Learning and Individual Differences 20 (2010) 639–643
Contents lists available at ScienceDirect
Learning and Individual Differences
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / l i n d i f
The mathematics skills of children with reading difficulties
Rose K. Vukovic a,⁎, Nonie K. Lesaux b, Linda S. Siegel c
a
b
c
Department of Teaching & Learning, New York University, 239 Greene St 2nd Floor, New York, NY 10003, United States
Harvard Graduate School of Education, 14 Appian Way, Cambridge MA, 02138, United States
Department of Educational and Counselling Psychology and Special Education, University of British Columbia, 2125 Main Mall, Vancouver, BC, V6T 1Z4, Canada
a r t i c l e
i n f o
Article history:
Received 16 December 2009
Received in revised form 11 August 2010
Accepted 12 August 2010
Keywords:
Dyslexia
Reading difficulties
Mathematics achievement
Arithmetic fact fluency
a b s t r a c t
Although many children with reading difficulty (RD) are reported to struggle with mathematics, little
research has empirically investigated whether this is the case for different types of RD. This study examined
the mathematics skills of third graders with one of two types of RD: dyslexia (n = 18) or specific reading
comprehension difficulty (n = 22), as contrasted to a comparison group (n = 247). Children's performance
on arithmetic fact fluency, operations, and applied problems was assessed using standardized measures. The
results indicated that children with dyslexia experienced particular difficulty with arithmetic fact fluency
and operations: they were 5.60 times and 8.54 times more likely than other children to experience deficits in
fact fluency and operations, respectively. Our findings related to arithmetic fact fluency were more
consistent with domain-general explanations of the co-morbidity between RD and mathematics difficulty,
whereas our findings related to operations were more consistent with domain-specific accounts.
© 2010 Elsevier Inc. All rights reserved.
1. Introduction
Research on learning disabilities (LD) has largely focused on reading
difficulty (RD) despite documentation that RD is more likely to co-occur
with mathematics difficulty (MD) than without (Dirks, Spyer, van
Lieshout, & de Sonneville, 2008; Rubinsten, 2009). For example, Badian
(1983) found that 56% of students with RD had poor mathematics
achievement; other research has found that, as a group, children with
RD had lower mathematics achievement than typical achievers (Fuchs,
Fuchs, & Prentice, 2004; Miles, Haslum, & Wheeler, 2001). Yet there is a
dearth of research that has systematically investigated mathematics
ability as it relates to RD. Responding to calls for such research (Gelman
& Butterworth, 2005; Rubinsten, 2009; Simmons & Singleton, 2008),
the present study examined the performance on arithmetic fact fluency,
operations, and problem solving for two types of RD: dyslexia and
specific reading comprehension difficulty.
those with specific reading comprehension difficulty have intact word
reading ability (e.g., Cain & Oakhill, 1999; Cain, Oakhill, & Bryant,
2000) and thus phonological processing is not implicated. This
reader's difficulties reflect deficits in higher-order cognitive processes,
such as integrating information in text, making inferences, and using
metacognitive strategies (Cain & Oakhill, 1999; Cain et al., 2000;
Oakhill, 1993). Such skills may also influence mathematics achievement, particularly in word problem solving. Preliminary evidence
suggests that children with RD broadly construed may not show
uniformly weak achievement across different mathematics domains
(e.g., Geary, Hamson, & Hoard, 2000; Hanich, Jordan, Kaplan, & Dick,
2001; Jordan, Hanich, & Kaplan, 2003; Miles et al., 2001; Simmons &
Singleton, 2009). Thus, a next step is to examine achievement across
different mathematics domains using samples that differentiate
dyslexia from specific reading comprehension difficulty.
