Mathematics learning disability in girls with Turner syndrome or fragile X syndrome

Brain and Cognition 61 (2006) 195–210 www.elsevier.com/locate/b&c

Mathematics learning disability in girls with Turner syndrome or fragile X syndrome Melissa M. Murphy a,b, Michèle M.M. Mazzocco a,b,c,¤, Gwendolyn Gerner b, Anne E. Henry b a

Johns Hopkins School of Medicine, Department of Psychiatry and Behavioral Sciences, Baltimore, MD, USA Kennedy Krieger Institute, 3825 Greenspring Avenue, Painter Bldg, Top Floor, Baltimore, MD 21211, USA c Department of Population and Family Health Sciences, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA b

Accepted 31 December 2005 Available online 24 February 2006

Abstract Two studies were carried out to examine the persistence (Study 1) and characteristics (Study 2) of mathematics learning disability (MLD) in girls with Turner syndrome or fragile X during the primary school years (ages 5–9 years). In Study 1, the rate of MLD for each syndrome group exceeded the rate observed in a grade-matched comparison group, although the likelihood of MLD persisting through the primary school years was comparable for all three groups. In Study 2, formal and informal math skills were compared across the syndrome groups, a normative group, and children from the normative group who had MLD. Few diVerences were observed between the Turner syndrome and normative groups. Despite having rote counting and number representation skills comparable to those in the normative group, girls with fragile X had diYculty with counting rules (e.g., cardinality, number constancy). However, this diYculty did not distinguish them from the MLD group. Overall, counting skills appear to distinguish the Turner syndrome and fragile X groups, suggesting that the speciWcity of math deWcits emerges earlier for fragile X than Turner syndrome. © 2006 Elsevier Inc. All rights reserved. Keywords: Fragile X syndrome; Turner syndrome; Mathematics learning disability

1. Introduction Turner syndrome and fragile X syndrome are two X-chromosome associated disorders, both of which are linked to poor math performance (Bennetto, Pennington, Porter, Taylor, & Hagerman, 2001; Bruandet, Molko, Cohen, & Dehaene, 2004; Grigsby, Kemper, Hagerman, & Myers, 1990; Kemper, Hagerman, Ahmad, & Mariner, 1986; Mazzocco & McCloskey, 2005; Rovet, 1993; Rovet, Szekely, & Hockenberry, 1994; Temple, Carney, & Mullarkey, 1996; Temple & Marriott, 1998). Indeed, children with either syndrome are more likely than children from age, grade, and IQ matched comparison groups to meet criteria for math learning disability (MLD) even as early as kindergarten (Mazzocco, 2001). However, it is unclear whether MLD in young children with

*

Corresponding author. E-mail address: [email protected] (M.M.M. Mazzocco).

0278-2626/$ - see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.bandc.2005.12.014

Turner syndrome or fragile X represents a persistent phenotypic characteristic or a short term delay. In addition, the nature of the math diYculties in either syndrome may vary substantially due to diVerences in their respective cognitive phenotype (Bennetto et al., 2001; Mazzocco, 1998, 2001; Mazzocco & McCloskey, 2005; Molko et al., 2003; Rivera, Menon, White, Glaser, & Reiss, 2002; Rovet, 1993, 2004; Rovet & Buchanan, 1999; Rovet et al., 1994). Although there is very little research comparing math performance in children with Turner syndrome versus fragile X, there is theoretical support for the notion that the nature of math diYculties may diVer because of these contrasting phenotypes. As such, exploring MLD and its manifestation in syndromes with known genetic causes may inform our understanding of variations in underlying sources of math diYculties. Towards that end, the present study was designed to examine both the persistence of early math diYculties during the primary school years and the nature of those diYculties among children with Turner syndrome or fragile X.

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1.1. Turner syndrome Turner syndrome results from the partial or complete loss of one of the two X chromosomes typically present in females. Its prevalence is approximately 1 in 1900 live female births (Davenport, Hooper, & Zeger, in press). One consequence of X monosomy is that the ovaries of females with Turner syndrome fail to develop, resulting in a lack of estrogen production (Ross & Zinn, 1999). Estrogen may inXuence performance, particularly on verbal and nonverbal memory tasks (Ross, Roeltgen, Feuillan, Kushner, & Cutler, 2000), and may contribute to the cognitive phenotype associated with Turner syndrome (McCauley, Kay, Ito, & Treder, 1987; Ross & Zinn, 1999; Ross et al., 2000). Females with Turner syndrome do not typically meet criteria for mental retardation; however, they may have learning disabilities, particularly in the area of mathematics (Rovet, 1993). 1.2. Fragile X syndrome Fragile X syndrome is the leading known cause of inherited mental retardation. It occurs in approximately 1 in 4000 to 1 in 9000 live births (e.g., Crawford, Acuna, & Sherman, 2001) as the result of a single gene mutation on the long arm of the X-chromosome (Verkerk et al., 1991; Yu et al., 1991). This mutation leads to impaired production of a protein (FMRP) that is important for neural development. Although there is much phenotypic variability in children with the syndrome, most males with fragile X meet criteria for moderate to mild mental retardation (i.e., IQ scores between 36 and 70; Bailey, Hatton, & Skinner, 1998). In contrast, »50% of females with fragile X will have mental retardation (Rousseau et al., 1994), whereas the remaining females may have less severe cognitive impairments including learning disabilities, or may have no noticeable eVects of the syndrome (Cronister, Hagerman, Wittenberger, & Amiri, 1991; Hagerman, Hills, Scharfenaker, & Lewis, 1999). To investigate the subtle aspects of the cognitive phenotype in fragile X, in the present study we limited participation to those individuals with fragile X without mental retardation, which included only females. 1.3. Prevalence and persistence of MLD DiYculties with mathematics in Turner syndrome or fragile X are seen throughout the life span, including the early primary school age years (Grigsby et al., 1990, 1996; Kovar, 1995; Mazzocco, 1998, 2001; Mazzocco, Pennington, & Hagerman, 1993; Miezejeski & Hinton, 1992; Rovet, 1993), the later school years (Buchanan, Pavlovic, & Rovet, 1998; Mazzocco, 1998; Rivera et al., 2002; Rovet, 1993; Rovet et al., 1994; Temple et al., 1996), and adulthood (Bennetto et al., 2001; Bruandet et al., 2004; Grigsby et al., 1990; Mazzocco et al., 1993). Rovet (1993) found that 55% of girls with Turner syndrome between the ages of 6 and 16 years met criteria for MLD, either alone or in combination with

reading disability (RD). (In Rovet’s study, MLD was deWned as performance below the 25th percentile on the Arithmetic subtest of the Wide Range Achievement TestRevised.) This percentage exceeded the rate observed in the comparison group of sixth graders, of whom only 26% met criteria for a learning disability in mathematics, reading, or both. Using an even more conservative criterion than Rovet (i.e., quotient scores below the 10th percentile on the Test of Early Math Ability-second edition; TEMA-2), Mazzocco (2001) reported that 43% of girls with Turner syndrome met criteria for MLD. This percentage was signiWcantly higher than the 10% observed among a gender, grade, age, and IQ matched comparison group of girls without Turner syndrome. There are fewer studies of MLD in girls with fragile X syndrome; however, among girls with fragile X, Mazzocco (2001) reported that Wve of the nine girls with fragile X (56%) included in her initial study met criteria for MLD, deWned as performance below the 10th percentile on the TEMA-2. This prevalence rate was not signiWcantly diVerent from the prevalence rate of 20% observed among an age, grade, and full scale IQ matched comparison group of girls without fragile X. However, when the MLD criterion was broadened to include performance below the 12th percentile on the TEMA-2, the group diVerence became signiWcant: 87% of the girls with fragile X scored below the 12th percentile compared to 20% in the comparison group. Although together these studies indicate greater prevalence of MLD in Turner syndrome or fragile X relative to comparison groups, the studies were cross-sectional, and were not designed to address whether MLD in girls with Turner syndrome or fragile X persists over time, or whether it reXects a transitory delay. Over the school age years, children can vary as to whether they meet criteria for MLD (Francis et al., 2005; Shalev, Manor, Auerbach, & GrossTsur, 1998; Silver, Pennett, Black, Fair, & Balise, 1999). For example, among a relatively normative sample of children recruited from a large suburban school district, Mazzocco and Myers (2003) found that approximately 44% of primary school age children meet investigator-deWned criteria for MLD during at least one of their primary school age years. However, approximately 66% continued to meet this criterion for one or more of the remaining primary school years, whereas the remainder did not meet the criterion more than once (Mazzocco & Myers, 2003). Individual variability over time has led some researchers to suggest that the criteria for MLD must be met at more than one point in time, if the classiWcation of MLD is to be valid (Geary, 2004; Geary, Hamson, & Hoard, 2000). Given the elevated prevalence of math diYculty in both Turner syndrome and fragile X, it is important to assess whether the early MLD observed in girls with Turner syndrome or fragile X persists over time at a rate that matches or exceeds the frequency reported for the general population (Mazzocco & Myers, 2003). Determining the persistence of MLD in Turner syndrome and fragile X will contribute to understanding the degree to which children

