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Skill Development in Different Components of Arithmetic and Basic
Cognitive Functions: Findings From a 3-Year Longitudinal Study of
Children With Different Types of Learning Difficulties
Ulf Andersson
Linko¨ping University
Arithmetic and cognitive skills of children with mathematical difficulties (MD-only), with comorbid
reading difficulties (MD-RD), with reading difficulties (RD-only), and normally achieving children were
examined at 3 points from Grades 3– 4 to Grades 5– 6 (age range, 9 –13 years). Both MD groups
displayed severe weaknesses in 4 domain-specific arithmetic components (factual, conceptual, proce-
dural, and problem-solving skills) during all 3 measure points. Telling time and approximate arithmetic
were also problematic for children with MD. Both MD groups displayed a small weakness related to
visual–spatial working memory, and the MD-RD group also displayed small weaknesses related to verbal
short-term memory, processing speed, and executive functions. The 4 groups developed at similar rates
within all domain-specific components as well as basic cognitive functions. These findings demonstrate
that children identified as having MD when they are 9 years old do not catch up with their normally
achieving peers in later school grades, when they are 13 years old. They also continue to lag behind their
peers with respect to the domain-general cognitive system.
Keywords: mathematical difficulties, problem-solving skill, conceptual knowledge, procedural knowl-
edge, cognitive functions
Supplemental materials: http://dx.doi.org/10.1037/a0016838.supp
Mathematics is a subject that consists of many different domains
(e.g., arithmetic, measurement, geometry, algebra, statistics). The
first domain that children start to learn is usually arithmetic. In
school, different aspects of arithmetic are taught and learned in a
hierarchical manner (Aunola, Leskinen, Lerkkanen, & Nurmi,
2004; Dowker, 2005; Geary, 1994). Learning and acquiring skills
in single-digit arithmetic (e.g., 2 ⫹ 3 ⫽ 5) is a main activity during
the beginning of formal schooling, followed by multidigit arith-
metic and arithmetic word problem solving (Baroody & Wilkins,
1999; Geary, 1994; Geary, Hamson, & Hoard, 2000; Jordan,
Hanich, & Uberti, 2003; Kilpatrick, Swafford, & Findell, 2001;
Mazzocco & Thompson, 2005).
Approximately 4%–7% of school-age children have difficulties
with acquiring age-adequate skills in the key aspects of arithmetic
introduced in the early years of schooling (Badian, 1999; Dirks,
Spyer & de Sonneville, 2008; Lewis, Hitch, & Walker, 1994; see
Mazzocco, 2007, for a review). As arithmetic is a complex skill,
these difficulties may result from weaknesses in several possible
components. For example, to develop and perform with a high skill
level in arithmetic, at least four domain-specific components of
knowledge and skills are required: conceptual knowledge, proce-
dural knowledge and skills, factual knowledge, and problem-
solving skills (Baroody & Dowker, 2003; Delazer, 2003; Dowker,
2005; Goldman & Hasselbring, 1997; Kilpatrick et al., 2001).
Conceptual knowledge refers to an understanding of mathemat-
ical concepts, operations, and relations between concepts (Gold-
man & Hasselbring, 1997; Hiebert & Lefevre, 1986; Kilpatrick et
al., 2001; McCloskey, Aliminosa, & Macaruso, 1991). In relation
to arithmetic, it is crucial to possess a conceptual understanding of
place value, the base-10 number system, and the relationships
within and between arithmetic operations (i.e., calculation princi-
ples), as it facilitates the development of single-digit and multidigit
arithmetic as well as word problem solving (Dowker, 2005; Geary,
1994, 2004; Jordan, Hanich, & Uberti, 2003; Kilpatrick et al.,
2001; Russell & Ginsburg, 1984). Procedural knowledge and
skills refers to knowledge of calculation strategies and procedures,
knowledge of how and when to use them, and the skill to apply
them in a flexible manner (Dowker, 2005; Jordan, Hanich, &
Uberti, 2003; Kilpatrick et al., 2001; Siegler, 1988). Factual
knowledge refers to memory representations regarding the associ-
ations between problems and answers to simple arithmetic problems
in long-term memory (Ashcraft, 1992; Geary, 1993; McCloskey,
Caramazza, & Basili, 1985). Arithmetic facts allow the child to solve
single-digit arithmetic by direct retrieval, thus providing the automa-
tion that is necessary to effectively solve complex arithmetic tasks
(Ashcraft, 1992; Baroody & Wilkins, 1999; Geary et al., 2000;
Mazzocco & Thompson, 2005). Problem-solving skill refers pri-
marily to the ability to identify the problem, construct a problem
UIf Andersson, Department of Behavioral Sciences and Learning,
Linko¨ping University, Linko¨ping, Sweden.