1.2. The multi-faceted nature of mathematics ability
1.1. Types of reading difficulty
Research that has suggested a relationship between RD and MD
has focused primarily on dyslexia, which refers to slow, laborious
decoding due to underlying deficits in phonological processes (e.g.,
Swanson & Siegel, 2001; Torgesen, 2000). Phonological processing has
also been shown to influence mathematics performance, particularly
arithmetic fact fluency (see Simmons & Singleton, 2008). In contrast,
⁎ Corresponding author. Tel.: +1 212 998 5205; fax: +1 212 995 4049.
E-mail addresses: [email protected] (R.K. Vukovic), [email protected]
(N.K. Lesaux), [email protected] (L.S. Siegel).
1041-6080/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.lindif.2010.08.004
Like the construct of reading, mathematics is multi-faceted in
nature. In elementary school, the target areas for mathematics
instruction are arithmetic fact fluency, operations, and word problem
solving (e.g., Fuchs et al., 2006; Gersten et al., 2009).
Arithmetic fact fluency refers to the automatic retrieval of simple
single-digit addition and subtraction facts and in later elementary
years, multiplication facts. There is evidence that children with
dyslexia experience difficulty in this area (see Simmons & Singleton,
2008, 2009). Domain-general explanations posit that deficient
phonological processing accounts for this relationship (e.g., Simmons
& Singleton, 2008, 2009). In contrast, the domain-specific account
640
R.K. Vukovic et al. / Learning and Individual Differences 20 (2010) 639–643
posits that numerical processing underlies the mathematics deficits
children with dyslexia experience (Gelman & Butterworth, 2005;
Landerl, Bevan, & Butterworth, 2004; Landerl, Fussenegger, Moll, &
Willburger, 2009). Little is known about the arithmetic fact fluency
skills of children with specific reading comprehension difficulty.
Given that these readers do not have phonological processing deficits,
in this study, we expected arithmetic fact fluency to be related only to
dyslexia.
Operations refers primarily to the ability to perform calculations
using algorithms and arithmetic. Some research shows that children
with RD perform operations more poorly than typically achieving
children (e.g., Jordan et al., 2003) while other research has not found
differences between children with RD and typically achieving children
(Hanich et al., 2001; Jordan & Hanich, 2000). These studies did not,
however, differentiate dyslexia from specific reading comprehension
difficulty, making it difficult to discern how children with specific
types of RD perform on operations. Neuropsychological evidence
suggests that operations and arithmetic fact fluency are processed in
different brain regions and that unlike arithmetic fact fluency,
operations do not make heavy demands on the language system
(e.g., Dehaene, Piazza, Pinel, & Cohen, 2003). Given the primarily
language-based nature of both dyslexia and specific reading comprehension difficulty, we expected minimal impairment on operations.
However, because arithmetic fact fluency is foundational for operations (e.g., Fuchs et al., 2006), we expected the dyslexia group would
show some impairment on operations.
Word problem solving refers to linguistically presented problems
involving mathematical relations and properties. Evidence suggests
that children with RD perform similarly to typical achievers on word
problems (e.g., Hanich et al., 2001; Jordan et al., 2003). These results
are somewhat surprising given the language-based nature of RD. To
account for these findings, researchers have speculated that children
with RD use their relative strengths in mathematics to compensate for
their low reading abilities. However, the children with RD in these
studies were selected to demonstrate at least average mathematical
ability. More research is needed with RD samples that are not selected
on the basis of their mathematics ability. Given that arithmetic fact
fluency is foundational for word problems (e.g., Fuchs et al., 2006), we
expected that children with dyslexia would perform lower than
children with specific reading comprehension difficulty and that both
groups would perform lower than a comparison group.
Table 1
Student demographic characteristics by reader group.
