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The Role of Executive Functioning Skills in the Academic Achievement of Children from Low-income Families

Permanent Link: http://ufdc.ufl.edu/UFE0024130/00001

Material Information

Title: The Role of Executive Functioning Skills in the Academic Achievement of Children from Low-income Families A Growth Curve Modeling Analysis
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Delucca, Teri
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: acheivement, children, cognitive, development, executive, functions, income, london, low, math, of, reading, skills, socioeconomic, status, tower
Psychology -- Dissertations, Academic -- UF
Genre: Psychology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Children from low-income families are at an increased risk for delays in cognitive development. Such delays may affect a set of basic underlying cognitive skills termed executive functions that are necessary for learning in academic environments. The primary goal of this study was to examine differences in the executive functioning skills of children from low-income families compared to their more affluent peers over time. A second goal of this study was to determine the role of family income in the relationship between children?s executive functioning and academic performance. Performance on the Tower of London (TOL) was measured in 174 low and middle-income children who were followed from kindergarten through fourth grade. Growth curve analyses were conducted using multilevel modeling techniques. Findings indicate that family income differences were associated with disparities in performance on each measure of executive functioning over the course of the study. Low-income children solved fewer problems correctly and of the problems solved they had longer solution times and made less efficient moves than middle-income children. Executive functions were found to mediate the relationship between family income and children?s reading and math achievement. Results are discussed in terms of implications for early intervention programs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Teri Delucca.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Berg, William K.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0024130:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024130/00001

Material Information

Title: The Role of Executive Functioning Skills in the Academic Achievement of Children from Low-income Families A Growth Curve Modeling Analysis
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Delucca, Teri
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: acheivement, children, cognitive, development, executive, functions, income, london, low, math, of, reading, skills, socioeconomic, status, tower
Psychology -- Dissertations, Academic -- UF
Genre: Psychology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Children from low-income families are at an increased risk for delays in cognitive development. Such delays may affect a set of basic underlying cognitive skills termed executive functions that are necessary for learning in academic environments. The primary goal of this study was to examine differences in the executive functioning skills of children from low-income families compared to their more affluent peers over time. A second goal of this study was to determine the role of family income in the relationship between children?s executive functioning and academic performance. Performance on the Tower of London (TOL) was measured in 174 low and middle-income children who were followed from kindergarten through fourth grade. Growth curve analyses were conducted using multilevel modeling techniques. Findings indicate that family income differences were associated with disparities in performance on each measure of executive functioning over the course of the study. Low-income children solved fewer problems correctly and of the problems solved they had longer solution times and made less efficient moves than middle-income children. Executive functions were found to mediate the relationship between family income and children?s reading and math achievement. Results are discussed in terms of implications for early intervention programs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Teri Delucca.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Berg, William K.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0024130:00001


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THE ROLE OF EXECUTIVE FUNCTIONI NG SKILLS IN THE ACADEMIC ACHIEVEMENT OF CHILDREN FROM LOW -INCOME FAMILIES: A GROWTH CURVE MODELING ANALYSIS By TERI L. DELUCCA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 1

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2010 Teri L. DeLucca 2

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My dissertation is dedicated to several pe ople. First is my husband, Sam, you are my endless love. You are su pportive beyond measure of all of my crazy endeavors, regardless of how unattainable they may seem. You have sh ared many uncertainties & challenges throughout this process and have sacrificed so much so that I could chase my dr eams. Second is my son Sammy, you are my sunshine and my joy thank you for helping me remember what really matters in life. Next to my Mom, for always encouraging me to chase after my dreams and for instilling in me a firm foundati on of faith through which I have the confidence to achieve my goals. I want to be just like you when I grow up. Also to Kim who provided immeasurable support in every way imaginable throughout this process. I could not survive graduate school without you. We survived many ch allenges together and I cheris h all of the crazy memories weve created. Getting to the top starts with a si ngle step, right? Also to Leslie, Andrea, & Cody, youve been my emotional anchors, my prayer warriors, and my cheerleaders throughout not only grad school but life. Each of you has such a unique and vital role in my life. Additionally, to my in laws, thank you for raisin g such an amazing man, for welcoming me into your family, and for all you have done to prov ide me with the opportunity to continue my education. Also to Hannah, the best little sister in the world, I am so thankful to have been matched with you. You are so strong and inspire me to be better in every way. I hope that you too will set big goals for yourself and dare to reac h them. Lastly, to Cristina, the piece of me I left in Guatemala. Te quiero mucho mi querida ni a. Siempre ests en mi corazn y apesar de la distancia haces parte de nuetra familia. Gracias po r todas tus enseanzas y el nimo que me has dado en los momentos difciles. Recuerda siempre la promesa que hicimos; algn da, se har re alidad. Espero que siempre tus as piraciones sean grandes, y estoy segura que sers una persona de mucho impacto para este mundo. 3

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ACKNOWLEDGMENTS I would like to thank Dr. Keith Berg for ma king my graduate career possible. Thank you for taking a chance on me and for your invaluable wisdom, guidance, and friendship. I admire your intellect, integrity, and approach to life. Your personal support and encouragement during stressful times kept me in the game, thank you fo r never letting me give up on myself. You truly are the best advisor a graduate student could ever hope for. I al so thank Dr. Joe McNamara for his friendship and contributions to this research and for collaborating with a new graduate student on such a major endeavor. I could not have made it through without my academic big brother. Thank you to Dr. Julie Graber whose comments brought clarity and expertise to my research studies and whose personal support an d guidance have carried me through difficult situations through the years. I have learned a great deal from you as my professor, committee member, and friend over the years and I am foreve r grateful. I thank Dr. Lauren Fasig for her personal support, encouragement, and direction provided in many areas. We still need to hang that plaque, white gloves included. To Dr. Bridget Franks, Dr. Pat Ashton, and Dr. Kate Fogarty: thank you for your invaluable contributions to my work. My dissertation is a better finished product because of your assistance and expertise and I am stronger professionally because of your confidence in me. Thank you to Jen Tamargo for her immeasurable support and statistical guidance through this process. Las tly, I thank the many research as sistants who have contributed to my research over the years and the parents and children who participated in my studies. 4

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TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4LIST OF TABLES ...........................................................................................................................7LIST OF FIGURES .........................................................................................................................8ABSTRACT ...................................................................................................................... ...............9 CHAPTER 1 INTRODUCTION ................................................................................................................ ..10Cognitive Development and Low-income Children ...............................................................10Development of Executive Functions .....................................................................................14Executive Functions in Low-Income Children .......................................................................20Goals of the this Study ............................................................................................................282 METHODS ..................................................................................................................... ........31Participants .................................................................................................................. ...........31Data Collection .......................................................................................................................32Measures ...................................................................................................................... ...........33Verbal IQ Assesment .......................................................................................................33The Tower of London ......................................................................................................34Academic Ac hievement ...................................................................................................37Family Income Status ......................................................................................................38Procedure ..................................................................................................................... ...........383 ANALYTICAL APPROACH TO EXAMINING CHANGE OVER TIME .........................42Describing and Predicting Change in Executive Functioning Skills ......................................42Multilevel Modeling for Change .....................................................................................42Evaluation of Random and Fixed Effects ........................................................................45The Level 1 Model ..........................................................................................................46The Level 2 Model ..........................................................................................................46Mediation Analyses ................................................................................................................484 RESULTS ..................................................................................................................... ..........52Preliminary Analyses .......................................................................................................... ....52Sample Means and Standard Deviations .........................................................................52Sample Selectivity: Attri tion Related to Drop Out ..........................................................52Age at Time of Testing ....................................................................................................53Application of the Multil evel Model for Change ............................................................54 5

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Major Analyses: Hypotheses Testing .....................................................................................56Describing and Predicting Change in Propor tion of Correctly Solved Problems ..................57The Unconditional Means Models ..................................................................................57The Unconditional Growth Models .................................................................................58Main Effects of Family Income .......................................................................................60Summary of Findings ......................................................................................................62Describing and Predicting Change in Solution Time .............................................................63The Unconditional Means Models ..................................................................................63The Unconditional Growth Model ..................................................................................65Main Effects of Family Income .......................................................................................67Moderation of the Rate of Change by Income ................................................................69Summary of Findings ......................................................................................................70Describing and Predicting Change in the Efficiency of Moves Made ...................................71The Unconditional Means Models ..................................................................................71The Unconditional Growth Models .................................................................................72Main Effects of Family Income .......................................................................................74Summary of Findings ......................................................................................................75Mediation of Academic Achievement ....................................................................................76Mediation Results for Reading Achievement ..................................................................78Mediation Results for Math Achievement ......................................................................795 DISCUSSION .................................................................................................................. .......95Findings from Growth Curve Modeling .................................................................................96The Effects of Family Income on the Devel opment of Executive Functioning Skills ....96Developmental Trajectories of Executive Functio ning Skills .......................................100Income Level Gap in Performance on Hard Problems ..................................................102Why might the growth curve models have give n different results than are suggested by the simple means in the first year? ....................................................................................105Mediation of Executive Functi ons in the Relationship between Family Income and Academic Ac hievement ....................................................................................................106Critical Evaluation of Findings and Future Implications .....................................................108Attrition due to Drop Out ..............................................................................................108Advantages and Disadvantages Related to Using the Tower of London Task .............110Design of the Study .......................................................................................................114Conclusion .................................................................................................................... ........117LIST OF REFERENCES .............................................................................................................122BIOGRAPHICAL SKETCH .......................................................................................................129 6

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LIST OF TABLES Table page 2-1 Number of participants tested and ac ademic achievement means for each year ...............414-1 Multilevel mode l building steps.........................................................................................814-2 Means and standard deviations for easy problems on the TOL .........................................814-3 Means and standard deviations for hard problems on the TOL .........................................814-4 Results of model tests for easy prob lems on the proportion solved measure ....................844-5 Results of model tests for hard pr oblems on the proportion solved measure ....................854-6 Results of model tests for easy pr oblems on the solution time measure ...........................864-7 Results of model tests for solution speed on hard problems ..............................................874-8 Results of model tests for m ove efficiency on easy problems ...........................................884-9 Results of model tests for m ove efficiency on hard problems ...........................................894-10 Associations between academic data, inco me, and executive functioning in Year 2 on easy problems .....................................................................................................................904-11 Associations between academic data, inco me, and executive functioning in Year 2 on hard problems .....................................................................................................................904-12 Associations between academic data, inco me, and executive functioning in Year 3 on easy problems .....................................................................................................................914-14 Regression coefficients estimated in academic achievement mediation models for hard problems on the move efficiency m easure for the second testing session .................92 7

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LIST OF FIGURES Figure page 2-1 Tower of London (TOL) presentation screen ....................................................................413-1 Conceptual diagram of the mediation analyses and estimated relationships for reading achievement ..........................................................................................................5 03-2 Conceptual diagram of the mediati on analyses representing the estimated relationships for math achievement ...................................................................................514-1 Proportion Solved Simple Means ......................................................................................824-2 Solution Speed Simple Means ...........................................................................................824-3 Move Efficiency Simple Means .........................................................................................834-4 Conceptual diagram of the mediati on analyses and standardized regression coefficients representing the estimated relationships for reading achievement in the second testing session ........................................................................................................934-5 Conceptual diagram of the mediati on analyses and standardized regression coefficients representing the estimated relationships for math achievement in the second testing session ........................................................................................................94 8

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Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE ROLE OF EXECUTIVE FUNCTIONI NG SKILLS IN THE ACADEMIC ACHIEVEMENT OF CHILDREN FROM LOW -INCOME FAMILIES: A GROWTH CURVE MODELING ANALYSIS By Teri L. DeLucca December 2010 Chair: W. Keith Berg Cochair: Bridget Franks Major: Psychology Children from low-income families are at an increased risk for delays in cognitive development. Such delays may affect a set of basic underlying cognitive skills termed executive functions that are necessary for learning in ac ademic environments. The primary goal of this study was to examine differences in the executive functioning skills of children from low-income families compared to their more affluent peers over time. A second goal of this study was to determine the role of family income in the re lationship between childr ens executive functioning and academic performance. Performance on the Tower of London (TOL) was measured in 174 low and middle-income children who were follo wed from kindergarten through fourth grade. Growth curve analyses were conducted using mu ltilevel modeling techniques. Findings indicate that family income differences were associated with disparities in pe rformance on each measure of executive functioning over the course of the study. Low-income children solved fewer problems correctly and of the problems solved they had longer solution times and made less efficient moves than middle-income children. Executive functions were found to mediate the relationship between family income and childre ns reading and math achievement. Results are discussed in terms of implications for early intervention programs. 9

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CHAPTER 1 INTRODUCTION An important educational challenge that faces our nation today is the gap in achievement between economically disadvantaged children and thei r majority peers. Even with the benefits of early intervention, children from low-income families begin school with poor academic skills and continue to have more academic problems when compared to more advantaged children (Brooks-Gunn, Kelbanov, & Duncan, 1996; Stipek & Ryan, 1997). This gap in achievement increases with each grade level. Children rais ed in low-income families have delayed cognitive development and deficits in skills such as la nguage, memory, and attentional capacities (Carlson, Mann, Merola, & Moylan, 1984; Craig, Connor, & Washington, 2003; Duncan, Brooks-Gunn, & Klebanov, 1994; Waber, Lupien, King, Meaney, & McEwen, 2001;). Delays in development of basic cognitive processes may place children from low-income families at an increased risk for academic failure and high school drop out, whic h limit their successes la ter in life (Buckner, Mezzacappa, Beardslee, 2003; Stevenson & Newm an, 1986). Thus, it is important to understand the role of family income in cognitive development, particularly the skills critical for learning. The developmental risks associated with econom ic disadvantage have been well documented, but few studies have explored the effects of pove rty on specific cognitive skills such as those involved in executive functioning. T hus the primary goal of this pape r is to examine the effect of family income level on the developmental trajectories of the executive functioning skills of children. A second goal of this paper is to determin e the role of both execu tive functioning skills and family income in childrens academic performance. Cognitive Development and Low-income Children Numerous studies have shown that low socio economic status (SES) is highly correlated with delayed cognitive development, even mo re so than social emotional development 10

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(Bendersky & Lewis, 1994; Donahue, Finnegan, Lutkus, Allen, & Campbell, 2001; Ryan, Fauth, and Brooks-Gunn, 2006). These SES disparities ar e not subtle. Low-income children are 1.3 times more likely to develop learning disabili ties and experience developmental delays than more advantaged children (Brooks-Gunn & Duncan, 1997). Large differences in cognitive skills have been found in childrens performance on tasks that involve basic reading and numeric skills, problem solving, creativity, memory, and language skills (Stipek & Ryan, 1997). In most studies of children from low-income families, socioeconomic status is the most consistent predictor of IQ, cognitive functioning, and school readiness, even more so than parent education level or occupation (Davis & Ginsburg, 1993; Stipek & Ryan, 1997). For instance, children with incomes less than half the poverty threshold scored between 6 and 13 points lower on several measures of verbal ability, IQ, cognitive abil ity, and achievement when compared to children from families with incomes between 1.5 and twice the poverty threshold (Brooks-Gunn & Duncan, 1997). The authors mention that from an educational viewpoi nt 6-13 points lower on these measures is quite large and could make the difference in being placed in special education classes or not. What is even more concerni ng is that these differences remain even after maternal age, education, ethnic ity, and marital status were controlled for. Similarly, family income level has been found to account for 20% of the variance associated with childrens IQ scores (Gottfreid, Bathurst, Gu erin, & Parramore, 2003). In fact one research group found that family income level at age three was a powerful pred ictor of IQ at age five, even when IQ at age three was controlled for (Duncan et al., 1994). This relationship between inco me and childrens cognition is fr equently been reported to be mediated by the degree of cognitive stimulation available to children in the environment provided by families (Yeung, Linver, & Brooks-Gunn, 2002). For example, Yeung et al. (2002) 11

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found that the association betw een family income level and childrens scores on the Woodcok Johnson Achievement Test was mediated by the fam ilys ability to provide a stimulating learning environment. In their review of the literature, Bradley and Co rwyn (2002) concluded that access to cognitively stimulating learning resources a nd provision of learning opportunities mediated the relationship between family income and cogn itive skills. Low-income children experience these effects through pathways i nvolving poorer health and nutri tion, less sensitive and poorer quality parent child interactions poorer physical conditions of the home, parental irritability and depression, and residence in neighborhoods characterized by social diso rganization and fewer resources than more affluent children (Br ooks-Gunn & Duncan, 1997). Thus, insufficient income greatly affects childrens cognitive development in that it results in much more stress and negative life events for parents which in turn re sults in psychological di stress that most likely decreases their abilities to provide responsive, sensitive, and stimula ting interactions with children (McLoyd, 1998). The timing and duration of poverty is dire ctly related to ch ildrens cognitive development. Researchers have found that childre n who live in persistent or chronic poverty have poorer physical and mental health and more problematic cognitive and social development than children in transitory poverty (NICHD Early Childcare Research Network, 2005). Children who live in persistently poor fa milies (defined as a four year span) showed poorer cognitive abilities on several assessments when compared to children who had never experienced poverty. Children of the National Longitudinal Survey of Youth (NLSY) who experienced long term poverty (derived from family income averaged over 13 years prior to testing) showed much greater effects of poverty on cognitive abilities than children who experienced short-term poverty (income in the year of observation) (S mith, Brooks-Gunn, & Klebanov, 1997). This poor 12

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cognitive development directly affects childrens academic performance. It is quite interesting that these effects remained from the ages of tw o to eight years old for children reared in long term poverty (Brooks-Gunn & Duncan, 1997). These findings suggest that the timing of poverty may have more deleterious effects when it occu rs earlier in development. Comparisons of siblings reared within the same families allo w researchers to examine the effects of family income at different times. For example, in an attempt to compare siblings achievement at the same age, researchers may examine family income in 2005 when the first born was five years old and in 2009 when the second born was five years old. A study using this a pproach indicated that differences in family income found at the time each siblings age was compared were related to differences in the number of school years co mpleted by the siblings (Duncan, Yeung, BrooksGunn, & Smith, 1998). Similarly, one report indicated that early family income (averaged from birth to five years) had stronge r effects on the number of school years completed than did family income measured 5 or 10 years later. An incr ease in family income of $10,000 before age 5 was associated with one year increase in complete d schooling where as income increases later in childhood did not have significant effects. These results highlight the cr itical role of income during early child developmen t (Duncan et al., 1998). The majority of studies focused on the tim ing of poverty corroborate the findings that early poverty is more detrimental to child outcom es than later poverty; however it is important to note that one study reported conf licting results. In the NIC HD Early Child Care Research Network (2005) examination of the effects of duration and timing of poverty on childrens development, researchers found that the poorer quality home environments with less cognitive stimulation provided by chronically poor familie s resulted in children with lower cognitive performance on a variety of measures and more be havioral problems when compared to children 13

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from families who were never poor, poor only du ring infancy (0-3 years) or poor only after infancy (4-9 years). In this study, any experience of poverty resulted in more problematic family situations and child outc omes than children who were never poor Thus, contrary to the previous results, being poor later had more negative effects on the child than early poverty (NICHD, 2005). The results of this study have been reported as an excepti on to the plethora of research indicating that early pov erty is the cause of less favorable developmental outcomes. It is important however to note that caution is need ed when generalizing these results given that children in the early and late poverty groups did not differ significantly on cognitive outcomes but rather all differences in the timing of poverty were found on the behavioral measures. Children who experienced poverty la ter in life had more internaliz ing and externalizing behavior problems than those in families who experienced ear ly poverty. It is possible that differences in the methods used to measure family income as well as the design of studies (sibling studies, longitudinal, and cross sectional are often used) may lead to different conclusions. Given that the home environment has the gr eatest effects on cognitive development during the early years since older children have access to alternate pathways for stimulation and learning via school, cognitive skills that devel op during these early years may be the most vulnerable to effects of income. One specific set of underlying cognitive skills, executive functions, have proven critical to learning in academic environments and everyday functioning. Development of Executive Functions The definition of executive functions is elusive. An ongoing debate remains in the literature regarding its specific de finition and whether the process is unitary or a collection of related functions. In general, however, the te rm executive function encompasses multiple higher order brain functions that are in terrelated and functionally dependent and that work together to control lower level cognitive processes. The typical list of these component processes includes 14

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planning, strategy use, working memory, attent ion, inhibition and cognitive flexibility. These problem solving skills are important in em otional control, cognitive functioning, and goal directed behavior. Planning, in some ways, may be considered an encompassing component of executive functioning in that the ability to plan effectively involves severa l aspects of executive skills. Complex cognitive skills such as recognizing differences between start and goal contexts, creating effective strategies to accomplish goals, the use of working memory to remember rules of tasks, monitoring actions taken to reach goals, switching between aspects of tasks or cognitive flexibility, and correcting errors are all aspects of planning (N ICHD, 2005) and, of course, of executive functioning. Competence in planning therefore requires sufficient development of working memory, cognitive flexibility or set sh ifting, inhibition of irre levant information or behaviors, and sustained atte ntion (Bull & Scerif, 2001). The development of executive functions is related to neuropsychological changes that occur in normal development. Evidence from neuroimaging studies of adults and children indicates that executive functions are predominantly influenced by the frontal lobes of the brain. For instance, in studies focused on damage to the frontal lobes, lesions in the prefrontal cortex were directly related to defic its in inhibition and working memo ry, which are both key aspects of executive functions (Dowsett & Livesey, 2000). From birth to 5 years, progressive myelinization, especially in the frontal lobes, results in rapid and more efficient neuronal connections between different areas of the brain. This maturation of frontal lobe circuitry creates increased effectiveness of information proces sing. The resulting increases in neuronal connections and myelinization thus allow for the integration of cognitive processes or, in other words, enhanced executive control (Anders on, 2002). Repeated exposure to novel tasks 15