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with either syndrome are at risk for MLD. In addition, the longitudinal design of the present study extends previous, cross sectional Wndings that suggest persistent diYculty with math across development. 1.4. Nature of MLD in Turner syndrome and fragile X syndrome Awareness of the prevalence and persistence of MLD serves to describe the extent to which math diYculties occur in a given population, but this information is insuYcient for uncovering the nature of MLD. Mathematical competence depends both on conceptual knowledge of mathematical domains and the relevant procedural knowledge that is used for problem solving in those domains (Geary, 2005). This knowledge is supported by multiple cognitive systems including executive controls (e.g., working memory function, such as attention and inhibition), and language and visuospatial systems (see Geary, 2005 for a detailed summary). As such, MLD could reXect deWcits in conceptual or procedural knowledge in mathematics, or it may reXect deWcits in the underlying cognitive domains (Geary, 1993, 2005). It is unclear which of these alternatives pertains to Turner syndrome and fragile X, because cognitive phenotypes for both disorders include diYculty with mathematics, and deWcits in working memory and visuospatial ability. Both alternatives need to be explored to establish the nature of MLD in Turner syndrome and fragile X. The present study was designed to examine whether girls with Turner syndrome or fragile X can be distinguished from each other, or from their peers with no known syndrome, on the basis of mastery of formal and informal math skills such as reading and writing numbers, judging magnitude, counting, and addition facts. Based on the limited studies presented to date, there is evidence that diVerent MLD proWles exist for these two groups, as summarized below. 1.4.1. Sources of variation in poor math performance Only speciWc aspects of mathematics appear to be problematic for girls with Turner syndrome. For example, simple arithmetic, number comprehension and production, counting, and some aspects of understanding quantity, such as number comparison and estimation, are intact among adults with Turner syndrome (Bruandet et al., 2004). Similarly, some basic aspects of number sense, including counting (Mazzocco, 2001), reading and writing numbers, and magnitude judgments are age appropriate among school age girls (Temple & Marriott, 1998). Yet individuals with Turner syndrome, as a group, perform more poorly on measures of mathematics achievement than do their peers (McCauley et al., 1987; Molko et al., 2003; Rovet, 1993). Girls with Turner syndrome also have signiWcantly lower performance on visual-perceptual and visual-motor tasks relative to their age and grade matched peers (Mazzocco, 2001; Rovet & Netley, 1982; Temple & Carney, 1995), which may be related to their math performance (Mazzocco, 1998; Rovet, 1993). Yet, Rovet et al. (1994) did not

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Wnd a consistent relationship between visual spatial processing and procedural knowledge or math fact retrieval, which lead them to conclude that poor math performance in Turner syndrome is independent of visual spatial abilities. Other researchers have observed a relationship between visual spatial and mathematical skills. Mazzocco (1998) found that, relative to girls with fragile X, girls with Turner syndrome made more errors associated with visual spatial ability, such as alignment errors on math calculation problems. In addition, Mazzocco (1998) found that visual spatial ability (as measured by the Judgment of Line Orientation test) was a strong predictor of math performance in girls with Turner syndrome, but was the weakest predictor of math performance in girls with fragile X. In a later study of MLD in kindergarteners, Mazzocco (2001) reported that girls with fragile X demonstrated lower performance relative to an age, grade, and IQ matched comparison group on the KeyMath-Revised Numeration subtest, which includes items ascertaining number sense, such as counting. Girls with Turner syndrome did not diVer from their comparison group on this subtest. These Wndings suggest that girls with fragile X can be distinguished based on mastery of basic numerosity concepts (Mazzocco, 2001). Taken together, the Wndings summarized above suggest that the two syndrome groups may be characterized by deWcits in speciWc areas of math, such as counting, or by diVerences in the cognitive processes that underlie speciWc math skills (Mazzocco & McCloskey, 2005). In the present study, we Wrst examine whether MLD observed in either syndrome persists over the primary school years. We then examine whether performance on speciWc individual items from three math measures, or composite scores from items reXecting one of several basic number concepts, diVers across these two syndrome groups. Of interest is whether either syndrome group represents a model of distinct math deWcits. 2. Study 1 Findings from earlier studies have demonstrated a higher incidence of mathematics learning disability (MLD) in children with Turner syndrome or fragile X, relative to children with neither disorder (e.g., Mazzocco, 2001; Rovet, 1993). The criteria for MLD that were used in these previous studies were based on one-time assessments. In the present study, we examined the frequency with which children met investigator-determined criteria for MLD at two time points during primary school years. One challenge associated with the study of MLD both in the general population and in genetic syndromes is related to the lack of a consistent, precise deWnition of MLD (Mazzocco & Myers, 2003; Murphy, Mazzocco, Hanich, & Early, 2005). Two approaches to deWning MLD predominate. Earlier deWnitions were based on a discrepancy between IQ and achievement scores, whereas more recent deWnitions rely on low achievement models, such as performance below a given

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criterion. Although both approaches have limitations (Francis et al., 2005), discrepancy-based deWnitions are particularly problematic (e.g., Fletcher et al., 1998; Francis, Fletcher, Shaywitz, Shaywitz, & Rourke, 1996). For example, discrepancy-based deWnitions of MLD are not reliable for distinguishing children with MLD from their peers (Mazzocco & Myers, 2003). Also, the appropriateness of IQ in the deWnition of learning disabilities is questionable (see Siegel, 1989), in part because of the potential impact of a learning disability on IQ scores. In MLD speciWcally, cognitive ability alone does not account for poor math performance (Jordan, Hanich, & Kaplan, 2003; Landerl, Bevan, & Butterworth, 2004). Currently, low achievement deWnitions of MLD that rely on a “cut-oV” criterion for determining “poor” performance are a common approach to deWning MLD (Geary, 2004; Murphy et al., 2005). Therefore, in the present study, we deWne MLD based on low achievement. 2.1. Method 2.1.1. Participants There were three groups of participants, all three of which were drawn from an ongoing longitudinal study of mathematics ability in primary school age children (Mazzocco, 2001; Mazzocco & Myers, 2003). Participants in the normative comparison group were drawn from one of seven public elementary schools in a large metropolitan school district, as described elsewhere in greater detail (Mazzocco & Myers, 2003). Participation in the normative study was open to all English-speaking children enrolled in a regular half-day kindergarten program in one of these seven public schools. Although not a completely random sample, the Wnal sample is representative of students from a large, socio-economically diverse public school district. Schools with high rates of children eligible for reduced or free lunch were excluded from the study as a means by which to exclude participants from the lowest socio-economic background from the study, because low socio-economic status is linked to poor mathematics achievement (Jordan, Kaplan, Nabors Olah, & Locuniak, in press; Leventhal & Brookes-Gunn, 2003). These participants in the longitudinal school-based study were seen annually from kindergarten through third grade. These participants comprised the sample for the present study. Participants with Turner or fragile X syndrome were recruited primarily from newsletters and websites associated with family support groups for either disorder. For children in these two syndrome groups, there were insuYcient data to examine performance at four annual assessments (as was done with the normative study), so performance over time was examined for two assessments only. Inclusion in one of the two syndrome groups was limited to children who had at least one assessment either during kindergarten or Wrst grade, or at the age of 5–7 years; and who also had at least one follow up assessment at second or third grade, or at the age of 7–9 years. Some of these participants were reported on in a previous study (Mazzocco, 2001). The Wnal groups of participants included 210