Variable
Sex
Female
Family background
Majority culture
First nations
Middle Eastern
Asian
Other
Comparison
(n = 247)
Specific reading
comprehension
difficulty (n = 22)
Dyslexia
(n = 18)
n
%
n
%
n
%
117
47.4
8
36.4
9
50
134
32
36
21
24
54.3
13.0
14.6
8.5
9.7
9
3
2
1
7
40.9
13.6
9.1
4.5
31.8
9
6
2
0
1
50.0
33.3
11.1
0
5.6
Note. There were no differences in gender distribution by reader group, Χ2 (2, N = 287)=
1.07, p N .05. The distribution of family background by reader group were not subjected to
chi-square analyses because 6 cells (40%) had frequencies less than 5 and in one instance
had a frequency of 0.
working class neighbourhoods characterized by high mobility rates
(43%–67%). Demographic characteristics of the sample are presented
in Table 1; there were no differences by demographic group on the
study measures.
2.1.1. Classification scheme
Children were classified into one of three groups: dyslexia, specific
reading comprehension difficulty, or comparison. Consistent with the
RD literature, dyslexia was defined as performance below the 15th
percentile on the Woodcock Johnson-III (WJ-III) Letter-Word Identification (LWID) test; and specific reading comprehension difficulty was
defined as performance below the 15th percentile on the Stanford
Diagnostic Reading Test (SDRT) Reading Comprehension subtest and
performance above the 50th percentile on LWID. The dyslexia group
included 18 children (6.3%), 22 children (7.7%) were classified with
specific reading comprehension difficulty, and the remaining 247
children (86.1%) were placed into the comparison group.
One-way analyses of variance (ANOVA) confirmed three distinct
reader groups: word reading, F(2, 284) = 59.13, p b .001; and reading
comprehension, F(2, 284) = 56.14, p b .001. The comparison and the
specific comprehension difficulty group had significantly higher word
reading scores than the dyslexia group; both RD groups had
significantly lower reading comprehension scores than comparison
children.
1.3. Present study
Previous research has not systematically investigated the mathematics profiles of children with different types of RD. This lack of
knowledge limits a comprehensive understanding of the etiology and
typology of dyslexia and specific reading comprehension difficulty,
which has implications for both research and practice. We sought to
advance the research base by examining the mathematics skills of
third graders with dyslexia or specific reading comprehension
difficulty, contrasted to a comparison sample. We chose third grade
to ensure variation in reading and mathematics achievement (Fuchs
et al., 2006). This study was guided by a single research question:
What are the differences among third graders with dyslexia, specific
reading comprehension difficulty, and comparison children on
measures of operations, arithmetic fact fluency, and word problem
solving?
2. Method
2.1. Participants
The participants were 287 third graders (mean age = 8.60 years,
sd = .31, range = 8.00- to 9.75-years) attending five elementary
schools in western Canada. The schools were located primarily in
2.2. Materials
Raw scores for tests were converted to standard scores based on
normative data. Standard scores were utilised in the tables and
analyses.1
2.2.1. Word reading
The LWID test of the WJ-III: Research Edition (Woodcock, McGrew,
& Mather, 1999) was used; in this task children identify and
pronounce isolated letters and words of increasing difficulty (e.g.,
cat, palm). The publisher reports reliability between .96 and .97.
2.2.2. Reading comprehension
The Reading Comprehension test of the SDRT (Karlsen & Gardner,
1994) was used; children have 45 min to read short passages and
provide responses to multiple-choice questions. The publisher reports
.91 reliability for third graders.
1
Standard scores for reading comprehension represent approximately equalinterval units that are particularly suitable for analyses (Karlsen & Gardner, 1994);
however, these scores are not norm-referenced, which makes interpretation difficult.
For ease of interpretation, we provide the percentile conversion of these scores where
appropriate.
R.K. Vukovic et al. / Learning and Individual Differences 20 (2010) 639–643
3. Results
Table 2
Descriptive characteristics and correlations among variables.
Correlations
Variable
M (SD)
1.
2.
3.
4.
5.
105.21
620.53
97.41
100.01
107.59
Word reading
Reading comprehensiona
Arithmetic fact fluency
Operations
Applied problems
(12.64)
(47.25)
(16.19)
(12.13)
(17.11)
641
1
2
3
4
5
–
.65
.41
.49
.55
–
.32
.44
.52
–
.57
.59
–
.51
–
Note. Means are presented in standard scores. Correlation coefficients are significant at
p b .001.
a
Percentile score = 38.55 (25.85).
