The trajectory of mathematics skills and working memory thresholds in girls with fragile X syndrome

Cognitive Development 24 (2009) 430–449

Contents lists available at ScienceDirect

Cognitive Development

The trajectory of mathematics skills and working memory thresholds in girls with fragile X syndrome Melissa M. Murphy a, Michèle M.M. Mazzocco b,c,∗ a

School of Education, College of Notre Dame of Maryland, Baltimore, MD, USA Math Skills Development Project, Kennedy Krieger Institute, 3825 Greenspring Avenue, Painter Building, Baltimore, MD, USA c Department of Psychiatry and Behavioral Sciences, Johns Hopkins University School of Medicine, 3825 Greenspring Avenue, Painter Building, Baltimore, MD, USA b

a r t i c l e

i n f o

Keywords: Mathematics skills Working memory Fragile X syndrome Mathematical learning disability

a b s t r a c t Fragile X syndrome is a common genetic disorder associated with executive function deficits and poor mathematics achievement. In the present study, we examined changes in math performance during the elementary and middle school years in girls with fragile X syndrome, changes in the working memory loads under which children could complete a cognitive switching task, and the association between these two areas of function, in girls with fragile X syndrome relative to their peers. Our findings indicate that the trajectory of math and executive function skills of girls with fragile X differs from that of their peers and that these skills contribute to predicting math achievement and growth in math performance over time. Also, changes in math performance were associated with incremental increases in working memory demands, suggesting that girls with fragile X have a lower threshold for being able to perform under increasing task demands. Still, we found improvement in executive function performance between 10 and 12 years in girls with fragile X rather than a performance plateau as has been reported in other studies. The findings implicate the importance of early

∗ Corresponding author at: Math Skills Development Project, Kennedy Krieger Institute, 3825 Greenspring Avenue, Painter Building, Baltimore, MD, USA. E-mail address: [email protected] (M.M.M. Mazzocco). 0885-2014/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.cogdev.2009.09.004

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

431

intervention in mathematics for girls with fragile X that addresses poor calculation skills, the supporting numerical skills, and deficits in executive functions, including working memory. © 2009 Elsevier Inc. All rights reserved.

The roles of executive functions and working memory in mathematical performance have received both theoretical and empirical support (Bull, Johnston, & Roy, 1999; Bull & Scerif, 2001; Mazzocco & Kover, 2007). Moreover, learning disabilities in mathematics have been linked to working memory deficits specifically (Andersson & Lyxell, 2007; Swanson & Beebe-Frankenberger, 2004), even if not exclusively. Collectively, studies of the cognitive characteristics of children with mathematical difficulties implicate a wide range of executive skills as underlying their performance deficits and support different theoretical constructs of working memory (as reviewed by Berch, 2008). Children with fragile X syndrome, an X-linked genetic condition, have a biologically predisposed increased risk for mathematical learning disability (MLD; Mazzocco, 2001; Murphy & Mazzocco, 2008) and select executive dysfunction (e.g., Mazzocco, Hagerman, Cronister-Silverman, & Pennington, 1992). Thus, the present paper builds upon findings across studies of fragile X syndrome by exploring the link between poor math achievement and these select deficits reported for children with fragile X syndrome, including deficits in executive functions and working memory. If we define working memory as the ability to maintain and access stored information while engaged in cognitive tasks, its relevance to math ability is evident for both higher mathematics and basic math skills such as counting and specific strategy use (e.g., finger counting and verbal counting; Geary, 2004). Implicit in this definition is the need to successfully inhibit competing responses, a deliberate executive function that can serve to either prevent a proponent response from occurring or to both prevent the response and substitute an alternative less automatic one. The working memory demands (or load) of a task may influence whether attempts to inhibit a prepotent response are successful; more effortful tasks may diminish the availability of cognitive resources needed, relative to those available for tasks with fewer cognitive demands. Individual differences in the “threshold” level that impedes a person’s efficient executive functions may partially underlie individual differences in mathematics achievement levels. In our earlier work, we reported lower thresholds among girls with fragile X syndrome, but that study did not include a parallel evaluation of math performance (Kirk, Mazzocco, & Kover, 2005). Studying specific etiologies of MLD, such as fragile X syndrome, may reveal aspects of the association between executive functions, working memory, and math achievement. In studies of working memory influences on math performance, researchers often rely on Baddeley’s model of working memory (Baddeley, 1996, 2003), although other models have been proposed as highly relevant (see Berch, 2008 for a discussion). Summarized briefly, Baddeley conceptualizes working memory as a central executive, with deliberate control of two core “slave systems;” these slave systems, the phonological loop and visuospatial sketchpad, store linguistic or visually represented information, respectively. Both storage systems are subject to the limitations of storage capacity and processing efficiency. Several recent studies have explored components of working memory deficits in children with MLD. Geary, Hoard, Byrd-Craven, Nugent, and Numtee (2007) demonstrated that the central executive, phonological loop, and visuospatial sketchpad were each associated with math performance, but the precise association depended on the math task being assessed. Still, these researchers identified the central executive as the “core component” underlying performance deficiencies in children with MLD, as did Andersson and Lyxell (2007), who also showed correlations between working memory and math achievement among children of all math achievement levels. Working memory deficits differentiated children with versus without MLD in both of these studies. However, in Geary’s study, working memory did not differentiate children with low average math achievement from those with age appropriate math outcomes. Instead, processing speed differentiated both those with MLD or low achievement from their typically achieving peers. The findings of Geary et al. (2007) are similar to those we reported earlier (Murphy, Mazzocco, Hanich, & Early, 2007),

432

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

suggesting profile variability among children who meet different criteria for math difficulties (e.g., MLD vs. low average achievement). In the present study, we explore how processing speed and thresholds for working memory demands change over time among children whose MLD is associated with fragile X syndrome. 1. Why focus on fragile X syndrome to study math-working memory associations? Fragile X syndrome is a common genetic disorder that is associated with cognitive deficits. Nearly all males, and 50% of females, with fragile X syndrome meet criteria for an intellectual disability;1 these individuals are not the focus of the present study. Among those females without intellectual disability, cognitive deficits have been reported in executive functions, including working memory (Bennetto, Pennington, Porter, Taylor, & Hagerman, 2001; Cornish et al., 2004; Lasker, Mazzocco, & Zee, 2007; Mazzocco, Pennington, & Hagerman, 1993) and in mathematics (Brainard, Schreiner, & Hagerman, 1991; Kemper, Hagerman, Ahmad, & Mariner, 1986; Mazzocco, 2001; Murphy, Mazzocco, Gerner, & Henry, 2006). Across studies, mathematics deficits are evident in primary school (Mazzocco, 2001; Murphy et al., 2006), elementary school (Murphy & Mazzocco, 2008), and adulthood (Bennetto et al., 2001; Mazzocco, 1998). Executive function deficits are also seen in both childhood and adulthood (Bennetto et al., 2001; Mazzocco, 1998), and may reflect a lower threshold for overcoming the demands of increased working memory loads during various types of problem solving (Kirk et al., 2005). It is unclear whether the mathematics deficits observed in fragile X are primary or secondary to the working memory deficits associated with the syndrome; the nature of both types of deficits remain topics of research. Understanding whether the working memory threshold in fragile X is comparable to that observed in the general population, its relationship to math performance, and how it changes over time, may contribute to developing models of the relation between math and working memory. In the present study, we address these questions with use of the Contingency Naming Test (CNT), which we have used in other studies of executive function to focus specifically on working memory demands. 2. Using the Contingency Naming Test to study the effect of incremental working memory demands on performance The CNT is a Stroop-like task that requires naming stimuli according to a one- or two-attribute contingency rule (Anderson, Anderson, Northam, & Taylor, 2000). The stimuli, which vary in shape and color, are comprised of a small, inner shape within a larger, outer shape. The one-attribute rule involves naming the color of a stimulus when the inner and outer shapes match, or naming the shape when the inner and outer shapes do not match. This one-attribute rule serves as the foundation for the two-attribute rule, except that the rule is reversed if an arrow appears above a given stimulus. Several researchers have used the CNT to demonstrate how working memory demands impair performance of individuals from various clinical populations, including females with fragile X (Bennetto et al., 2001; Kirk et al., 2005; Mazzocco et al., 1992). Use of the CNT demonstrates that females with fragile X have a lower threshold for working memory demands relative to IQ matched peers (Kirk et al., 2005; Mazzocco et al., 1992). That is, the one-attribute task is as challenging as for girls with fragile X as the two-attribute task is for their peers (Kirk et al., 2005). This working memory impairment in females with fragile X is evident even in the presence of spared verbal memory skills. To our knowledge, in only one normative study of the CNT has its association with math performance been explored (Mazzocco & Kover, 2007). In that longitudinal study, the association between CNT and math scores varied with age, but some degree of association was evident at each of three assessments, which occurred at approximately ages 6–7, 8–9, and 10–11 years (grades 1, 3, and 5). In terms of the changes in working memory efficiency from 6 to 11 years, the difficulty posed by the most challenging task at 11 years was comparable to the difficulty of the moderately challenging task at 9

1 Intellectual disability refers to limitations in intellectual and adaptive functioning; this term replaces the term mental retardation, thereby referring to identical constructs (Schalock et al., 2007).