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requiring the use of executive functions lik ely accelerates the acquisition of executive functioning skills by strengthening ne uronal connections in the asso ciated brain regions (Jurado & Rosselli, 2007). Given this proces s, it has been suggested that experience plays a critical role in the executive functioning ab ilities of young children (Zelazo, Reznick, & Pion, 1995; Zelazo & Reznick, 1991, Jurado & Rosselli, 2007). Executive functions begin to develop in early childhood and continue through early adulthood. There is some disagreement in the lite rature regarding the order in which each of these skills develops and thei r developmental trajectory. Researchers do agree that the overall rate of the progression of these functions parall els growth spurts and maturation of the frontal lobes and its connections with other brain areas (Jurado & Ross elli, 2007). An examination of EEG data indicated that periods of rapid growth reflective of an increase in neuronal connections in the frontal lobes are evident. Th e first growth spurt is from birt h to age 5, the second from 7 to 9 years of age, and third occurs between 11 and 13 years of age. Interestingly, each of these three periods is consistent with spur ts when rapid gains in specif ic executive skills are evidenced (Anderson, 2002). The development of executive func tions is preceded by the development of motor inhibition and selective attention after whic h the development of mo re complex skills such as complex working memory and the use of st rategies emerge (Klenberg, Korkman, & LahtiNuuttila, 2001). The development of these func tions does not occur all at once as different executive skills have been found to have different developmental trajectories with some components maturing much earlier than others (Jurado & Rosselli, 2007). Welsh, Pennington, and Grossier (1991) reported that basic execu tive functions such as attention and search behaviors can be observed by 6 years of age, and complex executive functions such as higher level planning skills, verbal fluency, and co mplex working memory can be observed after 12 16

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years of age. Other evidence suggests a much earlier onset, however. For example, children between the ages of 3to 6 years of age exhibit early forms of executive functioning skills when given developmentally appropriate tasks (Jur ado & Rosselli, 2007; McNamara, Byrd, DeLucca, & Berg, under review). Further complicating the interpretation of how ex ecutive skills develop is the fact that younger children are of ten able to verbalize the corre ct response yet continue to make an incorrect motor response, which indicat es that their impaired performance on executive functioning tasks is not due to a lack of understanding of th e rules but rather immature development of the skills required to perform the task correctly or to reliably translate their knowledge into corresponding motor actions. More recently, research from our laboratory has shown that preschoolers are far better problem solvers than previously reported. Pres choolers have solved prob lems that adults found challenging, although preschoolers solved fewer of these difficult problems and they did so with less efficiency (Berg, DeLucca, Case, Byrd, & Mc Namera, 2009, in press). Preschoolers show a great deal of variability in th eir use of strategies when solv ing problems both within individual sessions and within problems (Byrd, Van der Veen, McNamara, Berg, 2004). These age related effects on childrens executive functioning performa nce suggests immaturity of the brain regions associated with executive functioning skills (Anderson, 2002; Diamond & Taylor, 1996; Rennie, Bull, & Diamond, 2004). The development of these executive functions is not homogenous in that each component has a separate trajectory, but al l functions show improvement with an emergence in infancy or early childhood and progression th rough early adulthood. The first to emerge is the ability to inhibit external stimuli and task irrelevant in formation so that the child may have increased attentional control over the envi ronment. Attentional control is an aspect of executive function 17

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that includes selective attenti on, sustained attention, attenti on maintenance, and response inhibition. These skills have been found in infants as early as 9 months ol d with infants able to inhibit responses by 12 months old. The period with the greate st development of sustained attention and response inhibition occurs between 6 and 8 years old, and children seem to have mastered these skills by age 12 (Jurado & Rosselli, 2007), although one study reported an additional growth spurt in these skills and processing speed at age 15 (Anderson, 2002). Cognitive flexibility or set shifting refers to the ability to switch rapidly between response sets and begins to emerge between the ages of 3 and 5 years old (Jurado & Rosselli, 2007). These skills show a rapid increa se in development between the ages of 7 and 9 years old and continue to then improve gradually until adoles cence (Anderson, 2002). Verbal fluency is often included as an index of executive function as it involves successful inhi bition and retrieval of words and has been shown to be one of the most sensitive to frontal lobe dysfunction (Anderson, 2002). Verbal fluency is dependent on childre ns phonological awareness and processing speed; therefore it emerges in the early preschool period and has been shown to have two distinct periods of significant improvement at the ages of 8 and 12 years old (Brocki & Bohlin, 2004) with adult levels reached betw een the ages of 14 and 15 years old (Jurado & Rosselli, 2007). Planning is multifaceted and refers to the ab ility to identify, strategize, and organize actions needed to reach a goal. Planning is perhap s one of the last abilit ies developed by children although simple forms of planning have been evid enced in children as y oung as 3. Planning was found to have the greatest periods of developm ent between 5 and 8 years old, and development continually improves into adulthood. For instan ce, seven year old children exhibited more complex planning abilities as they developed stra tegies and reasoning skills and began to solve problems efficiently when compared to younger children (Jurado & Rosselli, 2007). It is 18

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necessary to note that a lack of consensus exists as to the exact ages for the trajectories of each executive skill and as such mixed results exist in the research. In addition to the developmental timetables of specific executive skills, anot her ongoing debate exists in the executive f unctioning literature. This fundamental debate is related to the organizational nature of executive functions. Se veral theorists view th e executive components as functioning independently of each other (the theory of non-unity) (Diamond, 1997; Juardo & Rosselli, 2007) whereas others assert the ex ecutive components as uni tary functions with partially dissociable component s (Huizinga, Dolan, & van der Mo len, 2006; Lehto, Juujarvi, Kooistra, & Pulkkinen, 2003; Mi yake, Friedman, Emerson, Witzki, Howerter, & Wager, 2000). Many authors have suggested the existence of a unifying central factor (i.e., working memory, planning, or intelligence) under wh ich the other executive compone nts function to organize goal directed behavior. Miyake et al. (2000) took a quite different approach to understanding the organizational nature of executive functions by studying shifting, updating, and inhibition, three functions often suggested as unifying components. Their analyses revealed that although the three components were distinguishable and func tioned separately, they also shared underlying commonality. The authors postulated that executive functions are separable but moderately correlated constructs, (p. 87) and thus contain both unitary and non unitary components. A number of recent papers have begun to explore the relevance of this issue to the development of executive functioning skills. Lehto et al. (2003) extended the Miyake framework to a ch ild sample of 8-13 year old children and found three in terrelated components that resemble d the same factor structure and results reported by Miyake and colleagues. Le hto et al. (2003) labe led their three factors working memory, inhibition, and shifting and repo rted that the relations hips among the latent 19

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executive components were str onger than those found for each of the eight sepa rate executive tasks. Their results provided good agreement to the uniformity and diversity of executive functions in children. Although Lehto et al. replicated th e original work of Miyake et al. in children, their study did not shed much light on our understanding of how executive skills develop. Huizinga and colleagues (2006) elaborated on the Miyake and Lehto work with a more developmental approach by testing the same three skills in four different age groups, ages 7, 11, 15, and young adults. Interestingly, although their repo rt supports both unified and diverse nature of executive functions in children they did not fi nd the same factor structure. Their results indicated prominent roles of updating (working memory) and shifting (cognitive flexibility) but unlike Miyake and Lehto, did not find any effects of inhibition. Further, Huizinga et al. suggested that the developmen t of both updating and shifting occur more gradually than previously expected. They found that while shifting matures by adolescence working memory continued to develop into young adulthood thus providing strong support that components of executive functions develop at different rates. Clearly, the research of Huizinga et al. (2006) expanded the Miyake line of re search by elaborating on develo pmental change and confirming the unity and diversity of ex ecutive functions in children. Executive Functions in Low-Income Children Children from low-income families perform more poorly on measures of intelligence, language proficiency, and academic achievement (Bradley & Corwyn, 2002). Family income, particularly for young children, has a greater effect on childrens cognitive and academic outcomes than do health and behavior (Duncan & Brooks-Gunn, 1997). In recent years there has been growing interest in the role of socioeconomic status (S ES) in the development of executive functioning skills given the important role of these skills in cognitive development. Only recently has research begun to emerge on the development of executive functioning skills specifically in 20

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low-income children, and the existing behavioral and physiological research is consistent with the proposition that low-income children are more susceptible to delays in the development of executive skills when compared to their affluent p eers. The data available are as yet limited and have only begun to examine the various aspects of executive functions. For example, one of the major features of executive func tioning, planning, has yet to be ex amined from the perspective of income effects and thus only indi rect conclusions may be derived. The research reported in this section is an attempt to relate findings on the in fluence of family income on specific aspects of executive functioning development with the ultimate goal of dem onstrating strong associations between SES and executive skills. A number of studies have reported evidence of the influence of SES on attention and selfregulation, both of which are cognitive processe s closely related to executive functions. For example, Mezzacappa (2004) reported that children from low-income families made more errors and performed significantly worse on tasks assessing speed and accuracy on measures of executive attention and inhibition skills when compared to their affluent peers. Other investigators who examined the self-regulation skills of children found that low-income children were less successful at regulati ng their task attention and have overall poorer self-regulation when compared to their more affluent peer s (Blair, 2003; Howse, Lange, Farran, & Boyles, 2003). In other work, Buckner, Mezzacappa, an d Beardslee (2003) compared resilient and nonresilient children living in pove rty and found that resilient ch ildren possessed more effective self-regulation silks. Furthermore, data from the NICHD study of early child care (2003) revealed that the quality of the home environment, which is highly correlated with socioeconomic status, predicted childrens pe rformance on tasks of sustained attention and inhibition. Sustained attention a nd inhibition, in turn, mediated the relationship between the 21

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home environment and school readin ess. It is clear that conv erging evidence suggests that SES greatly impacts the development of childrens atte ntion regulation skills, skills that play a key role in learning environments. Childrens self-regulation and attention skills are not the only cogniti ve systems affected by SES. The effects of family income are clearly seen in childrens approach to problem solving. Davis and Ginsburg (1993) found that children from low-income families developed problemsolving skills slower than children from middle class families. Low-income children performed more poorly than middle class children in both fo rmal and informal mathematic cognition at all age levels in the study (3 to 8 years old). Problem solving skills are greatly affected by the ways in which children process information. Waber a nd colleagues (1984) reported that low-income children used very different cognitive styles in problem solving situations. The cognitive styles these authors identified, global ve rsus analytic, have been associ ated with the broadly different functions of the cerebral hemispheres; the right hemisphere has been related to a more global processing approach that is associated with the early stages of problem solving whereas the left hemisphere has been related to a more analytic pr ocessing style. The latter is often used in the later stages of efficient problem solving that involve choice of strategy and planning. In their study Waber et al (1984) found th at low-income children relie d more heavily on the global problem solving processes associated with the right hemisphere and less on analytic processing, whereas children from more affluent families uti lized more of the analytic processes related to the left hemisphere. The authors noted that surprisingly the two inco me groups of children showed only marginal differences in their overall performance levels on the task but clearly used different processing methods and brain regions to solve proble ms. Planning abilities play a critical role in effective problem solving and ar e thus important executive skills. Research from 22

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our laboratory focused on the development of pl anning abilities found that low-income children solved significantly fewer problems on a planni ng task and of the problems they solved correctly, low-income children took longer and solved problems less efficiently. Perhaps even more important, low-income children did not cat ch up to their more advantaged peers with repeated practice on the task (Ber g et al., 2009, in press). Taken together, these findings indicate that the SES disparities in childrens problem so lving skills are not subtle. Children growing up in low-income environments perform significantly worse on tests of prob lem solving abilities. One research group has extended the literature by consistently showing that executive functions are one of the primary cognitive syst ems affected by social inequalities in early experiences. Noble and her colleagues (2005, 2006, 2007) conducted a program of research based on the effects of SES on executive functioning sk ills. The first in their series of studies was conducted on low-income kindergartners and exam ined five developing neurocognitive systems. SES was related disproportionally to both the left perisylvian (la nguage) center and the prefrontal (executive) system, with low-income children perfo rming more poorly than middleincome children in both areas on measures of go/no go tasks, spatial working memory tasks, false alarms, card sorting, and the PPVT (Noble, Norman, Farah, 2005). In a subsequent study they extended this work by examining a larger sa mple of first graders. They used SES as a continuous variable and rather th an considering the prefrontal/executive system a single system they divided it into three subsystems to study. The results showed that with the increase in power gained from using SES continuously, strong effects of SES were found for the lateral prefrontal/working memory and anterior cingulat e/cognitive control areas whereas there were no effects of SES on ventromedial prefrontal/rew ard processing (Noble, McCandliss, & Farah, 2007). Finally, the authors examined these same neurocogntive systems in older children using 23

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middle school students who were matched for ag e, gender, and ethnicity. Again, significant effects of SES were found for language and the executive subsystems of working memory and cognitive control as well as memory ( Farah et al., 2006) Although the outcomes of each study varied slightly, their results point to the large effects of income on executive skills related to working memory, memory, cognitive control, and language as well as hi ghlights the importance of intervening with children at risk for execu tive delays. Early intervention will allow lowincome children to get the invaluable training an d practice with executive functions that is so clearly needed as early as po ssible thus allowing them to be gin school ready to learn. Physiological research on the effects of in come on executive functions corroborates the results found in behavioral studies. EEG measures of young childre n raised in poverty revealed less activation of the frontal l obes when compared to children in the control group (Otero, Pliego-Rivero, Fernandez, and Ricardo, 2003). Child ren were first tested at 18 months followed by 4, 5, and 6 years of age. The SES disparities pe rsisted even at age 6 and the authors concluded that insufficient environmental s timulation is a major contributor to this developmental lag in brain maturation. Similarly, Kishyama et al ( 2009) examined EEG recordings in high and low SES children looking specifically at event related potentials (ERPs) in an attention task. The authors found that prefrontal-d ependent ERP responses were reduced in the low SES group and low SES children performed more poorly on beha vioral tasks of working memory, cognitive flexibility, and semantic fluency, all clea rly aspects of executive functioning. Moreover, the physiological ev idence is not limited to asse ssment of brain function. Measures of heart rate variability (HRV) have provided additional support for SES disparities in physiological processes related to executive f unctioning. One such cogniti ve process important to executive functioning is at tentional effort, and a comm on way to measure executive 24

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functioning performance and attentional effort is by measuring changes in heart rate variability during a task (Friedman, Allen, Christie & Santu cci, 2002). Of particular interest, a reduction in heart rate variability, also known as vagal suppression or suppression of respiratory sinus arrhythmia, is associated with increased effortful attentional c ontrol. Blair (2003) and Blair and Peters (2003) found that Head Start children showed a negati ve relationship between vagal suppression and on-task behavior during tasks of executive functi on that required children to inhibit a prepotent response while remembering rules and executing a correct responses. In other words, as judged from heart rate variability m easures low-income children were less able to focus their attention and display on-task behaviors during an executive task when compared to their peers. Lastly, research from our laborator y found that low-income children displayed higher heart rate variability, suggesting that they exhibited lower levels of sustained attention during difficult problems on a planning task (DeLucca, McNamera, & Berg, 2006). In short, many limitations of behavioral research on the executi ve skills of low-income children have been addressed by studies employing physiological measures of neural activity and heart rate. The physiological research on this topic substant iates the strong relationship between executive functioning and family income status. A number of studies have examined the pa thways through which aspects of low-income family environments might influence the develo pment of executive functioning skills. There are multiple aspects of the rearing conditions in these families that may impact executive skills. Low-income children receive much less cogniti ve stimulation and opportunities for learning when compared to their more advantaged p eers (Bradley & Corwyn, 2002; Kishiyama, Boyce, Jimenez, Perry, and Knight, 2009 in press). Income status alone predicts that these children will have fewer books and educational toys in their ho mes as well as less exposure to zoos, museums, 25

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and other such learning activities (Bradley & Corwyn, 2002). To add to the lack of cognitive stimulation in homes, low-income children receiv e less attention from adults, hear significantly fewer words in their homes per year, and have le ss favorable quality interactions with caregivers (McLoyd, 1998). Low-income children also experience higher levels of chr onic stressors as they live in much more stressful neighborhoods and ha ve caregivers who suffer more from depression and other psychological and physical effects of stress (Dearing, 2008; Duncan et al., 1994; McLoyd, 1998). These factors combined contribute gr eatly to diminished quality and quantity of family and other social interactions for low-income children. Given the distressed environments of lowincome children, and given the likelihood of these children entering school with less proficient executive skills, perhaps the most important question we should be addressing is if delays in the development of executive skills could be reduced with executive skills training. Despite the abundant work on the importance of executive skills in learning environments and the evidence of a strong relationship between SES and these skills, little is known regardi ng executive skills training with low-income children. The few studies focused on training low-income children in these important skills have demonstrated the crucial role of repeated expos ure and practice as both have served to substantially improve childrens executive functioning abilities (Thorell, Lindqvist Bergman Nutley, Bohlin, & Klingberg, 2009). Such improvements may be important mediators in the academic success of these children. Nevertheless, appr opriately studying a problem is a crucial step in understanding it; thus this study is an attempt to better unders tand the development of executive skills in lowincome children so that effective tr aining strategies may be developed. Although the general developm ent of executive functions has been well studied, the degree to which socioeconomic factors may play a role in the relationship between executive 26

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skills and academic achievement is, for the mo st part, unexplained. Little is known about the relationship between academic success and executiv e functions, specifically in relation to children from low-income families. Academic success is in part dependent on these underlying cognitive skills that are involved in aspects of se lf-regulation. Cognitive ta sks require skills such as inhibition, working memory, pl anning, the use of strategies, and cognitive flexibility. In order to learn basic literacy and math skills, it is necessary for children to inhibit inappropriate behaviors, hold instructions in working memory, use strategies to reach goals, and plan exactly how they will accomplish their goals. It is in this way that executive skills serve as the foundational skills needed for learning in academic environments. Children with delays and deficits in executive skill development would naturally find it more difficult to succeed in school. Socioeconomic status may thus be a central moderating factor of the effect of executive functioning on academic achievement. The existing re search on the role of executive functions in academic achievement proves that academic success is dependent on efficient coordination of executive functioning skills. For instance, performance on measures of executive functions rather than general intelligence has been shown to predict math and reading achievement at the end of kindergarten (Blair and Razza, 2007). Although st udies of this nature are becoming more common, most of this research is conducted within low-income populations rather than comparing low-income children to their more affluent peers (e.g., Blair & Razza, 2007). Additionally, even fewer studies have investigated the influence of income on planning abilities as related to academic achievement. Given the critical role that execu tive functions play in academic functioning, it is important to examine th e role of family income in the relationship between delays in executive functioning developm ent and success in school. It is expected that 27

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measures of executive functioning skills would be found to mediate the relationship between family income and childrens reading and math achievement. In summary, the environment that a child grows up in influences cognitive development. Early childhood is the developmental period when the effects of family income are most influential in development. Family income is more strongly correla ted with early childhood achievement and abilities than are measures of hea lth and behaviors in part because of the effects of income on experiences provide d within the home (Duncan et al., 1998). Variations in the learning experiences provided in the home e nvironment results in differential cognitive stimulation. Although research on the developmen t of executive functi ons in low-income children is only now beginning to emerge, existing data suggest that income specifically affects at least some aspects of executive functioning. Similarly, there is littl e specific data on the effects of income on planning abilities and most of these data are rather indirect; however work related to other executive skills suggests this is an important and fruitful area to explore. There are also very few studies invest igating the development of execu tive functions specifically in low-income children over time. It is impera tive that we understand how their development differs from their more advantaged peers so that we may better intervene. Further, few studies have focused on the role of execu tive functioning skills in childrens academic performance with particular attention to the role of income in this relationship Goals of the this Study A plethora of research exists on the achievement gap between the academic success of low-income children and their more affluent peer s; however far less is kno wn about the role of executive functions in this gap. Gi ven the critical role that execu tive functions play in academic functioning, it seems likely that delayed c ognitive development may result from an underdeveloped set of basic executive functioning skills. It is quite possible that delays in 28