2.2.3. Arithmetic fact fluency
The WJ-III Math Fluency test (Woodcock, McGrew, & Mather,
2001) was used; children are instructed to complete as many simple
single-digit addition, subtraction, and multiplication facts within
3 min, skipping those they do not know. The publisher reports
reliability between .90 and .93.
2.2.4. Operations
The Calculation test of the WJ-III: Research Edition (Woodcock et al.,
1999) was used; children read and complete a series of written
calculation problems of increasing difficulty, presented in horizontal
or vertical format. The publisher reports reliability between .80 and
.87.
2.2.5. Applied problem solving
The Applied Problems test of the WJ-III: Research Edition
(Woodcock et al., 1999) was used; children solve practical problems
in mathematics read aloud by the experimenter. Because many of
these questions require solving applied problems (e.g., telling time or
temperature, counting money) instead of word problems, this test is
better conceived as a measure of applied problem solving. The
publisher reports reliability between .91 and .93.
2.3. Procedure
The assessments occurred between January and March. Children
were individually assessed (15 min per child) on word reading and
applied problems; arithmetic fact fluency, operations, and reading
comprehension were group administered in classrooms (60 min).
Table 2 displays the sample characteristics and the correlations
among the variables.
To address the research question, a series of one-way ANOVAs
were conducted; significant differences were followed-up with
Scheffé post-hoc tests. To interpret the magnitude of the group
differences, we used Cohen's d (Cohen, 1988). Given that fact fluency
has been found as a determinant of operations and word problems
(e.g., Fuchs et al., 2006), we included it as a covariate in one set of
models with operations and applied problems as dependent variables.
Table 3 summarizes the results.
The groups differed significantly on arithmetic fact fluency,
F(2, 284) = 17.19, p b .001, operations, F(2, 284) = 23.80, p b .001,
and applied problems, F(2, 284) = 17.56, p b .001. The comparison and
specific reading comprehension difficulty groups did not differ
significantly on arithmetic fact fluency or operations, and both groups
performed significantly higher than the dyslexia group. On applied
problems, the RD groups performed lower than comparison children.
When fact fluency was included as a covariate, the pattern did not
change on either operations, F(2, 283)= 9.66, p b .001, or applied
problems, F(2, 283) = 6.53, p = .002, although the performance of the
dyslexia group approached the average range.
To further investigate the relationship between RD and mathematics, we conducted chi-square analyses to examine the extent to
which RD corresponded with deficits in fact fluency, operations, and
applied problems. Children's performance was categorized into deficit
(i.e., ≤15th percentile) or no-deficit (i.e., ≥16th percentile) in each
mathematics domain; this cut-score scheme is consistent with how
deficits are defined in recent MD research (e.g., Geary, Hoard, ByrdCraven, Nugent, & Numtee, 2007; Murphy, Mazzocco, Hanich, & Early,
2007).
The chi-square analyses were significant for all mathematics
measures: arithmetic fact fluency, Χ2 (2, N = 287) = 51.22, p b .001,
operations, Χ2 (2, N = 287) = 36.72, p b .001; and applied problems,
Χ2 (2, N = 287) = 21.44, p b .001. Table 4 displays the relative risk
ratios and Table 5 presents descriptive data on the mathematics
deficits of the 18 children with dyslexia.
4. Discussion
Little research has examined how children with different types of
RD perform on mathematics skills. The purpose of this study was thus
to explore the arithmetic fact fluency, operations, and applied
problems skills of children with two different types of RD: children
with dyslexia and children with specific reading comprehension
difficulty. Our findings indicated that mathematics achievement was
differentially associated with type of RD. Specifically, children with
dyslexia demonstrated weaker mathematics performance and they
Table 3
Mean standard scores and effect sizes by reader group on measures of reading and mathematics.