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

433

years (Anderson et al., 2000; Mazzocco & Kover, 2007). Performance floors on the most difficult task were evident at first grade, whereas performance ceilings for the moderate task occurred at about age 12 years (Anderson et al., 2000). On the basis of earlier findings of lower working memory thresholds in girls with fragile X, relative to the general population, we hypothesized that these floor and ceiling effects would emerge at a later age for girls with fragile X. A curious finding from the normative study was that performance efficiency on a “warm-up” task for the CNT (i.e., a basic color and shape naming task designed to familiarize children with the task stimuli and procedure, but not designed to measure working memory) was associated with mathematics performance at first grade (6–7 years of age) in much the same way that efficiency on the working memory task was associated with math performance at ages 9 and 11 years. Mazzocco and Kover suggested that the warm-up task posed a working memory component at the children’s first assessment only; that is, developmental characteristics determine whether a task measures working memory. Herein we further explore this notion of a developmental threshold for working memory; and thus, include basic naming tasks that may have a working memory component at only the youngest age assessed. In the present study, we examined the developmental trajectory of math and working memory skills in girls with fragile X relative to peers from a normative sample during the elementary and middle school years. We predicted that level of math achievement and working memory performance will continue to distinguish girls with fragile X from their peers at all ages examined (approximately 7–13 years of age); and that the rate of growth of math and working memory skills from 7 to 13 years also differentiates girls with fragile X from their peers, as do lower thresholds for working memory demands. Because the CNT is used primarily to assess reactive flexibility and working memory, we included a digit span task as a one-time measure of verbal memory span, specifically phonological loop capacity. Finally, we hypothesize that working memory is predictive of 6th grade math achievement, and growth in math performance over time, when working memory is assessed with developmentally appropriate indices. 3. Method 3.1. Participants All participants were selected from an ongoing study of mathematics ability and disability in school-age children (Mazzocco, 2001; Mazzocco & Myers, 2003; Mazzocco & Thompson, 2005). A normative sample of children from the general population were enrolled at kindergarten and tested yearly through middle school: working memory was measured at grades 1, 3, 5, and 7. The math calculation measure was administered at grades 1, 3, 5, and 6. There were 54 girls from this normative group who participated in the ongoing study at each of these five grades, from which an IQ-matched comparison group was drawn (n = 19; see Table 1). Unlike the normative group who began the study at the same grade, girls with fragile X syndrome (n = 24) varied in the grade at which they were enrolled. As a result, four girls were seen at all four time points, four were seen at three of the four points, and nine were seen at two points (see Table 1). Eighteen of the 24 girls with fragile X syndrome (75%) had poor math performance levels during most or all of the grade levels at which they were assessed; of the remaining six girls, four consistently received age-appropriate scores across math achievement assessments, and the performance of two girls was inconsistent during this same time period (i.e., showing evidence of both poor and ageappropriate math performance at different grade levels, with no clear increasing or decreasing trend). In contrast, six of the 19 girls in the normative matched sample (32%) had consistently poor math performance levels across assessments, while eight consistently performed at age appropriate levels over time and five had inconsistent math performance over the same time period. 3.2. Materials and tasks 3.2.1. General cognitive ability A full scale IQ score (FSIQ) was obtained for each participant from the four-subtest Wechsler Abbreviated Scale of Intelligence (WASI, Wechsler, 1999) administered at 3rd or 5th grade. WASI

434

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

Table 1 Demographic information for the analyses: mean (SD). Comparison group

Fragile X Na

b

Na

FSIQ

98.16 (6.28)

19

88.54 (12.04)

24

Age in years Grade 1 Grade 3 Grade 5 Grade 6c Grade 7d

6.69 (.26) 8.66 (.32) 10.67 (.27) 11.80 (.24) 12.80 (.27)

19 19 19 19 19

7.43 (.51) 9.20 (.70) 11.29 (.52) 12.60 (.47) 12.92 (.58)

12 16 15 10 10

a Comparison group selected such that each child was seen at all four time points. This was not possible for the group with fragile X. Sample size reported reflects number of eligible participants at each grade. Of these, 17 were seen two or more times. b FSIQ was based on Wechsler Abbreviated Scale of Intelligence (n = 22) or the Standford-Binet-Fourth Edition (n = 3) (see Section 3 for details). c Woodcock Johnson-Revised Calculation scores only. d Contingency Naming Test and Rapid Automatized Naming subtests only.

scores are based on M = 100 (SD = 15), with internal consistency reliability coefficients > .95 for children in the study age range. Four-subtest WASI scores were unavailable for six girls with fragile X, so FSIQ was instead based on the two-subtest WASI (n = 3) or the Stanford Binet, fourth edition (SBIV, Thorndike, Hagen, & Sattler, 1986) (n = 3). The age-relevant correlation between the twoand four-subtest WASI is .93. FSIQs from the SB-IV were adjusted to conform to distribution on the WASI (because SB-IV composite scores are based on a SD = 16). FSIQ was included in the study as a covariate in selected analyses, and provided descriptive information about our participants (see Table 1). 3.2.2. Mathematics achievement Because the study concerned acquired math knowledge, rather than math processing, we focused on performance during an untimed math assessment. This is because the performance of children, with or without math difficulties, is typically weaker during timed versus untimed math assessments (Kellogg, Hopko, & Ashcraft, 1999). We used an untimed standardized academic achievement test, the Woodcock Johnson Psycho-Educational Battery-revised Calculations subtest (Woodcock & Johnson, 1990), to assess performance on paper-and-pencil calculations. The internal consistency reliabilities range from r = .89 to r = .93 for the age range of participants in this study. 3.2.3. Executive function skills: Working memory and verbal memory span We included two measures of executive function. The Contingency Naming Test was administered over multiple time points to assess reactive flexibility (i.e., the efficiency of cognitive switching) under increasing working memory demands. Use of the CNT allowed us to test for hypothesized group differences in the developmental trajectory of executive function skills and how these trajectories varied as working memory load demands increased. Digit Span was used to assess verbal memory span. It was administered only once because of the limited growth trajectory anticipated for either the forward or backward span (Wechsler, 1991, p. 267). 3.2.3.1. The Contingency Naming Task. The CNT (Anderson et al., 2000; Taylor, Albo, Phebus, Sachs, & Bierl, 1987) was used to assess cognitive flexibility under different threshold levels. The CNT is described previously and in detail elsewhere (Anderson et al., 2000), and so is only briefly described here. This Stroop-like task involves naming a series of 27 colored stimuli (small shapes embedded in large shapes) according to a set of rules. The rules for the first two warm-up trials are single-attribute naming rules; for each, the participant is to name the items by the color (Trial 1) or outer shape (Trial 2). The third and fourth trials incrementally add to the working memory demands of the task by asking the participant to switch back and forth between rules based on one or two attributes of the items. For example, on Trial 3 (one-attribute switching rule), the child names the color if the two shapes