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executive functioning development of low-in come children may mediate the relationship between family income and academic achievement. Therefore, a major focus of this research is to examine multiple aspects of the planning abilities over time of children from lower and hi gher income families on their performance on an executive functioning task. Specifically, the Tower of London (TOL) task will be used to examine childrens executive functioning including their general succes s at problem solving, their speed of solving problems, and their move efficiency at problem solving. Four specific research questions will be a ddressed within this study: (1) What differences can be discerned in childrens executive functioning performance from kindergarten through fourth grade? (2) What role does family income play in childrens executive functioning development? (3) Does family income affect childrens pe rformance on easy problems differently than more difficult problems? (4) Does executive functioning ability mediat e the relationship between family income and academic achievement? The statistical analyses of these data will ma ke use of growth curve modeling given that children were followed from kindergarten through fourth grade. The multilevel modeling approach will be employed so that developmen tal trajectories of child ren may be compared. Growth curve models allow for both fixed (overall effects that are averag ed across all subjects) and random (an estimation of the size of individual differences in th e strength of effects) effects to be examined so that the effect of family income on the way childre ns executive functioning skills change over time may be measured. It is expected that low-income children will have poorer executive functioning skills over time wh en compared to their more affluent peers. Mediation analyses will be utilized to exam ine the relationship between family income, 29

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executive functioning, and academic achievement Based on the earlier review, the following specific hypotheses will be tested: Hypothesis 1. Children will improve their performa nce with maturation but the rate at which they improve may vary for each component of executive functioning. Children will show individual differences in their growth trajectories of each executive skill. Hypothesis 2. Low-income children will perform more poorly overall on multiple components of executive functions reflected in th e task, such as accuracy, speed, and efficiency. These patterns will remain over time and middle-income children will improve performance each year at a faster rate than low-income childre n and the performance gap will increase with each year. Hypothesis 3 Low-income children will perform comparably to middle-income children on easy problems but discrepancy in performan ce will be found for more difficult problems. Hypothesis 4. Executive functions will mediate the relationship between family income and reading and math achievement. 30

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CHAPTER 2 METHODS Participants One hundred and eighty-five participants were initially tested once in kindergarten and then followed through fourth grade. Six childre n were excluded from the study because of computer program failure during the testing se ssion which resulted in a loss of data, three children were excluded based on low Peabody Pi cture Vocabulary Test (PPVT) scores (see criteria below), one child was excluded because of an age outside the limits set, and one child was fearful of the electrodes. Thus, the final sample for the first year of tes ting included a total of 174 children which consisted of 102 low-income and 72 middle-income children (see Table 2-1 for Ns). The number of subjects included in th e second and final testing sessions was reduced further due to attrition. More low-income children were initially test ed than middle-income children since it was anticipated that low-income children would have hi gher attrition rates. The final sample includes an even distribut ion of males and females in the kindergarten testing session (78 girls and 96 boys) and it is ethnically diverse. The ethnic composition of the first year of the study was 48.9% Caucasion, 42.5% African American, 5.7% Hispanic, and 2.9 % of other origin. All participan ts were English speaking. Low-income children ranged from 5 years and 4 months old to 7 years old (M = 5.98, SD = .44) and middle-income children ranged from 4 years and 5 months old to 6 years and 8 months old (M = 5.69, SD = .54) at the time of the kindergarten (first) testing se ssion (see Table 2-1 for ages in additional testing sessions). The differing age ranges during this first year result ed from the inclusion of children who were repeating kindergarten a second time, typically more common in lower income children. 31

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Data Collection All procedures were approved by the Institut ional Review Board. Participants were recruited from Head Start centers, preschools and public and private elementary schools that were considered low-income or middle-income based on the percentage of children on free and reduced lunches. Schools that were considered low-income schools were Head Start programs or schools in which 90% percent or more of children were enrolled in the free or reduced lunch program. Schools which were considered non lo w-income schools were private schools and public schools in which 89% or less were enrolled in the free a nd reduced lunch program, which in the case of public schools re sulted in an average of 10 to 15% of children within these non low-income schools enrolled in the free and reduced lunch program. It was expected that some changes in schools would occur over the course of this study but children we re for the most part expected to remain in the same type of school such that low-income children would remain at low-income schools even upon transferring and vice versa. Data collection took place in multiple schools wi thin four counties in the state of Florida from the 2004-2010 school years. Participants were enrolled at the end of preschool or start of kindergarten and where possible followed in two additional data collections. The goal for the second testing was to complete it within one year and six months after th eir first testing session. Eighty-nine children were included in the second testing session, but this timing related goal was accomplished for 79 of those participants (see Ta ble 2-1 for Ns of each testing session). The remaining ten children were unable to be located within that time period but were tested within the next six months. Children w hose second testing session occurred two years and one day after their first session was then considered to ha ve a missing second session and when the second testing occurred, it was recorded as their third session. This occurr ed for eight of the ten children and as such they were grouped within the thir d testing session. Subse quently, the goal for the 32

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third testing session was again to test children within one year and six months from the second testing session. This goal for the third testing session was acc omplished for 79 participants. Fourteen children requiring follow up were unable to be located within this time frame and were thus tested within the next six months. Any child whose second testing session occurred two years and one day after their se cond session was then considered a lost subject for the third session and their data from this final session was excluded from the study. Lastly, 40 children were unable to be followed up for either the second or final testing session. As the above data make clear, not all child ren participated in each testing session. The low-income children had higher attrition rates th an the middle-income children as 55 % of lowincome children were lost the second year as compared to only 40 % of middle-income children. Surprisingly the Ns for the third testing session remained closely similar to those of the second testing session for both groups of children. The r easons for attrition differed by family income given that most middle-income children were located but most had moved out of the area, transferred schools, or the parent s declined participation in the follow-up years. The majority of low-income children who were not tested in subsequent sessions however were never able to be located despite extensive efforts. Many transfer red schools multiple times within the school year and new contact info was not available. Several low-income children entered foster care and a few were transferred to corrective schools that di d not allow testing. Analyses that tested for selected attrition confirmed that children who did not continue in the study were not different from children who remained in th e study (see preliminary analyses of results section for details). Measures Verbal IQ Assesment The Peabody Picture Vocabulary Test III (PPVT) was used to assess children's verbal IQ abilities and their receptive langua ge skills. Any children with a verbal IQ score of 60 or less on 33

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the PPVT were excluded from the study. The sta ndard PPVT was modified so that it could be administered on a laptop computer. These comput erized images were close approximations to the paper versions and al l other aspects of testing followed the standardized instructions and procedures for this task (Dunn & Dunn, 1997). As with the paper versio n of the Peabody, four pictures were presented on the screen in a two-by-two array. The researcher read a word aloud and the child was instructed to use the computer mouse to click on the picture that best represented the word. For example, the researcher said "show me ball" a nd the child pointed to the picture of a ball. Words were presented in a sequence that became progressively more challenging as the child continue d with the task. The program e nded and calculated a final score after children answered 7 incorrectly out of a sequence of 13 problems. The Tower of London The TOL was used to assess children's ex ecutive functioning skill s (Shallice, 1982). The TOL was presented on a laptop computer (see Fi gure 2-1 for an example of the presentation screen). All participants were given a maximum of 60 seconds to arrange the three colored balls (red, blue, and green) on their own board to match the three colored balls on a cartoon character's (Sesame Streets Ernie) goal board. Each boa rd had three pegs arranged in decreasing size order. The small peg could hold one ball, the mi ddle peg could hold two balls, and the tall peg could hold three balls. Children were allowed to m ove only one ball at a time and they were not allowed to move a ball if it had another ball on top of it. A ball could only be moved to one of the pegs that had an available space. TOL problems were presented in a sequence of increasing difficulty levels that allowed children to become more experienced with the task and allowed for them to build familiarity and confidence on easier problems before the more difficult problems were presented. More difficult problems are defined here to mean problems that require more moves to complete an optimal 34

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solution. Children were given three sets of 10 TOL problems. Each set contained two problems of each difficulty level; three-move, four-move five-move, six-move, and seven-move. After each problem children received feedback on their pe rformance from an animated cartoon figure. Feedback was given according to the number of moves the child used to solve the problem. Children received a "high five guy" (solved the problem in the fewest number of moves possible), a "dancing guy" (solved the problem quickly, but in an extra move or two), "good job" (solved problem, but took three or more extra moves to solve), or "the cloc k" (tried really hard but ran out of time). TOL data were available on a move by move basis that allowed the following measures of performance to be analyzed. Performance wa s assessed in three categories as suggested by Berg and Byrd (2002): solution success rate, mo ve efficiency, and speed measures. Solution success rates were analyzed usi ng two measures: the proportion of correctly solved problems and the proportion of problems solved perfectly (problems solved usi ng the fewest number of moves possible). Move efficiency has often been assessed by determining the number of moves required beyond the minimum required to solve the problem typically referred to as extra moves. The Berg laboratory has adopted a more effective measure that determines the number of optimal moves less the number non-optimal move s and then divides the result by the total of optimal and non-optimal moves. An optimal move is one that takes th e participant one move closer to the goal and a non-optimal move is one that does not. This approach generates a scale with a range from -1 to +1, with more positive numbers indicating proportionally more optimal move selections. A score of 1 indicates every move was optimal and corresponds to a perfectly solved problem. Negative sc ores indicate more non-optimal than optimal moves were made. The advantage of this score is that it can be determined even when a problem is not solved 35

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whereas extra moves cannot. For problems that ar e solved, this score correlates very highly with extra moves (Berg, Byrd, McNamara, & Case, 2010) The third category of performance, speed measures were analyzed in terms of the time used to initiate the first move and then the time needed to correctly solve problems after the first move was made, referred to as solution time. Although all TOL variables were assessed using multilevel modeling, variables were reduced to include only one variable from each of the th ree categories suggested by Berg and Byrd (2002). Subsequently, the primary analyses will incl ude the number of problems solved correctly, solution time, and move efficiency in terms of the optimal move score. Stability of the Tower of London Measures Although the TOL task has gained in popularity in recent year s, the majority of studies employing the task have been focused on college students and older adults Since studies that have used the TOL with children are much less frequent in the li terature, and none have reported reliability scores for children, reliability coefficients from Schneiderman, Welsh, and Retszlaff (1998)s study on college students are reported. The authors measured the number of problems solved perfectly or in the fewest number of move s. Their participants r eceived 30 problems that were selected as the most reliable (correlated most highly with the overall groups scores) from a larger selection of 69 problems. Problems cons isted of 4-move, 5-move, and 6-move problems. Reliability was reported for two studies by Schneiderman et al. The authors computed an interitem reliability in which problems were compared with other problems of a similar type and reported strong reliability ( = 0.79). In a second study the author s also computed test-retest reliably for participants on two occasions and reported a within-subject correlation of r =. 70. The stability of the TOL for the sample used in this study was measured by calculating the ICC (intraclass correlations) of each meas ure on both easy and hard problems of the course 36

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of the study. These results are reported in detail in the results sec tion of this paper. It is important to note however, that these reliability estimates include childrens maturation over three years and thus are not as clean of a measure of reliability as would be optimal. Academic Achievement Childrens reading and math scores on th e Florida Comprehensive Assessment Test (FCAT) and the Stanford Achievement Test Series 10 (SAT-10) were collected as indicators of childrens reading and math achie vement. Neither test was admini stered in kindergarten thus results were only available for the second and third testing sessions. It wa s initially planned for both the reading and math scores to be incorporat ed in to the multilevel models however the lack of consistency in the numbers of children with a cademic data for each year as a result of some variation in test administration by schools and th e lack of correspondence between children who remained in this study and had academic data for bot h years resulted in a loss of too much data to be included in modeling. Consequently, the academ ic data was used to test for mediation of executive functioning skills in the relationship between childrens family income status and academic success. It was not necessary to use scores from both assessments so the Stanford 10 was selected because it is more comprehensive in scope than the FCAT. The SAT-10 is a leading standard ized achievement test in the United States and is used to assess educational progress by measuring read ing comprehension, mathematical problem solving, and science, although science scores were not included in these dissertation analyses. The tests include multiple choice, short answer, and extended response questions. Reliability for the Reading portion of the SAT-10 is .87 and th e Math portion is .80-.87 (American Educational Research Association, 1999). Test scores are available in a va riety of formats and the normbased scores are reported in this paper. Scor es may range from 200 to 800 with higher scores representing better performance. 37

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Family Income Status A revised version of the Holli ngshead Four Factor Index ( Hollingshead, 1975 ) was originally used to obtain a more accurate estima te of the childs socioeconomic status. The revised Hollingshead index included a ll of the original questions (e.g., what is your current education level; what is your occupation ) as well as additional questions that were added to provide a more detailed description of families income levels (e.g., did the child receive any type of formal preliminary education such as preschool, mothers day out program, or early head start program ). This revised Hollingshead index wa s given to primary care takers during a phone interview that was conducted after the tes ting session. Primary care takers received 25 dollars for their particip ation in the study and children receiv ed a 5 dollar gift card to Walmart stores. Unfortunately although extensive effort was put into collection of this information, not all parents completed the Hollingshead index and as such income data was available for only 124 children. Given these challenging circumstances, ch ildrens eligibility for enrollment in the free and reduced lunch program at kindergarten was us ed to group children by family income status. Eligibility for the free and redu ced lunch program was based on family income collected from public school district records. Children who were not enrolled in public schools but were tested at private schools were placed in the middle-income group. Given th e challenges associated with determining income status within low-income samples, this method is becoming more widely accepted (Blair, 2003; Waber et al, 2006). Procedure Children were generally tested at their school and testing se ssions were video taped. In the few cases when parents did not want child ren removed from their classrooms children were tested in the laboratory or their home. Two rese archers were present at each testing session. The testing session began with the rese archer asking the child to draw a picture of a "smiley face" as 38

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a warm up task and to determine handedness. The researcher then explained the procedure to the child. Following this, the children completed the PPV T and were given a sticker for their effort. Heart rate was obtained during all testing. Fo r this the researcher then explained how the "special stickers" (electrodes) were used to watch what thei r heart was doing while they played the game for the EKG and asked the child's permi ssion to adhere them. Electrodes were placed on children as they watched a cartoon in an attemp t to distract them. It was expected that this distraction might prevent children from moving a nd result in children fee ling less fearful of the electrodes. Though recorded, the heart rate data w ill not be presented as part of the dissertation to keep the length of the re sults more manageable. Once the accuracy of the EKG signal was confirmed, the researcher then explained the TOL "game" to the children and the children were given four practice problems to ensure an understanding of the rules. All practice proble ms consisted of one move problems that the children solved without assistance. After the first practice problem was presented, the researcher reminded the children which game board was theirs and which was Ernies and that the goal was to match their game board to Ernies goal board. If the child appeared to not understand the directions or rules, the research er reiterated the rules of the game. The child was required to solve all four practice pr oblems correctly and without help from the researcher to proceed to the experimental TOL. If the child was unable to solve the one move practice problems, the problems were repeated up to three times. The child then received three sets of 10 problems each. After each set children were given a sticker and a 5 minute cartoon break to accommodate their short attention spans. Ca rtoons breaks consisted of clips from popular childrens movies (e.g., Shrek, Toy Story, and Finding Nemo). During the last cartoon break, when testing was complete, the re searchers removed the electrodes. Testing sessions ended with 39

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the researcher thanking children for helping them learn about "how kids think" and walked the child back to class. The time required to complete all parts of a single session was approximately 1 hour. Children and their parents or guardia ns were recruited with an in-class ice cream incentive in that any child who returned the consent form got an ice cream cup, regardless of whether consent was given. Some schools also received 100 dollars for the kindergarten teachers to use at their discretion if more than 50% of the consent forms within the classroom were returned. The childrens primary care takers were contacted after the testing session was completed and asked to participate in a phone in terview. The phone interview consisted of the revised Hollingshead index as well as several ot her questionnaires related to their childrens development and behaviors that ar e not included as part of this study. It was originally expected that the majority of parents or guardians would complete the parent interviews; however this was not the case. One hundred and thirty three parent /guardian interviews were completed with the majority of these being completed by average-in come parents. Considerab le effort was expended to try to reach more of the pare nts for this interview, unfortunate ly with less success than hoped for, especially among low-income parents. Give n this situation the children were ultimately grouped according to their free and reduced lunch program status at the time of initial testing. This method has been used in the literature and is accepted as appropriate given the common high attrition with this populat ion (Liew, Chen, & Hughes, 2009). 40

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Figure 2-1. Tower of London (TOL) presentation screen Table 2-1. Number of participants tested a nd academic achievement means for each year Testing Session N Mean Age SAT-10 Reading SAT-10 Math N Mean Std DevN Mean Std Dev Low-Income Year 1 1025.98 Year 2 467.14 62.00535.0249.0163.00536.1343.18 Year 3 488.46 75.00577.6978.3973.00562.2345.06 Middle-Income Year 1 725.69 Year 2 436.99 29.00594.4551.4729.00596.0035.76 Year 3 458.41 40.00627.2548.6640.00619.6344.71 41

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CHAPTER 3 ANALYTICAL APPROACH TO EXAMINING CHANGE OVER TIME Growth curve models were used to test previously describe d hypotheses 1 through 3 regarding the developmental trajec tories of executive functioning sk ills: (1) the differences that may be discerned in children s executive function ing performance from kindergarten through fourth grade (2) the role of family income in childrens executive func tioning development over time and (3) the effect of family income on childrens performance on easy problems versus more difficult problems. For the final and f ourth hypothesis correlati ons and hierarchical regressions were employed to examine if executi ve functions mediate the effect of family income on childrens academic achievement. Describing and Predicting Change in Executive Functioning Skills Multilevel Modeling for Change Growth curve models using multilevel modeling (MLM) for change (Singer & Willet, 2003) were used to explore the re search questions and to examine the developmental trajectories of specific aspects of executive functioning as indicated by the various measures of TOL performance. Researchers who wish to analyze longitudinal data typically choose between using structural equation modeling (SEM) or MLM to examine data. Both methods clearly have advantages and disadvantages. Although SEM a llows for more specific modeling of time spacing, MLM was determined most appropriate for this study given that it better handles missing data at varying time points when compared to SEM. An additional advantage of the MLM approach is that the change function over time is fit to the entire sample, and the parameters are allowed to freely vary so that the best model fit may be achieved. The MLM approach for change i nvolves a sequence of mu ltiple models in which each model is nested within each prior model with the goal of determining which statistical 42

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model best fits the data. Models are statistical ly compared using three different methods: the deviance statistic (-2 Log Likeli hood or deviance test), the Ak aike Information Criterion (AIC) and Bayesian Information Criterion (BIC). These are typically jointly utilized to select the model with lowest deviance that is statistica lly significant and has the lowest AIC and BIC. Although there is no statistical test for significant differences in f it for the AIC and BIC, all three criteria of model fit will be used to make co mparisons and the most parsimonious model will be reported. This nested model approach allows re searchers to examine change in the group as a whole as well as the examination of individual growth trajectories Given the complicated nature of the model building process, the particular nesting sequence used in these analyses will be described here. This outline is strictly for expl anatory purposes as not a ll steps were included for each TOL measure, but generally speaking most cases followed these steps which are shown in Table 4-1. These model building steps were repeated twice for each dependent variable so that childrens performance on easy problems and hard problems could be examined separately. The first model of the sequence, the null model or unconditional means model (UMM), estimates the variability in participants scores su ch that variation over ti me is treated as error variance. The UMM is tested as a precondition to further analyses and thus in the analyses to follow was always the first tested. The UMM is ev aluated for several purposes. It is examined for fit statistics to test if th e assumption of no change over time was a poor fit. The UMM is also evaluated for significant within -person variance and if found would be in dicative that timevarying predictors may improve model fit. And finally, it is also evaluated for significant between person variations, which if found would indicate that i ndividual differences in initial status (starting values) may be due to time-invari ant predictors. If the U MM criteria was a poorly fitting model, then further model building would c ontinued with the addition of predictors. In the 43

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few cases reported where UMM criteria were no t met in the present study, model building continued based on theoretical e xpectations and assumptions of floor effects that prevented significance in initial status. The second step of model building involved th e nesting of the uncondi tional growth model (UGM) within the UMM. The UGM estimates the average within-person variance in initial status and rate of change with time as the only pr edictor included in the model. It is imperative that the effect of time is stat istically significant since without a significant effect of time modeling results would be meaningless. On th e other hand, a significant main effect of time indicates that growth curve modeling may be esti mated. It is also imperative that significant change in chi square, or the deviance statistic, is found along with reductions in the AIC and BIC fit indices for model building to continue since these provide confidence in the significance of the fit. The variances and covariances are then used to determine the relationship between the initial status and rate of change such that low in itial status with a steep slope is indicative of stronger gain over time and high initial status and shallow slope are i ndicative of fewer gains over time. It is also necessary for the UGM with in-person variance to be reliably different from zero for model building to conti nue. Significant within-person vari ance suggests that the addition of more predictors to the model may clarify the nature of the change in the dependent variable over time. The subsequent models test the major theo retical variables, an d if appropriate, the interactions. This step includes the addition of either time varyi ng or time invariant predictors. Following the modeling of the varying time factor, a model was tested in which family income was added as a predictor variable. Significant variance in the family income variable as well as the effect of time was required for further model building. Fit statistics were compared again and 44