children from the school-based normative sample, 24 girls with Turner syndrome, and 15 girls with fragile X. Using these selection criteria, participants in either the Turner syndrome or fragile X groups were well aligned with the normative group, at least in terms of age or school grade. Karyotype test results were available to conWrm the diagnosis of Turner syndrome, whereas DNA test results were used to conWrm the presence of full mutation for all of the girls in the fragile X group. Previous studies with these two syndromes suggest that girls with fragile X have lower IQ than girls with Turner syndrome. Therefore, no attempt was made to match the two syndrome groups on FSIQ. Of interest was the prevalence and persistence of MLD in each group of participants. 2.1.2. Procedure During each year of the study, the Test of Early Mathematical Ability—second edition (TEMA-2) was administered as part of the overall testing battery. The TEMA-2 is a test of formal and informal mathematics skills, such as counting, number facts, place value, magnitude judgment, cardinality, reading and writing numbers, and mental calculation. The TEMA-2 is normed for children between the ages of two and eight years. The age-referenced standard scores are based on a mean of 100 and a standard deviation of 15. Investigator-established criteria for MLD were based on TEMA-2 scores that fell below the 10th or 25th percentile for the comparison group, as described elsewhere in detail (Mazzocco & Myers, 2003). Children were grouped according to whether they met criteria for (a) MLD, based on a relatively restrictive 10th percentile cut-oV score from the TEMA-2, (b) borderline risk for MLD, based on whether their TEMA-2 score was higher than the 10th percentile, but below the 25th percentile, and (c) not at risk for MLD, based on whether their TEMA-2 score was above the 25th percentile. 2.1.3. Analyses Of interest was the frequency with which children in either the Turner syndrome or fragile X group met criteria for MLD, relative to children in the normative group. Of particular interest was how often children with Turner syndrome or fragile X had persistent MLD, demonstrated by meeting criteria for MLD at two assessments during their primary school age years. These frequencies have already been reported for children in the normative group, in an earlier report (Mazzocco & Myers, 2003), but not for children with Turner syndrome or fragile X. We used two sets of 2 statistics to compare these frequencies. When expected values fell below Wve in any given cell, we used the Fisher’s Exact statistic as an alternative to the 2. 2.2. Results 2.2.1. Turner syndrome Children with Turner syndrome were more likely than children in the normative group to meet either criteria for MLD during one of their primary school age years, 2 (1, n D 233) D 10.66, p D .0011. Among the 24 girls with Turner

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syndrome, 19 (79%) met criteria for MLD at least one time between grades K and 3, versus 44% of children from the normative group. Among those children who did meet MLD criteria at one point in time, the rate at which children continued to meet MLD criteria during another primary school age year was comparable among the Turner syndrome and comparison groups (84 and 70%, respectively), p D .19. Among those children who met criteria for persistent MLD, there was a signiWcant diVerence in the frequency with which the restrictive or less restrictive criteria of MLD were met, 2 (1, n D 74) D 4.61, Fisher’s Exact p D .04. Most (66%) children with persistent MLD who were from the normative group met only the least restrictive criteria (25th percentile cut-oV), as expected, given that the 25th verses 10th percentile cut-oV criteria will lead to diVerent sample sizes. However, among children with MLD who had Turner syndrome, 70% met the 10th percentile criterion during both evaluations. In summary, girls with Turner syndrome are more likely to have MLD than girls without Turner syndrome; and although they are no more likely than their MLD peers to have a persistent MLD, they are signiWcantly more likely to meet stricter criteria for MLD. 2.2.2. Fragile X syndrome Children with fragile X were more likely than children in the normative group to meet criteria for MLD during one of their primary school age years, 2 (1, n D 224) D 10.22, p D .002. Among the 15 girls with fragile X, 13 (87%) met MLD criteria at least one time, versus 44% of children in the normative group. Among those children who did meet MLD criteria at one point in time, the rate at which children continued to meet MLD criteria during another primary school age year was comparable between the fragile X and normative groups (77 and 70%, respectively), Fisher’s Exact p D .75. However, all 10 of the children with fragile X who had a persistent MLD met the more restrictive MLD criteria (10th percentile), whereas most of the children from the normative group who had persistent MLD met the less restrictive cutoV, 2 (1, n D 74) D 15.18, Fisher’s Exact p < .0001. In summary, girls with fragile X are more likely to have MLD than girls without fragile X. Even though no more likely than MLD peers to have a persistent MLD, they are more likely to meet stricter MLD criteria. 2.3. Discussion The frequency of children never meeting MLD criteria was much lower among children with Turner syndrome or fragile X (21 and 13%, respectively), relative to the frequency observed in the normative group (56%). Among children who met criteria for MLD during any one of their primary school years, MLD was more likely to persist than to not persist, for all three groups. There was no signiWcant increase in this frequency for either the Turner syndrome or fragile X group, relative to the normative group, ps > .09. However, the published frequencies reported by Mazzocco and Myers (2003) for the normative group are based on performance

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criteria during at least two out of four assessments, since each child in the normative group received annual assessments during grades K through 3. It is remarkable, then, that the children with Turner syndrome or fragile X who met criteria for MLD continued to do so, with as great a frequency as children from the normative group, despite having only one opportunity (versus three) to once again score in the MLD range. These longitudinal follow up data demonstrate that math diYculties are common, and persistent, among children with Turner syndrome or fragile X. 3. Study 2 Although both Turner syndrome and fragile X are associated with poor math performance (Bennetto et al., 2001; Rovet, 1993; Temple & Marriott, 1998), the causes of this math diYculty may distinguish these syndrome groups from each other and from the general population. The present study was designed to compare the performance of girls with Turner syndrome or fragile X to their peers, during the early primary school years. Based on previous Wndings that girls with Turner syndrome have intact number processing skills relative to peers (e.g., Temple & Marriott, 1998), but lower performance on visual-perceptual tasks that may be related to math performance (Mazzocco, 1998), we predicted that girls with Turner syndrome would show a relative strength on items measuring number processing, such as reading and writing numbers, counting, and magnitude comparisons. We also predicted that areas of challenge would include items that rely on visual spatial ability. In contrast, based on Wndings suggestive of deWcits in mastery of number processing relative to peers (e.g., Bennetto et al., 2001; Mazzocco, 2001), girls with fragile X were expected to have relative strengths on items involving recognizing numbers, but relative weaknesses on items measuring number sense, such as counting and magnitude comparisons. Although both visual spatial ability and number sense are complex constructs, this study represents an initial attempt to broadly address these constructs and their relationship to math performance in Turner syndrome and fragile X. The overall pattern of deWcits associated with Turner syndrome and fragile X may also distinguish them from children in the general population who meet criteria for MLD. Despite lack of a consensus deWnition of MLD, there is agreement in the Weld that children with MLD are deWcient in conceptual or procedural knowledge associated with areas of mathematics, such as counting, adding, or retrieving number facts from memory (Geary, 2005). Comparing the math performance of children with Turner syndrome or fragile X to children with MLD may reXect whether the math performance proWles associated with these syndromes are consistent with characteristics of children with MLD, or whether either or both of these two syndrome groups represent a potential model of a subtype of MLD. Toward that end, the present study also included participants from the general population who met criteria for MLD during the primary school years, and compared these children to girls with Turner syndrome or fragile X.