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

435

match, or switch to naming the outside shape if the two shapes do not match. Trial 4 (two-attribute switching rule) requires applying the one-attribute rule, but reversing the rule for any item above which a backwards arrow appears. For each CNT trial, response time (RT) to name the stimuli was recorded via a handheld stopwatch. The RTs for the two single-attribute naming rules (Trials 1 and 2) were added to form a single RT for the “warm-up” trial based on 54 stimuli. For all trials, a shorter response time indicates faster (i.e., “better”) processing speed. In first grade, several girls in the fragile X (n = 4) or comparison group (n = 3) had difficulty with the one-attribute trial. In these cases the scores were imputed following the procedures detailed by Kirk et al. (2005, pp. 764–766). In brief, imputed scores were determined by comparing an individual’s observed scores to the poorest observed score reported for her group (i.e., fragile X or comparison). Consistent with reported normative data on this task, most participants had so much difficulty with the two-attribute trial in first grade that they either did not attempt or complete the task. Because most first graders in both groups (9 of 12 participants and 15 of 19 in the fragile X and comparison groups, respectively) were unable to complete the two-attribute trial due to task difficulty, performance on this trial is not considered further. 3.2.3.2. Digit span. The Memory for Digits subtest from the Stanford Binet, 4th edition (SBIV, Thorndike et al., 1986) was used to assess verbal memory span (forward digits) and working memory (backward digits) in 5th grade (at approximately 10 years of age). On both forward and backward digit recall, each participant is asked to recall progressively longer strings of digits, but backward recall requires recalling the strings in the reverse order from which they were presented. For each task, two trials are presented for each length of digit strings. In addition to separate span lengths that we used for select analyses, an age-referenced standard score was calculated based on the forward and backward total score. Phonological processing speed: Our final set of measures includes a contrast to the effortful processing required during the CNT. Specifically, Rapid Automatized Naming (RAN; Denckla & Rudel, 1976) was included to measure the phonological processing speed and retrieval. Although this task was selected in part because of the “automatic” response processing with which it is generally associated, the RAN was an attractive measure also because of reports of incremental difficulties in naming colors versus numbers and letters, at least for school-age children (Stringer, Toplak, & Stanovich, 2004). As reading becomes more automatic with advancing age, the processing demands associated with number- and letter-naming may diminish to a greater degree than the demands associated with color-naming. Moreover, as a means to delineate the development and function of working memory thresholds in our participant groups, the RAN is particularly well suited to be compared with the CNT because of similarities in test administration and response expectations (i.e., naming items as quickly as possible). Three RAN subtests were included: Colors, Numbers, and Letters. During each subtest, the participant is first presented with a warm-up trial consisting of five unique stimuli—colored squares, numerals, or letters. A timed experimental trial follows, during which the participant must name 50 of these five, recurring stimuli—presented in a 5 × 10 array—as quickly as possible. Reaction time is explicitly recorded with a handheld stopwatch. Because of the stimuli simplicity, errors occur very infrequently and are not evaluated in either the original version of the RAN (Denckla & Rudel, 1976) or subsequent adaptations (e.g., Wagner, Torgesen, & Rashotte, 1999). Instead, children who fail the warm-up trial are not administered the experimental trial. This did not occur for any of the participants in the present study. 3.3. Procedure All participants were individually assessed at approximately one-year intervals. For both groups, testing took place during a single day or over a period of several days at a meeting room in the child’s school, the examiner’s offices, or a commonly agreed upon location within the community (e.g., a public library).

436

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

For girls with fragile X, we included results of assessments that occurred at the age closest to (or slightly older than) the mean age for children in a given grade, as determined by the overall sample mean from the ongoing longitudinal study. Thus, as reflected in Table 1, girls with fragile X were slightly older than girls in the comparison group, which gave this group a performance advantage in view of the positive correlations between age and performance on the RAN or CNT. Also of note, although school grade is used in the analyses, not all girls with fragile X were necessarily in the school grade indicated. In most cases, the girl with fragile X was in a higher grade than the grade targeted (e.g., a 6th grader included with the 5th grade analyses). However, in a few cases, a girl had repeated a school grade more than once (n = 3). In these cases, the age associated with a given grade was used to determine the grouping (e.g., an 11 year-old who was repeating 4th grade for the third time would be included with the 5th grade analyses because of her age).

3.4. Variables and analyses Growth curve models were used to examine the trajectory of growth in cognitive flexibility and mathematics during the elementary and middle school years. These models are robust enough to handle missing data (Burchinal, Nelson, & Poe, 2006, p. 71) and are described elsewhere in detail (Jordan, Hanich, & Kaplan, 2003). Following the growth curve models, differences are examined at each grade. Such an approach allowed us to assess differences in growth between girls with fragile X and their peers, as well as the point at which any group differences emerged or disappeared during that time period. The primary outcome variables were raw scores used as indices of performance accuracy on mathematics achievement (calculations) and the digit span tests, reaction time on the RAN, and both reaction time and accuracy from the CNT. Age-referenced standardized FSIQ scores were included as a covariate. In addition to assessing overall performance fluency, we were interested in group differences in the degree to which increases in working memory demands across tasks diminished relative cognitive performance. To assess these working memory threshold differences across groups, we calculated difference scores using procedures from the Stroop color word interference task as a guide. We calculated a commonly used difference score, Memory for Digits Forward minus Backward (Sattler, 2001, pp. 278–279; Wechsler, 1991, p. 268), to compare performance on the more effortful digit span task (digits backwards) with the more automatic task (digits forward). Next we calculated a series of difference scores on the RAN and CNT to evaluate potential response time (RT) differences relative to degree of working memory demands, or load. Specifically, we compared RTs between the CNT warm-up and one-attribute trials to assess the incremental addition of working memory demands associated with the one-attribute subtest. Next, we considered the possibility that the CNT warm-up trial demands executive control only for individuals who have a low threshold for working memory demands. Thus, we compared RTs across the RAN Colors and CNT warm-up subtests, both of which require participants to name stimuli according to a single attribute (e.g., color), but differ in the degree of response inhibition required. For example, stimuli on RAN Colors differ by only a single salient attribute (color), whereas the stimuli on the CNT may differ on two or three salient attributes (color, outer shape, and inner shape), which must be inhibited when naming the stimuli by a single attribute. In summary, the differences we calculated were designed to reflect the progressive increase in working memory demands across the RAN and CNT as measures of the difference in RT between: (1) Colors and Numbers; (2) CNT warm-up and Colors; and (3) CNT one-attribute and warm-up trials. For the latter, to make the trials comparable, the RT for the warm-up trials was the average of Trial 1 and 2. No adjustment was made for the warm-up-Color difference because both were based on comparable numbers of items, 54 and 50 items, respectively. For the RAN subtest difference score, only one of the subtests (Numbers and Letters) was selected to avoid the redundancy resulting from the similar performance on the two subtests (discussed subsequently); the Numbers subtest was selected as the test more consistent with the aims of the study.

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

437

Fig. 1. Growth in raw score group means (and standard errors) on Calculations from primary to middle school.

4. Results 4.1. Growth curve models Each set of growth curves compared the two groups in growth rate and score at the final time point (i.e., 7th grade for CNT and RAN, 6th grade for Calculations). Two linear models were calculated separately for each dependent variable: Calculations raw score, CNT and RAN subtest RTs, and RT difference scores. The first model (Model 1; Table 2) shows the effects of diagnostic group (i.e., fragile X, comparison) on each measure. The second model (Model 2; Table 2) was calculated to take into account the possible effects on performance of FSIQ and working memory (for the Calculation measure only). For Model 2, the predictors were centered around the sample mean. As a result, the final time point and growth rate represent the average performance of the reference group (i.e., fragile X), with the predictors at the mean of the sample on the given measure. For all models, girls with fragile X are used as the reference group. For ease of interpretation, we take the reader through the interpretation of the growth curve models using Calculations and CNT. Other measures are discussed in less detail but are summarized in Table 2. 4.1.1. WJ-R calculations Model 1 (Table 2) indicates that the average raw score on Calculations is 22.16 problems for girls with fragile X in 6th grade. As reflected in Fig. 1, this average score is 5.66 problems lower than the average score among girls in the comparison group. The rate of growth also distinguishes the two groups. Girls with fragile X are growing at a rate of 3.51 problems per year, which is 0.70 problems per year slower than the growth rate in the comparison group. We consider influences of FSIQ and executive function working memory on this trajectory (Model 2), at the end of this results section, after presenting the analyses by grade. 4.1.2. CNT Fig. 2 summarizes the CNT RTs from 1st to 7th grades for girls with fragile X (a) and the comparison group (b). Note that for the fragile X and comparison groups, 1st grade CNT scores for the one-attribute trial contain both observed and imputed scores. The analyses were first run on the observed, and then on the combined observed and imputed scores. No differences in the pattern of results were found. Below we report on the findings of the analyses using the combined observed and imputed scores.