Measure
Word reading
Reading comprehension
Arithmetic fact fluency
Operations
Controlling for fact fluency
Applied problems
Controlling for fact fluency
Comparison (1)
M (SD)
107.13
630.43a
99.08
101.23
100.60
109.75
108.79
(11.31)a
(40.72)a
(15.62)a
(11.09) a
(0.62)a
(16.64)a
(0.87)a
Specific reading
comprehension
difficulty (2)
M (SD)
105.27
571.45b
95.14
100.82
101.67
98.55
99.85
(4.97)a
(16.03)b
(15.19)a
(10.38) a
(2.07)a
(12.92)b
(2.90)b
Dyslexia (3)
M (SD)
78.83
544.61c
77.28
82.28
89.82
88.94
100.50
(4.20)b
(51.03)b
(10.92)b
(14.34) b
(2.41)b
(12.88)b
(3.38)b
Effect size
1 vs 2
1 vs 3
2 vs 3
0.15
1.25
0.24
0.03
0.09
0.66
0.52
2.25
1.82
1.35
1.57
0.89
1.22
0.49
2.15
0.58
1.13
1.57
1.00
0.58
0.04
Note. Analyses based on n = 247 in the comparison group, n = 22 in the specific reading comprehension difficulty group, and n = 18 in the dyslexia group. Means in the same row
that do not share subscripts differ at p b .001 based on post hoc analyses. Cohen's d (Cohen, 1988) was used as the measure of effect size: .2 (small effect), .5 (medium effect), and .8
(large effect). Effect size calculations were computed using the standard deviation of the whole group as a scale.
a
Percentile score = 43.43 (24.46).
b
Percentile score = 8.91 (3.74).
c
Percentile score = 7.78 (9.05).
642
R.K. Vukovic et al. / Learning and Individual Differences 20 (2010) 639–643
Table 4
Chi-square and relative risk analyses of likelihood of mathematics deficits by reader group.
Arithmetic fact fluency
No-deficit
Deficit
Operations
No-deficit
Deficit
Applied problems
No-deficit
Deficit
Comparison
(1)
Specific reading
comprehension
difficulty (2)
Dyslexia
211 (85.4%)
36 (14.6%)
18 (81.8%)
4 (18.2%)
234 (94.7%)
13 (5.3%)
230 (93.1%)
17 (6.9%)
Relative risk dyslexia vs
1+2
1
2
3 (16.7%)
15 (83.3%)
5.60 (3.94–7.97)
4.58 (1.84–11.39)
5.72 (3.9 7– 8.24)
21 (95.5%)
1 (4.5%)
10 (55.6%)
8 (44.4%)
8.54 (4.13–17.65)
8.44 (4.03–17.69)
9.78 (1.35 – 71.06)
18 (81.8%)
4 (18.2%)
11 (61.1%)
7 (38.9%)
4.98 (2.45–10.13)
5.65 (2.70–11.83)
2.14 (0.74–6.17)
Note. Analyses based on n = 247 in the comparison group, n = 22 in the specific reading comprehension difficulty group, and n = 18 in the dyslexia group.
were also more likely to have mathematics deficits: compared to other
reader groups, children with dyslexia were 5.60 times, 8.54 times, and
4.98 times more likely to show deficits in fact fluency, operations, and
applied problems, respectively. These findings clarify and extend
previous research to show that it is children with dyslexia in
particular, not children with reading difficulties broadly construed,
who have associated mathematics deficits, especially in arithmetic
fact fluency.