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3.1. Method 3.1.1. Participants Four groups of children, including two comparison groups, participated in the present study. The Wrst comparison group was comprised of a normative group of 226 kindergartners (111 boys) recruited from a large, urban public school district (Mazzocco & Thompson, 2005). This sample included 210 children from the normative group in Study 1 and an additional 16 participants enrolled in the longitudinal study for whom fewer than four assessments were completed. The second comparison group was limited to children from the overall normative sample who met criteria for MLD (n D 23). MLD was assessed a priori for the normative group, so children were not screened for mathematics ability prior to entering the study. Instead, MLD was determined retrospectively, based on performance below the 10th percentile on the TEMA-2, after the children completed third grade. SpeciWcally, children were classiWed as having MLD if their performance was below the 10th percentile on the TEMA-2 for at least two years from kindergarten through third grade. Table 1 contains descriptive information for each of the participant groups. As an index of socioeconomic status, parents were asked to indicate the level of education they had attained by the onset of their child’s participation in the study. Note that children with MLD were drawn from six of the seven schools participating in the normative study. Also, since a thorough investigation failed to reveal signiWcant gender diVerences on the math or math-related tasks administered in the present study (Lachance & Mazzocco, 2006), both boys and girls from the comparison groups were included (to maximize statistical power, especially the number of participants in the MLD group). The two syndrome groups included all of the children from Study 1, and additional children who had received only one evaluation during their early school years. Girls with Turner syndrome (n D 28) or fragile X (n D 21) were included in the present study if they were in kindergarten or Wrst grade at the time of assessment, or were within the age range of kindergartners in the normative comparison group (5.03–6.99 years).

istered to obtain a standardized intelligence quotient (IQ) score (see Table 1). A full scale IQ was obtained using eight subtests, which were selected because of their appropriateness for children in kindergarten and Wrst grade. The full scale IQ was based on a mean of 100 and a standard deviation of 16. 3.2.2. Mathematics measures Items were selected from each of three math measures for analysis of individual items and speciWc item combinations. Items one through ten from the SBIV Quantitative subtest (Thorndike et al., 1986) were administered, along with items from the Kindergarten appropriate subtests of the KeyMath-revised (KM-R; Connolly, 1998), including Numeration (items one to Wve), Geometry (items one to ten), Addition (items one to three), and Measurement (items one to six). The SBIV Quantitative subtest measures understanding of numbers less than ten, and basic computational skills. The KM-R Numeration and Geometry subtests measure basic concepts, such as quantity, order, and place value in the former, and spatial attributes and relationships in the latter. The KM-R Addition subtest measures understanding of adding sets of pictured items. The items in the KM-R Measurement subtest focus on determining the relative height, length, and weight of pictured objects in a set, or the length or amount of single items or item sets. Items 1 through 28 from the Test of Early Mathematics Ability—second edition (TEMA-2; Ginsburg & Baroody, 1990) were also administered to measure formal and informal mastery of early mathematical skills (discussed previously). In addition to these math measures, Wve a priori composite scores were calculated. These composite scores were sums of items that required (1) rote counting (e.g., counting aloud to 50, counting backwards), (2) written representation (e.g., reading and writing numbers), (3) knowledge of counting rules (e.g., number constancy, cardinality), and (4) enumeration (e.g., one-to-one correspondence). Note that individual items that comprised the composite scores are indicated in Appendix A. 3.3. Results

3.2. Materials 3.2.1. Cognitive ability For descriptive purposes, the Stanford Binet, fourth edition (SBIV; Thorndike, Hagen, & Sattler, 1986) was admin-

Preliminary analyses revealed that the two syndrome groups did not diVer from each other, or from the normative sample, in the percentage of children whose mother completed high school versus those with at least some

Table 1 Characteristics of participants in Study 2 Group

Sample size No. in kindergarten (No. in grade 1) No. of girls Mean age at testing (range) Stanford Binet IV, full scale IQ (range)

Normative

Turner syndrome

Fragile X

Math learning disability

n D 226 226 (0) 115 5.77 (5.03–6.99) 99.07 (76–133)

n D 28 20 (8) 28 6.49 (5.37–7.78) 90.79 (64–108)

n D 21 14 (7) 21 6.47 (5.10–7.78) 80.57 (58–108)

n D 23 23 (0) 9 5.85 (5.08–6.99) 88.57 (76–106)

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college enrollment (ps > .07). Twenty-eight percent of mothers in the normative sample had an education level of high school level or less, relative to 14 and 10% in the Turner syndrome and fragile X groups, respectively. The MLD group did not diVer from the remaining children in the normative sample in the percentage of children whose parents completed high school versus college enrollment (p D .94). Nor were diVerences found between the MLD and syndrome groups in maternal education level (ps > .24). Twenty-nine percent of mothers in the MLD group had an education level of high school level or less. Separate univariate ANOVAs were conducted on age at testing to compare the two syndrome groups to the normative group, and to children with MLD. Girls with Turner syndrome and fragile X were older than the normative group (ps < .001), and the MLD group (ps 6 .001), but did not diVer from each other. These results are not surprising given the inclusion of Wrst graders in the two syndrome groups. To address the possible confound of age, any analyses that favored a syndrome group were repeated including only kindergarteners in the syndrome group. When appropriate, we report the analyses on both the total sample and the kindergarten-only subsample. Chi square analyses were used to assess the relative frequency with which participants from diVerent groups successfully passed each of the individual items. For a priori determined item sets, scores of 1 (pass) or 0 (fail) were summed across items, and the composite scores were compared across groups using analysis of variance (ANOVA). As with Study 1, 2 analyses were used along with Fisher’s Exact test, when appropriate, to make comparisons between each of the syndrome groups and the normative group for each math measure. To ensure that our results were statistically signiWcant given the number of comparisons conducted, the signiWcance level was adjusted to .01. For ease of interpretation, individual items comprising each of the math skill areas examined are summarized in Appendix A (syndrome versus normative group comparisons) and Appendix B (syndrome versus MLD comparisons), along with the percentage of children who passed each of the items. Group diVerences in frequencies were examined for results of potential clinical signiWcance. In the following sections we discuss primarily those results that are either statistically or clinically signiWcant. 3.3.1. Syndrome group comparisons to the normative sample 3.3.1.1. Item analyses: Turner syndrome. Few remarkable diVerences were found between girls with Turner syndrome and children in the normative group (ps 7 .040). Consequently, the performance of girls with Turner syndrome overall is at a level comparable to that of their peers on the SBIV Quantitative subtest, KM-R Numeration, Geometry, Addition, and Measurement subtests, and the TEMA-2. A statistically signiWcant diVerence was found on one item from the TEMA-2 that measures understanding of one-toone correspondence when counting, 2 (1, n D 254) D 16.96, Fisher’s Exact p D .004; however, this result is not clinically

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signiWcant because the majority of girls with Turner syndrome (89%) passed this item. No single item served to illustrate the nature of early math diYculties in girls with Turner syndrome. 3.3.1.2. Item analyses: Fragile X syndrome. In contrast, several diVerences emerged for girls with fragile X that distinguish their performance from that of their peers, primarily on the KM-R and TEMA-2. Only one signiWcant diVerence was found on the SBIV Quantitative subtest, on an item that required counting two dots (p D .005); however, 90% of girls with fragile X passed this item, and no diVerences were seen on the subsequent (and more diYcult) item, which required counting Wve dots. Thus, the clinical signiWcance of this diVerence is minimal. More meaningful group diVerences, of varying clinical and statistical signiWcance, emerged from select items from the KM-R. On the Geometry subtest of the KM-R, only 1 of 19 girls (5%) with fragile X was able to identify which similar shapes diVered in size compared to 37% of children from the normative group, 2 (1, n D 218) D 7.83, p D .005. Nevertheless, the majority of children from both groups failed this item. Also, although not statistically signiWcant (Fisher’s Exact p D .019), only about half (57%) of girls with fragile X syndrome were able to correctly identify which of several shapes were rectangles compared to 82% of peers. On the Measurement subtest, fewer girls with fragile X (67%) correctly identiWed the longest and shortest items in a set, relative to their peers (88%), 2 (1, n D 245) D 7.28, Fisher’s Exact p D .015. Other noteworthy diVerences were observed on the Addition subtest. Although not statistically signiWcant (Fisher’s Exact p D .017), fewer girls with fragile X syndrome (71%) were able to combine two sets totaling less than Wve than children in the normative group (91%). Not surprisingly, this diYculty extended to combining sets less than ten a task on which 57% of girls with fragile X and 82% of children in the normative group succeeded. However, this group diVerence was not signiWcant, Fisher’s Exact p D .019. It is possible that diYculty with addition is related to diYculty with the application of counting skills. At Wrst glance, it may appear that counting applications are not diYcult for girls with fragile X, because the majority (81%) of girls with fragile X were able to form a set of three or Wve objects. However, nearly all (98%) of children from the normative group could form a set of 3, 2 (1, n D 246) D 18.21, Fisher’s Exact p D .002; and, although not statistically diVerent (p D .017), nearly all (96%) of children from the normative group could form a set of 5. Similarly, when asked to count all of the items in a pictured set of 8, 55% of girls with fragile X correctly counted the items, compared to 84% of children from the normative group, 2 (1, n D 243) D 10.63, Fisher’s Exact p D .003. Also, group diVerences observed on the TEMA-2 suggest diYculty with mastery of counting skills and understanding of quantity among girls with fragile X compared to their peers.