438

Table 2 Growth curve results (Models 1 and 2). Calc

CNT Warm-up RTa

CNT one-attribute RTa

RAN Num RTa

RAN Lett RTa

RAN Col-Num RT

CNTWarm-up-Color RT

CNT oneattribute—1/2(Warm-up) RT

41.83*** −3.16*** −12.65***

29.75*** −1.39*** −9.34***

26.03*** −2.40*** −6.67**

16.11*** −0.57* −5.59**

8.39*** 0.24 −2.91

25.13*** −2.30*** −7.23**

−0.73

−0.98*

−0.16

−0.74

−2.96**

40.47*** −2.73*** −9.34***

30.33*** −1.03*** −9.84***

25.47*** −2.27*** −5.12*

14.85*** −0.41 −2.27

7.58*** −0.24 −1.11

23.61*** −2.39*** −3.80

−1.58*

−1.62***

−0.40

−0.10

0.36

−2.82**

0.02 0.07**

−0.19 0.02

−0.40*** 0.01

−0.19 −0.12**

−0.37*** −0.01

−0.37*** 0.09*

0.12

Note: Reference group = Fragile X; Calc = Woodcock Johnson-Revised Calculations; CNT = Contingency Naming Test; RT = response time (in seconds); RAN = Rapid Automatized Naming, Col = Colors, Num = Numbers, Lett = Letters; Response times were rounded to the nearest second. a On response time measures, a negative growth rate indicates that RT is getting faster across grades. b “Intercept” reflects the fragile X mean at final time point. c d e * ** ***

“Slope” reflects the fragile X growth rate from initial to final time point. “Intercept on comparison group” reflects the difference between fragile X and comparison mean at final time point. “Slope on comparison group” reflects the difference between fragile X and comparison growth rate from initial to final time point. p < .05. p ≤ .01. p ≤ .001.

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

Model 1 (diagnostic group differences) 22.16*** 45.52*** 49.65*** Interceptb 3.51*** −4.14*** −3.68*** Slopec Intercept on 5.66** −13.09*** −14.76*** comparison groupd Slope on 0.70* −0.61 −3.57*** comparison groupe Model 2 (diagnostic group differences with predictors) 24.80*** 44.43*** 47.15*** Interceptb 3.86*** −4.10*** −3.79*** Slopec Intercept on 0.14 −8.87** −9.07** comparison groupd −0.08 −0.34 −3.33** Slope on comparison groupe Predictor variables −0.57*** −0.63*** Intercept on IQ 0.36*** Slope on IQ 0.04* −0.05 −0.03 Intercept on −0.19** Gr. 3 CNT Trial 1 RT Slope on Gr. 3 −0.05** CNT Trial 1 RT

RAN Color RTa

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

439

Fig. 2. Growth in response times on two RAN subtests and two CNT subtests (see Tables 3 and 4 for standard deviation values). On the most demanding of these four subtest (the CNT one attribute task), girls with fragile X have shorter (better) response times, but at a cost for accuracy relative to the comparison group.

Model 1 (Table 2) indicates that the average RT in 7th grade for girls with fragile X on the warm-up is 45.52 s. The average growth rate for girls with fragile X across grades reflects a decrease in response time by 4.14 seconds per year from 1st to 7th grade. By 7th grade, the average RT for girls with fragile X is 13.09 s slower than the RT in the comparison group X. However, no difference was found between the groups in rate of growth. On the one-attribute switching subtest, a significant difference between groups in the average RT in 7th grade and growth rate are reported. For girls with fragile X, the average RT in 7th grade is 49.65 s, which is 14.76 s slower than the comparison group. Response times among girls with fragile X got faster at a rate of 3.68 seconds per year, which is 3.57 s slower per year than the average growth rate in the comparison group whose average performance on the task got faster at a rate of 7.25 seconds per year. Model 2 (Table 2) reflects the addition of FSIQ as a predictor in Model 1. When FSIQ is held constant, the 7th grade RT on the warm-up for girls with fragile X is 44.43 seconds per year and the growth rate (i.e., improvement in RT) is 4.10 seconds per year. The comparison group had RTs that were 8.87 seconds per year faster than the fragile X group when controlling for FSIQ; and no group differences in growth rate emerged. FSIQ was a significant predictor of RT in 7th grade on both the warm-up and one-attribute trials, but not growth rate. Specifically, children with higher FSIQ scores have faster RTs on the warm-up and one-attribute subtests than children with lower FSIQ scores. No changes in the pattern of group differences in RT on either trial were observed after controlling for FSIQ. In summary, these results suggest that the trajectory of cognitive flexibility and working memory skills varies for girls with fragile X relative to peers, even after controlling for group differences in FSIQ. Specifically, by 7th grade, the performance of girls with fragile X lags behind that of their peers on warm-up and cognitive switching trials. As a point of comparison, we were interested in the

440

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

developmental trajectory of performance on the RAN, which is generally associated with “automatic” response processing. 4.1.3. RAN Fig. 2 depicts RAN RTs from 1st to 7th grades for girls with fragile X (a) and the comparison group (b). According to Model 1 (Table 2), the average RT on Colors in 7th grade for girls with fragile X is 41.83 seconds per year and the average response time gets faster by a rate of 3.16 seconds per year. These results reflect longer RTs in the group with fragile X than the comparison group at 7th grade. There was no difference in growth rate of RT between groups. Likewise, on the Numbers and Letters subtests, girls with fragile X had longer RTs than girls in the comparison group (see Table 2). Relative to the comparison group, the RTs of the girls with fragile X were decreasing at a slower rate on Numbers (p ≤ .01), but not Letters. Model 2 (Table 2) presents the results of the RAN subtests with FSIQ as a predictor. FSIQ significantly predicted RT in 7th grade on the Colors subtest, but not the Numbers or Letters subtests. Specifically, children with higher FSIQ scores have faster RTs on RAN Colors than children with lower FSIQ scores. FSIQ predicted growth rate in RT on Colors and Numbers, such that children with higher FSIQ scores decrease at a slower rate than children with lower FSIQs. FSIQ did not predict growth in RT on Letters. Once FSIQ is entered into the model, there were no changes in the pattern of group differences in RT on the Numbers or Letters subtests. However, a group difference in the growth rate of RT on the Colors subtest at 7th grade emerged after controlling for FSIQ. In summary, these findings indicate differences between girls with fragile X and their peers in the developmental trajectory of phonological retrieval fluency. Even after controlling for FSIQ, group differences in RTs were present on all three RAN subtests, suggesting that the tasks continue to be less automatic for girls with fragile X than their peers into middle school. Together, the pattern of findings across the CNT and RAN subtests led us to assess performance based on three sets of difference scores that reflected the progressive increase in processing demands across the RAN and CNT. 4.1.4. Difference scores Fig. 3a–c depicts the difference in RT between (1) Colors and Numbers (3a), (2) CNT warm-up and Colors (3b), and (3) the CNT one-attribute and warm-up (3c). Model 1 (Table 2) indicates that the average difference for naming Color versus Numbers is 16.11 s for girls with fragile X; which exceeds the difference for the comparison group (10.52 s). Similarly, the average RT difference for oneattribute versus warm-up naming is 25.13 s for girls with fragile X; which exceeds the difference for the comparison group (17.90 s). No group difference was observed for the warm-up-Colors difference. Group differences in growth rate were observed for the one-attribute—warm-up difference, but not the warm-up-Color difference or Color–Number difference. Specifically, the difference in RT for the one-attribute—warm-up trial is decreasing at a slower rate for girls in the comparison group than girls with fragile X (Fig. 4). Model 2 (Table 2) reflects the addition of FSIQ as a predictor. FSIQ is a significant predictor of the Color–Number RT difference and the one-attribute—warm-up RT difference. Specifically, children with higher FSIQs have smaller difference scores than children with lower FSIQs, suggesting that they experience comparable automaticity when naming colors and numbers, unlike children with lower FSIQs. FSIQ did not predict the warm-up-Colors difference. FSIQ is also a significant predictor of growth rate for the CNT warm-up-Colors difference (p ≤ .01). Children with higher FSIQ scores have difference scores that decrease at a faster rate than those with lower FSIQs, suggesting that the naming subtest of the CNT is becoming “as automatic” as Colors at a faster rate for those with higher versus lower FSIQs. Once the predictor variables were entered into the model, the groups are no longer distinguished based on the Color-Number RT difference or the CNT one-attribute—warm-up difference. The groups continue to be distinguished on the rate of growth of the CNT one-attribute—warm-up difference. In summary, these findings suggest that the groups are distinguished based on their responsiveness to the degree of working memory load associated with a task. More specifically, even basic naming tasks may pose some degree of executive load for girls with fragile X, thereby decreasing their pro-