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if criteria were met other models might be tested. Consequently, th e next and final model included the interaction of family income status and time as a predictor variable. This interaction model was included to determine if the rate of change in the dependent variable differed according to family income status or if all ch ildren changed over time in a similar manner. MLMs for change are advantageous for longitudinal data given that models may still be estimated when the data set is not perfectly balanced (Singer & Willet, 2003). Missing data in this study was addressed using the Full Informat ion Maximum Likelihood (FIML) procedure in SPSS which allowed individuals to vary in the spacing of data points and the number of data points contributed. In other words, FIML can estimate parameters using all available information, with appropriate corrections made to stochastic terms for both fixed and random effects. Variance components were selected for both repeated and random error structures based on the SPSS default choices for the FIML. Evaluation of Random and Fixed Effects For each of the MLM models to be presented below, the MLM approach addresses research questions by including at least two levels, the individu al subject level and the group level within modeling. It is possible to include more levels (eg., school le vel) however given the sample size of this study, only two levels were analyzed in an effort to limit the number of parameters estimated within analyses. As a resu lt, the growth curve models for change address two types of research questions in that each nest ed model addresses these two aspects of change. The level 1 model describes how individuals change over time and therefore includes withinperson time varying predictors. The level 2 m odel describes how such changes vary across individuals and for this reason includes betw een-person time invariant predictors (Singer & Willet, 2003). These levels of model detail s are referred to in the MLM literature as submodels. 45

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The Level 1 Model The level 1 submodel describes the random effects within the model with an estimation of within-person change over time expected in the outcome variable and how time varying predictors may affect such change. Clearly, th e current analyses addr ess the question of how childrens problem solving (e.g., solution time used to complete TOL problems) changed over time. It was expected that such change would be explained by the interac tion of family income and time (age at each testing session). Singer a nd Willets (2003) notation was used to form the following model: Yij = 0i + 1i TIMEij + 2i X 2ij+ ij This sub model asserts that Yij is the predicted outcome for person i at time j. TIMEij is the value of time for person i at time j and X2 is the time varying pr edictor (income by time interaction) at the within-person level. initial status) is the value of Y when time is zero (mean for kindergarten age) and li is the slope of the lin ear trajectory of person i. 2i, is the unique effect of X2 on Y and ij is the within-person error term of which the variance is estimated from in this model. If the interaction term is dropped from the equation or is unable to be included, the resulting equation is in essence the unconditional growth model (UGM) and is represented as this: Yij = 0i + 1i TIMEij+ ij The Level 2 Model Provided the random effect of time is found significant, the level 2 sub model would be examined. This model describes the fixed eff ects or estimated betweenperson differences in the outcome variable over time as well as the predictors. This level includes the outcome 46

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parameters estimated at Level 1 (the main effect of time along with the income by time interaction) as the outcomes of new equations that include the time invariant variable (income) as a predictor. The analyses included below examin e if childrens growth trajectories differ across the group and whether the change varies by family income status. Given the time vector is age at each testing session and children range in age at each session, age at the first test session was included as the time predictor that was expected to moderate the relationship between family income and the outcome measures. The following equations were used for the Level 2 analyses with two time invariant predic tors (age and income): 0i = Y00 + Y01 + Si + 0i, 1i = Y10 + Y11 + Si + 1i, The model at this level asserts that S is the variable for the time invariant predictor of family income (low = 0; middle = 1). The Level 2 intercepts Y00, Y10, Y20 are the estimates of the Level 1 parameters 0i 1i when all time invariant predictors are zero. Thus with the time invariant predictor of family income where S = 0 represents low-income children the above equations are estimates of the two Level 1 para meters for low-income children. The coding of the family income variable indicates how each of the Level 1 parameters changes for low-income children as compared to their more affluent peers. The Level 2 intercepts Y01, Y11, Y21 are thus the effects of time, family income, and the ti me by income interaction. The error terms 0i and 1i, represent the individual differences in the Level 1 parameters that are not explained by the time invariant Level 2 predictors. 47

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Mediation Analyses In addition to MLM analyses, mediational an alyses were conducted to examine hypothesis 4 related to academic achievements. Mediation is said to occur when a variable accounts for a substantial or significant portion of the relationship between the independent variable (IV) and the dependent variable (DV) (Baron & Kenny, 1968) Based on the research reviewed in the introduction, it was predicted that executive f unctioning skills would me diate the relationship between family income status a nd childrens academic achievement. Mediation was tested using Baron and Kennys (1986) causal steps approach with the Sobel test (1982) due to its higher power and higher control over Type 1 error rate (H ayes, 2009; MacKinnon, Lockwood, & Williams, 2004). Baron and Kenny (1986) outlined four steps th at are used to establish mediation: (1) the relationship between the IV and DV must be significant, (2) the relationshi p between the IV and mediator variable must be significant, (3) the rela tionship between the medi ator and DV must be significant, and most critically, (4) the effect of the IV on the DV is substantially reduced with the addition of the mediator to the model. St eps three and four are estimated in the same equation. Full mediation is achieve d if all four of these steps ar e met and the relation between the IV and DV is reduced to zero after controlling for the relation between the mediator and the DV. Consequently, partial mediation then occurs when a significant relationship exits between the IV and DV variables but this relationship is reduced, but not to zero, when the relationship between them is controlled. Meet ing these four steps does not conc lusively prove that mediation has occurred. Even so, these crite ria are most often all that are used to informally judge if mediation has occurred. Although mediation is frequently ex plored in research, formal significance tests are rarely conducted and thus misleading conclusions are often reported (Preacher & Hayes, 2004). MacKinnon & Dwyer (1993) and MacKinnon, Wa rsi, & Dwyer (1995) have argued for 48

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statistically based methods of which mediati on should be formally assessed. The commonly reported Sobel test was selected for the analyses reported below over th e increasingly popular alternative of bootstrapping. The reason is that th e Sobel test is more conservative. The Aroian version of the Sobel test was selected because the Aroian version does not make the unnecessary assumption of the Sobel test regarding the product of the error terms for paths a and b, which is extremely small (Preacher & Hayes, 2004). Thus, a significant test statistic (z ) confirmed with the Sobel test establishes mediation. Mediation models are path models that spec ify causal relationships between the DV, IV, and mediation variables (MacKinnon, 2008). Fo r illustrative purposes, Figures 3-1 and 3-2 depict the general mediation model used for all of the mediation analyses employed here, illustrating the hypothetical paths through which ex ecutive functioning skills were expected to mediate the relationship between family income status and childrens academic achievement. It is important to distinguish between effects and their corresponding paths for interpretation of the following results. Figures 3-1a and 3-2a illustrate the total effect of the IV (family income) on the DV (academic achievement) which is path c. Figures 3-1b and 3-2b re presents the effect of the IV (family income) on the mediator (e xecutive functioning) re presented in path a. Path b represents the effect of the mediator on the DV (i.e., reading and math achievement) when the effect of the IV (family income) is entered in to the model. Each path is quantified with standardized regression coefficients. The indirect effect of the IV on the DV through the mediational path is the product of a and b. The indirect and the direct effects (i.e., path c') of the IV on the DV when the mediator is controlled results in the total effect of the IV on the DV when summed. In other words, significant mediation is dependent on not only indirect effects but also a direct effect (i.e., path c') that is smaller in magnitude when compared to the total effect (i.e., 49

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path c ). Path c was thus estimated by entering academic achievement as the dependent variable and executive functioning as the independent vari able to determine if the effect of executive functioning on academic achievement existed. A Reading Family Achievement Income c B Executive Functioning Figure 3-1. Conceptual diagram of the media tion analyses and estimated relationships for reading achievement (A) Direct effect of family income on reading achievement (B) Indirect effect of family income on reading achievement Family Income Reading Achievement a b c 50

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A B Figure 3-2. Conceptual diagram of the medi ation analyses representing the estimated relationships for math achieve ment (A) Direct effect of family income on math achievement (B) Indirect effect of family income on math achievement Family Income Math Achievement c Executive Functioning Family Income Math Achievement c a b 51

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CHAPTER 4 RESULTS Preliminary Analyses Sample Means and Standard Deviations Although the major analyses for the present study involve ML M, interpretation of these models can be enhanced by providing the simp le means and standard deviations for each outcome measure at each age, and for easy and hard problems for both low and middle income children. Table 4-1 and 4-2 show this informati on and therefore provide a general idea of the mean trajectories and variances for each outcome variable at each testing session. The means and standard deviations were all in the expected direction and followed hypothesized patterns (see (Figures 4-1 through 4-3). Means suggested a ll children showed improvement related to maturation with each year. In general, middle-income children had better performance on all aspects of executive functioning as measured by th e TOL. They solved more problems correctly, took less time to solve problems, and solved pr oblems more efficiently than low-income children. Problems identified as ea sy and hard showed substantial mean differences in the expected direction indicating thes e labels were appropriate. Lastl y, gender effects were examined for all variables and no significan t effects of gender were found. Sample Selectivity: Attrition Related to Drop Out Preliminary analyses were conducted to examin e the sample selectivity related to subject drop out. It was expected that a large number of low-income children would be lost due to attrition and this was indeed the case. From the first to second testi ng session, 40% of middleincome children and 55% of low-income children were lost to attrition with an additional 37% of middle-income and 47% of low-income lost between the second and thir d testing sessions. A major concern was that the children who dropped out of the study were from families with the 52

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lowest income status when compared to other families within this study and following the theoretical basis of this study, the children might well be th ose with the p oorest executive functioning skills. Thus, mean comparisons were conducted to test for at trition bias. Analyses examined differences in performance during on e ach variable during the first year of the study between those who remained in the study and thos e who dropped out. Contrary to expectations, the only variable in which ch ildren that dropped out performe d differently than those who remained in the study was the measure of genera l success. Students who remained in the study and thus did not experience any drop out perf ormed significantly better on easy problems on the proportion solved variable t (1, 161) = -2.622, p < .010) ( M= .78, SD = .192 vs. M= .84, SD = .125). Mean comparisons were also conducted sepa rately within each income group to determine if lower performing children within each income group dropped out at a hi gher rate than better performing children but no income differences were found. Thus, most results were not significantly affected by attrition regardless of where or how long the dropout occurred or family income. These findings suggest that, for the most part, the children who dropped out of the study did not differ significantly from thos e who remained in the study. Age at Time of Testing Another concern was that childrens age at te sting session may bias results if one income group was younger on average than the other income group at the time of testing. Participants ages were compared for each of the three testin g sessions to ensure that the participants who dropped out after the first and second testing session were not si gnificantly different from those who completed the entire study. A MANOVA show ed no differences between the two income groups at any of the three testi ng sessions in regards to their age when tested at each session, F (1, 71) = 1.08, p > .05. These findings argue against the possibility that childrens age when tested may have been biased by income group. 53

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Application of the Multilevel Model for Change The MLM of change over time analyses were conducted at both easy and difficult problem levels for each of the three TOL outcome variables that were measured each year and expected to change over time. TOL problems were divided into easy versus hard problems based on their difficulty level such that 3 and 4 move problem s were considered easy and 5, 6, and 7 move problems were considered hard. TOL data we re collapsed across th e three sets of TOL problems1. The proportion of correctly solved probl ems was used as a measure of solution success rate or problem solving success, the solu tion time used to solve problems was used as a measure of timing or speed, and a measure of th e efficiency used when making moves was used as an estimate of move efficiency. Subsequently results will be presented in order of models described in chapter 3 for both easy and hard problems and for each dependent variable followed by the mediation analyses conducte d with the academic data. MLMs included four variables consisting of the outcome variable (TOL performance variable), a measure of time, a time-invariant pred ictor (income status), and an interaction term (income by time). This study examined the linear e ffects of time as a predictor variable with age at each testing session as the indicator of time. It was not possible to examine curvilinear effects (quadratic and cubic) of time give n that this study only included three time points. In order for the intercept of analyses to be meaningful, the first measure of tim e, that is, the childrens age at the kindergarten testing session, was centered on the mean age at th at time (M= 5 years and eight months, SD= .50) Thus time was centered at the beginning of the study to facilitate interpretations of coefficients such that the kindergarten testing session was the baseline against which other testing sessions we re compared. The time-invariant predictor of family income 1 Separate analyses were conducted that isolated performa nce in the first set as well as the third set of problems, although this method did not provide enough power for significant results. 54

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status (low versus middle) was coded as 0 for low-income and 1 for middle-income such that the estimates reported in SPSS for the income mode l represented low-inco me childrens scores. High-income childrens scores were then calculated based on coe fficients corresponding to this level. Lastly, to avoid problems with multicolin earity, the income by time interaction term was created by first calculating a product term of inco me by time and then using logistic regressions to calculate the unstandardized residuals from the income by time product term that was then entered as the interaction term in the interaction model. As previously outlined, MLMs for change were used to test 4 models for each TOL variable labeled A-D (see Ta ble 4-1 for model building steps). Model A was the UMM null model and Model B was the unconditional growth model (UGM) in which the linear effect of time was examined without the inclusion of pr edictor variables. This model added the time variable as a Level 1 predictor of Y such that Y became a function of an error term, the intercept, and individual growth. The UGM added no predictors at Level 2 so that the Level 2 model is left unconditional. In the Level 2 model there were tw o random effects, the in tercept (initial status) and slope (rate of change). It is important to remember that th e age at the first testing session (kindergarten) was centered to facilitate inte rpretation of the parame ters. A significant main effect of time indicates that random effects of time are evident in th is model so that growth curve modeling may be estimated. Predicto r variables of this change were added in subsequent models if significant within-person change was present over time in this model. If the random effects of time could not be estimated due to a lack of power the Level 1 e ffect of time and intercept were removed from the model and only the Level 2 effect of family income was investigated further. Models C and D were thus nested theoretical mode ls in which the main effect of family income status and its interaction with time were then tested. Model C examined the main effect of family 55

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income on the initial status in the first testing se ssion as well as linear change in the dependent variables over time. Significant variance in the family income variable at Level 2 as well as the random effect of time at both Level 1 and 2 was re quired for model building to continue. If this was not achieved model building ceased. Model D was thus contingent on both random and fixed effects of time and a fixed e ffect of family income. Initially the time varying predictor of childrens PPVT scores at each test date were entered as an additional step but once this indicator of receptive language was entered the effect of family income became non significant so PPVT was in the end dropped from all models. The inclusion of the PPVT rendered all effects of fa mily income non significant, a finding that is consistent in the litera ture given the strong re lationship between family income and childrens language abilities (Noble et al., 2005). The result is that if eith er of these variable s is statistically removed in models such as these, the other va riable is, for all practic al purposes, removed along with it. Major Analyses: Hypotheses Testing Each of the study hypotheses were examined below. The first three hypotheses were addressed using multilevel modeling for change. Given the complex nature of the MLM analyses, the three hypotheses were examined with in the same series of nested models conducted on each outcome measure (i.e., proportion of co rrectly solved problems, solution speed, and move efficiency). Results for each of the outcomes will be presented separately following the order in which models were nested. A summary section that specifically addresses hypotheses will be included at the end of each section. The final hypothesis related to the mediating role of family income in the relationship between executive functioning skills and academic achievement was examined using hierarchical regressions. Results for these analyses are presented after the MLM results. 56

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Describing and Predicting Change in Prop ortion of Correctly Solved Problems The Unconditional Means Models The UMM (null model) for proportion correct was first tested in Model A with results for easy problems shown in Table 4-4 and results of hard problems shown in Table 4-5. Model A indicated that forcing the data to show no change over time resulte d in a poorly fitting model for both easy and hard problems. The grand mean (a fixed effect) for easy problems was Y00 =.849, p< .001 and was Y00 = .667, p< .001 for hard problems. These intercepts indicate that on average children solved five of the six easy problems and six of the nine hard problems. The more important effects for both models were found in the random effects. For both easy and hard problems, the estimated Level 1 within-person variances were 2 =.017 and 2 =.059, ps <.001 for easy and hard problems, respectively. Th e significant variance i ndicates that time could be a significant factor in proportion of correctly solved problems. The estimated Level 2 between-person variance for easy proble ms also differed reliably from zero 0 2=.007, p< .001 indicating significant individual di fferences in the proportion of co rrectly solved easy problems. A lack of significant estimated between-person vari ance is sometimes indicative of a floor effect. In essence, there was no variation in where chil dren began on the more difficult problems since the problems were difficult for all children. It was expected that additional predictors at both Level 1 and Level 2 would improve the fit of each model. A better understanding of the UMM varian ces is often obtained by computing the intraclass correlation coefficient (ICC) (Singe r & Willett, 2003). The ICC used in multilevel modeling as a measure of the proportion of UMM variation compared to the total UMM variation in the dependent variab le. The ICC is calculated by ( = 0 2 /( 2 + 0 2 )),thus the ICC for easy problems was = .98 and for hard problems was = .919. The ICC indicates that 57

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approximately 98% of total varia tion in correctly solved easy pr oblems and 91% of variation in correctly solved hard problems was due to indi vidual differences between children. The ICC is also used as a measure of the average autocorrel ation or stability of th e dependent variable over time. Thus, the estimated average stability for easy problems was .98, while the estimated stability for hard problems was .92. Clearly more stability was found in the easy problems when compared to hard problems over time. The Unconditional Growth Models Next, the UGM (growth models) were tested as Model B. Results of Model B are shown in the Model B column of Table 4-3 for easy prob lems and Table 4-4 for hard problems. No random effects estimate could be calculated for individual differences in the rate of change which suggests that these diffe rences were neglible, a term used in MLM nomenclature indicative of effects too small to be estimated or a lack of power. For this reason, the random effects of time and intercept were then dropped from the model such that results include only fixed effects. The positive linear effect of time indicated that child ren did improve their performance over the course of the study for bot h easy and hard problems. For easy problems, the intercept (initial status) was Y00 = .522, p < .001 and the slope, or rate of change, was Y10 =.049, p < .001. The significance of th e intercept and slope indicate that on average children solved approximately half of the easy problems (t hree out of the six possi ble) when tested in kindergarten and improved by solving approximately one third of a problem more each year. The Level 1 residual variance or random effects measur es the overall scatter of childrens data around their change trajectory. The incl usion of change over time in the model increased the residual (random) variance from 2 =.017 in Model A to 2 =.019 in Model B for easy problems. Given that the Level 2 intercept and slope for easy problems remained significantly different 58

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from zero with the inclusion of time in the model, as did this within-person variance, it was expected that additional predictors would help to improve model fit a nd explain random effects left in the model. Results for the hard problems did not show th e same pattern of performance that was found for the easy problems. It was expected that ch ildren would have more difficulty solving the harder problems and thus solve fewer problems corre ctly as compared to the easy problems and indeed this was the case. As with the easy problems, no estimate could be calculated for individual differences in the rate of change Consequently the random effects of time and intercept were then dropped fr om the model so that results include only fixed effects. The intercept (initial status) for hard problems was Y00 = -.177, p < .002 and the slope, or rate of change, was Y10 =.124, p < .001. Given that the intercept for this measure is calculated as a proportion (between zero and one), a negative in tercept indicated that the proportion correct at Time 1 estimated from the linear change over ti me was close to zero and thus indicates that children struggled to solve any problems correctly. Since there we re nine hard problems that children solved, this slope can be roughly calculated by multiplying the estimate by nine to improve interpretation. The positive slope t hus indicates that children improved their performance each year by solving almost 1 more problem each year. The inclusion of time in the model significantly reduced the Level 1 random effects residual variance from 2 =.059 in Model A to 2 =.036 in Model B. As with the eas y problems, the Level 2 within-person variance remained significantly different from zer o, which indicated that additional predictors would help to improve model fit and expl ain fixed effects left in the model. Fit statistics of the UMM and the UGM in bot h models were statistically compared to determine if the addition of the time variable as a predictor improved the fit of the model. For the 59

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easy problems, the inclusion of time resulted in a significant reduction in deviance ( (1) = 55.172, p < .001) along with reductions in AIC and BIC. Similarly, the hard problems also resulted in a significan t reduction in deviance ( (1) = 119.992, p < .001) AIC and BIC. Based on the values, the addition of time resulted in a gr eater improvement in fit and explained much more within-person variance for the hard problems as compared to the easy problems. Taken together, the fixed effects of time along with the l ack of a random effect of time for both models indicates that children did improve their perfor mance each year albeit there were not significant individual differences in this rate of change. This result is cont rary to expectations as it was hypothesized that middle-income children would cha nge faster and show more growth over time compared to low-income children. Main Effects of Family Income The main effect of family income was tested in Model C for both easy and hard problems to explain additional fixed effects left in the model. Only the fixed effects of income were estimated given that income is a between subjects variable and as such it can not vary within subjects, rendering meaningless the possibility of any random effects. It was expected that the addition of the income variable would affect the initial status, which is the intercept, of the number of problems solved in the first session a nd more clearly describe linear change in the proportion of both easy and hard problems solved. Results of Model C are shown in the Model C columns of Table 4-4 and Table 4-5. The intercept Y00 = .503, p < .001 is the average estimated proportion of easy problems solved by low-income ch ildren at their first testing session (Table 41). The intercept was again, not surprisingly, differe nt from zero indicating that thus low-income children solved on average about half of the easy problems at the first session. The intercept for middle-income children may be calculated by addi ng the income coefficient to the intercept ( Y01 = .038, p < .013) thus revealing that mi ddle-income children were also estimated to solve solved 60