4.1. Arithmetic fact fluency
The finding that children with dyslexia, not specific reading
comprehension difficulty, showed fact fluency deficits provides
indirect support for domain-general accounts, which posit that
phonological processing underlies the development of arithmetic
fact fluency in children with dyslexia (e.g., Simmons & Singleton,
2008, 2009). Indeed, Landerl et al. (2004) suggested that underlying
verbal or phonological deficits might affect performance on mathematics tasks that place demands on such systems; arithmetic fact
fluency is one such mathematics task (see Dehaene et al., 2003). Of the
18 children with dyslexia in the present study, only three did not have
deficits in arithmetic fact fluency. Thus, dyslexia and deficits in
arithmetic fact fluency appear to share a common etiology. Phonological deficits might constitute the common underlying etiology,
given that phonological deficits are involved in dyslexia and not in
specific reading comprehension difficulty.
However, it is important to note that in our overall sample,
mathematics deficits were more likely to occur without corresponding
reading difficulties, suggesting that domain-specific processes also
contribute to arithmetic fact fluency deficits (Gelman & Butterworth,
2005). Landerl et al. (2004, 2009) found preliminary evidence that
dyslexia and MD stemmed from dissociable underlying deficits, namely,
phonological processing in dyslexia and numerical processing in MD.
More research using direct measures of phonological and numerical
processing is needed to understand whether failure to automatize
number facts reflects an impaired phonological system, deficits in
mathematical understanding, or both. In addition, more research is
needed to understand whether the arithmetic fact fluency deficits
experienced by children with dyslexia differ from the arithmetic fact
Table 5
Type of mathematics deficit in the 18 children with dyslexia.
Type of mathematics deficit
n
Arithmetic fact fluency deficits
Fact fluency only
Fact fluency and operations
Fact fluency and applied problems
Fact fluency, operations, applied problems
Operations-only deficits
No mathematics deficits
15
6
2
2
5
1
2
fluency deficits experienced by children without word reading deficits.
Such research is critical to inform targeted instruction and intervention.
4.2. Operations
We found that children with dyslexia performed significantly lower
on operations than other children, even when controlling for arithmetic
fact fluency; this suggests that operations is influenced by more than
fact fluency – and phonological processes implicitly – which is
consistent with domain-specific accounts (e.g., Landerl et al., 2004,
2009). These findings indicate that in the case of dyslexia, operations
deficits might reflect mathematical problems that are independent of
reading-related processes. Geary (1993) suggested that deficits in
operations reflect developmental delays in mathematical understanding and thus these deficits may be more amenable to intervention than
are arithmetic fact fluency deficits. More research is necessary to
examine the sources of operations deficits in children with dyslexia.
That operations deficits might reflect domain-specific processes also
has implications for MD identification, given the inconsistencies in how
“mathematics deficit” is defined in MD research. Specifically, we have
yet to determine whether children with MD have primary difficulties in
arithmetic fact fluency, operations, word problems, or other areas such
as measurement, geometry or algebra. One of the most consistent
findings in MD research is that these children tend to be characterized
by deficits in arithmetic fact fluency (Geary, 1993; Vukovic & Siegel,
2010). However, the current results suggest that it is not clear whether
arithmetic fact fluency deficits are underpinned by domain-general or
domain-specific processes. In contrast, performance on operations
appeared less contingent upon domain-general processes. Together,
these results suggest that operations may represent a more suitable
construct to identify MD than arithmetic fact fluency.
In this study, children with specific reading comprehension difficulty
did not experience operations deficits. Yet these children are thought to
have deficits in higher-order cognitive processes, such as inference
making and using metacognitive strategies (e.g., Cain & Oakhill, 1999;
Cain et al., 2000). This may indicate that general reasoning and
metacognitive skills do not underlie operations, supporting a domainspecific account of operations; however, without direct measures of
general or number-specific reasoning skills our conjecture is speculative
and should be tested. More research is needed to examine the cognitive
abilities of children with RD and how different cognitive processes
impact mathematics achievement.