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

441

Fig. 3. Growth in response time group means (and standard errors) for difference scores on two pairs of tasks administered from primary to middle school. The difference scores are calculated on the basis of subtracting the response time for the less challenging task from the response time for the more challenging task, such that a larger difference score implicates a stronger influence of the increased demands between the two tasks.

cessing efficiency; whereas it is questionable whether these tasks even pose an executive load for their peers. We now turn to performance at each grade level to address the point at which any group differences emerged or disappeared during the study time period. 4.2. Analyses by grade For the CNT, RAN, and Calculations measures, a series of Mann-Whitney tests was conducted at each of first, third, and fifth grades to examine group differences in RT. Sixth and seventh grade performance was not considered in order to avoid redundancy with the growth curve models, which evaluated group differences relative to performance at the final time point included in the study (i.e., in 6th or 7th grade). Nonparametric statistics were used because the relatively small sample sizes in both groups made normality of the data difficult to determine. For each test, Z scores are reported based on two-tailed p-values. When necessary, these scores are adjusted for tied rankings. Also, in order to avoid further redundancy with the growth curve models, FSIQ was not included as a covariate except in the case of Digit Span (discussed subsequently). For all of the following analyses, see Tables 3 and 4 for a summary of the means and standard deviations in the fragile X and comparison groups, respectively. Overall, the results are consistent with our previous reports from 3rd grade (Kirk et al., 2005) and 5th grade (Murphy & Mazzocco, 2008). WJ-R calculations: Girls with fragile X had lower standard scores on calculations than girls in the comparison group at first (Z = −3.53, p < .001, d = 1.27), third (Z = −3.70, p < .001, d = 1.12), and fifth grades (Z = −3.32, p = .001, d = 1.07). 4.2.1. Digit span In 5th grade, a 2 (fragile X, comparison) × 2 (digit span direction: forward, backward) analysis of covariance (ANCOVA) with repeated measures on the second factor was conducted on the number

442

Table 3 Descriptives for fragile X syndrome group. Variable

Assessment age 6–7 years n = 12

10–11 years n = 15

12–13 years (grade 6) 12–13 years (grade 7) n = 10 n = 10

SD

M

SD

M

SD

M

SD

M

SD

Age at testing WJ-R Calc raw WJ-R Calc SS SBIV MD digit span forward SBIV MD digit span backward SBIV MD forward-backward discrepancy

7.43 5.50 84.67*** – – –

0.51 3.53 14.91 – – –

9.20 10.75 75.31*** – – –

0.70 5.70 31.62 – – –

11.29 18.93 83.27*** 4.47 2.73 1.73

0.52 6.79 19.55 1.13 1.49 1.53

12.60 22.40 83.90*** – – –

0.47 7.21 19.33 – – –

12.92 22.50 82.50 – – –

0.58 7.23 18.82 – – –

CNT Warm-up RT total Errors

67.08 2.92

15.59 5.58

62.69* 1.44

12.86 2.07

52.67** 2.27*

12.61 4.56

– –

– –

52.20*** 0.30

15.22 0.48

One-attribute RT totala Errors

83.33 8.67

22.38 10.75

68.50 4.88*

21.98 5.44

52.80 4.73*

11.23 8.09

– –

– –

50.50** 0.50

15.56 0.85

Response time differences RAN Color RT–RAN Number RT Warm-up RT–RAN Color RT One-attribute RT–Warm-up/2 RT

19.42* 6.08 49.79

12.52 14.96 21.93

17.50 9.44 37.16

10.24 11.71 21.46

19.60* 6.20 26.47

12.99 9.73 8.69

– – –

– – –

16.20 9.10 24.40*

61.00 41.58 40.92

13.88 9.19 11.48

53.25* 35.75 35.13*

13.67 10.71 9.37

46.47** 26.87 27.00

11.67 8.90 7.97

– – –

– – –

43.10*** 26.90** 28.30**

RAN Colors RT RAN Numbers RT RAN Letters RT

9.13 9.63 9.15 9.80 8.03 10.11

Note: WJ-R = Woodcock Johnson Revised; SS = Standard Score; SBIV MD = Stanford Binet IV Memory for Digits; CNT = Contingency Naming Test; RT = Response Time in seconds; RAN = Rapid Automatized Naming; Response times were rounded to the nearest second. Note: CNT, RAN based on grade 7; Calc based on grade 6. a Scores at 6–7 years reflect the combination of observed and imputed scores. No imputing was necessary at the remaining age ranges so these scores reflect only observed values. * p < .05. ** ***

p ≤ .01. p ≤ .001.

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

M

8–9 years n = 16

Table 4 Descriptives for comparison group. Variable

Assessment age (n = 19) 6–7 years

CNT Warm-up RT total Errors One-attribute RT totala Errors Response time differences RAN Color RT–RAN number RT Warm-up RT–RAN Color RT One-attribute RT–warm-up/2 RT RAN Colors RT RAN Numbers RT RAN Letters RT

M

10–11 years

SD

6.69 6.84 106.21*** – – –

0.26 2.65 12.33 – – –

8.66 15.00 106.84*** – – –

67.53 2.05

17.70 3.33

50.84* 0.58

8.73 1.31

101.21 7.37

34.36 7.07

61.89 1.63*

10.53* 11.79 67.45

8.49 14.09 33.01

55.74 45.21 38.32

17.31 16.35 9.14

12–13 years (grade 6) 12–13 years (grade 7)

SD

M

SD

0.32 3.23 13.70 – – –

10.67 23.37 103.74*** 4.74 3.42 1.32

0.27 3.75 13.27 0.99 0.84 1.06

M 11.80 28.05 105.68*** – – –

SD 0.24 4.42 14.72 – – –

M

SD

12.80 – – – – –

0.27 – – – – –

40.58** 0.26*

6.66 0.73

– –

– –

35.89*** 0.16

7.58 0.50

14.71 2.39

47.32 0.68*

9.78 1.06

– –

– –

36.47** 0.53

7.82 0.96

14.89 6.21 36.47

8.55 9.06 13.46

11.21* 5.11 27.03

4.38 4.74 9.37

– – –

– – –

10.26 6.11 18.53*

3.75 4.12 6.27

44.63* 29.74 28.26*

11.07 6.39 5.31

35.47** 24.26 24.53

7.21 5.77 4.98

– – –

– – –

29.79*** 19.53** 20.26**

5.38 3.55 4.28

Note: WJ-R = Woodcock Johnson Revised; SS = Standard Score; SBIV MD = Stanford Binet IV Memory for Digits; CNT = Contingency Naming Test; RT = Response Time in seconds; RAN = Rapid Automatized Naming; Response times were rounded to the nearest second. Note: CNT, RAN based on grade 7; Calc based on grade 6. a Scores at 6–7 years reflect the combination of observed and imputed scores. No imputing was necessary at the remaining age ranges so these scores reflect only observed values. * p < .05. ** ***

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

Age at testing WJ-R Calc raw WJ-R Calc SS SBIV MD digit span forward SBIV MD digit span backward SBIV MD forward-backward discrepancy

8–9 years

M

p ≤ .01. p ≤ .001.