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Girls with fragile X were less likely than their peers to succeed on items that measured one-to-one correspondence, number constancy, and mental number line concepts. Only 81% of girls with fragile X successfully counted the examiner’s Wve Wngers compared to 99.6% (all but one) of the children in the normative group, 2 (1, n D 247) D 33.54, Fisher’s Exact p < .001. Although this diVerence may represent more statistical than clinical signiWcance, diYculty with applied counting was evident on other items as well. All but one (99.6%) of the children in the normative group correctly used one-to-one correspondence when counting 10 objects along with the examiner, whereas only 71% of girls with fragile X passed this item, 2 (1, n D 247) D 55.21, Fisher’s Exact p < .001. When counting independently, 55% of girls with fragile X correctly counted sets of scattered dots, whereas 85% of children from the normative group correctly counted these sets of dots, 2 (1, n D 246) D 11.43, Fisher’s Exact p D .003. In addition, slightly more than half of the girls with fragile X (62%) recognized that merely rearranging objects did not change the total number of objects (number constancy), compared to 90% of peers, 2 (1, n D 247) D 13.65, Fisher’s Exact p D .002. When asked to provide the next number in a series of sequential consecutive single digit numbers (e.g., “6, 7, 8, and then comesƒ?”), 76% of girls with fragile X passed this item compared to 96% of children in the normative group, 2 (1, 247) D 14.13, Fisher’s Exact p D .003. Finally, when asked to determine which of 2 one-digit numbers was closest to a target number, 44% of the girls with fragile X succeeded, in contrast with 73% of children in the normative group, 2 (1, n D 236) D 6.79, p D .009. It is possible that diYculty with addition and counting applications is based upon diYculty with basic rote counting, and that diVerences in such skills would also be apparent. Yet no signiWcant diVerences were observed on items involving rote counting, such as counting aloud from one, counting backward by ones, or counting by tens. Girls with fragile X were also as successful as their peers at reading one digit numbers, and over 90% of girls in each group succeeded on this task. Although fewer girls with fragile X (81%) successfully represented quantities using either tallies or numerals, the diVerence between their rate of success and that of their peers (96%) did not reach statistical signiWcance with our adjusted alpha, Fisher’s Exact p D .017. Despite relatively intact rote counting skills, another area that distinguished girls with fragile X and their peers was perception of quantity. Fewer than half (48%) of girls with fragile X made correct magnitude judgments given verbally presented numbers, compared to 81% of the normative group, 2 (1, n D 247) D 13.02, Fisher’s Exact p D .001. This group diVerence was especially pronounced with verbally presented quantities. For example, when asked to judge which of two visually presented quantities was larger, 86% of girls with fragile X made accurate judgments. Although signiWcantly lower than the 99% rate observed for children in the normative group, 2 (1, n D 246) D 13.54, Fisher’s Exact p D .009, this combination

of Wndings suggests an advantage to presenting information visually for girls with fragile X. In summary, consistent with our study predictions, girls with fragile X demonstrated mastery of number recognition at a level comparable to their peers. For example, girls with fragile X, as a group, were able to read and write one and two digit numbers. Areas of relative weakness were also consistent with predictions. Multiple aspects of number sense appear to be challenging for girls with fragile X, including forming sets less than 10, understanding one-toone correspondence when counting, counting scattered dots, and verbal magnitude judgments. 3.3.1.3. Item analyses: Turner syndrome versus fragile X. Only one statistically signiWcant diVerence emerged between the two syndrome groups. More girls with Turner syndrome (93%) were able to count eight pictured objects on the KM-R Numeration subtest than girls with fragile X (55%), 2 (1, n D 48) D 9.47, Fisher’s Exact p D .004. It is noteworthy that girls with fragile X, but not girls with Turner syndrome, had more diYculty on this item than the normative group (discussed previously). Several other diVerences emerged that may be clinically relevant despite not reaching statistical signiWcance with alpha adjusted to .01. On the KM-R Geometry subtest, a higher percentage of girls with Turner syndrome (86%) correctly identiWed rectangles among a set of shapes relative to girls with fragile X (57%), 2 (1, n D 49) D 5.03, p D .025. As stated earlier, the performance of girls with Turner syndrome on this item was comparable to that of the normative group (82% of whom passed this item), whereas signiWcantly fewer girls with fragile X succeeded on this item. Of note, however, is that diVerences were not found on later (more diYcult) items that involved selecting circles or triangles. Likewise, on the KM-R Measurement subtest, when asked to select the longest and shortest items from a set of four objects, the frequency of correct responses was higher among girls with Turner syndrome (93%) relative to girls with fragile X (67%), but the diVerence did not reach statistical signiWcance, 2 (1, n D 49) D 5.49, Fisher’s Exact p D .028. Comparisons discussed previously indicated that girls with fragile X diVered from the normative group on this item (88% of whom passed the item), but girls with Turner syndrome did not. As with judging similar rectangles, statistically signiWcant diVerences were not found on other relative measurements, such as selecting items that were the hottest or the coldest. Other items of note include two items from the TEMA2. Although the diVerence was not statistically signiWcant (p D .05), 87% of girls with Turner syndrome, versus 62% of girls with fragile X, recognized that merely rearranging objects does not change the total number of objects (number constancy). For magnitude judgments with verbally presented numbers, the overall diVerence between Turner syndrome and fragile X was not statistically signiWcant (p D .02). However, a higher percentage of girls with Turner syndrome (79%) made accurate magnitude judgments

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relative to girls with fragile X, less than half of whom (48%) successfully completed this task. In sum, few statistically signiWcant diVerences emerged between girls with Turner syndrome or fragile X, on individual items, despite marked diVerences in the phenotypic characteristics associated with each syndrome. Based on the pattern of results, however, girls with Turner syndrome may have strengths in several mathematical domains, such as mastery of some aspects of counting and verbally presented quantity judgments, relative to girls with fragile X. 3.3.1.4. Comparisons across item sets. The pattern of results emerging from the item analysis suggests that girls with fragile X are comparable to their peers on rote counting and written representation of numbers, but that they may have diYculty applying counting rules, such as one-to-one correspondence and number constancy. This notion was examined further, using the a priori composite scores. Univariate ANOVAs were conducted on the rote counting, written representation, counting rules, and enumeration composite scores to compare the two syndrome groups and the normative group (see Table 2 for means and ranges). No group diVerences were observed on the rote counting or written representation composite scores; however, a main eVect of group was found for the counting rules and enumeration scores, F (2, 274) D 5.75, p D .004 and F (2, 266) D 7.26, p D .001, respectively. Follow up contrasts using Fisher’s least signiWcant diVerence (LSD) indicated that girls with fragile X had lower composite scores for counting rules than did girls with Turner syndrome (p D .015), or children from the normative group (p D .001), who did not diVer from each other (p > .10). Follow up contrasts also indicated that girls with fragile X had lower enumeration scores than did the normative group (p < .001), although no diVerences were found between the two syndrome groups (p D .06) or between girls with Turner syndrome and the normative group (p D .14). 3.3.2. Syndrome group comparisons to children with MLD 3.3.2.1. Item analyses: Turner syndrome. Comparisons between girls with Turner syndrome and children with MLD revealed a number of diVerences on speciWc math

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items, all of which favored the girls with Turner syndrome, even after limiting the sample to only kindergarteners. Therefore, for clarity, the values and statistics reported in this section reXect kindergarten-only results. Appendix B summarizes all relevant items, including the percentage of children who passed each item. On the SBIV Quantitative subtest, girls with Turner syndrome were favored on the Wrst and last of three items that required simultaneously matching two sets of dots to those of the examiner, 2 (1, n D 40) D 6.60, Fisher’s Exact p D .013, and 2 (1, n D 40) D 8.21, p D .004, respectively. Ninety-four and 89% of girls with Turner syndrome passed the Wrst and third matching item, respectively, whereas 59 and 45% of children with MLD passed these items. Although no statistically signiWcant diVerence was observed on the second of these three matching items (Fisher’s Exact p > .10), 89% of girls with Turner syndrome passed this item, compared to only 59% of children with MLD who passed. No diVerences were observed on these items between girls with Turner syndrome and children in the normative group, suggesting that girls with Turner syndrome have mastered this concept at a level consistent with their peers without MLD. Two diVerences were observed on the KM-R subtests between girls with Turner syndrome and children with MLD. On the Numeration subtest, more girls with Turner syndrome (67%) were able to correctly order three single digit numbers, compared to about a quarter (27%) of children with MLD, 2 (1, n D 40) D 6.21, Fisher’s Exact p D .013. On the Measurement subtest, more girls with Turner syndrome were able to order objects by size (67%), than the comparison group of which only 24% could order by size, 2 (1, n D 39) D 7.24, p D .007. No statistically signiWcant diVerence was observed between groups on ordering objects by weight or volume, although 39% of girls with Turner syndrome correctly ordered pictured objects by weight and 44% ordered by volume, compared to 22 and 12% of children with MLD who passed each item, respectively. An additional diVerence was found between girls with Turner syndrome and children with MLD on the TEMA-2. Although not statistically signiWcant (p D .021), more girls with Turner syndrome (67%) made correct