This research was supported by Grant 2003– 0158 to me from the
Swedish Council for Working Life and Social Research. During the revi-
sion of this article, I held a postdoctoral research fellowship at the Swedish
Institute for Disability Research at Linko¨ping University.
Correspondence concerning this article should be addressed to Ulf
Andersson, Department of Behavioral Sciences and Learning, Linko¨ping
University, SE-581 83 Linko¨ping, Sweden. E-mail: ulf.andersson@liu.se
Journal of Educational Psychology © 2010 American Psychological Association
2010, Vol. 102, No. 1, 115–134 0022-0663/10/$12.00 DOI: 10.1037/a0016838
115
representation, and develop a solution plan for the problem (Geary,
1994; Kilpatrick et al., 2001; Kintsch & Greeno, 1985; Mayer &
Hegarty, 1996).
Geary (2004; Geary & Hoard, 2005) has proposed a conceptual
framework for guiding future research on MD in children. Accord-
ing to the framework, a specific mathematic domain—for exam-
ple, arithmetic—is dependent on conceptual knowledge (e.g.,
place value) and procedural knowledge and skills (e.g., calculation
strategies) that support the actual arithmetic performance (see
above). The domain-specific competencies, in turn, are supported
by an underlying cognitive system that, to simplify things, can be
divided into working memory and long-term memory. Working
memory and executive functions support arithmetic skills in chil-
dren by providing a flexible and efficient mental workspace that
can handle the different processes involved in arithmetic perfor-
mance (Fu¨rst & Hitch, 2000; Logie, Gilhooly, & Wynn, 1994;
McLean & Hitch, 1999; Seitz & Schumann-Hengsteler, 2000;
Swanson, 2004; Swanson & Beebe-Frankenberger, 2004). This
support involves not only the capacity to process and store infor-
mation simultaneously but also the ability to inhibit irrelevant
information from gaining access to working memory and the
ability to shift from one strategy or operation to another. When a
child solves arithmetic problems by means of counting strategies,
which involve working memory, associations should eventually
form between problems and a generated answer. Because counting
engages phonological and semantic information from long-term
memory (e.g., understanding the quantity associated with number
words), deficits in relation to these memory systems should result
in difficulties in forming problem–answer associations during
counting (Geary, 1993; Logie & Baddeley, 1987). The conse-
quences would include difficulties in learning arithmetic facts and
in retrieving those facts that do become represented in long-term
memory.
Based on Geary’s (2004; Geary & Hoard, 2005) conceptual
framework, the present study had two purposes: (a) to examine
from a longitudinal developmental perspective whether children
with MD display a deficiency related to one domain-specific
knowledge and skill component or a global deficiency involving
all four domain-specific components; (b) to examine from a lon-
gitudinal developmental perspective whether children with MD
display deficiencies related to basic cognitive functions. This study
builds directly on the study of Andersson (2008), in which third
and fourth graders with MD-only, MD-RD, and RD-only were
compared to age-matched controls on tasks tapping domain-
specific components and basic cognitive functions. The findings
from Andersson (2008) and other relevant research are reviewed
below.
