in addition to their demonstrated impact on WJ III Calculation subtest scores. Future Directions and Implications In our attempt to better understand MLDs , it is clear we are scratching the surface in terms of our knowledge of the underlying cognitive processes associated with math a chievement . Results of the present study provide some directions in terms of next steps. First, to better clarify the relationship between working memory and math skills , fu ture research should utilize both current and varied measures of working memory . In addition, it might be helpful to investigate if the association between working memory varies based on demographics such as age and race/ethnicity. Additionally, to better understand the relationship between executive function s and mathematical achievement, future studies may benefit from examining other areas of math ability beyond calculation (i.e., math fluency or wo rd problem solving). Lastly, just as dyslexia is not pur ely defined by the inability to read, dyscalculia or MLD should not be solely measured

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21 The findings of the present study may be helpful in informing assessment practices with regard to MLDs. The use o f multiple assessment measures reduces the number of falsely identified students with MLD. It has been recommended that students with math difficulties be administered a comprehensive battery of tests, including assessment of basic cognitive processes asso ciated with MLD (Watson & Gable, 2012). Consistent with previous research, the present study indicated that working memory is involved in math calculations and thus measures of this cognitive process may be helpful in correctly identifying children with ML D. Given the relationship between working memory and math calculation ability, recommended interventions for a child with difficulty in math calculations should likely include interventions and accommodations to address any working memory difficulties o r deficits as well. By intervening and/or supporting the underlying cognitive processes involved in working memory, practitioners may be able to help children compensate for deficits and/or improve the cognitive skills needed to perform math calculations. Meta analyses have indicated that working memory training programs are associated with significant short term improvements in working memory; however, there is no convincing evidence of generalization of working memory training to other abilities, such a s mathematics (Melby Lervag & Hulme, 2013). Furthermore, working memory capacity is highly stable from the ages 6 to 19 years (Flanagan, Ortiz, & Alfonso, 2013) and is unlikely to change during that time frame. Thus, it would seem to be a poor use of resou rces to attempt to working memory abilities. Instead, it may be more beneficial to provide the child with accommodations to support working memory deficits. For example, avoiding multi step directions, reducing demands, and providing temp lates for step by step procedures are all

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22 calculation problems. Finally, t he findings of the present study also suggest that children lacking organizational skills may have greater difficulty accurately completing math problems. As such, teaching a child to organize his or her belongings can be a useful, concrete tool for teaching greater task organization and may also be helpful in supporting improved math achievem ent.

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