Mathematics learning disability in girls with Turner syndrome or ...

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
b
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
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: mazzocco@kennedykrieger.org (M.M.M. Mazzocco).
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.
0278-2626/$ - see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.bandc.2005.12.014

196 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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 Test-
Revised.) 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, & Gross-
Tsur, 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

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210 197
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 pro-
Wles 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
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

198 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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 χ 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 χ 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, χ 2 (1,
n D233)D10.66, pD.0011. Among the 24 girls with Turner

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210 199
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), pD .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,
χ 2 (1, nD74) D4.61, 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, χ 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, χ 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
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.

200 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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 (nD28) or fragile X (nD21) 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).
3.2. Materials
3.2.1. Cognitive ability
For descriptive purposes, the Stanford Binet, fourth edition
(SBIV; Thorndike, Hagen, & Sattler, 1986) was administered
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
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
Normative Turner syndrome Fragile X Math learning disability
Sample size n D 226 n D 28 n D 21 n D 23
No. in kindergarten (No. in grade 1) 226 (0) 20 (8) 14 (7) 23 (0)
No. of girls 115 28 21 9
Mean age at testing (range) 5.77 (5.03–6.99) 6.49 (5.37–7.78) 6.47 (5.10–7.78) 5.85 (5.08–6.99)
Stanford Binet IV, full scale IQ (range) 99.07 (76–133) 90.79 (64–108) 80.57 (58–108) 88.57 (76–106)

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210 201
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, χ 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, χ 2 (1, n D 254) D 16.96,
Fisher’s Exact p D .004; however, this result is not clinically
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, χ 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%), χ 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, χ 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, χ 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.

202 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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, χ 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,
χ 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, χ 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, χ 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,
χ 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, χ 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, χ 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, χ 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%), χ 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%), χ 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, χ 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 TEMA-
2. 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

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210 203
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
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, χ 2 (1,
n D 40) D 6.60, Fisher’s Exact p D .013, and χ 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, χ 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, χ 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 groups a : means (ranges)
Composite type
Group
Normative
(n D 226)
Turner syndrome Fragile X Math learning
All participants Kindergarteners All participants Kindergarteners
disability (n D 23)
(n D 28)
only (n D 18) (n D 21)
only (n D 14)
Rote counting (out of 6) 4.51 (0–6) 4.64 (0–6) 4.20 (0–6) 5.06 (3–6) 5.09 (4–6) 2.86 (0–6)
Written representation (out of 4) 3.28 (0–4) 3.38 (0–4) 3.06 (0–4) 3.79 (3–4) 3.75 (3–4) 2.00 (0–4)
Counting rules (out of 2) 1.85 (0–2) 1.82 (0–2) 1.78 (0–2) 1.52 (0–2) 1.43 (0–2) 1.43 (0–2)
Enumeration (out of 7) 6.26 (2–7) 5.92 (2–7) 5.63 (2–7) 5.32 (3–7) 5.50 (3–7) 5.20 (2–7)
Application composite (out of 2) — .93 (0–2) .78 (0–2) .53 (0–2) .43 (0–2) .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).

204 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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, χ 2 (1,
nD32)D9.41, Fisher’s Exact pD.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, χ 2 (1, nD37)D7.70,
pD.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 ANO-
VAs were conducted on the rote counting, written representation,
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

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210 205
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 dysfunction
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

206 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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) 195–210 207
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.
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 children a 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)
Rote counting b
Count aloud from 1 to 21 82.3 78.6 76.2
Count aloud from 1 to 41 52.0 57.7 60.0
Count backward by 1s 75.2 74.1 70.0
Count by 10 s 58.7 53.8 82.4
What number comes next? (1 digit) 96.0 85.7 ¤ 76.2 ¤¤
What number comes next? (2 digits) 83.6 78.6 80.0
Written representation b
Read 1 digit numbers

208 M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210
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 98.7 96.4 85.7 ¤¤
Verbal magnitude judgments 81.4 78.6 9 47.6 ¤¤¤,9
Mental number line
What is closer to X? (610) 73.4 68.0 44.4 ¤¤
Sequencing one digit numbers (

M.M. Murphy et al. / Brain and Cognition 61 (2006) 195–210 209
Appendix B (continued)
Item
Group
Math learning disability (n D 23) Turner syndrome (n D 18) Fragile X (n D 14)
Quantity and sequence of dot pattern (2–1) 59.1 94.4 ¤ —
Quantity and sequence of dot pattern (5–2) 59.1 88.9 —
Quantity and sequence of dot pattern (6–4) 45.5 88.9 ¤¤ —
Verbal magnitude judgments 30.4 66.7 ¤ —
Sequencing 1 digit numbers ( .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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