Domain-Specific Arithmetic Components and MD
Consistent with prior research, Andersson (2008) demonstrated
that fact retrieval deficit is a cardinal characteristic of MD in
children (Bull & Johnston, 1997; Geary, Brown, & Samaranayake,
1991; Jordan & Montani, 1997; Ostad, 1997, 1998; Siegler, 1988).
These problems have been observed from first to seventh grade
(Andersson & Lyxell, 2007; Geary et al., 1991; Geary, Hoard, &
Hamson, 1999; Ostad, 1997, 1998; Russell & Ginsberg, 1984;
Siegler, 1988; Temple, 1991). Furthermore, the persistence of fact
retrieval problems has been documented by longitudinal studies
performed on children in Grades 2–3 (Jordan & Hanich, 2003;
Jordan, Hanich, & Kaplan, 2003).
It is well established that children with MD have procedural
problems, as they are slow, commit many errors, and employ
immature counting procedures when solving single-digit problems
by means of different counting strategies (i.e., finger, verbal, and
silent counting; Geary et al., 1999, 2000; Hanich, Jordan, Kaplan,
& Dick, 2001; Russell & Ginsburg, 1984; see also Geary, 2004 for
a review). Moreover, second to fourth graders with MD have
difficulties with written multidigit calculation (Hanich et al., 2001;
Jordan & Hanich, 2000; Jordan, Hanich, & Kaplan, 2003; Jordan,
Kaplan, & Hanich, 2002; Russell & Ginsburg, 1984). Children
with MD-RD appear to have more severe procedural problems
than the MD-only children, as demonstrated by their poor reason-
ing and judgment skills, whereas the MD-only children’s proce-
dural problems consist of calculation bugs (e.g., minor miscalcu-
lations, wrong operations) that are also found in normally
achieving children (Andersson & Lyxell, 2007; Geary et al., 2000;
Hanich et al., 2001; Jordan & Hanich, 2000; Jordan & Montani,
1997). An important finding obtained by Andersson (2008) was
that the lower performance of the MD children was eliminated
when the influence of fact retrieval was controlled for, suggesting
that deficits with fact retrieval are an underlying factor in the MD
children’s problems with multidigit calculation.
Studies show that second and third graders with MD have poor
conceptual knowledge related to understanding of place value and
calculation principles, whereas fourth graders with MD display
normal understanding of place value (Andersson, 2008; Hanich et
al., 2001; Jordan & Hanich, 2000; Jordan, Hanich, & Kaplan,
2003; Russell & Ginsburg, 1984). Thus, children with MD may
catch up on this fundamental aspect of arithmetic as they grow
older (i.e., Grade 4). However, studies focusing on the understand-
ing of calculation principles have generated mixed results. Anders-
son (2008) found that third and fourth graders with MD and
MD-RD have a less developed understanding of calculation prin-
ciples, whereas Russell and Ginsburg (1984) found no such weak-
ness in fourth graders with MD-RD.
Consistent with previous research, Andersson (2008) found that
both groups with MD performed poorly on simple one-step arith-
metic word problems and even worse on complex multistep word
problems (cf. Fuchs & Fuchs, 2002; Hanich et al., 2001; Jordan &
Hanich, 2000; Jordan et al., 2002; Jordan, Hanich, & Kaplan,
2003; Jordan & Montani, 1997; Parmar, Cawley, & Frazita, 1996;
Russell & Ginsburg, 1984). An important finding obtained by
Andersson (2008) was that the MD children’s poor performance
on the one-step problem-solving task was completely accounted
for by their conceptual, procedural, and factual knowledge prob-
lems, whereas their problems with multistep problems remained
when the influence related to deficits in conceptual, procedural,
and factual components were controlled for. This latter finding
suggests that children with MD have a deficit in the problem-
solving skill component. Prior studies have also shown that chil-
dren with MD-RD have more severe difficulties in solving word
problems than children with MD-only, probably due to their poorer
problem comprehension skills (e.g., Fuchs & Fuchs, 2002; Hanich
et al., 2001; Jordan & Hanich, 2000; Jordan, Hanich, & Kaplan,
2003; Jordan & Montani, 1997).