443

444

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

Fig. 4. Growth in response time group means (and standard errors) for difference scores on a working memory and warm-up task of the CNT, administered from primary to middle school. A larger difference score implicates a stronger influence of the increased working memory demands between the CNT one attribute versus warm-up tasks.

of digits recalled in the forward condition and backward condition. We included FSIQ as a covariate because no corresponding growth curve models using FSIQ as a predictor were computed. No group difference or Group × Digit Span Direction interaction was found. The main effect of Digit Span Direction reflected that, for both groups, forward digit span was longer than backward digit span, F(1, 31) = 4.23, p = .048). 4.2.2. CNT On the warm-up trial, no group differences in RT were found in first grade. By 3rd grade, girls with fragile X took significantly longer than girls in the comparison group to name the items (Z = −2.49, p = .013, d = 0.97). This difference in RT continued into 5th grade (Z = −2.97, p = .003, d = 1.07). Together with the results of the growth curve model (discussed previously), these findings suggest that by 3rd grade, girls with fragile X fall behind their peers on basic naming tasks. On the one-attribute trial in 1st grade, analyses were conducted first using the observed scores and then using the combined observed and imputed scores, as recommended by McCartney, Burchinal, and Bub (2006, pp. 52–53). As with other analyses involving imputed scores, no outcome differences emerged with the use of imputed scores, so only the results from the combined observed and imputed scores are reported here. RTs on the one-attribute trial did not differ between girls with fragile X and their peers at 1st, 3rd, or 5th grades. Thus, when combined with the results of the growth curve model (discussed previously), these findings suggest that differences in RT do emerge for girls with fragile X relative to the comparison group, but not until 7th grade (d = 1.10). 4.2.3. RAN No differences in RT were apparent at first grade for naming Colors; however, by third grade, girls with fragile X had slower RTs than the comparison group (Z = −2.31, p = .021, d = 0.67). This group difference in RT persisted at 5th grade (Z = −2.87, p = .004, d = 1.02) and 7th grades (see growth curve model, d = 1.40). Differences in RT for naming Letters were only apparent at 3rd grade (Z = −2.26, p = .02,

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

445

d = 0.90) and 7th grade (see growth curve model, d = 1.04). In contrast, no differences in RT for naming Numbers were apparent until 7th grade (as indicated in the growth curve model). 4.2.4. Difference scores The difference in RT on Colors—Numbers was larger for girls with fragile X than in the comparison group at 1st and 5th grades (p = .028, d = 0.81 and p = .043, d = 0.84, respectively), and in 7th grade (see growth curve model). However, no group difference was found in 3rd grade. RT differences for the Warm-up—Colors difference did not distinguish girls with fragile X and their peers at 1st, 3rd, or 5th grades. However, results from the growth curve analyses indicate that a difference does emerge for girls with fragile X relative to the comparison group by 7th grade (d = 0.89). A similar pattern was apparent for the one-attribute—Warm-up difference scores. Specifically, no group differences were found at 1st, 3rd, or 5th grade, but were apparent by 7th grade (see growth curve model, d = 0.76). 4.3. Predictors of math achievement: FSIQ and CNT The findings reported above implicate the CNT warm-up RT score as a candidate predictor for mathematics achievement. Thus, 3rd grade RT on the CNT warm-up was added to Model 1 (presented earlier) for the Calculations task, as was FSIQ. Model 2 (Table 2) reflects the addition of these two predictors to Model 1 presented at the beginning of the results section. When the predictors are held constant, the 6th grade score on Calculations for girls with fragile X is 24.80 problems correct and the growth rate is 3.86 problems per year. FSIQ and CNT warm-up RT are each significant predictors of 6th grade Calculation scores and growth rate. Specifically, children with higher FSIQ scores or faster warm-up RTs have higher Calculation scores in 6th grade and grow at a faster rate than children with lower FSIQ scores or slower warm-up RTs. Once FSIQ and CNT are in the model, there is no longer a difference between groups in scores at 6th grade or in rate of growth. 5. Discussion The present study was a quasi-longitudinal investigation of mathematics achievement and working memory thresholds among girls with versus without fragile X syndrome. Consistent with our predictions, the findings support the notion that poor math performance among girls with fragile X is a stable characteristic and that efficient executive functions accounts for some growth in math achievement over time. Moreover, the trajectory of executive function skills among girls with fragile X distinguishes them from their peers. Finally, the performance levels impacted by increases in working memory load suggest that, relative to peers, girls with fragile X have a lower threshold for being able to overcome the demands imposed by increased executive load. Nevertheless, we observed improvement in executive skills of girls with fragile X after 10 years of age rather than the performance plateau as has been reported for more general cognitive tasks (Fisch et al., 1999). 5.1. Working memory Consistent with earlier reports of spared verbal memory (Bennetto et al., 2001), digit span did not distinguish girls with fragile X from their peers. Both groups of participants have longer forward than backward digit spans, to comparable degrees. The lack of an interaction between digit span direction (forward, backward) and group suggests that memory span per se is not problematic for girls with fragile X, particularly for verbal stimuli. Instead, working memory constraints and processing speed differentiated the two groups of participants. By 7th grade (approximately age 12–13 years), girls with fragile X have longer RTs than girls in the comparison group on the one-attribute CNT subtests and even the warm-up trial, the latter of which is typically not considered a working memory task. No group differences are evident in growth rate over time on the warm-up trial; but over time, the RTs of girls with fragile X on the CNT’s one-attribute subtest improve (i.e., decrease or get faster) at a slower rate than for the comparison group.

446

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

The grade at which group differences emerge varies as a function of the working memory demands, such that group differences in RT are apparent by 3rd grade (8–9 years) on the warm-up trial, and at 5th grade for the one-attribute trial (10–11 years). As reflected by the moderate effect size (Cohen’s d = 0.52), the mean difference in RT is greater for girls with fragile X (M = 52.80 s) relative to the comparison group (M = 47.32 s). Although not statistically significant (p = .068), the greater mean RT differences among girls with fragile X may be of clinical significance. That is, relative to their peers, the same task may be more effortful for girls with fragile X and therefore may place demands on working memory over and above those expected based on age or grade when completing school work or other tasks. Lack of differences in RT at earlier grades may reflect difficulty with this trial in both groups of children (Kirk et al., 2005). Together these findings are consistent with the notion that working memory tasks are even more challenging for girls with fragile X than expected given their chronological age and level in school, despite the reported strength in verbal skills (further supported by a forward and backwards digit span comparable to their peers). In normative studies of the CNT, efficient cognitive flexibility on the two-attribute subtest at 5th grade is comparable to efficiency on the one-attribute subtest at 3rd grade (Mazzocco & Kover, 2007). A similar comparison is evident for girls with fragile X between the warmup and one-attribute subtests between 3rd and 7th grades, reflected in both their RTs and performance accuracy. The slower RT on the warm-up subtest at 3rd grade suggests that the warm-up remains effortful for girls with fragile X at about age 8 years (3rd grade) when their performance is as accurate as that of their peers despite longer RTs. This trade of speed for accuracy is in contrast to 3rd grade performance as the working memory demands of the task increase: Third graders with fragile X are less accurate than their peers on the one-attribute trial, despite comparable RTs, thus trading accuracy for speed. Differences in the speed/accuracy trade-off as a function of the working memory demands of the task could reflect the relative difficulty of each subtest. Thus, an automatic task for 3rd graders without fragile X (e.g., CNT warm-up) may be a working memory task for those with fragile X; and a working memory trial (e.g., one-attribute trial) in 3rd grade may exceed the working memory abilities of girls with fragile X in the same way that the two-attribute task (excluded in the present study) is too challenging for 59% of typically developing first graders (as reported by Mazzocco & Kover, 2007). If performance of girls with fragile X reflects a similar developmental shift in RT slowing in response to working memory demands between 3rd and 5th grades (relative to children in the general population), at 5th grade their performance on the one-attribute task should be comparable to their warm-up performance at 3rd grade. That is, RT on the one-attribute trial should be longer among girls with fragile X relative to the comparison group, but their performance should be just as accurate. In 5th grade, however, the warm-up remains a difficult task, as evidenced by longer RTs with minimal improvement in accuracy. On the warm-up in 5th grade, the number of errors made by girls with fragile X is significantly higher than that of their peers because of improvements made by the latter in both RT and accuracy between 3rd and 5th grades. Additionally, elevated error rates on the one- and twoattribute trials, but comparable RTs, may reflect continued difficulty of girls with fragile X on these tasks. Indeed, there is little change in the frequency of errors between 1st and 5th grades for girls with fragile X. An increased number of errors among girls with fragile X relative to peers supports the notion that girls with fragile X have a lower threshold for working memory demands than their peers (Kirk et al., 2005; Rivera, Menon, White, Glaser, & Reiss, 2002). By 7th grade, the speed/accuracy shift occurs on the one-attribute trial for girls with fragile X, relative to previous grades. On both the one- and two-attribute tasks, RTs are longer for girls with fragile X, but their performance is now as accurate as their peers’. This switch in speed-accuracy trade off between 5th and 7th grades may reflect executive function gains that permit task completion. It also reflects the performance ceiling reached by the comparison group such that significant group differences are unlikely. To further explore the notion that tasks considered to be automatic for children in the general population are more effortful for girls with fragile X, consider performance on the RAN, which is a measure associated with relatively “automatic” processing and retrieval. Girls with fragile X have longer RTs than their peers on all three RAN subtests. Longer RTs in 7th grade suggest that processing and retrieval on these subtests require more effort for girls with fragile X as expected based on their