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on average somewhat over half of the easy problems that were presented correctly at the first testing session. Family income had a signifi cant positive effect on the proportion of easy problems solved such that middle-income childre n solved more problems correctly at the first testing session. The linear effect of time (slope) remained statistically significant and positive as in Model B ( Y10 = .049, p < .001) thus indicating that the nu mber of correctly solved problems increased over the course of th is study by one third of a problem. For the random effects, the addition of income as a Level 2 predictor redu ced the within-person variance in easy problems from 2 =.019, p < .001 in the UGM to 2 =.018, p < .001. Although this was only a slight drop, the extent of the decline wa s still statistically different from zero thus suggesting that additional predictors may further improve model fit. Because the UGM had no Level 1 predictors, it was not surprising that the inclusion of income at Level 2 did not result in a difference in the rate of change. Since the varian ce in Model C was still reliably different from zero the inclusion of additional Level 2 predictors may improve model fit more. Family income also had a significant positiv e effect on the proportion of hard problems solved in Model C ( Y01 = .056, p < .009) such that middle-income children solved more hard problems correctly at the first testing se ssion (Table 4-5). The negative intercept ( Y00 = -.204, p < .001), similar to the findings for hard problems in the unconditional growth model (Model B), indicated that on average low-income children were not able to solve any of the most difficult problems. Middle-income children also seemed to struggle with the difficult problems in that they on average only solved a little more than one hard problem correctly. Even though their performance appeared comparable to low-income children it is important to note that this difference was significant. The linear effect of time remained statistically significant and positive as in Model B ( Y10 = .124, p < .001) thus indicating that th e number of correctly solved 61

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problems increased linearly over the course of this study by a little over one problem per year for all children. For the random effects, the addition of the Level 2 income predictor did not reduce the within-person variance in hard problems from ( 2 = .037, p < .001). As with easy problems, although a dramatic change was not found the variance was still statistically different from zero thus suggesting that additional predictors may further improve model fit. Because the UGM had no Level 1 predictors of random effects, it was no t surprising that the inclusion of income at Level 2 did not reduce the variance in rate of change. Model C fit the data for both easy ( (1) =6.12, p < .05) and hard problems ( (1) = 6.802, p < .001) provided a better fit than did the UGM in Model B. This model also resulted in reduc tions in AIC and BIC. La stly, it is preferable to examine the time by income interaction wh enever possible but the interaction was not warranted given that the random effect of time was dropped from modeling in Model C. Summary of Findings Generally speaking, the nested models for th e most part partially supported predictions related to the first three hypothese s. It was first predicted that childrens executive functioning skills would improve each year as they matured a nd that these differences would be more drastic for some childrens individual growth trajectorie s when compared to ot hers. Childrens problem solving success did improve each year with matura tion yet, contrary to expectations, significant differences were not found in relati on to individual grow th trajectories, most likely due to a lack of power. The random effects of time had to be dropped from modeling and as such individual differences in trajectories coul d not be examined. The income hypothesis was fully supported by these results. Family income explained a sign ificant amount of additional variance for both easy and hard problems. Middle-income children solved more problems correctly at the first testing session and this difference in performance was main tained each year. It is preferable to examine 62

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the time by income interaction whenever possibl e but the interaction was not warranted given that the random effect of time was dropped from modeling in Model C. Childrens PPVT scores were examined as a final predic tor, however results are not in cluded as this step was dropped from model building. The inclusion of PPVT did explain additional varian ce however it rendered all effects of income non significant and did not significantly improve mo del fit. Lastly, the prediction that low-income ch ildren would perform comparably to middle-income children on easy and more poorly on hard problems was only partially supported. Low-income children solved fewer problems correctly than middle-inco me children; regardless of difficulty level but the income level performance gap was greater for more difficult problems. In Model B, the negative intercept found for hard problems was indicative of a floor effect. Subsequent model building showed this floor effect was present because of the low-income childrens difficulty with these most difficult problems. Model C i ndicated that middle-income children performed better than low-income children on the more diff icult problems, however it was expected that differences would be more pronounced. An income disparity or increment of difference was found between the easy and hard problems, as the income coefficient that indicates the difference between the two groups was larger for hard problems ( Y01= .056) compared to easy problems ( Y01=.038). Clearly the effects of family income were most pronounced for the more difficult problems. Describing and Predicting Change in Solution Time The Unconditional Means Models The UMM (null model) for the solution time va riable was first tested in Model A with results for easy problems shown in Table 4-6 an d results of hard problems shown in Table 4-7. Model A did prove that forcing or restricting the data to show no change over time was resulted in a poorly fitting model for both easy and hard problems. The grand mean (a fixed effect) for 63

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easy problems was Y00 =15.791, p< .001 and was Y00 = 26.396, p< .001 for hard problems indicating that children took 16 seconds to solve easy problems co rrectly and 26 seconds to solve hard problems correctly. The rando m intercept on hard problems was not significantly different from zero and was thus dropped from model build ing. The more important effects were found in the random effects portion of the model. The estimated Level 1 within-person variance for both easy and hard problems was 2 =31.470, 2 =42.728, respectively, both ps <.001. The significant variances indicate that time could be a significant factor in solution time scores for both easy and hard problems. The estimated Le vel 2 between-person variance for easy problems also differed reliably from zero 0 2= 6.994, p <.022 thus indicating significant individual differences in the speed of solving easy problem s. The Level 2 between-person variance for hard problems was not calculated since the random inte rcept was excluded from the model. Given that the models included significant within and between-person variance it was expected that additional predictors at both Level 1 and Leve l 2 would improve the fit of the model. As with the proportion of correct problems solved variable, the ICC was used to understand the relative magnitude of the variances in the solution times for easy problems. The ICC for easy problems was = .334 and = .382 and for hard problems. The ICC results reveal that approximately 33% of total variation in correctly solved easy problems and 40% of variation in correctly solved hard problems was due to in dividual differences between children. The ICC is also used as a measure of the average autoco rrelation or stability of the dependent variable over time. Thus, the estimated average stability for easy problems was .33, while the estimated stability for hard problems was .38. Clearly more stability was found in the hard problems when compared to easy problems over time. 64

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The Unconditional Growth Model Results of growth models are shown in the Model B column of Table 4-6 for easy problems and Table 4-7 for hard problems. No ra ndom effects estimate could be calculated for individual differences in the rate of change for easy problems which suggested that these differences were either negligible or there was not enough power. The random effects of time and intercept were then dropped fr om the model for easy problems such that results include only fixed effects. Interestingly, the random effect of time was present in Level 1 for hard problems which will be discussed below. The negative linear effect of time for easy problems indicated that children did improve their pe rformance over the course of the study. That is, the time needed to find a solution was decreased as children matu red. For easy problems, the intercept (initial status) was Y00 = 29.019, p < .001 and the slope, or rate of change, was Y10 = -1.957, p < .001. The significance of the interc ept and negative slope indicate that on average children took thirty seconds to correctly solve problems in the firs t testing session and improved by solving problems approximately 2 seconds faster each year. Th e Level 1 residual variance or random effects indicated that the inclusion of time in the model increased the residual variance from 2 = 31.470, p < .001 in Model A to 2 = 31.884, p < .001 in Model B. This within-person variance remained significant with the addition of time thus indicating that the add ition of predictors may explain random effects left in the model. Given that the Level 2 within-person variance remained significant with the inclus ion of time in the model thus it was expected that additional predictors would help to improve model fit and expl ain random effects le ft in the model. Results for the hard problems showed the sa me pattern as the easy problems with one slight exception. In the results presented up to this point, the random effect of time was not estimated due to a lack of power. This was not the case for hard probl ems on the solution time 65

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variable. The positive random ef fect of time was estimated ho wever it was not significant (1 2 = .062, p < .111) indicating there were not significant within-person diffe rences in the Level 1 rate of change over time. This finding indicates that children did not show in dividual differences in how long they took to solve problems. The Level 1 intercept was not estimated given that it was excluded from model building in the UMM wh en no random effects were found. The Level 2 intercept (initial status) for hard problems was Y00 = 45.875, p < .001 and the slope, or rate of change, was Y10 = -2.835, p < .001. The linear effect of time for hard problems was negative indicating that children showed the same overall showed gains in their rate of change over time as their solution times improved with maturation. The intercept and slope were both significantly different from zero thus indica ting that on average ch ildren needed 46 sec onds to solve hard problems in the first testing session and impr oved by solving problems approximately three seconds faster each year. The inclusion of time as a predictor in the model reduced the Level 1 random effects residual vari ance a great deal from 2 = 42.728 p < .001 in Model A to 2 =26.094, p < .001 in Model B. As with the easy probl ems, this within-person variance was still significant with the addition of ti me thus indicating that the addi tion of predictors may explain random effects left in the model. As with the ea sy problems, the Level 2 within-person variance, remained significant which indicated that additiona l predictors would help to improve model fit and explain fixed effects left in the model. Fit statistics of the UMM and the UGM in bot h models were statistically compared to determine if the addition of the time variable as a predictor improved the fit of the model. For the easy problems, the inclusion of time resulted in a significant reduction in deviance ( (1) = 55.547, p < .001) along with reductions in AIC and BIC. Similarly, the hard problems also resulted in a significan t reduction in deviance ( (1) = 130.024, p < .001) AIC and BIC. Clearly 66

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the addition of time resulted in a greater improvement in fit and e xplained much more within and between-person variance for th e hard problems as compared to the easy problems. Taken together, the fixed effect of time along with the la ck of a random effect of time for both models indicates that children did improve their perfor mance each year albeit there were not significant individual differences in this ra te of change. The addition of pred ictors at both Level 1 and Level 2 should not only improve the model fit but tell a better story of how childrens solution times changed over time. This result is contrary to expectations as it wa s hypothesized that middleincome children would change faster and show mo re growth over time compared to low-income children. Main Effects of Family Income The main effects of family income were test ed in Model C for both easy and hard problems to explain additional fixed effects left in the model. Results of Model C are shown in the Model C columns of Table 4-6 and Table 4-7. It was exp ected that the addition of the income variable would not only affect the initial st atus of the speed in which child ren solved problems in the first testing session but also more cl early describe the negative linear improvement in speed for both easy and hard problems solved. The intercept Y00 = 30.001, p < .001 is the average estimated time low-income children used to solve easy problems at their first testing se ssion. The intercept was again, not surprisingly, di fferent from zero indica ting that low-income children on average took 30 seconds to solve easy problems at the first te sting session. The inter cept for middle-income children may be calculated by adding the income coefficient ( Y01 = -1.959, p < .002) to the intercept thus revealing that middle-income children took on average 27 seconds to solve easy problems at the first testing session. Family inco me thus had a significant negative effect on solution times such that middle-income children solved easy problems faster at the first testing session. The linear effect of time remained statis tically significant and negative as in Model B 67

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( Y10 = -1.973, p < .001) thus indicating that the speed at which children solved problems increased linearly over the course of this st udy by two seconds each year. For the random effects, the addition of income as a Le vel 2 predictor reduced the with in-person variance in easy problems from 2 = 31.884, p < .001 in the UGM to 2 = 30.936 p < .001. The significance of this variance suggests that additional predic tors may further improve model fit. Because the UGM had no Level 1 predictors, it was not surprising that the inclusion of income at Level 2 only slightly reduced the fixed effects variance in rate of change but because the variance in Model C was still reliably different from zero th e inclusion of additional Level 2 predictors may improve model fit more. Family income also had a significant negative effect on the solu tion speed of hard problems solved ( Y01 = -2.095, p < .001) such that middle-income children solved hard problems faster at the first test ing session. The intercept ( Y00 = 46.875, p < .001) indicated that on average low-income children took 47 seconds to solve hard problems while middle-income children took on average 44 seconds to solve hard problems at th e first session. The Leve l 2 linear effect of time remained statistically significant and negative as in Model B ( Y10 = -2.849, p < .001) thus indicating that the speed at which children solv ed problems increased linearly over the course of this study for all children by three seconds. For the random effects, the addition of the Level 2 income predictor reduced the within-p erson variance in hard problems from 2 =26.094, p < .001 in the UGM to 2 = 25.941, p < .001 in this conditional model. As with easy problems this was only a slight drop yet the propor tion was still statistically differe nt from zero thus suggesting that additional predictors may further improve model fit. The Le vel 1 random effect of time (1 2 = .043, p < .232) did not become significant with the addition of family income to the model. 68

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Both the between subjects and within subjects variance in Model C was reliably different from zero thus the inclusion of additional Level 1 a nd Level 2 predictors may improve model fit. Lastly, Model C fit the data for both easy ( (1) = 10.014, p < .001) and hard problems ( (1) = 11.282, p < .001) better than the UGM in Model B a nd also resulted in reductions in AIC and BIC. Family income did explain a significant amount of between and within-person variance for both hard problems. Given that the random effect of time was estimated in this model and a fixed main effect of family income was found it is highly likely that moderation of the rate of change by income may exist. Thus, the time by income interaction was tested in a final Model D. Moderation of the Rate of Change by Income The moderation of the linear rate of change over time by income interaction was examined in Model D for hard problems only given that the random effect of time was dropped from the easy problems. It was expected that the intera ction of time and family income would not only improve model fit but also provide a more clear description of how children s individual growth trajectories for solution time when problem solv ing changed over the course of the study. The interaction was created using th e product term of the interaction and then regressing out the unstandardized residual. Moderati on by income would exist if the linear trend in the change over time in speed of problem solving differed for low and middle-income children; however the interactions for both fixed and random effects in Model D on hard problems were not significant. The model would not converge when the residualiz ed interaction term wa s entered in the random effects portion of the model most likely due to a lack of power gi ven the number of data points estimated. The model did successfully converge when the residualized interaction term was entered in the fixed effects portion of the m odel but surprisingly th e interaction was not significant. It is again highly likel y the lack of significance is relate d to a lack of power given the low sample size of this study. 69

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Summary of Findings Taken together the results found for the solu tion time variable partially support the first three hypotheses. The first hypothesis was genera lly supported given that children did reduce their solution times each year although as will the proportion solved variable there was not evidence of individual variation in growth trajectories over time. The fixed effects of time along with the lack of a random effect of time indicates that children differed in their their performance each year albeit the manner in which they cha nged appeared to be si milar and did not differ according to problem difficulty level. All childr en changed at the same rate such that higher income children began the study solving problems fa ster than low-income children and this rate of performance was maintained throughout the st udy, even with maturation effects. Childrens individual growth trajectories did not reveal that some children cha nged at a faster rate than other children. It was expected that the income by time interaction would indicate that this was not the case but rather middle-income children showed greater improvement over time when compared to low-income children but unfortunately the fixed effects of the interaction term (Model D) were not significant and the random effects c ould not be estimated. The second hypothesis was also supported by these nested models given the main effect of family income. Middle-income children solved problems faster at the first testing session though results from the conditional growth model in Model C indicate they change d at the same rate as low-income children throughout the course of the study. Thus, middl e-income children began the study solving problems faster than low-income children and th is speed difference was maintained each year. Lastly, the third prediction was also supported given that low-in come children took longer to solve both easy and hard problems. Similar to th e findings from the propo rtion solved variable, differences were smaller than expected yet the income coefficient for solution time showed an 70

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income level performance gap given that a larg er difference between the income groups was found for the harder problems. Describing and Predicting Change in the Efficiency of Moves Made The Unconditional Means Models The UMM (null model) was first tested for th e efficiency of moves measure in Model A with results for easy problems shown in Table 48 and results of hard problems shown in Table 4-9. Model A proved that forcing the data to show no change ove r time resulted in a poorly fitting model for both easy and hard problem s. The random intercept for hard problems approached significance ( p < .06) but was not truly significant thus the random effect of intercept was dropped from the model for hard problems such that results include only fixed effects. The grand mean (a fixed effect) for easy problems was Y00 =.656, p< .001 and was Y00 = .393, p< .001 for hard problems. Essentially 85% of childrens moves made on easy problems were optimal (got them closer to a problem solution) and 70% of moves made on hard problems were optimal. The more important effects for both models were found in the random effects. For both easy and hard problems, the estimated within-person variances were 2 =.032 and 2 =.040, respectively, both ps <.001. The significant variances indicate that time could be a significant factor in efficiency scores for both easy a nd hard problems. The estimated between-person variances also differed reliably from zero for the easy problems 0 2=.012, p <.003 thus suggesting significant individual di fferences in the extent to which time influenced the move efficiency scores. Given that the models incl uded statistically significant within and betweenperson variance when time was not included as a variable it was exp ected that additional predictors at both Level 1 and Level 2 of the UGM would improve the fit of each model. 71

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The ICC was again calculated to better unde rstand the magnitude of the variances. The ICC for the move efficiency score on easy problems was = .952 and for hard problems was = .908. The ICC results reveal that approximately 95% of total variation in correctly solved easy problems and 90% of variation in correctly solved hard pr oblems was due to individual differences between children. The ICC is also used as a measure of the average autocorrelation or stability of the dependent va riable over time. Thus, the estimated average stability for easy problems was .95, while the estimated stability for hard problems was .90. Clearly more stability was found in the easy problems when comp ared to hard problems over time. The Unconditional Growth Models Next, the UGM (growth models) for the move ef ficiency scores were tested as Model B. Results of Model B are shown in the Model B column of Table 4-8 for easy problems and Table 4-9 for hard problems. No estimate could be calcul ated for individual differe nces in the rate of change, consequently the random effect of time wa s then dropped from both the easy and hard models such that results include only fixed effects of time. The positive linear effect of time indicated that children did improve their move ef ficiency scores over the course of the study for both easy and hard problems. For easy probl ems, the intercept (initial status) was Y00 = .399, p < .001 and the slope, or rate of change, was Y10 =.038, p < .001. The significance of the intercept and slope indicate that on aver age 70% of moves made on easy problems were optimal and thus took them closer to the goal in the first te sting session. Children im proved their proportion of optimal moves by approximately 2% each year. The Level 1 residual variance measures the overall scatter of childrens data around their change trajectory. The inclusion of time in the model increased the residual variance from 2 =.033, p < .001 in Model A to 2 =.042, p < .001 in Model B. Even with the slight increase th is within-person varian ce was still significant 72

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with the addition of time thus indicating that th e addition of other predic tors may explain random effects left in the model. Given that the Level 2 intercept and slope for easy problems remained significantly different from zero with the inclusion of time in the model, as did this within-person variance, it was expected that additional predicto rs would help to improve model fit and explain fixed effects left in the model. Results for the hard problems showed the same pattern as performance on the easy problems and the same pattern found for hard pr oblems on the solution time variable. It was expected that children would have difficulty solving the harder probl ems and thus make less efficient moves while problem solving and the MLM results confirmed this hypothesis. The intercept (initial status) for hard problems was Y00 = -.187, p < .001 and the slope, or rate of change, was Y10 =.085, p < .001. The negative intercept indicate s that children were making more non optimal moves away from the goal and thus so lving hard problems in a less efficient manner than easy problems. Essentially only 40% of childrens moves were optimal for hard problems when tested in the first testing session. The positive slope indicates that children improved the proportion of optimal moves by approximately 4% each year. The inclusion of time in the model reduced the residual vari ance a great deal from 2 =.042, p < .001 in Model A to 2 = .029, p < .001 in Model B. As with the easy problems, the Level 2 within-per son variance remained significantly different from zero, which indicated that additional predictors would help to improve model fit and explain fixed effects left in the model. Fit statistics of the UMM and the UGM in bot h models for the move efficiency variable were statistically compared to determine if the time variable as a predictor significantly improved the fit of the model. For the easy pr oblems, the inclusion of time resulted in a significant reduction in deviance ( (1) = 8.504, p < .001), along with reductions in AIC and 73

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BIC. Similarly, the hard problems also result ed in a significant reduction in deviance ( (1) = 114.426, p < .001) AIC and BIC. Clearly the addition of tim e resulted in a greater improvement in fit and explained much more within-person va riance for the hard problems as compared to the easy problems. Taken together, the fixed effects of time along with th e lack of a rand om effect of time in both easy and hard models indicates that children did improve their move efficiency scores each year though childre n did not show a significant di fference in their individual variation in the rate at which they improved their scores each year. Main Effects of Family Income The main effects of family income on the move efficiency measure were tested in Model C for both easy and hard problems. Results of Model C are shown in the Model C columns of Table 4-8 and Table 4-9. The in tercept for easy problems was Y00 = .369, p < .001 which indicates that in general 68% of moves made on easy pr oblems brought low-income children closer to a solution at their first testing session. The slope or rate of change for easy problems ( Y10 = .039, p < .001) indicated that on average childrens optimal moves increased by 2% each year. The intercept for middle-income children may be calculated by adding the income coefficient ( Y01 = .060, p < .008) to the intercept thus revealing that middle-income children made on average 72% of moves that were optimal at their first test ing session. Family income thus had a significant positive effect on the move efficiency scores of easy problems solved such that middle-income children solved problems more efficiently at the first testing session than low-income children. The linear effect of time remained statistically significant and positive as in Model B thus indicating that childrens move efficiency scores increased over the cour se of this study even after income was accounted for. For the random e ffects, the addition of the Level 1 predictor reduced the within-person va riance in easy problems from 2 =.042, p < .001 in the UGM to 74