4.3. Applied problems
Consistent with our prediction, we found that both RD groups
performed lower than comparison children on applied problems,
although we did not find a difference in performance between the RD
groups. Interestingly, once fact fluency was controlled, children with
dyslexia performed in the average range on applied problems. This
is consistent with previous research that children with RD do not
R.K. Vukovic et al. / Learning and Individual Differences 20 (2010) 639–643
struggle with word problems (e.g., Hanich et al., 2001; Jordan et al.,
2003). However, the language demands inherent in word problems
and the language-based nature of both dyslexia and specific reading
comprehension difficulty begs further investigation of this finding.
These results suggest that context – in the form of stories and
practical scenarios – assists in the completion of mathematics problems,
which has implications for how to teach mathematics to children with
RD, particularly dyslexia. That said, we remind the reader that in the
current study, as well as in previous studies (Hanich et al., 2001; Jordan
et al., 2003), the problems were read to the children, effectively
eliminating the reading demands of the task. However, both RD groups
performed lower than typical achievers, suggesting that their performance was at least somewhat impaired, presumably due to reading or
language-related processes. Examining the performance of children
with RD on word problems when the problems are read aloud vs
when the children have to read the problems on their own is a necessary
next step. Additionally, cognitive clinical interviews, which provide
richer insights into children's mathematical thinking than traditional
standardized achievement tests (Ginsburg, 1997), would enrich the
evidence base.
4.4. Summary
The body of research focused on the relationship between RD and
mathematics remains very small, and there exists very little specific
information about the nature of the mathematics deficits of children
with different types of RD. Although it is the case that all children's
mathematics needs are consistently overlooked in favour of literacy
skills and support, we know this is especially the case for children with
dyslexia, for whom reading interventions are much more common than
is mathematics support. Yet mathematics difficulties can have serious
long-term consequences on job performance, employability, and daily
living (Statistics Canada & OECD, 2005). In order to design effective
models of identification and intervention, research must continue to
generate a robust understanding of the nature of mathematics deficits
in children with dyslexia and whether these deficits reflect domainspecific or domain-general processes.
Acknowledgements
This research was supported in part by grants from the Social Sciences
and Humanities Research Council of Canada, the William T. Grant
Foundation, and the Natural Sciences and Engineering Research Council
of Canada. The authors would also like to thank Richard Woodcock, Kevin
McGrew, and Laurie Ford, for the generous use of the Woodcock–Johnson
tests. Thank you to Helen Bent, Jessica Krajicek, Sabrena Jaswal, Elenita
Tseng, Maggie Vukovic, and Laura Zajac, for their assistance with data
collection and entry.
References
Badian, N. A. (1983). Dyscalculia and nonverbal disorders of learning. In H. R. Myklebust
(Ed.), Progress in learning disabilities, Vol. 5. (pp. 235−264). New York, NY: Stratton.
Cain, K., & Oakhill, J. (1999). Inference making ability and its relation to comprehension
failure in young children. Reading and Writing, 11, 489−503.
Cain, K., Oakhill, J., & Bryant, P. (2000). Phonological skills and comprehension failure: A test
of the phonological processing deficit hypothesis. Reading and Writing, 13, 31−56.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences, 2nd ed. Hillsdale,
NJ: Lawrence Erlbaum.
643
Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number
processing. Cognitive Neuropsychology, 20, 487−506.
Dirks, E., Spyer, G., van Lieshout, E. C., & de Sonneville, L. (2008). Prevalence of
combined reading and arithmetic disabilities. Journal of Learning Disabilities, 41,
460−473.
Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., et al.
(2006). The cognitive correlates of third grade skill in arithmetic, algorithmic
computation, and arithmetic word problems. Journal of Educational Psychology, 98,
29−43.
Fuchs, L. S., Fuchs, D., & Prentice, K. (2004). Responsiveness to mathematical problemsolving instruction: Comparing students at risk of mathematics disability with and
without risk of reading disability. Journal of Learning Disabilities, 37, 293−306.
Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and
genetic components. Psychological Bulletin, 114, 345−362.
Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and arithmetical
cognition: A longitudinal study of process and concept deficits in children with
learning disability. Journal of Experimental Child Psychology, 77, 236−263.
Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive
mechanisms underlying achievement deficits in children with mathematical
learning disability. Child Development, 78, 1343−1359.
Gelman, R., & Butterworth, B. (2005). Number and language: How are they related?
Trends in Cognitive Sciences, 9, 6−10.
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., et al. (2009).
Assisting students struggling with mathematics: Response to Intervention (RtI)
for elementary and middle schools (NCEE 2009-4060).Washington, DC: National
Center for Education Evaluation and Regional Assistance, Institute of Education
Sciences, U.S. Department of Education Retrieved from http://ies.ed.gov/ncee/
wwc/publications/practiceguides/
Ginsburg, H. P. (1997). Mathematics learning disabilities: A view from developmental
psychology. Journal of Learning Disabilities, 30, 20−33.
Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001). Performance across different
areas of mathematical cognition in children with learning difficulties. Journal of
Educational Psychology, 93, 615−626.
Jordan, N. C., & Hanich, L. B. (2000). Mathematical thinking in second-grade children
with different forms of LD. Journal of Learning Disabilities, 33, 567−578.
Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical
competencies in children with specific mathematics difficulties versus children with
comorbid mathematics and reading difficulties. Child Development, 74, 834−850.
Karlsen, B., & Gardner, E. (1994). Stanford Diagnostic Reading Test. San Francisco:
Harcourt Brace.
Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic
numerical capacities: A study of 8–9-year old students. Cognition, 92, 99−125.
Landerl, K., Fussenegger, B., Moll, K., & Willburger, E. (2009). Dyslexia and dyscalculia:
Two learning disorders with different cognitive profiles. Journal of Experimental
Child Psychology, 103, 309−324.
Miles, T. R., Haslum, M. N., & Wheeler, T. J. (2001). The mathematical abilities of dyslexic
10-year-olds. Annals of Dyslexia, 51, 299−321.
Murphy, M. M., Mazzocco, M. M. M., Hanich, L. B., & Early, M. C. (2007). Cognitive
characteristics of children with mathematics learning disability (MLD) vary as a
function of the cutoff criterion used to define MLD. Journal of Learning Disabilities,
40, 458−478.
Oakhill, J. (1993). Children's difficulties in reading comprehension. Educational Psychology
Review, 5, 223−237.
Rubinsten, O. (2009). Co-occurrence of developmental disorders: The case of
developmental dyscalculia. Cognitive Development, 24, 362−370.
Simmons, F. R., & Singleton, C. (2008). Do weak phonological representations impact on
arithmetic development? A review of research into arithmetic and dyslexia.
Dyslexia, 14, 77−94.
Simmons, F. R., & Singleton, C. (2009). The mathematical strengths and weaknesses of
children with dyslexia. Journal of Research in Special Educational Needs, 9, 154−163.
Statistics Canada and Organization for Economic Cooperation and Development
(OECD) (2005). Learning a Living: First Results of the Adult Literacy and Life Skills
Survey.
Swanson, H. L., & Siegel, L. S. (2001). Learning disabilities as a working memory deficit.
Issues in Education, 7, 1−48.
Torgesen, J. K. (2000). Individual differences in response to early interventions in
reading: The lingering problem of treatment resisters. Learning Disabilities Research
& Practice, 15, 55−64.
Vukovic, R. K., & Siegel, L. S. (2010). Academic and cognitive characteristics of persistent
mathematics difficulty from first through fourth grade. Learning Disabilities Research &
Practice, 25, 25−38.
Woodcock, R. W., McGrew, K. S., & Mather, N. (1999). Woodcock-Johnson III Tests of
Achievement: Research Editions. Itasca, IL: Author.
Woodcock, R. W., McGrew, K. S., & Mather, N. (2001). Woodcock-Johnson III Tests of
Achievement. Itasca, IL: The Riverside Publishing Company.