Few studies have examined MD children’s ability to quickly
provide an approximate answer to an arithmetic problem (i.e.,
116
ANDERSSON
approximate arithmetic). Findings reported by Andersson (2008),
Hanich et al. (2001), and Jordan, Hanich, and Kaplan (2003)
indicate that children with MD have problems with approximate
arithmetic, whereas Russell and Ginsburg’s (1984) study suggests
that fourth graders with MD-RD display normal skills in approx-
imation. Because of the mixed results and the fact that approxi-
mation requires a visual–spatial representation in the form of a
mental number line that appears to be independent of language, an
approximate arithmetic task was included in the present study
(Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; see also
Dehaene & Cohen, 1991; Hanich et al., 2001).
A new and important finding provided by Andersson (2008) was
that both groups with MD demonstrated substantial problems with
telling time and that these problems were equal for analog and
digital clocks. As most normally developing children are able to
identify almost all analog and digital times at the age of 8 to 10
years, one would expect that the third and fourth graders partici-
pating in the Andersson study could manage to correctly identify
most of the clocks used in the study (Case, Sandieson, & Dennis,
1986; Friedman & Laycock, 1989; Siegler & McGilly, 1989;
Vakali, 1991). The poor time-telling skills of the children with MD
can be explained by the fact that this skill in many respects
resembles basic arithmetic (counting procedures and fact retrieval;
Case et al., 1986; Friedman & Laycock, 1989; Siegler & McGilly,
1989; Vakali, 1991).
With respect to arithmetic skill development, longitudinal stud-
ies have indicated that children with MD-only and MD-RD display
rates of growth similar to normal-achieving children among all
components and aspects of arithmetic (Jordan et al., 2002; Jordan
& Hanich, 2003; Jordan, Hanich, & Kaplan, 2003); and even if
they do progress faster, they do not catch up with their normal-
achieving peers (Jordan et al., 2002).
Reading difficulties have a negative impact on specific aspects
of arithmetic, as children with MD-RD are worse at arithmetic
word problem solving and exact calculation of arithmetic combi-
nations compared with children with MD-only (Fuchs & Fuchs,
2002; Hanich et al., 2001; Jordan, Hanich, & Kaplan, 2003; Jordan
& Montani, 1997). Thus, children with MD-RD seem to have a
disadvantage compared to children with MD-only on tasks that are
mediated by language but not on tasks that depend on numerical
magnitude and automaticity (e.g., approximate arithmetic, fact
retrieval). However, the negative impact of RD on mathematics
might also include children with RD-only, as studies have found
that these children may have weaknesses in word problem solving,
multidigit calculation, and addition fact retrieval, and a vague
understanding of the concept of place value (Geary et al., 2000;
Hanich et al., 2001; Jordan & Hanich, 2003; Jordan, Hanich, &
Kaplan, 2003). It should, however, be noted that in Andersson
(2008), the two MD groups performed on a par on all arithmetic
tasks, and the RD-only children did not show any signs of weak-
nesses in arithmetic.
To summarize: Children with MD have deficits related to all
four domain-specific arithmetic components that are important to
the development of arithmetic skills. They do, however, progress
at rates of growth similar to normal-achieving children, suggesting
that MD in children is fairly persistent during the first 3 years of
elementary school. However, the question is whether MD in chil-
dren is primarily due to deficits related to one specific component,
a number of components, or more basic cognitive functioning. For
example, the procedural component involved in multidigit calcu-
lation is supported by conceptual knowledge, and the development
of conceptual knowledge appears to be facilitated by procedural
knowledge and skills and vice versa (Baroody & Dowker, 2003;
Byrnes & Wasik, 1991; Geary, 1994; Kilpatrick et al., 2001;
Rittle-Johnson & Alibali, 1999). Thus, to tap into a child’s proce-
dural component, it is important to control for the conceptual
component (cf. Andersson, 2008). To account for the arithmetic
word problem-solving difficulties that children with MD display is
especially problematic, as this task not only draws upon the
problem-solving skill component but also the procedural and con-
ceptual components (Fuchs & Fuchs, 2002; Geary, 1994; Kil-
patrick et al., 2001). Thus, to target the problem-solving skill, it is
important to control for possible deficits related to these two
components (cf. Andersson, 2008).