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

447

age or grade. Although RAN performance generally reflects automaticity, the relative degree of effort required across subtests is relevant to the issue of working memory if we consider the difference in RT and growth across subtests. The series of RAN-CNT difference scores implicates the increase in RT associated with increasing working memory demands. Larger difference scores reflect a greater discrepancy in automaticity between the subtests. As expected, adding to working memory demands of a task increases the RT required to complete the task, in both the fragile X and comparison groups (based on visual inspection of the data in Fig. 3). However, the difference scores suggest that the degree to which RT increases is more pronounced for girls with fragile X than for their peers. For example, as task demands change from naming numbers to colors, 7th grade girls with fragile X have longer RTs than their peers. The peers show decreases in amount of effort required for naming colors versus numbers such that there is no noticeable difference by 7th grade; but 7th grade girls with fragile X continue to experience color naming as more effortful than number naming. Further, we hypothesized that differences in the complexity of the stimuli on the CNT warm-up versus RAN Colors would add incrementally to the processing demands of the Color subtest (as reported in Section 3), and lead to the warm-up trial serving as a measure of working memory performance for those with a low threshold of working memory. The warm-up-Color difference score is not significantly different between the two groups, perhaps because both groups had longer response times on the CNT warm-up than on Colors, suggesting an increase in the effort required for the warm-up versus Colors for all participants. More importantly, although difference scores were comparable between groups, both Colors and CNT warm-up are effortful for girls with fragile X, as indicated by longer RTs. The pattern of group differences from 3rd to 7th grade may reflect how much harder both tasks are for girls with fragile X in contrast to the relative ease of both tasks for children in the comparison group. The significant group difference observed for the one-attribute—warm-up difference score further supports the notion of a lower threshold for working memory in girls with fragile X relative to their peers. Specifically, as the working memory demands increase from the warm-up to the one-attribute trial, 7th grade girls with fragile X again have longer RT difference scores. Not only is the one-attribute task more effortful for girls with fragile X relative to the warm-up, the rate at which the one-attribute trial is becoming easier relative to the warm-up is slower among girls with fragile X relative to peers. In contrast, group differences in growth rate were not apparent for the other two differences scores. Such a pattern suggests that tasks that are not typically considered to be working memory tasks for children in the comparison group, such as the warm-up, are working memory tasks for girls with fragile X. Of note, however, the difficulty of girls with fragile X with traditional working memory tasks may reflect a lower working memory threshold than that their peers, not that they are unable to complete working memory tasks. The notion of a lower working memory threshold among girls with fragile X is consistent with findings from previous studies (Kwon et al., 2001; Rivera et al., 2002). However, despite converging evidence, the precise mechanisms of this difficulty remain unclear. Although the construct of working memory is still widely debated and explored (Miyake, Friedman, Emerson, Witzki, & Howerter, 2000), one framework for interpreting the present findings is that of Roberts and Pennington (1996), who suggest that formulating a response requires both inhibition of alternate responses and the working memory resources to generate the appropriate response. Interpreted within this framework, difficulty with the CNT warm-up may reflect difficulty inhibiting extraneous information, such as the stimuli color when naming the shape, which was not intended to be part of the original task. Their model further predicts that as working memory demands of a task increase, the ability to inhibit alternate responses decreases, resulting in more errors than on tasks with fewer working memory demands. Elevated rates of errors among girls with fragile X on the working memory subtests relative to peers are consistent with this prediction. However, the working memory subtest (one-attribute trial) may be difficult because of its cognitive switching demands or because both cognitive switching and response inhibition are required. Regardless, the findings in the present study suggest that the use of the CNT among girls with fragile X (and possibly other populations with deficits in executive function) may be expanded upon by including the RAN subtests as a means by which to assess incremental changes in working memory demands and by paying more attention to performance on the warm-up trial.

448

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

5.2. The association between mathematics and working memory Exploring the factors that may contribute to difficulty in mathematics is critical for understanding pathways that may lead to math difficulty and routes to effective intervention. Lower FSIQ among girls with fragile X is not the sole predictor of math performance. Both FSIQ and 3rd grade CNT warm-up contributed significantly to the trajectory and growth rate on Calculations, highlighting the contribution of working memory to math achievement as early as 3rd grade. Despite a growing literature supporting the association between mathematics performance and working memory (Passolunghi, Mammarella, & Altoe, 2008), the precise nature of that association remains unclear. Our findings suggest that the tendency of girls with fragile X to trade accuracy for speed as the working memory demands of a task increase may contribute to poor math performance. If there is a direct relationship between math and working memory, the relative improvements in working memory among girls with fragile X noted by 7th grade suggests the potential for improvements in math performance rather than a plateau during the middle school years. Together the findings in the present study implicate the importance of early intervention in mathematics for girls with fragile X. Third grade is an important time by which to target intervention because it is the grade by which group differences in working memory and processing demands emerge. Of note, however, is that WJ-R Calculations is designed to measure performance on only a subset of specific math skills, including single and multi-digit calculations. Other more basic math skills, such as counting and magnitude judgments, may contribute to competency with calculations. Difficulty in these basic math skills are documented among primary school age girls with fragile X (Murphy et al., 2006) and may contribute to poor performance on calculations. As such, successful intervention efforts for girls with fragile X may require addressing poor calculation skills, the supporting numerical skills, and deficits in working memory. Acknowledgements This research was supported by NIH grant R01 HD-034061-01 to 09, awarded to Dr. Mazzocco. Additional support was provided by a National Fragile X Foundation Summer Rosen Fellowship awarded to Stacy Chung. The authors wish to thank the many people who contributed to this research, including the participants and their families, the faculty and staff at our participating schools in the Baltimore County Public School District, former research coordinator Gwen F. Myers, and former research assistants Stacy Chung and Anne Henry. The authors also acknowledge research assistant Munisa Bachu for assistance preparing figures. References Anderson, P., Anderson, V., Northam, E., & Taylor, H. G. (2000). Standardization of the contingency naming test (CNT) for schoolaged children: A measure of reactive flexibility. Clinical Neuropsychological Assessment, 1, 247–273. Andersson, U., & Lyxell, B. (2007). Working memory deficit in children with mathematical difficulties: A general or specific deficit? Journal of Experimental Child Psycholology, 96, 197–228. Baddeley, A. (1996). Exploring the central executive. Quarterly Journal of Experimental Psychology A: Human Experimental Psychology, 49, 5–28. Baddeley, A. (2003). Working memory: Looking back and looking forward. Nature Reviews Neuroscience, 4, 829–839. Bennetto, L., Pennington, B. F., Porter, D., Taylor, A. K., & Hagerman, R. J. (2001). Profile of cognitive functioning in women with the fragile X mutation. Neuropsychology, 15, 290–299. Berch, D. B. (2008). Working memory and mathematical cognitive development: Limitations of limited-capacity resource models. Developmental Neuropsychology, 33, 427–446. Brainard, S. S., Schreiner, R. A., & Hagerman, R. J. (1991). Cognitive profiles of the carrier fragile X woman. American Journal of Medical Genetics, 38, 505–508. Bull, R., Johnston, R. S., & Roy, J. A. (1999). Exploring the roles of the visual-spatial sketch pad and central executive in children’s arithmetical skills: Views from cognition and developmental neuropsychology. Developmental Neuropsychology, 15, 421–442. Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of children’s mathematics ability: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293. Burchinal, M., Nelson, L., & Poe, M., IV. (2006). Growth curve analysis: An introduction to various methods for analyzing longitudinal data. Monographs of the Society for Research in Child Development, 71, 65–87. Cornish, K., Swainson, R., Cunnington, R., Wilding, J., Morris, P., & Jackson, G. (2004). Do women with fragile X syndrome have problems in switching attention: Preliminary findings from ERP and fMRI. Brain and Cognition, 54, 235–239.