Table 2 Composite scores according to participant groupsa: means (ranges) Composite type

Group Normative (n D 226)

Rote counting (out of 6) Written representation (out of 4) Counting rules (out of 2) Enumeration (out of 7) Application composite (out of 2)

4.51 (0–6) 3.28 (0–4) 1.85 (0–2) 6.26 (2–7) —

Turner syndrome

Fragile X

All participants (n D 28)

Kindergarteners only (n D 18)

All participants (n D 21)

Kindergarteners only (n D 14)

4.64 (0–6) 3.38 (0–4) 1.82 (0–2) 5.92 (2–7) .93 (0–2)

4.20 (0–6) 3.06 (0–4) 1.78 (0–2) 5.63 (2–7) .78 (0–2)

5.06 (3–6) 3.79 (3–4) 1.52 (0–2) 5.32 (3–7) .53 (0–2)

5.09 (4–6) 3.75 (3–4) 1.43 (0–2) 5.50 (3–7) .43 (0–2)

Math learning disability (n D 23)

2.86 (0–6) 2.00 (0–4) 1.43 (0–2) 5.20 (2–7) .09 (0–1)

Note. Dash indicates data for application composite was not available from all participants in the normative sample. a The stated n reXects the largest sample size per group; sample size varies for composite scores. Ranges for group sample sizes are as follows: normative (221–226); Turner syndrome, all participants (25–28); Turner syndrome, kindergarteners only (15–18); fragile X, all participants (17–21); fragile X, kindergarteners only (11–14); math learning disability (20–23).

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magnitude judgments with verbally presented numbers than children with MLD (30%). In sum, the diVerences observed between girls with Turner syndrome and children with MLD suggest a few skills distinguish girls with Turner syndrome from their MLD peers, including matching, ordering series of numbers, and verbal magnitude judgments. In addition, lack of a clear pattern of diVerences on items involving visual spatial skills, suggests that the visual spatial deWcits associated with Turner syndrome do not impact math performance more than among children with MLD. 3.3.2.2. Item analyses: Fragile X syndrome. No remarkable diVerences were found between girls with fragile X syndrome and children with MLD on items from the SBIV Quantitative subtest or the KM-R subtests (ps > .05). Thus, suggesting that although girls with fragile X have more diYculty than their peers on items (discussed previously), such as counting dots, identifying shapes based on similarity, and judging longest and shortest, they do not have more diYculty on these items than children with MLD. On the TEMA-2, the majority of signiWcant diVerences favored the girls with fragile X, even after limiting the sample to only kindergarteners. Except where indicated, the values and statistics reported here reXect kindergarten-only results (see Appendix B for a summary of all relevant items). The MLD group was favored on a measure of oneto-one correspondence when counting. Because the MLD group was favored, this analysis was not rerun with the kindergarten-only subsample of girls with fragile X. Almost all (96%) of children with MLD demonstrated an understanding of one-to-one correspondence by correctly counting ten objects along with the examiner, compared to only 71% of girls with fragile X who passed this item. This diVerence, although not statistically signiWcant (p D .042), is noteworthy because the fragile X group was not adjusted for age diVerences. Thus, one-to-one correspondence may be an area of challenge for girls with fragile X, even relative to their peers with MLD. Although girls with fragile X had diYculty with one-toone correspondence relative to their peers with MLD, the performance of girls with fragile X exceeded that of the MLD group on reading and writing numbers. More girls with fragile X (75%) than children with MLD (20%) were able to correctly read two digit numbers between 11 and 20, 2 (1, n D 32) D 9.41, Fisher’s Exact p D .004. Also, the percent of girls with fragile X who were able to correctly write one digit numbers (86%) was signiWcantly higher than the 39% of children with MLD who passed this item, 2 (1, n D 37) D 7.70, p D .006. No diVerences were observed on these two items between girls with fragile X and children in the normative group, or on any items involving reading and writing numbers, suggesting that girls with fragile X have mastered these skills at a level consistent with their peers without MLD. 3.3.2.3. Comparisons across item sets. Univariate ANOVAs were conducted on the rote counting, written represen-

tation, counting rules, and enumeration composite scores to compare the two syndrome groups (including all participants) and the MLD group (see Table 2 for means and ranges). No diVerences were observed across groups on the counting rules or enumeration composite scores, suggesting that the three groups did not diVer in these areas. As described previously, diVerences did emerge for girls with fragile X, but not girls with Turner syndrome, in these areas relative to the normative group. In the MLD comparisons, a main eVect of group was found for the rote counting and written representation composite scores, but the follow up contrasts favored both the syndrome groups, so the analyses were rerun with the kindergarten-only subsample of girls with Turner syndrome or fragile X. Both the rote counting and written representation scores remained signiWcant with the kindergarten-only subgroup, F (2, 47) D 8.19, p D .001, and F (2, 47) D 7.82, p D .001, respectively. Follow up contrasts using Fisher’s LSD indicated no diVerences between the two syndrome groups on either rote counting or written representation scores (ps D .16). However, both girls with Turner syndrome and girls with fragile X had higher rote counting scores and higher written representation scores than children with MLD (ps D 0.01, and ps < .001, for Turner syndrome and fragile X, respectively). Given the relative strength of the syndrome groups in rote counting, we wanted to determine whether the groups were distinguishable when asked to apply counting principles. A univariate ANOVA was conducted on a Wfth composite score that required the application of counting principles (e.g., identifying the 4th object in an array), and was comprised of two items not included in the main testing battery. Several children did not complete one of the items because it was beyond their performance ceiling. In such cases, because of the exploratory nature of the analysis, the child was assigned a score of zero. There was a signiWcant main eVect of group on the application of counting principles composite score, even after limiting the syndrome groups to the kindergarten-only subsample, F (2, 53) D 5.34, p D .008. Girls with Turner syndrome had higher application of counting principles composite scores than children with MLD (p D .002), but did not diVer from girls with fragile X (p D .14). Despite stronger rote counting skills than their peers, girls with fragile X syndrome did not diVer from children with MLD on the application of counting principles composite score (p D .14). Considered together, these results suggest that girls with fragile X have a relative strength in rote counting skills that distinguishes them from children with MLD, despite comparable performance in other skill areas, such as written representation of numbers, understanding of counting rules, enumeration, and application of counting principles. 3.4. Discussion The research presented in this paper is an initial attempt to diVerentiate the aspects of mathematics that contribute

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to poor achievement in girls with Turner syndrome or fragile X. Few such studies have been carried out for girls with Turner syndrome, and results from those few studies have been either inconclusive or limited to scores on standardized tests or tests of mathematics calculations (e.g., Mazzocco, 1998, 2001; Rovet, 1993; Rovet et al., 1994). Even fewer such studies have been carried out for children with fragile X. By assessing the formal and informal mathematical ability associated with Turner syndrome or fragile X during the early primary school years, we are able to address potential delays or deWcits in basic concepts that may underlie poor mathematics achievement in children with these disorders. 3.4.1. Syndrome group comparisons to the normative group Consistent with our predictions, girls with Turner syndrome did not diVer from their peers on items involving number processing, such as reading and writing numbers, and ordering numbers. Contrary to study predictions, girls with Turner syndrome also showed no group diVerence from their peers on several items with overt visual spatial components, such as determining spatial-relationships, location, deciphering visual patterns, relative measurements, or ordering objects by size or weight. The apparent lack of systematic diYculty on items with strong spatial components is consistent with Rovet’s Wndings (Rovet et al., 1994) that poor math performance is independent of spatial abilities in Turner syndrome. Alternatively, the relationship between math and visual spatial skills may be limited to speciWc mathematics skills not evident during the early primary school years, such as later emerging math concepts or procedures, such as place value or regrouping. Consistent with our predictions, although girls with fragile X demonstrated mastery of number recognition and rote counting at a level comparable to their peers, multiple aspects of number sense were more challenging for these girls. These challenges included forming and combining sets less than 10, understanding one-to-one correspondence when counting, counting scattered dots, and verbal magnitude judgments. Contrary to our predictions, girls with fragile X also had diYculty with some items that had a visual spatial component, such as identifying rectangles from an array of varying shapes, and judging relative lengths of pictured objects. On the one hand, this relationship between visual spatial skills and performance on speciWc math concepts is consistent with the Wndings of Mazzocco, Bhatia, and Lesniak-Karpiak (in press), which suggest that poor math performance in fragile X is not restricted to diYculty with counting skills or basic calculation. On the other hand, it is worth noting that the two visual spatial items implicated as more challenging for girls with fragile X were both initial items within a speciWc domain set, and that subsequent, more diYcult items in those domain sets did not pose the same degree of challenge. It is possible that this poor performance on initial items reXects diYculty orienting to a new task or expectation, which would implicate executive dys-