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2 =.041, p < .001. Although this was only a slight dr op, the proportion was still statistically different from zero thus sugges ting that additional pr edictors may further improve model fit. Because the UGM had no Level 1 predictors, it was not surprising that the inclusion of income at Level 2 did not reduce the variance in rate of ch ange and because the variance in Model C was still reliably different from zero the inclusi on of additional Level 2 predictors may improve model fit more. Family income also had a significant positive effect on the move efficiency score on hard problems ( Y01= .051, p < .007) such that middle-income children made more efficient moves when solving hard problems at the first testing session. The intercept ( Y00 = -.212, p < .001) indicated that on average only 38% of moves made by low-inco me children on hard problems were optimal while 42% of the moves made by middle-income children were optimal moves at the first testing session. The rate of change remained significant ( Y10 = .085, p < .001) thus indicating that children made 4% more optimal moves each year of this study. For the random effects, the addition of the Le vel 1 income predictor also di d not change the within-person variance in hard problems from the UGM ( 2 = .029, p < .001). Again given that this proportion was still statistically different from zero it is possible that additional predictors may further improve model fit. Lastly, Mode l C fit the data better for both easy ( (1) = 7.061, p < .001) and hard problems ( (1) = 7.271, p < .001) than the UGM in Model B and the addition of income also resulted in re ductions in AIC and BIC. Summary of Findings In sum, modeling results from the move effi ciency measure followed the same pattern as the other TOL measures and part ially supported the first three hypot heses. It was first predicted that childrens executive functioning skills would improve each year as they matured and that 75

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these differences would be more drastic for some childrens individual grow th trajectories when compared to others. Childrens move efficiency did improve each year with maturation yet, contrary to expectations, signifi cant differences were not found in relation to individual growth trajectories. The random effects of time had to be dropped from modeling, most likely due to a lack of power, and as such individual differences in trajectories could no t be examined. In terms of the second hypothesis, family income explained a significant amount of between and withinperson variance for both easy and hard problems. Middle-income children made more efficient moves at the first testing sessi on and results from the conditio nal growth model in Model C indicate that there was no significant evidence to suggest that the rate of change over time differed from that of low-income children. T hus, the same pattern was found for the move efficiency measure as the other TOL measures examined with the exception of hard problems on this move efficiency measure. Middle-income children began the study solving with greater move efficiency than low-income children and the level of this performance difference was maintained each year. It was expected that the m oderation of the rate of change by income would be further examined in a final Model D; however once again the lack of a random effect of time prevented testing of the time by income interaction. Lastly, the prediction that low-income children would perform similarly to middle-in come children on easy and more poorly than middle-income children on hard problems was not supported. Low-income children actually made less efficient moves than middle-income ch ildren on all problems regardless of difficulty level. Unlike the other variables, the perf ormance gap was similar for both easy and hard problems. Mediation of Academic Achievement The final research question regarding the role of executive functioning skills in the relationship between family income status a nd childrens academic achievement was examined 76

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using mediation analyses. It was expected that executive functioning performance would mediate or explain the relationship between family income status and their readi ng and also their math achievement. Results for reading achievement are presented first followed then by math achievement. Results for the second testing session (the first time the reading and math scores were available) will be presented in Table 4-10 (easy problems) a nd in Table 4-11 (hard problems). Results for the final session will be presented in Table 4-12 (easy problems) and in Table 4-13 (hard problems). As de scribed earlier, the general test for mediation first involved a series of bivariate correlations with the goal of establishing significant relationships between the criterion, predictor, and mediator variables. If all three relationshi ps were found to be significant, hierarchical regressions were then used to esta blish a mediation effect and the Sobel test was conducted to examine the degree of me diation (full versus partial). The three TOL performance variables, as measures of executive functioning, were the mediator variables, family income status was the predictor variable, and childrens reading and math scores were the criterion variables in this study. Mediation analyses were conducted for the last two testing sessions only si nce formal reading and math testing was not conducted by the school board in kindergarten. To begin, bivariate Pearsons correlations we re conducted for all of the variables to be used in medi ation analyses: family income st atus, the TOL variables, and the academic achievement variables. Measures that me t the criterion for mediation were then entered into hierarchical regression analyses. Regression results indicative of mediation were evaluated using the Sobel test to determine if partial or full mediation existed. Pearson correlations were conducted separately for each of the three TOL va riables for both easy and hard problems using reading and math data from both the FCAT and the Stanford 10 tests for the last two years of the study. Results for the FCAT and Stanford 10 were virtually identical so for the sake of 77

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parsimony only Stanford 10 results are reported here given that more children had these scores available Mediation Results for Reading Achievement Correlation results for reading achievement in the second testing session revealed a similar pattern for each TOL measure a lthough only the hard problems on the move efficiency measure met the mediation criteria (see Table 4-11). Correlations for ea sy problems on all three TOL measures indicated significant relationship s only between reading and family income ( ps <.05). Correlations for hard problems showed the same pattern for the solution time measure but the proportion solved measure indicated significance in the relationshi p between family income and reading along with the relations hip between reading and TOL performance. The move efficiency measure (hard problems) was the only measure met the correlation criteria for mediation analyses in the second testing session. The hier archal regression performed indicated that executive functions did in fact partially medi ate the relationship between TOL performance on this move efficiency measure and reading achievement (see Table 4-14 for regression coefficients). The Aroian version of the Sobel test confirmed partial mediation (z = 1.90, SE= 2.54, p< .05) revealing that the relationship betw een family income status and reading achievement was mediated by childrens efficiency of moves when problem solving (see Figure 4-4). Correlation results for reading achievement in the final testing session did not follow the same pattern as in the previous testing session (see Tables 4-12 and 413). Correlations for both easy and hard problems indicated significant re lationships only between reading and family income ( ps <.05). The exception to this was the hard problems on the move efficiency measure. Results on these problems also showed that TO L performance and income were significantly 78

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related. The lack of a significant relationship between TOL perf ormance and reading prevented testing of the mediation models. Thus, none of the measures for reading in the final session met the mediation criteria. Generally speaking, the reading achievement mediati on results were somewhat surprising. Mediation was not expected to be found for read ing achievement simply based on the fact that executive functioning has been more closely rela ted to mathematical abilities than reading abilities in the literature. It wa s anticipated that none of the TO L variables would be significantly associated with reading and thus would not meet the criteria for mediation, however as reported above this was not the case with the move efficiency measure. Mediation Results for Math Achievement The correlation results for math achievement differed from the pattern found for reading achievement. As expected, all TOL measures were significantly related to math achievement at both sessions for easy and hard problems w ith the exception of the easy problems on the proportion solved measure in the final session (see Tables 4-10 and 4-11). Correlations for hard problems on the move efficiency measure for both years indicated significance between all three variables and thus met the mediation criteria. It was expected that propor tion correct and solution speed would have also met the mediation criteria but this was not the case and as such mediation models were only conducted on the move efficiency measure. The hierarchal regression performed for the second testing session indicated that performance on this move efficiency measure mediated the relationship between fami ly income status and math achievement (see Table 4-14 for regression coefficients). The Aroian version of the Sobel test confirmed mediation (z = 1.96, SE= 2.11, p< .05) revealing that rela tionship between family income status and math achievement was mediated by childrens efficien cy of moves when solving hard problems (see 79

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Figure 4-5). The hierarchal regr ession performed on the move e fficiency measure in the final testing session also indicated that childrens e fficiency when solving hard problems mediated the relationship between family income and math achievement (see Table 4-14 for regression coefficients). Interestingly, the Aroian version of the Sobel test confirmed that mediation was actually not achieved (z = 1.80, SE= 1.53, p< .07). The Sobel test was close to significance however results do not permit us to c onclude that media tion occurred. As with the results for the mediation of reading achievement, the results for math achievement were not as expected. Each TOL variable was expected to meet mediation criteria and as such executive functioning performance on each TOL measure was expected to mediate the relationship between family in come status and childrens math achievement but this was only the case for the move efficiency measure. In both the second and fina l testing sessions the income and math relationship and the math and TOL relationships were significant for both easy and hard problems on all of the TOL measures (with the exception of easy problems on the proportion solved measure). In all of these it wa s the lack of a signifi cant relationship between income and TOL that prevented mediation. Review of the data suggested a lack of power given the floor effects found on the most difficult proble ms for low-income children. Missing data was handled by employing list wise deletion in the correlation analyses and thus the number of subjects who contributed data for analyses was greatly diminished. Most likely with larger Ns this final correlation would have been found and mediation achieved. In sum, mediation effects for both reading a nd math in the second testing session and for math in the final testing session were limited to hard problems on the move efficiency measure. Each regression model indicated that the rela tionship between family income status and academic achievement was explained by efficien cy when problem solving, however mediation 80

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was only confirmed in the second testing session. Thus, executive functioning as measured by move efficiency mediated the e ffect of family income on both r eading and math achievement in the second testing session. Table 4-1. Multilevel m odel building steps Nested Steps ModelVariables addedDescription Unconditional Means Model (UMM)ANone Null model Unconditional Growth Model (UGM)BTime Examines change over time in DV without inclusion of predictors Conditional Intercept Model CIncome Examines main effect of income on change over time in DV Conditional Intercept and Slope ModelDIncome X Time Examines income by time interaction as a predictor of change in DV Note. Level 1 and Level 2 submodels are included w ithin each nested model. These aspects of the models estimate the within and between-pers on change over time as a component of each modeling step. Table 4-2. Means and standard devi ations for easy problems on the TOL Year 1 Year 2 Year 3 N Mean Std DevN Mean Std DevN Mean Std Dev Proportion Solved Low-income 940.78 0.18430.870.12460.93 0.09 Middle-income 690.820.16 36 0.910.10 43 0.940.09 Solution Speed (s) Low-income 9518.73 6.134315.445.794612.61 5.19 Middle-income 6917.145.79 36 13.775.45 43 11.174.37 Move Efficiency Low-income 950.59 0.25430.670.20460.71 0.15 Middle-income 690.660.21 36 0.710.18 43 0.740.14 Table 4-3. Means and standard devia tions for hard problems on the TOL Year 1 Year 2 Year 3 N Mean Std DevN Mean Std DevN Mean Std Dev Proportion Solved Low-income 950.50 0.21440.750.20460.86 0.14 Middle-income 690.510.22 36 0.810.17 43 0.910.12 Solution Speed (s) Low-income 9530.73 4.574425.605.264621.68 6.07 Middle-income 6929.044.77 35 24.444.96 43 20.036.54 Move Efficiency Low-income 950.27 0.19440.440.15460.52 0.12 Middle-income 690.280.19 35 0.520.14 43 0.570.11 81

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0.40 0.50 0.60 0.70 0.80 0.90 1.00 123 Testing SessionProportion Solved Low-income Easy Middle-income Easy Low-income Hard Middle-income Hard Figure 4-1. Proportion Solved Simple Means 5.00 10.00 15.00 20.00 25.00 30.00 35.00 123 Testing SessionSeconds Low-income Easy Middle-income Easy Low-income Hard Middle-income Hard Figure 4-2. Solution Speed Simple Means 82

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0.20 0.30 0.40 0.50 0.60 0.70 0.80 123 Testing SessionOptimal Move Score Low-income Easy Middle-income Easy Low-income Hard Middle-income Hard Figure 4-3. Move Efficiency Simple Means 83

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Table 4-4. Results of model tests for easy problems on the proportion solved measure ParametersModel AModel BModel CModel DFixed Effects Initial Status InterceptY00.849*(.01).522*(.04).503*(.04) IncomeY010.038*(.02)Rate of Change Time (linear)Y100.049*(.01)0.049*(.01) X IncomeY30Random Effects Level 1 Within Person s0.017*(.01)0.019*(.01)0.019*(.01)Level 2 In Initial Status 00.007*(.01) In Rate of Change 1 Fit Statistics Deviance-310.621-365.793-371.913 AIC-304.621-359.793-363.913 BIC-304.547-348.387-348.705 Note. *p<.001 84

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Table 4-5. Results of model tests for hard problems on the proportion solved measure ParametersModel A Model BModel CModel D Fixed Effects Initial Status Intercept Y00.667*(.01)-.177*(0.06)-.204*(.06) Income Y01.056*(.02) Rate of Change Time (linear) Y10.124*(.01).124*(.01) X Income Y30Random Effects Level 1 Within Person s .059*(.01).038*(.01).037*(.01) Level 2 In Initial Status 0 .004*(.01) In Rate of Change 1 Fit Statistics Deviance 27.299-147.291-154.093 AIC 33.299-141.291-146.093 BIC 44.723-129.866-145.971 Note. *p<.001 85

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Table 4-6. Results of model tests for easy problems on the solution time measure ParametersModel AModel BModel CModel DFixed Effects Initial Status InterceptY0015.791*(.36)29.019*(1.66)30.001*(1.66) IncomeY01-1.959*(.61)Rate of Change Time (linear)Y10-1.957*(.24)--1.973*(.24) X IncomeY30Random Effects Level 1 Within Person s31.470*(3.40)31.884*(2.47)30.936*(2.40)Level 2 In Initial Status 06.99*(3.05) In Rate of Change 1 Fit Statistics Deviance2147.1362091.5892081.575 AIC2153.1362097.5892089.575 BIC2164.5512109.0042104.796 Note. *p<.001 86

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Table 4-7. Results of model tests for solution speed on hard problems Parameters Model AModel BModel CModel DFixed Effects Initial Status InterceptY0026.396*(.36)45.875*(1.56)46.875*(1.57) IncomeY01-2.095*(.61)Rate of Change Time (linear)Y10-2.835*(.23)-2.849*(.22) X IncomeY30Random Effects Level 1 Within Person s42.73*(3.32)26.094*(2.58)25.941*(2.54)Level 2 In Initial Status 0.062(.04).043(.04) In Rate of Change 1 Fit Statistics Deviance2188.7842058.7602047.478 AIC2192.7842066.7602057.478 BIC2200.3942081.9802076.504 Note. *p<.001 87

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Table 4-8. Results of model tests fo r move efficiency on easy problems ParametersModel AModel BModel CModel DFixed Effects Initial Status InterceptY00.656*(.01).399*(.06).369*(.06) IncomeY01.059*(.02)Rate of Change Time (linear)Y10.038*(.01).039*(.01) X IncomeY30Random Effects Level 1 Within Person s.033*(.01).042*(.01).041*(.01)Level 2 In Initial Status 0.012*(.01) In Rate of Change 1 Fit Statistics Deviance-105.743-114.247-121.308 AIC-99.743-108.247-113.308 BIC-88.328-96.832-98.087 Note. *p<.001 88

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Table 4-9. Results of model tests fo r move efficiency on hard problems ParametersModel AModel BModel CModel DFixed Effects Initial Status InterceptY00.393*(.01)-.187*(.05)-.212*(.05) IncomeY01.051*(.02)Rate of Change Time (linear)Y10.085*(.01).085*(.01) X IncomeY30Random Effects Level 1 Within Person s.040*(.01).030*(.01).029*(.01)Level 2 In Initial Status 0.002*(.01) In Rate of Change 1 Fit Statistics Deviance-109.606-224.032-231.303 AIC-103.606-218.032-223.303 BIC-92.191-206.617-208.083 Note. *p<.001 89

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Table 4-10. Associations between academic data, income, and ex ecutive functioning in Year 2 on easy problems Reading Math Income SAT-10Income SAT-10 Easy Problems Proportion Solved SAT-10.50** .57** TOL 0.170.24 0.17.48** Solution Time SAT-10.50** .57** TOL -0.15-0.27 -0.15-.43** Move Efficiency SAT-10.50** .57** TOL 0.120.11 0.120.25 Note. **p<.001, *p<.05 Table 4-11. Associations between academic data, income, and ex ecutive functioning in Year 2 on hard problems Reading Math Income SAT-10Income SAT-10 Hard Problems Proportion Solved SAT-10.50** .57** TOL 0.16.31* 0.16.43** Solution Time SAT-10.50** .57** TOL -0.11-0.26 -0.13-.29* Move Efficiency SAT-10.50** .57** TOL .26*.3* .26*.39** Note. **p<.001, *p<.05 90

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Table 4-12. Associations between academic data, income, and ex ecutive functioning in Year 3 on easy problems Reading Math Income SAT-10Income SAT-10 Easy Problems Proportion Solved SAT-10.32** .53** TOL 0.04-0.12 0.040.19 Solution Time SAT-10.32** .53** TOL -0.15-0.13 -0.15-.37** Move Efficiency SAT-10.32** .53** TOL 0.090.07 0.09.25* Note. **p<.001, *p<.05 Table 4-13. Associations between academic data, income, and executive functioning in Year 3 on hard problems Reading Math Income SAT-10Income SAT-10 Hard Problems Proportion Solved SAT-10.32** .53** TOL 0.170.13 0.17.34** Solution Time SAT-10.32** .53** TOL -0.13-0.14 -0.13-.33** Move Efficiency SAT-10.32** .53** TOL .20*0.15 .20*.27* Note. **p<.001, *p<.05 91

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Table 4-14. Regression coefficients estimated in academic achievement mediation models for hard problems on the move efficiency m easure for the second testing session Reading Math NbSE NbSE Mediation Steps Second Session Step 19059.4311.20.49**9159.879.2.57** Step 2780.080.030.26*780.080.030.26* Step 342 43 Model 1 Income 67.5816.10.55** 60.8512.81.59** Model 2 Income 62.7217.610.50** 53.4313.62.52** TOL 41.5359.060.1 66.6545.50.19 Note. Step 1 = SAT-10 entered as criterion and fa mily income as predictor; Step 2= Move efficiency on hard problems entered as criterion and family income as predictor; Step 3= SAT-10 as criterion and family income entered as first predictor with move effi ciency on hard problems as the second predictor. **p<.001, *p<.05 92

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A Family Income Reading Achievement c = 0.30 B Executive Functioning Family Income Reading Achievement c = 0.13 a = 0.26 b = 0.50 Figure 4-4. Conceptual diagram of the medi ation analyses and standardized regression coefficients representing the estimated relationships for reading achievement in the second testing session (A) Direct effect of family income on reading achievement (B) Indirect effect of family income on reading achievement 93

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A Family Income Math Achievement c = 0.39 B Figure 4-5. Conceptual diagram of the medi ation analyses and standardized regression coefficients representing the estimated relationships for math achievement in the second testing session (A) Direct effect of family income on math achievement (B) Indirect effect of family income on math achievement Executive Functioning Family Income Math Achievement c = 0.14 b = 0.52 a = 0.26 94

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CHAPTER 4 DISCUSSION The developmental risks associated with economic disadvantage have been well documented, yet relatively few studies have ex plored the effects of poverty on the specific cognitive skills involved in executive functioning. Executive functioning skills are critical for learning in academic environments and as such delays in the development of these basic cognitive processes may place children from lowincome families at an increased risk for academic failure and high school drop out, which limit their successes later in life (Buckner et a., 2003; Stevenson & Newman, 1986). This study was de signed to evaluate the effects of income on multiple aspects of executive functioning with th e hope of better targeting intervention efforts geared towards children at risk for developmen tal delays in these skills. This study expanded previous research by examining performance on multiple aspects of a complex executive functioning task, the TOL. These aspects in clude executive functions such as success, efficiency, and timing, as were identified by (Berg & Byrd, 2002). The major focus of this research was to thus examine the effect of family income level on the developmental trajectories of the executive functioning skil ls of children using growth curve modeling. A secondary goal of this research was to determine the role of executive functioning skills in the relationship between family income status and childrens academic performance. Findings indicate that family income differences were associated with disparities in perfor mance on each measure of executive functioning over the c ourse of the study. Thus, low-income children were found to have poorer planning abilities than middle-income children and results suggest that the construct of planning may in fact have a slower deve lopmental time table than other components of executive functioning. Furthermore, childrens e fficiency when problem solving was found to mediate the relationship between their family in come and their reading and math achievement. 95

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Specific findings are discussed in detail belo w in two sections: (1) a discussion of results from growth curve modeling and (ii) the mediatin g role of executive skills in the relationship between family income status and academic achievement. Study limitations and future directions are then discussed fo llowed by a final section that pr ovides implications of the study and overall conclusions. Findings from Growth Curve Modeling The Effects of Family Income on the Deve lopment of Executive Functioning Skills Planning is a critical aspect of executi ve functioning skills that involves multiple components such as attention, inhibition, work ing memory, and strategy use. Although a few recent studies have begun to explore the developm ent of planning in children, a careful review of the literature revealed that rese arch specific to low-income childr ens planning abilities is not yet available. Given the variety of components of executive skills that have been shown to be affected by SES, the goal of this study was to extend the existing literature by examining the effects of family income status on planning. Growth curve models allow researchers to study the shape of developmental trajectories while testing the effect of predictors on the rate of growth. These analyses examine the impact of predictors on initial and ending values of trajectories and essentially answer questions about developmental timing that traditional statistical methods can not address (Singer and Willett, 2003). The findings from the sequence of models tested for each measure of executive functions in this study are best summarized by the final models. Final models show similarities in the trajectories and predictors of each construct (see Tables 4-1 through 6-2). Linear change trajectories were found for the measures of proportion solved, solution speed, and move efficiency. This was coupled with the finding that family income status predicted childrens performance on both easy and hard problems for each measure. Highly significant SES differences were found on all measures of executive 96