M.M. Murphy, M.M.M. Mazzocco / Cognitive Development 24 (2009) 430–449

449

Denckla, M. B., & Rudel, R. G. (1976). Rapid “automatized” naming (R. A. N): Dyslexia differentiated from other learning disabilities. Neuropsychologia, 14, 471–479. Fisch, G. S., Carpenter, N., Holden, J. J., Howard-Peebles, P. N., Maddalena, A., Borghgraef, M., et al. (1999). Longitudinal changes in cognitive and adaptive behavior in fragile X females: A prospective multicenter analysis. American Journal of Medical Genetics, 83, 308–312. Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 1343–1359. Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74, 834–850. Kellogg, J. S., Hopko, D. R., & Ashcraft, M. H. (1999). The effects of time pressure on arithmetic performance. Journal of Anxiety Disorders, 13, 591–600. Kemper, M. B., Hagerman, R. J., Ahmad, R. S., & Mariner, R. (1986). Cognitive profiles and the spectrum of clinical manifestations in heterozygous fra (X) females. American Journal of Medical Genetics, 23, 139–156. Kirk, J. W., Mazzocco, M. M. M., & Kover, S. T. (2005). Assessing executive dysfunction in girls with fragile X or Turner syndrome using the Contingency Naming Test (CNT). Developmental Neuropsychology, 28, 755–777. Kwon, H., Menon, V., Eliez, S., Warsofsky, I. S., White, C. D., Dyer-Friedman, J., et al. (2001). Functional neuroanatomy of visuospatial working memory in fragile X syndrome: Relation to behavioral and molecular measures. American Journal of Psychiatry, 158, 1040–1051. Lasker, A., Mazzocco, M. M. M., & Zee, D. (2007). Ocular motor indicators of executive dysfunction in females with fragile X or Turner syndrome. Brain and Cognition, 63, 203–220. Mazzocco, M. M. M. (1998). A process approach to describing mathematics difficulties in girls with Turner syndrome. Pediatrics, 102, 492–496. Mazzocco, M. M. M. (2001). Math learning disability and math LD subtypes: Evidence from studies of Turner syndrome, fragile X syndrome, and neurofibromatosis type 1. Journal of Learning Disabilities, 34, 520–533. Mazzocco, M. M. M., Hagerman, R. J., Cronister-Silverman, A., & Pennington, B. F. (1992). Specific frontal lobe deficits among women with the fragile X gene. Journal of the American Academy of Child and Adolescent Psychiatry, 31, 1141–1148. Mazzocco, M. M. M., & Kover, S. T. (2007). A longitudinal assessment of executive function skills and their association with math performance. Child Neuropsychology, 13, 18–45. Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematics learning disability in the primary school-age years. Annals of Dyslexia, 53, 218–253. Mazzocco, M. M. M., Pennington, B. F., & Hagerman, R. J. (1993). The neurocognitive phenotype of female carriers of fragile X: Additional evidence for specificity. Journal of Developmental and Behavioral Pediatrics, 14, 328–335. Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten Predictors of Math Learning Disability. Learning Disabilities Research & Practice, 20, 142–155. McCartney, K., Burchinal, M. R., & Bub, K. L. (2006). Best practices in quantitative methods for developmentalists. Monographs of the Society for Research in Child Development, 71. Miyake, A., Friedman, N. P., Emerson, M. J., Witzki, A. H., & Howerter, A. (2000). The unity and diversity of executive functions and their contributions to complex ‘frontal lobe’ tasks: A latent variable analysis. Cognitive Psychology, 41, 49–100. Murphy, M. M., & Mazzocco, M. M. M. (2008). Mathematics learning disability in girls with Turner syndrome or fragile X syndrome during late elementary school. Journal of Learning Disabilities, 41, 29–46. Murphy, M. M., Mazzocco, M. M. M., Gerner, G., & Henry, A. E. (2006). Mathematics learning disability in girls with Turner syndrome or fragile X syndrome. Brain and Cognition, 61, 195–210. Murphy, M. M., Mazzocco, M. M. M., Hanich, L., & Early, M. (2007). Cognitive characteristics of children with Mathematics Learning Disability (MLD) varies as a function of criterion used to define MLD. Journal of Learning Disabilities, 40, 467–487. Passolunghi, M. C., Mammarella, I. C., & Altoe, G. (2008). Cognitive abilities as precursors of the early acquisition of mathematical skills during first through second grades. Developmental Neuropsychology, 33, 229–250. Rivera, S. M., Menon, V., White, C. D., Glaser, B., & Reiss, A. L. (2002). Functional brain activation during arithmetic processing in females with fragile X Syndrome is related to FMR1 protein expression. Human Brain Mapping, 16, 206–218. Roberts, R. J., Jr., & Pennington, B. F. (1996). An interactive framework for examining prefrontal cognitive processes. Developmental Neuropsychology, 12, 105–126. Sattler, J. M. (2001). Assessment of children: Cognitive applications (4th ed.). San Diego, CA: Jerome M. Sattler, Publisher, Inc. Schalock, R. L., Luckasson, R. A., Shogren, K. A., Borthwick-Duffy, S., Bradley, V., Buntinx, W. H. E., et al. (2007). The renaming of mental retardation: Understanding the change to the term intellectual disability. Intellectual and Developmental Disabilities, 45, 116–124. Stringer, R., Toplak, M. E., & Stanovich, K. E. (2004). Differential relationships between RAN performance, behaviour ratings, and executive function measures: Searching for a double dissociation. Reading and Writing: An Interdisciplinary Journal, 17, 891–914. Swanson, H. L., & Beebe-Frankenberger, M. (2004). The relationship between working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471–491. Taylor, H. G., Albo, V. C., Phebus, C. K., Sachs, B. R., & Bierl, P. G. (1987). Postirradiation treatment outcomes for chidlren with acute lymphocytic leukemia: Clarification of risks. Journal of Pediatric Psychology, 12, 395–411. Thorndike, R. L., Hagen, E. P., & Sattler, J. M. (1986). Stanford-Binet intelligence scale: Guide for administering and scoring the fourth edition. Chicago, Illinois: The Riverside Publishing Company. Wagner, R. K., Torgesen, J. K., & Rashotte, C. A. (1999). Comprehensive test of phonological processing (CTOPP). Austin, TX: Pro-Ed. Wechsler, D. (1991). Wechsler intelligence scale for children-III. San Antonio, TX: The Psychological Corporation. Wechsler, D. (1999). Wechsler Abbreviated Scale of Intelligence: WASI. San Antonio, Texas: The Psychological Corporation. Woodcock, R. W., & Johnson, M. B. (1990). Woodcock-Johnson psycho-educational battery—Revised. Allen, Texas: DLM Teaching Resources.