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function rather than visual spatial diYculties. Indeed, in the present study, girls with fragile X were better at magnitude judgment tasks if the task was presented visually compared to verbally. As such our Wndings fail to demonstrate a consistent diYculty with spatial aspects of mathematics in girls from either syndrome group, but do indicate diYculty with applied counting and other aspects of numerosity comprehension in girls with fragile X. Further investigation of the fragile X phenotype may provide insight into the precise nature of the relationship between these two domains. Although the proWle of relative strengths and challenges varied between the Turner syndrome and fragile X groups, in relation to their peers, syndrome speciWc diVerences on individual items were minimal. One interpretation of this Wnding is that girls from either syndrome group simply have proWles consistent with any children who have a mathematics learning disability (MLD), a possibility not supported by the syndrome group versus MLD comparisons (discussed subsequently). An alternative possibility is that performance on individual items was not powerful enough to detect discrete diVerences between syndrome groups, at least during the early primary school years. Indeed, when looking at the a priori item sets, which combined items of a given type, distinct patterns emerged that distinguished the syndrome groups on understanding of counting rules, but not rote counting or enumeration skills. As such, the results from this composite score analysis support the notion that the underlying sources of math diYculties in each syndrome group diVer from each other. 3.4.2. Syndrome group comparisons to children with MLD Relative to children with MLD, similar levels of performance were evident for both syndrome groups. This was particularly apparent for girls with fragile X, who showed minimal diVerences from their peers with MLD. Girls from both syndrome groups showed stronger rote counting and written representation skills than did their peers with MLD. However, only girls with fragile X did not diVer from children with MLD on applied counting skills, despite an apparent advantage in rote counting skills. Together these Wndings suggest that, despite similar performance levels, the proWle of math diYculty among girls with Turner syndrome or fragile X during the early primary school years may not be simply a function of having MLD. Rather, that the groups can be distinguished, even as early as kindergarten, on the basis of counting related skills. Of note, however, is that the Turner syndrome or fragile X groups were not selected to meet criteria for MLD. Thus, both groups included children whose math performance was above the 10th percentile, which was used to deWne MLD in the normative sample. As such, diVerences between the syndrome and MLD groups may reXect the inclusion of girls with Turner syndrome or fragile X with relatively higher overall math performance. However, the majority of girls in both syndrome groups did meet criteria for MLD based on the 10th percentile criterion, and so group diVerences are not

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likely to solely reXect better overall math performance in one group versus another. 4. General discussion The Wndings from the present study support previous reports of poor math performance in females with Turner syndrome or fragile X. The results of Study 1 indicate a higher frequency of MLD among girls with either syndrome, relative to the rate of MLD in the general population. Children from either syndrome group who meet the criteria for MLD will continue to do so, at least during their primary school age years, at a rate at least comparable to that observed in the normative group (»70%). The conclusions drawn from these data are conservative, in view of the fact that children from the syndrome groups were seen for two separate assessments, whereas children from the normative group were seen annually over a four-year period. Despite this diVerence in available data, the overall rates of persistent MLD—deWned as meeting criteria for MLD during two years of primary school—were higher in the Turner syndrome and fragile X groups (84 and 77%, respectively) than in the normative comparison group (70%). However, these diVerences were not statistically signiWcant. It is possible that a signiWcant diVerence would emerge if data for children in the syndrome groups were available for four (versus two) assessment periods. Nevertheless, the present study demonstrates that a signiWcantly heightened risk of persistent MLD is associated with Turner syndrome and fragile X. 4.1. Implications for the cognitive phenotypes of Turner syndrome and fragile X The results of Study 2 provided information concerning the nature of MLD in the syndrome groups. Based on the Wndings from the present study, it appears that although girls with fragile X were able to count, and read and write numbers, they had considerable diYculty relative to their peers and girls with Turner syndrome on counting rules, such as cardinality and number constancy. Girls with Turner syndrome do not diVer from their peers to the same extent as girls with fragile X. Only girls with fragile X had a dissociation between their strong rote counting and weak applied counting skills, because children with MLD were relatively weak on both types of skills, and girls with Turner syndrome had stronger rote skills and comparable applied counting skills relative to children with MLD. Mastery of counting and related skills is important for the subsequent development of math skills, including arithmetic operations. Together these results suggest that all three groups may beneWt from additional instructional emphasis on applied counting skills. Moreover, interventions that draw on the relative strength of girls with fragile X in rote counting skills must be administered with caution, as it appears that rote skills in this group may not reXect appropriate levels of corresponding conceptual mastery.

These diVerences were among the few that emerged from our study. The lack of more pronounced syndrome diVerences may be due, in part to limited statistical power. Alternatively, the presence of another learning disability or disorder, such as attention deWcit hyperactivity disorder (ADHD), may contribute to the variability in math performance in both groups, thereby masking potential group diVerences. Indeed, ADHD is a phenotypic feature of both Turner and fragile X syndrome (Hagerman et al., 1999; Russell et al., 2005), and may also be associated with MLD (Marshall, Hynd, Handwerk, & Hall, 1997). Although our limited sample sizes did not allow us to address whether comorbid ADHD inXuenced group characteristics, this potential confound is evident in both syndrome groups and, therefore, is unlikely to contribute to syndrome group diVerences. However, the diVerences that did emerge are consistent with previous Wndings of intact number sense skills in Turner syndrome (e.g., Temple & Marriott, 1998), but not fragile X (e.g., Mazzocco, 2001). Also, the signiWcant Wndings emerged for the fragile X group, which had the smallest sample size. In addition, our Wndings are preliminary and limited to the early school years, and cannot be generalized across the lifespan—particularly in view of the increase in complexity of mathematics skills over the school age years. Indeed, the lack of diYculty on spatially oriented mathematics items in girls with Turner syndrome, and the relatively higher rate of diYculty on some of these (initial) items by girls with fragile X, are inconsistent with the Wndings from a study of mathematics calculation errors by older, school age girls (Mazzocco, 1998). In that study, a higher percentage of girls with Turner syndrome made operation and alignment errors relative to girls with fragile X, despite the fact that the overall number of girls making errors was comparable in both groups (Mazzocco, 1998). The important diVerences between the present study and this study of calculation error—in addition to participant age—include the nature of mathematics assessments used and the way in which visual spatial components are attributed to math performance. In the present study, conclusions about the relationship between mathematics and visual spatial abilities are also limited to the early primary school years and to the grade-appropriate test items included in the study. Although most of the girls with Turner syndrome or fragile X met criteria for MLD, it is unclear whether any of the children who did not meet these criteria, including those from either syndrome group, will continue to show age appropriate gains in math over time. Indeed, little is known about the manifestation and trajectory of MLD, and about the percentage of children who will not demonstrate poor math achievement until late in their elementary or middle school. As additional aspects of mathematics emerge in and beyond the fourth grade, cognitive strengths and deWciencies reported for girls with Turner syndrome may become more apparent. For example, girls with Turner syndrome (but not girls with fragile X) show deWcits in Xuency across a wide range of tasks, including oral Xuency and rapid automatized naming, such that math fact retrieval deWcits—which are not typi-

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210

cally measured in the kindergarten years—may become more apparent by third or fourth grades. Strong rote skills among girls with fragile X suggest that math fact retrieval per se will not be deWcient in this group, but that correct comprehension and application of math facts may be deWcient. With time, phenotypic diVerences may become more apparent in these groups. Based on the data available in the present study, there is suYcient evidence to conclude that children with Turner syndrome or fragile X have poor math performance, that this poor math performance persists during the primary school years, and that the speciWcity of mathematics deWcit emerges earlier for girls with fragile X syndrome than for girls with Turner syndrome.