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functioning and were maintained throughout the study. Low-income children solved fewer problems correctly and the problems they did solve correctly were solved less efficiently and these children needed longer to solve the problems when compared to middle-income children. Low-income children began the study performing worse on all three measures at the ini tial testing session and this performance gap was maintained throughout the course of the study. Low-income childrens executive functioning skills did improve with matu ration in that they showed performance gains each year, but their performance never reached th at of the middle-income children. Additionally, the effect of family income was much stronger for hard problems than easy problems. Results from the final models the current study are thus consis tent with existing research and confirm that low-income children have less developed executive functioning skills when compared to average and high-income children (Davis & Gi nsburg, 1993; Noble et al., 2005). The impressive pattern of results found for each of the three measures of executive functions are in line with the few studies that have examined the effects of family income on other executive skills. The prefrontal brain region involved in executive functions undergoes a prolonged period of development which provides an extended opportunity for the different environmental experiences of low versus middle-income children to influence development (Farah et al., 2006). The findings of the present study support existi ng work revealing that the most robust SES disparities are found in the executive component s of working memory and cognitive control (Farah et al., 2004; Noble et al., 2005). Lowincome children in the present study performed worse in terms of overall success when problem so lving and the efficiency of the moves they made. Success on the TOL is dependent upon efficient strategy use which involves holding plans in working memory and the ability to resist th e urge to move balls into what looks like a good move for the moment, but in actu ality would take one farther fr om the goal. For instance, an effective strategy for solving more difficult problems requires children to decide not to move a 97

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ball into an obvious position that matches a goal ball but rather to temporaril y place the ball on a different peg where it remains out of the way while other balls are manipulated. This particular strategy requires a great degree of inhibitory control as children resist the obvious move and it also depends on working memory while sequentia l moves are planned. Similarly, the ability to solve problems correctly in the fewest numbe r of moves within the sixty second time frame, requires a great deal of cognitive flexibility and self regulation. Children must not only plan the most efficient strategies effectively but also atte nd to the task at hand and make use of multiple cognitive functions within a short time frame. The longer the childrens problem solving session continues the more likely it is that fatigue will de teriorate their sustained attention and control. This is supported in the literature related to the effect of income differences on childrens executive skills. Economically disadvantaged ch ildren are at a greater risk for poor self regulation abilities and, as a resu lt, basic cognitive functions such as cognitive flexibility, effortful control, and executive attention are delayed when compared to children from higherincome families (Blair, 2003; Buckner et al ., 2003; Chang & Burns, 2005; Kishiyama et al., 2009; Mezzacappa, 2004; NICHD, 2003). This combin ed research thus explains the strong SES effects found in the final growth curve models. These modeling results only partially supported the hypothesis related to income differences in that low-children did perform more poorly overall when compared to middleincome children yet significant income differences were not f ound for the rate at children improved their executive functioning performance. It is quite possible th at low-income children in this study were less able to remain focuse d on the task and less able than middle-income children to inhibit initia l responses to problems which would t hus explain their in ability to reach the same performance levels on difficult problem s. Previous analyses of the present data 98

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conducted only on the first testing session (DeLu cca, 2007) revealed that low-income childrens performance was greatly diminished by the third set of problems when compared to the middleincome children who improved with each set of problems. Results for that study demonstrated that low-income children lacked the same levels of sustained attention as middle-income children which resulted in them being less focused at the end of the task. This conclusion was confirmed with analyses of the heart rate variability data (DeLucca et al., 2006). Given this, it is very possible that for this study, the significant income effects found in the growth models were at least in part related to this l ack of sustained attention. Although the three measures of executive functioning were derived from a planning task, the measures of general success, speed, and efficiency each involve multiple components of executive skills. The pattern of results found for the success and efficiency measures may very well be explained by the literature focused on the attention and self-regulatory aspects of executiv e functions. Based on this it is reasonable to expect that the lower level of executive functioni ng skills that low-income children seen initially occurred because they are less able to remain focu sed on the task and as a result are less able to develop efficient strategies than middle-incom e children. Similarly, chil drens performance on the solution speed measure may be explained in terms of planning a nd inhibition abilities. Effective planning and strategy use should transl ate to fewer moves being made and less time needed to correctly solve problems, which is exactly what was found for middle-income children. Conversely, children with well developed inhibitory skills should be able to inhibit the urge to move balls until they have developed an effective strategy for solving problems. Such skills would then be reflected in the move efficiency score in that failure to inhibit a non optimal move which would take the child away from the goal position would re sult in a lower move efficiency score. Thus, in addition to planni ng, not only strategy selec tion but also inhibition 99

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may well have played a role in the pattern of results. Although inhibi tion is distinct from attention, it is evident that less inhibitory control may result in less task focus and thus the interaction of the two skills may also account for the varied performance of lower-income children (Bull, Espy, & Senn, 2004). Developmental Trajectories of Executive Functioning Skills The developmental trajectories of specific executive functioning skills are a major focus of the cognitive development literature, however research on the specific trajectories involved in the multistage development of executive functions have yielded inconsistent results. The inconsistencies in the literature make sense when the individual nature of executive skills and the plethora of factors that combine to affect development are considered. Clearly the development of executive functioning is one of the domains for which longitudinal data and growth curve analyses are important. With the plethora of factors that may impact the course of developmental trajectories these sophisticated statistical analyses can paint a clearer picture of how development is impacted by SES. The period of development selected for this study was expected to capture a time when income effects on planning may be the most pronounced and have largest impact. Growth curve models simultaneously exam ined two aspects of the developmental trajectories of executive functions. In addition to examining SES diffe rences in childrens rate of change, analyses examined both fixed effects (the overall effect averaged across all subjects) and random effects (an estimation of the size of individual variation of the effect) of how childrens executive functioning trajectories differed by family income. It was unclear how low-income childrens development of executive functions would compare to middle-income childrens development. It was expected that income differences would be found in childrens rates of change such that middle-income children woul d improve faster than low-income children; however this was not the case. Childrens ex ecutive functioning skil ls did improve with 100

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maturation but generally speaking, all children cha nged at the same rate given that no individual variation in rates of change were found for any of the executive function measures. In other words, low-income childrens rate of change did not fall behind that of the middle-income children nor did it catch up. It is po ssible that there simply are no income differences in the rate of change of executive skills but it is expected th at the lack of statistical power that resulted from the small sample size prevented the random effects of time from being estimated. The linear developmental trend of growth found for all children was particularly interesting. Existing research repor ts a rather rapid development of most executive skills during early and middle childhood however findings from this study are in star k contrast (Anderson, 2002; Garon et al., 2008; Lehto et al., 2003). Si gnificant change and thus growth in the development of executive skills was found but this change was minimal which was rather unexpected. This pattern was cons istently found for developmental trends on all three executive functioning measures. Children solved problems only a few seconds faster each year and made a small percentage of more efficient moves each year which resulted in them solving on average between only one and two additional problems w ith each testing session. The minimal rates of change are suggestive of a slow er maturation for planning abi lities than othe r aspects of executive functioning, a finding that is supported by existing research. Researchers within the NICHD Early Child Care Research Network (200 3) found that the quality of the early home environment was not related to childrens planning abilities in first grade and suggested the late biological maturation of planning as an explanation. Huizinga et al (2006) determined that planning abilities measured by th e number of additional moves a nd the time used to make the first move on the TOL did not reach maturation until age 11. When these investigators looked at the number of perfect solutions on the TOL, maturation actually did not occur until adolescence. 101

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Together these studies suggest that the specific skills utilized when solving TOL problems such as inhibition, working memory, attention, strategy use, and cognitive flexibility may develop rapidly in early childhood as previous research has evidenced but that the skills do not become fully integrated as a unified construct of planning until much later in development. Some proponents of the unified and dive rsified nature of executive skill s have argued that executive functions may not actually be structured the same way during the preschool period as they are during later childhood. Miyake et al (2000) attributed the overlapping variance of executive components to common processes such as atte ntion. Thus, given what is known about the development of executive functions in general and the findings of this study, it is quite possible that executive skills involved in planning are not yet fully integrated by this stage of development. The use of latent growth curve mo deling in the future w ould provide interesting insight into this speculation. Lastly, it is important to note that the linear effects of time discussed above should be interpreted with caution. Non linear effects could not be exam ined within this study although it is highly likely that non linear change may exist and would have been detected had more time points been included. Similarly, an interesting re search question could have been addressed had children been tested over a longer developmental period. It is possible low-income childrens performance may have eventually caught up to middle-income childrens with time and it is equally likely their performance may have diverged with greater income differences found. It is expected however that regardless of the effects of income, eventually developmental effects would asymptote for all child ren (Garon et al., 2008). Income Level Gap in Performance on Hard Problems Largely as predicted, low-income children performed more poorly than middle-income children on the easy problems for each measure of executive functioning; however there was 102

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very little difference in these performance le vels (e.g., children solved between one and two more problems correctly with each test session). This was not the case for difficult problems as the income coefficient revealed a greater disparity between the groups on more difficult problems. For instance, the increas e in the income gap was a little less than two times that of the easy problems on the proportion solved variable. In general, low-income children were unable to maintain a similar level of performance on both the proportion solved and solution time measures. Although low-income children showed im provement in their performances with each year, their performance was never fully comparab le to their more affl uent peers on difficult problems. All children in the study solved fewe r hard than easy problems. Although the lowincome children solved significantly fewer easy problems correctly than middle-income children, their performance was comparable to the middleincome children in that both income groups solved approximately half of the easy problems. The unconditional growth model (Model B) indicates that all children solved on average half of the easy problems yet virtually no hard problems were solved correctly. This difference in performance for easy and hard problems was explained when income was entered as a predic tor in the final growth model. Middle-income children were able to solve on average a few of the hard problems correctly while the lowincome children were on average unable to solve the majority of hard problems. Thus it was the low-income children who had the most difficulty solving the hard problems correctly and the final model made clear that the floor effect on the hard problems found in Model B was a resulted mostly from the low-income children. It is interesting that this gap in performance was not apparent for the easy problems but became evident for the hard. The hard TOL problem s were much more difficult in that they required increasingly more moves and complex pl anning and strategy development. Thus income 103

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had the strongest effect at th e problems that relied most heav ily on planning abilities, a finding supported by previous research. Wa ber et al (1984) determined that stylistic differences in childrens cognitive approach to problem solving are dependent on economic background. The authors showed that children from high-income families used the left cerebral hemisphere when problem solving which resulted in faster proces sing and the use of more analytical processes when compared to low-income children. Convers ely, low-income children were found to depend more on their right cerebral hemisphere while solving problems and as a result they solved problems slower, were more impulsive, and relied more heavily on global processes than highincome children. With these results in mind it is reasonable to expect that the performance gap found in this study on this measure of general success when problems solving resulted from differences in childrens overall stylistic approach to problems so lving. Such stylistic differences may elucidate why low-income children in our study struggled to solve very few hard problems correctly. Significant income differences were also found for hard problems on the solution speed and move efficiency measures. Results from these two measures can aid the interpretation of the results from the proportion solved measure. Low-income children did take significantly longer to solve problems and they made signific antly fewer efficient moves than middle-income children. They seemed to be w andering their way through the pr oblem searching for a solution (and therefore making many inefficient moves) wh ereas middle-income children clearly planned their moves more carefully resulti ng in more efficient moves and fa ster solutions. The end result was more effective information processing by middle income children and many more hard problems solved correctly. Thus, in line with the findings of Waber et al (1984), stylistic differences in problem solving provide a valid ex planation for this performance gap. The results 104

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suggest that low-income children in this study appeared to use a more global approach to problem solving whereas middle-income children a ppeared to use a more analytical approach. Why might the growth curve mode ls have given different results than are suggested by the simple means in the first year? The performance gap described above was perh aps the most interes ting result yielded by the growth curve models. This finding was rather unexpected given the simple means reported in Tables 4-2 and 4-3. Figures 4-1 through 4-3 portray the plots of the means for each measure of executive functioning in the first year of the study and the income level performance gap is not quite as evident and in some instances questionabl e. It is important to remember that estimates from the growth curve models are based on the assumption of a linear ti me trend. The intercept or initial level at the first testing session in the growth models is calculated using the slope which is based on all three years of data combined. T hus, the implications of the growth models for each measure are quite different than what is represented in the simple means tables (Tables 4-2 and 4-3). This discrepancy in the data suppor ts the value of using multilevel modeling over traditional statistical analyses because it tells a clearer story that would not have been evident otherwise. It could be argued th at the simple means better repr esent change over time and that the MLM approach takes away from the actual data. The MLM approach actually gives a more general prediction of the result s over time as it imputes the missing data for children. For the simple means this developmental data would be lost or those subjects with missing data (104 children) would be excluded from the analyses. Given that the growth curve models are based on the assumption of best linear fit it is surely possible that the linear fi t is not the best representation of the data. The non linear trends suggested in Figures 4-1 through 43 suggest this may be the case, however in the case of growth curve modeling calculations more than three time points of data are necessary to statistically 105

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model curvilinear functions. Either way, the findings from growth curve models raise an interesting question of a possible performance ga p on the hard problems that would not have been evident using means alone. Also, though it would have been preferable to have had fewer subjects lost to attrition, high drop out is extremely common within low-income samples. The question for the researcher then becomes difficult: Is it better to avoid data imputation and analyze the raw data that includes subject loss or is it best to use multilevel modeling and impute missing data? For this study, given the number of subjects who dropped out, the greatest statistical power was gained by estimating values for the subjects who dropped out. Nonetheless, results should be interpreted with caution given the discrepancy between model results and simple means from the first year of testing. Mediation of Executive Functions in the Relationship between Family Income and Academic Achievement Mediation analyses examined whether ex ecutive functioning was important in the pathway between academic success and family income Indeed, results of this analysis provided support for the idea that executive skills mediat e the relationship between family income and childrens reading and math achie vement. However, this support was evident only for the move efficiency measure and then only for more difficu lt problems. In contrast, evidence of mediation was not found for the proportion solved and solu tion speed variables. That is, instead of executive functioning performance on all meas ures acting as mediators, only childrens efficiency when solving problems mediated the relationship between childrens family income and their reading and math achievement for the second year of the study. The executive skills used to solve problems efficiently are likely to be among the skills critical for success in reading and math. Efficient problem solving is heavily dependent on several aspects of effortful control 106

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such as attention, inhibition, working memory, st rategy use, and planning. Children who have poor executive control have probl ems paying attention in class, inhibiting impulsive behaviors, and completing assignments correctly (Blair and Diamond, 2008). As was reviewed in the introduction section of this paper, children fr om low-income families enter school with less developed executive control and the mediation re sults from this study demonstrate that skills involved in successful planning when problem solv ing mediated the relationship between family income and childrens readi ng and math achievement. It is interesting that result s did not follow the expected pattern. Mediation was not found for the proportion solved and solution time measur es and mediation also did not occur in the final year of the study. Examination of the correla tion results showed that in most cases income was related to reading and math achievements and this achievement was related to TOL. The relationship that was not significant was the relationship between the TOL and income. One explanation for the unexpected resu lts is very simply that family income may have not played a mediating role. A more plausible caveat to accepting the null hypothesis that there is no mediation, however, is the critical role that missing data played in this study. Unlike with the modeling data, missing data in th ese relationships in the regres sion analyses were addressed using list wise deletion. The second and third te sting sessions clearly had significantly fewer children included in the sample. The amount of data included in the analys es was reduced further since children who did not solve pr oblems correctly did not receive a score when the statistical program excluded their data from analyses. Fort unately, the information gathered in this study from multilevel modeling can aid in the interpre tation of this finding. The lack of mediation results for the proportion solved and solution time variables may be considered in relation to the income effects found in the growth curve mode ls. As such, it seems most likely that the 107

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statistical lack in power is a vi able explanation for the inability to find significant mediation in most cases. This provides addi tional support for the use of multilevel modeling as the primary statistical approach for this study. In sum, mediation results from this study, to the extent they were found, are in line with previous work centered on the ex ecutive functions of children rais ed in poverty. A longitudinal study of low-income first through third graders s howed that effortful control predicted reading achievement two years later (Liew, McTigue, Barrois, and Hughes, 2008). Correspondingly, the relations between executive functions and the acqui sition of early mathematical skills have been documented in low-income children (Blair and Razza, 2007). Since the majority of existing research in this area has only been conduc ted within low-income samples, this study purposefully compared low-income children to th eir more affluent peers and as such made a significant contribution to existing research. The implications of these mediation results have implications for school readiness and early intervention which will be discussed in the conclusion section below. Critical Evaluation of Findi ngs and Future Implications A critical evaluation of the findings from this study will be discussed in terms of limitations and future directions. Suggestions for how the limitations may be addressed in future research will be discussed below. These limitations were related to attrition due to drop out, issues within the TOL task, and concerns with the design of the study. Implications and an overall summary and concluding remarks will follow this section. Attrition due to Drop Out Although the study of income ef fects on childrens development is crucial to expanding our understanding of development, longitudinal st udies that include low-income samples are rather difficult given that it is notoriously difficult to keep low-income children enrolled in a 108

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longitudinal study. High attrition rate s were expected for this study and to offset the attrition 30 additional low-income children were tested ini tially. Although almost equal numbers of low and middle-income children were enrolle d in the study at the final te sting session, attrition overall was much higher than expected. A greater num ber of both low and middle-income children dropped out for a variety of reasons but these reasons tended to differ according to family income status. The majority of middle-income children who did not remain in the study did so because their parents did not agree for them to be tested in follow up testing sessions or their family moved out of the area. The most common reason for drop out in the low-income sample was because the children simply could not be found for additional testing. Low-income families in this study moved within the area more freque ntly than middle-income families and as a result children transferred schools more often and possibly primary care takers as well. Preliminary analyses were conducted to show that at the initial testing session children who did not subsequently remain in the study were not significantly different from those who did complete the study. It is not possible, however, to know if the obtained sa mple is different from children who received recruitment letters for the initial session but did not participate. Recruitment efforts were quite different based on income. At the beginning of the study approximately 70% of children who received recruitment letters in middle-income schools participated compared to only 20% of those in low-income schools. In an effort to increase enrollment, an incentive of a free ice cream cup was offered to any child from a low-income school who returned consent forms, regardless of whether permission was received. This incentive increased the enrollment of low-income children to 50 percent. This is still well below the level for middle-income schools and demonstrates that despite the special effort with the low-income group, half of the eligible children were never enrolled. We would recommend, therefore, that when working with a low-income group in a longitudinal study, 109

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plan special incentives for the group, and then plan to recruit from a larger pool to account for reduced enrollment in the study. The lack of statistical power that resulted from missing data was mentioned several times previously. This loss of statistical power surely contributed to the failure to detect significant random effects of time in the growth curve models and thus to estimate the interaction between family income and childrens rate of change. Imputation methods were employed to account for missing data since the data loss was assumed to be for random reasons more than relevant systematic ones. However, the number of iterati ons involved in calculatin g random effects is far greater than those for fixed effects given that analyses of random aspects considers individual growth trajectories. Of course, it is quite possible that there simply is not be an interaction of family income and childrens rate of change In their study on the relationship between childhood attention problems and later executive functions, Friedman et al. (2007), found that differences in childrens attentional skills were stable over time and it was childrens initial levels of attention problems rath er than change over time that predicted later ex ecutive functions. The authors did not investigate family income as a predictor, however. Their results suggest that it is possible that childrens initia l levels of executive skills in this study may be more important to later functioning than the rates at which they changed over time. In any case, future research should employ the use of power analyses to determ ine the appropriate number of participants to include in the study so th at a loss of statistical power is not an issue. Advantages and Disadvantages Related to Using the Tower of London Task Although a great deal is known about the development of skills involved in executive functioning in childhood, there is limited knowledge of how the co mponents of executive skills function as an integrated and complex cogniti ve process. Planning involves multiple cognitive processes and as such the TOL task is consid ered primarily a planning task because success on 110

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the task is essentially dependent on the integr ation of attention, processing speed, working memory, strategy use, cognitive flexibility, and re sponse inhibition. The TOL is one of the most widely used tasks to assess planning and problem solving in adu lts and children in the last decade, and is a valuable tool to explore the effects of SE S. Perhaps one of the largest advantages of the TOL, especially for low-in come children, is that the task is minimally dependent on language proficiency or knowledge, so the language environment of the home or school is unlikely to be reflected in task performance. The TOL was selected for use in this study primarily because it allows for the exam ination of multiple com ponents of executive functioning within one task. Executive functioning researchers agree that one of the major problems with the assessment of executive skills is the so called task impurity issue. That is, that a task that may be identified as, say, testing working memory, may act ually require a variety of other skills as well as working memory. In fact task impurity is mo re likely the than not to be the case. The majority of research on executive functions in both children and adults employs multiple tasks (i.e., Stroop, WCT, Trails, etc) within one study yet most studies report very low correlations among tasks. Executive functions typically mani fest themselves by operating in conjunction with other cognitive processes and thus one exec utive task measures other cognitive processes that are not directly involved in the executive component of interest (Miyake et al., 2000). The task impurity issue is also complicated by the fact that most executive tasks yield low internal reliability most likely because individuals rely on different strategies to complete tasks in different situations and repeated exposure to a task reduces th e novelty of the task and greatly reduce the tasks effectiveness in measuring the intended execu tive component (Miyake et al., 2000). The use of the TOL greatly diminished th e task impurity issue within this study. 111