207

Acknowledgments This work was supported by NIH grant HD R01 03461, and by a grant from the Spencer Foundation, both awarded to Michèle Mazzocco. Additional support was received through the National Fragile X Foundation Rosen Summer Fellows program. The data presented and the views expressed are solely those of the authors. The authors thank the children who participated in the study, their parents and teachers, the staV at participating Baltimore County Public School elementary schools, and research assistant Stacy Chung. The authors acknowledge the outstanding contribution made by Gwen F. Myers, Project Coordinator for throughout the primary school years of this research.

Appendix A Summary of mathematical skill areas, and the percentage of childrena passing individual test items within each area, examined for each syndrome group relative to the normative group Item

Group Normative (n D 226)

Turner syndrome (n D 28)

Fragile X (n D 21)

b

Rote counting Count aloud from 1 to 21 Count aloud from 1 to 41 Count backward by 1s Count by 10 s What number comes next? (1 digit) What number comes next? (2 digits)

82.3 52.0 75.2 58.7 96.0 83.6

78.6 57.7 74.1 53.8 85.7¤ 78.6

76.2 60.0 70.0 82.4 76.2¤¤ 80.0

Written representationb Read 1 digit numbers <6 Read 2 digit numbers (teens) Write 1 digit numbers Read 1 digit numbers

93.8 58.6 78.8 92.4

92.9 69.2 78.6 89.3

95.2 78.9 90.5 95.2

Counting rulesb Cardinality rule (1–5 items) Number constancy (3–5 items)

95.1 89.8

96.4 85.7

90.5 61.9¤¤

Enumerationb Form set of three objects Form set of Wve objects Form set of 19 objects Count sets of 1–5 items Count three items Count eight pictured objects Count sets of scattered dots 610

98.2 96.0 64.9 95.1 99.6 84.3 85.0

96.4 89.3 50.0 85.7 100.0 92.999 70.4

81.0¤¤ 81.0¤ 52.6 81.0¤ 100.0 55.0¤¤,99 55.0¤¤

99.6 99.6 97.8 99.6 99.6 96.0 100.0 91.2

100.0 96.4 89.3¤ 96.4 89.3¤¤ 85.7¤ 100.0 85.7

100.0 95.2 85.7¤ 81.0¤¤¤ 71.4¤¤¤ 81.0¤ 90.5¤¤ 76.2¤

87.4 91.2 90.7 77.2 78.1 76.2

100.0 100.09 100.09 89.3 92.99 92.9¤,9

Other counting and number representation Recognizes quantity (1 and 2) Hold up Wngers (1 and 2) Hold up Wngers (3 and 4) Count 5 Wngers One-to-one correspondence when counting with examiner Representing quantity with tallies or numerals Count 2 dots Count 5 dots Quantity matching Dot pattern (1) Dot pattern (3) Dot pattern (6) Quantity and sequence of dot pattern (2–1) Quantity and sequence of dot pattern (5–2) Quantity and sequence of dot pattern (6–4)

90.5 81.09 81.09 66.7 66.79 66.79 (continued on next page)

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Appendix A (continued) Item

Group Normative (n D 226)

Turner syndrome (n D 28)

Fragile X (n D 21)

Quantity judgment (e.g., Which is more?) Visual magnitude judgments Verbal magnitude judgments

98.7 81.4

96.4 78.69

85.7¤¤ 47.6¤¤¤,9

Mental number line What is closer to X? (610) Sequencing one digit numbers (<10)

73.4 64.3

68.0 78.69

44.4¤¤ 47.69

Addition Combine sets totaling <5 Combine sets totaling <10 Combine sets (one operand obscured) Add with manipulatives (68) Add with manipulatives (<10)

90.6 81.7 52.7 64.2 85.6

89.3 82.1 46.4 64.3 88.0

71.4¤ 57.1¤ 35.0 47.6 66.7

Addition continued Mental addition number 68 Add 3 numbers, sum D 3 Add 3 numbers, sum D 9

57.5 89.5 61.2

40.0 89.3 60.7

38.9 85.7 63.2

Spatial attributes and relationships Location (in/on) Location (outside) Identify rectangles in array of shapes Identify circles in array of shapes Identify triangles in array of shapes Color pattern (1-attribute) Color pattern (2-attribute) Alike/diVerent Identify same shape/size Identify same shape/diVerent size

91.5 90.6 81.7 81.8 85.4 70.2 64.1 62.5 39.3 37.2

96.4 100.0 85.79 82.1 85.7 75.09 60.7 67.9 46.4 25.9

90.5 85.7 57.1¤,9 66.7 70.0 47.6¤,9 42.1 42.9 26.3 5.3¤¤

Measurement Identify longest/shortest items Identify tallest/shortest items Identify hottest/coldest items

87.9 83.9 88.4

92.99 85.7 82.1

66.7¤,9 66.7 90.5

Measurement continued Order shapes by size Order objects by weight Order objects by volume

49.1 41.9 35.3

75.0¤¤,c 46.4 53.8

52.4 23.8 33.3

a The stated n reXects the largest sample size per group; sample size varies for individual items. Ranges for group sample sizes are as follows: normative (199–226), Turner syndrome (25–28), fragile X (17–21). b These items were used to calculate the corresponding composite score used in Study 2. c When limited to only kindergarteners, the percentage of girls with Turner syndrome who passed dropped to 66.7%, and was no longer statistically diVerent from children in the normative group (p D .15). ¤ p 6 .05, relative to normative group. ¤¤ p 6 .01, relative to normative group. ¤¤¤ p 6 .001, relative to normative group. 9 p 6 .05, relative to syndrome groups. 99 p 6 .01, relative to syndrome groups.

Appendix B Summary of mathematical skill areas, and the percentage of childrena passing individual test items within each area, examined for each syndrome group relative to the math learning disability group Item

Count aloud from 1–41 Count by 10 s What number comes next? (one digit) Read 2 digit numbers (teens) Write 1 digit numbers One-to-one correspondence when counting with examiner

Group Math learning disability (n D 23)

Turner syndrome (n D 18) Fragile X (n D 14)

18.2 30.4 47.8 20.0 39.1 95.7

— — 66.7 56.3¤ — —

61.5¤ 72.7¤ 84.6¤ 75.0¤¤ 85.7¤¤ 71.4¤,b

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210

209

Appendix B (continued) Item

Group

Quantity and sequence of dot pattern (2–1)

Math learning disability (n D 23) 59.1

Turner syndrome (n D 18) Fragile X (n D 14) — 94.4¤

Quantity and sequence of dot pattern (5–2) Quantity and sequence of dot pattern (6–4) Verbal magnitude judgments Sequencing 1 digit numbers (<10) Add with manipulatives (68) Location (outside) Color pattern (1-attribute) Order objects by size Order objects by weight Order objects by volume

59.1 45.5 30.4 27.3 17.4 72.7 35.0 23.8 22.2 11.8

88.9 88.9¤¤ 66.7¤ 66.7¤ 50.0¤ 100.0¤ 66.7¤ 66.7¤¤ 38.9 43.8

8.7 0.0

44.4 33.3

Application of counting principlesc Identify the fourth object in an array Identify the ninth position

— — — — — — — — — — 28.6 14.3

Note. The percentage of children passing is reported for any item where the signiWcance level reached .05, although we adjusted alpha to .01 due to multiple comparisons. Dashes indicate percentages that were omitted due to p > .05. a Sample is limited to participants in the kindergarten-only sample. The stated n reXects the largest sample size per group; sample size varies for individual items. Ranges for group sample sizes are as follows: math learning disability (17–23), Turner syndrome (16–18), fragile X (12–14). b This percentage reXects all children in the fragile X group. The kindergarten-only percentage is not reported because the diVerence favored the math learning disability group. c These items were used to calculate the corresponding composite score used in Study 2, and were only examined as a composite score. ¤ p 6 .05. ¤¤ p 6 .01.

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