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However, it is somewhat of a trade off in that with the reduced impurity that results from the use of multiple measures come the issu e of mulitcolinearity am ong measures become a concern. Multicolinearity refers to a situation in which data from two or more variables are perfectly or near perfectly corre lated (Pedhazur, 1997). Since the measures in this study were obtained within the same task it was possible that mulitcolinearity would ex ist and bias results. Bivariate correlations of the pred ictor variables investigated in preliminary analyses revealed variables were correlated but not to the degree that required the data to be orthoganalized so that only the unique contributions of variables would be compared. Similarly, the fact that the income level performance gap was found for the measure of general success when problem solving but not for the more specific measures of speed and move efficiency provides a clear advantage for using the TOL to measure change in the development of executive functions. One of the unique advantag es of the TOL is the ability to measure both overall problems solving as well as more speci fic executive skills. The clear income level performance gap found for hard problems on the m easure of general succe ss at problem solving may not have been as evident had a more focused executive task been selected. Additionally, preliminary an alyses indicated that at least some of the executive functioning measures examined in this st udy were relatively stable over time. The intercorrelations confirmed by the ICCs in the un conditional growth models (Model A) indicated that the proportion solved and move efficiency m easures were comparable and much more stable over time than the measure of solution time. Th e poorer stability of solution time may well be attributed to the fact that ma turation is well known to have strong effects on motor speed, and if this rate of maturation varies substantially across participants, than it will reduce the stability of this measure. Differences were within each measure were found for difficulty levels with the 112

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hard problems on the solution time and move efficiency measures showing more stability over time than easy problems. This makes sense because all children consistently struggled more with the most difficult problems as compared to the easy problems. The proportion solved measure showed the opposite pattern in that the easy prob lems proved slightly more stable than hard problems. A second important decision was to collaps e the TOL data across the three sets of problems. This was advantageous as it allowed for a general overview of task performance to be examined but it was not without a cost. Important information related to th e benefit of repeated task exposure in childrens problem solving skil ls can be gleaned from examining performance across the three sets of data. Analyses conducted on the first year of this data as part of a preliminary investigation indicated income differences relate d to this repeated task exposure such that middle-income children were better ab le to learn from the first set of problems and greatly improved task performance. Low-income children also improved their performance but to a much smaller degree. Their performance at th e end of the task was far from comparable to the middle-income children and was shown to be re flective of differences in sustained attention (DeLucca, 2007). This data clearly has impor tant implications for childrens executive functioning capabilities in lear ning environments. Unfortunately, the decision to collapse over sets in this study prevented analyses of the effects of repeated task exposure over time. Future research that includes a much la rger sample may wish to exam ine childrens performance across sets as a third level of nested modeling. It is important to note that decisions made within this study regarding the handling of TOL data greatly impacted conclusions made regarding the development of executive skills over time. One of these decisions was to limit the study to just three measures of task performance. 113

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But the TOL can generate a plethora of data, espe cially when computerized, and it is possible to examine many more research questi ons with these data. Thus, another aspect of this study that may be extended in future research is related to the specific measures of executive functioning included within analyses. Initial analyses of this study examined childrens performance on seven available measures however on ly three were reported as part of this dissertation given time and space limitations. The decision regarding whic h measures to include has a bearing on an important general theoretical issue in the field of executive functioning research. This issue, perhaps the most debated issue within the executive functioning literature, is that of the unified verses diverse nature of executive functions that is discussed within th e introduction section of this paper. Recently several theorists have adopted the framework of Miyake et al (2000) that executive functions are both separable but integrated constructs. A few research groups have extended examination of this framework by testin g children and early adol escents (Huzzinga et al., 2006; Letho et al., 2003) however both have us ed the same constructs of working memory, inhibition, and shifting. As of yet, the latent c onstructs involved in planning have yet to be examined. An interesting extension of the current study would be to conduct latent growth curve modeling to examine what latent constructs of planning are found within the TOL and how the predictor of family income affects development of the constructs over time. Doing so would contribute to the debate regarding the unity a nd diversity of executive functions by providing insight on the unique nature of planning abilities. Design of the Study The longitudinal design of the study was of great importance to understanding the development of executive functions over time and is much preferred over cross sectional data analyses. However, there were improvements to our longitudinal design that would have added to its value. For example, the study was desi gned to measure childrens development over three 114

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testing sessions within early chil dhood but, as noted earlier, at the testing at just three time points did produce some limitations. Curvilinear effects of change over time within the MLM analyses may only be estimated with more than three waves of data. The inclusion of more waves of data would not only allow nonlinear rates of change to be evaluated but would also improve the precision of individuals rates of change. Thus additional waves of data points allow researchers to better judge if fluctuations within the data ar e due to true change or random error (Singer and Willet, 2003). Another design improvement would be to include additional age groups, both younger and older. Testing children at the start of preschool woul d provide valuable information on the development of planning as an indicator of school readiness. Similarly, research that extended the results of the exis ting study by examining children as they progressed in school and entered adolescence would add to the understand ing of the planning trajectory. Lastly, the examination of children twice a ye ar, once at the start of school a nd then again at the end of the school year would allow resear chers to determine the influe nce of schooling on low-income childrens executive functioning de velopment throughout each year. E ach of these suggestions is idealistic given the large investment of time a nd finances that would accompany endeavors such as these. A final limitation of the design of this study is related to the colle ction of demographic information. A more complete measurement of income variables such as background and contextual variables. This would have allowe d for a better understand ing of the ecological processes influencing family income and it effects on childrens executive functioning trajectories. This was in fact th e intent of the initial design of the study. That design involved the use of the Hollingshead Four Fact or Index (1975) as a measure of socio economic status that was accompanied by multiple background questions (i .e., measures of the stability the home 115

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environment, childrens enrollment in early in tervention programs, etc.). Because complete Hollingshead data was not available, these chal lenging circumstances prompted the adoption of childrens enrollment in the free and reduced lunc h program as a method of determining income status. Given the challenges associated with determining income status within low-income samples, this method is becoming more widely accepted (Blair, 2003;Waber et al, 2006). Some authors have even suggested this more simple approach to the measurement of SES as preferable given how outdated the Hollingshead scale is, whereas others have argued for much more precise measurements of economic well being such as an income to needs ratio (Cirno, Chin, Sevcik, Wolf, Lovett, Morris, 2002; Gottfreid, Gottf reid, Bathurst, Guerin, and Parramore, 2003; McLoyd, 1998). Future research should consider th e distinction between SES and income status and then employ the conceptual distinction that is of greatest importance to the research questions examined and overall goals of the study. Despite the limitations of using the school-based percentage of school lunch recipients, all of the random effects variance components in the final models for each measure were still significantly different from zero which indicated that the between-person predictor of family income reduced unexplained variance and improved m odel fit. The results from final models still suggest a need for additional predictors at both the within and between-person levels. To further understand the effects of income, receptive vocab ulary as measured by the PPVT was entered after income. The addition of this predictor di d not improve model fit and essentially removed the effect of family income (see results sect ion for detailed explanat ion), yet the additional variance was left begging to be explained. Thus, future research should examine additional predictors such as family background variables to further explain the de velopmental trajectories of executive skills. Contextual measures th at may explain aspects of childrens home 116

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environment such as (i.e., parent interaction, st ress level) would extend the contribution of this study. Such contextual data was co llected in the parent intervie w portion of this study however the data was not used in analyses given the lo w number of low-income parents who completed interviews. For future studies, the inclusion of a measurement of th e quality of the home environment (e.g., the HOME Observation Scale) would allow for more rich understanding of the effect of family income. Although a measure of the home environment could be interpreted as a measure of SES, it would allow more careful examination of the specific aspects of the lowincome environment contribute to the perfor mance gap. Existing research suggests parent education level, occupation, and other factors have all played a ro le in SES disparities while the quality of the home environment was not found to effect the development of planning. Lastly, future research could benefit in im portant ways by including physiological data to provide additional support of be havioral findings. Although child rens behavioral performance may appear to be comparable with changes in a condition, examination of physiological data can provide more comprehensive information which ma y indicate that despite this appearance, one child may have cognitively struggled much more than another to arrive at the same end result. As noted above, heart rate variabil ity or vagal tone is one commo n index of self-regulation and attention in children (Blair & Peters, 2003). Given that such da ta was collected for this study, additional analyses which examine the behavioral and physiological data together over time would provide valuable additi onal information related to low-income childrens executive functioning skills. Conclusion In sum, this study extended the existing re search by adopting a multifaceted approach to executive functioning using childrens performa nce on a planning task. The results of the current study shed light on three aspects of ex ecutive functioning performance in low-income 117

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children: general success at problem solving, effi cient use of moves when solving problems, and speed or time needed to correctly solve probl ems. The study examined childrens executive functioning by employing multilevel growth curve m odeling to provide an explanation of the effect of family income on the developmental trajectories of executive skills. Findings from this study demonstrated that family income signifi cantly affects the development of executive functioning skills in middle childhood. Low-income children have poorer executive functioning skills in kindergarten and, although they improved their executive function ing performance with maturation, they were not able to perform at the same levels as middle-income children. It is important to note that the income level performan ce gap in these deficits was larger for the more difficult problems on the measure of general succ ess when problem solving. Successfully solving a problem required the ability to develop and swit ch between new strategies and a great deal of inhibitory control. Thus when problem solvi ng involved the integration of executive functioning components in complex planning low-income child ren fell behind. Findings also suggested that individual children in this st udy changed at approximately the same rate over time. That is, individual variation in rates of change were not significantly pr esent which suggests that income influenced executive functioning performance through differences in performances levels for each measure rather than through differences in their individual rates of change. Thus, in line with previous research, results from this study confirm that low-income children enter schools without the same levels of executive functioning skills as their more affluent peers. Deficits in the executive functions of low-income children are not biological but rather a result of growing up in economically di sadvantaged environments (Waber et al., 2006). This study demonstrated that in some circum stances executive functioning, as measured by childrens efficiency when problem solving, was found to mediate the relationship between 118

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family income and their reading and math ach ievement. Variations in the early learning experiences provided in home environments pl ace children from low-income families at an increased risk for delays in the development of executive functioning skills (Waber et al., 2006). Numerous studies have demonstr ated that low-income children begin school at disadvantage with delayed cognitive skills and with less deve loped reading and math skills (Stipek & Ryan, 1997). This study not only confirmed the strong a ssociation between family income and reading and math skills but demonstrated that executive functions can likely mediate the relationship. That is, a critical aspect of school success is th e ability to plan effectively and these findings show that planning skills are less developed fo r low-income than for middle-income children. These results have important implications fo r early intervention programs targeted at lower-income children. Such programs should cons ider school readiness as encompassing not only academic cognitive factors su ch as letter and number knowle dge, but more basic cognitive skills, namely executive functions, which serve as foundational skills in academic environments. The preschool years are a time in which the pr olonged period of development for executive functioning skills is at an elevated period of vulne rability from the effects of poverty (Stipek and Ryan, 1997). These findings demonstrated a generally slower timetable for the development of planning abilities than previously expected. But there are probably multiple trajectories of planning depending on what aspect of planning the task focuses on. This, like the different trajectories of the executive functioning components allow for multiple impacts of family income differences to impact childrens development of these skills. Low-income children were found to lack the same levels of the executive skills involved in plan ning in kindergarten and the differences in performance were maintained th roughout the course of the study. Low-income children clearly lack experience with executive functioning skills befo re they enter kindergarten for variety of reasons and as a result they begin school at a disadvantage. To compound this, our study also provided 119

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evidence that low-income childrens performance did not catch up to that of middle-income children after they have had several years of schooling and thus suggests low-income children could benefit from additional practice with using these skills efficiently. The findings from this study have important im plications for intervention programs. Taken together, results support the need for early training of executi ve functioning skills which could help low-income children begin school at more comparable levels to middle-income children. Similarly, repeated practice with problems and situations requiring the us e of executive planning abilities in the early elementary years coul d help close the performance gap in academic achievement. Since low-income children are at an increased risk for deficits in these skills the results of this study suggest it is important that programs incor porate early training of planning skills. Planning is a foundational cognitive ski ll necessary for academic success. Children with poor planning abilities typically complete easie r problems on planning tasks, but struggle with more difficult, multi-step planning tasks (B yrd & Berg, 2002). Planning deficits can, for example, affect childrens abil ities to organize strategies for completing assignments, their ability to maintain attention th roughout task completion, and their ability to inhibit inappropriate behaviors. The inclusion of training which would teach children how to inhibit initial responses and plan effective strategies towards reaching goals would not only improve attention regulation but perhaps improve low-income children s overall potential for academic success. Although there is still much work to be done educational intervention programs geared towards these children have succeeded in narrowing the gap in achievement. The few intervention studies that have trained childre n in executive skills have reported marked improvement in childrens abilities (Dowse tt & Livesey,2000; Polderman, Posthuma, De Sonneville, Stins, Verhulst, and Boomsma, 2007; Thorell, Lindqvist, Bergman Nutley, Bohlin, and Klingberg, 2009). It is imperative that inte rventions focus on closing the gap in cognitive 120

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abilities that has been repeated ly observed in low-income childr en especially given the impact found for reading and math achievement. Interventions must target more specific cognitive abilities in order to be the most efficient. Given the results of this study a nd previous research, it is clear that the inclusion of ex ecutive skills training in early intervention programs would likely have far reaching implications for low-income children. Such programs would improve their intellectual development and thus give these children a better chance at academic success and improved opportunities in life. 121

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Blair, C. (2003). Behavioral inhibition and behavioral activa tion in young children: Relations with self-regulation and adaptation to preschool in ch ildren attending Head Start. Developmental Psychobiology, 42 (3) 301-11. Blair, C. (2002). School Readiness: Integrating Cognition and Emotion in a Neurobiological Conceptualization of Childrens Functioning at School Entry. American Psychologist, Vol 57 (2) pp 111-127. Bradley, R. H. and Corwyn, R. F. (2002). Socioeconomic Status and Child Development. Annual Reviews Psychology. 53, pp. 371. Brocki, K. C., and Bohlin, G. (2004). Executive functions in children aged 6 to 13: A dimensional and developmental study. Developmental Neuropsychology, 26(2), pgs. 571. Brooks-Gunn, J., Klebanov, P. K., and Duncan, G. J. (1996). Ethni c differences in childrens test scores: Role of economic depriv ation, home environment, and maternal characteristics. Child Development, 67(2), pp. 396-408. Brooks-Gunn, J. and Duncan, G. J. (1997). The Effects of Poverty on Children: The Future of Children. Children and Poverty Vol. 7, (2), pp. 55-71. Byrd, D. L., van der Veen, T., McNamara, J. P., and Berg, W. K. (2004). Preschoolers Dont Practice What They Preach: Preschoolers Planning Performances with Manual and Spoken Response Requirements Journal of Cognition and Development, 5(4), 427-449. Buckner, J. C., Mezzacappa, E., and Beardslee, W.R. (2003). Char acteristics of resilient youths living in poverty: The role of self-regulatory processes. Development and Psychopathology, Vol 15(1), pp. 139-162. Bull, R. and Scerif, G. (2001). Executive functio ning as a predictor of childrens mathematics ability: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, pp. 273-293. Bull, R., Espy, K.A., & Senn, T.E. (2004). A Comparison of Performance on the Towers of London & Hanoi in Young Children. Journal of Child Psychology and Psychiatry, 45, 743-754. Chang, F. and Burns, B.M. (2005). Attention in pres choolers: Associations w ith effortful control and motivation. Child Development, 76(1), pp. 247-263. Craig, H., Connor, C., and Washington, J. (2003). Early positive predicto rs of later reading comprehension for African American St udents: A preliminary investigation. Language, Speech, and Hearing Services in Schools, 34, 31-43. Cirino, P.T., Chin, C.E., Sevcik, R.A., Wolf,M ., Lovett, M. and Morris, R. D. (2002). Measuring Socioeconomic Status: Reliability and Preliminary Validity for Different Approaches Assessment 9, pp. 145-155. 123

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Friedman, N.P., Haberstick, B.C.,Willcutt, E.G ., Young, Miyake, A.,Corley, S.E., & Hewitt, J.K. (2007). Greater attention problems dur ing childhood predict poorer executive functioning in late adolescence. Psychological Science, Vol 18 (10), pgs. 893900. Friedman, B.H., Allen, M.T., Christie I.C., a nd Santucci, A.K. (2002). Validity concerns of common heart-rate variability indices: Addressing quantification issues in timeand frequency-domain measures of HRV. IEEE Engineering in Medi cine and Biology, 21(4) 35-40. Garon, N., Bryson, S. E., Smith, I.M. (2008). Executive function in preschooler s: A review using an integrative framework. Psychological Bulletin Vol. 134( 1) ,pgs. 31 Gottfreid, A.W., Gottfreid, A.E., Bathurst, K., Guerin, D.W., and Parramore, M.M. (2003). Socioeconomic status in childrens development and family environment: Infancy through adolescence. In M.H. Born stein & R. H. Bradley (Eds.) Socioeconomic Status Parenting, and Child Development (p. 189-207). Mahwah: Lawrence Erlbaum. Hollingshead, A.B. (1975). Four factor index of social status. Unpublis hed manuscript, Yale University, New Haven, CT. Howse, R. B., Lange, G., Farran, D. C., and Boyles, C. D. (2003). Motivation and selfregulationa as predictors of academic achie vement in economically disadvantaged young children. Journal of Experimental Education, Vol 71(2), pp. 151-174. Huizinga, M., Dolan, C.V., van der and Molen, M.W. (2006). Age-related change in executive function: Developmental trends a nd a latent variable analysis. Neuropsychologia, 44 pgs. 2017. Kishiyama, M. M., Boyce, W.T., Jimenez, A.M., Perry, L.M., and Knight, R.T. (2009 in press). Socioeconomic disparities affect prefrontal function in children. Journal of Cognitive Neuroscience,X:Y, pp.1-10. Klenberg, L., Korkman, M., and Lahti-Nuuttila, P. (2001). Differential development of attention and executive functions in 3to 12-year-old Finnish children. Developmental Neuropsychology, 20(1), pp. 407-428. Lehto, J.E., Juujarvi,P., Kooistra, L, and Pu lkkinen, L. (2003). Dimensions of executive functioning: Evidence from children. British Journal of Developmental Psychology 21 pgs 59. Liew, J., Chen, Q., & Hughes, J.N. (2009). Child e ffortful control, teacher-student relationships and achievement in academically at-risk children: Additive and interactive effects. Liew, J., McTigue, E. M., Barrois, L., & Hughes, J. (2008). Adaptive/effortful control and academic self-efficacy beliefs on literacy and math achievement: A longitudinal study on 1st through 3rd graders. Early Ch ildhood Research Quarterly, 23, 515-526. 125

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BIOGRAPHICAL SKETCH Teri DeLucca completed her undergraduate studi es at the University of North Florida where she graduated Summa Cum Laude with a B achelor of Arts Degr ee in Psychology. During an internship with the Heart of America Foundation in Washington, D.C., Teri had the opportunity to witness the problem s of economically disadvantaged children in some of the most under resourced schools in our nation. This bega n her dedication to the study of low-income children. As an immediate action, she helped to establish a Books fr om the Heart program, through the Heart of Am erica Foundation, in Jacksonville, Fl. She decided, however, that a bigger contribution would be to draw on her cap abilities and understandi ng of the strength of empirical research to discover the source of such disparities. It was th is realization that she would impact children in a broa der sense, through policy-impacti ng research that drove her to further her formal education. Teri was accepted into the developmen tal psychology and educational psychology dual degree doctoral program at the University of Florida where her primary research interest remained focused on the welfare and development of children from low-income families. She completed her masters degree in 2007 and her thesis investigated family income differences in kindergarteners problem solving skills using bo th behavioral and physiological methods. Teri has collaborated on several diffe rent research projects while in graduate school involving the development of executive functioning skills in gift ed children as well as a collaborative project focused on the longitudinal inves tigation of the effect s of prenatal cocaine exposure on the cognitive, social, and emotional aspects of development. Teri has published her research and presented results at multiple conferences. The resu lts and the reception of her research, combined with her volunteer experiences with low-income children and families, have confirmed for her the importance of early intervention research. Her future career plans involve an applied 129

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130 application of her research. She hopes to co ntinue her program of work focused on the development of executive functioning skills in low-income children by incorporating training and early intervention methods. She plans to continue to research the achievement gap that exists in our nation eventually within in a social policy setting. She is confident that the training and experiences gained from her time at the Univ ersity of Florida will provide her a necessary foundation that will prove inva luable in her career.