Basic Cognitive Functions and MD
A growing body of research shows that children with MD have
weaknesses related to basic cognitive functions that support the
performance of arithmetic. Children with MD perform poorly on
complex working memory tasks that require simultaneous storage
and processing of information (Andersson & Lyxell, 2007; Geary
et al., 1999; Geary, Hoard, Byrd-Craven, & DeSoto, 2004; Hitch &
McAuley, 1991; Swanson, 1994; Van der Sluis, van der Leij, & de
Jong, 2005; Wilson & Swanson, 2001). Furthermore, children with
MD-RD seem to have some problem with verbal and visual-spatial
short-term memory functions, whereas these functions appear to be
intact in children with MD-only (Geary et al., 1991, 1999; McLean
& Hitch, 1999; Passolunghi & Siegel, 2001, 2004; Siegel & Ryan,
1989; Swanson, 1994; Swanson & Beebe-Frankenberger, 2004).
However, a recent study by Schuchardt, Maehler, and Hasselhorn
(2008) showed that children with MD-only displayed a weakness
in visual-spatial short-term memory, whereas the MD-RD children
had weaknesses related to both verbal and visual-spatial short-term
memory. Both groups demonstrated normal verbal working mem-
ory functions. Children with MD also appear to have problems
with executive functions (e.g., inhibition control and shifting abil-
ity) and retrieving semantic and phonological information from
long-term memory (Andersson & Lyxell, 2007; Bull & Johnston,
1997; Bull, Johnston, & Roy, 1999; Bull & Scerif, 2001; Landerl,
Bevan, & Butterworth, 2004; McLean & Hitch, 1999; Passolunghi
& Siegel, 2001, 2004; Swanson & Beebe-Frankenberger, 2004;
Temple & Sherwood, 2002). Finally, studies have indicated that
children with MD might have poor general processing speed
(Andersson & Lyxell, 2007; Bull & Johnston, 1997).
In summary, these findings indicate that children with MD have
deficits related to basic cognitive functions such as working mem-
ory, short-term memory, executive functions, long-term memory,
and processing speed. It is possible that these cognitive problems
contribute to these children’s mathematical difficulties.
The Present Study
The aim of this study was to examine domain-specific knowl-
edge and skill components and basic cognitive functions in chil-
dren with different types of learning difficulties from a longitudi-
nal developmental perspective. This type of research design is
required to examine whether children with MD continue to have
117
DEVELOPMENT IN ARITHMETIC
problems related to different components of arithmetic knowledge
and skills or basic cognitive functions, or whether some of these
difficulties disappear with increasing age and further mathematics
instruction.
In this study, children were defined as having MD, MD-RD, or
RD if they received special education instruction in mathematics
or reading or both. This approach to classifying children as having
MD is somewhat different than most studies, where scores at or
below the 20th–35th percentile on a standardized mathematics
achievement test are used as a criterion for MD (Geary, 2004; see
Mazzocco, 2007, for a review). At first, these two criteria might
appear very different; however, both are in fact poor achievement
criteria (Mazzocco, 2007). The difference is that with the stan-
dardized mathematics achievement test approach, the classification
is based on a single measure point; whereas, to receive special
instruction, the child must have displayed poor achievement and
poor skill development during a period of time (see Participants
for more details). Thus, the present criterion may be considered a
more reliable indicator of poor achievement than the commonly
used cutoff score criteria.
The present study addressed the following hypotheses and re-
search questions:
1. As a defective factual knowledge component is a cardinal
characteristic of children with MD, it was predicted that both
groups with MD would continue to have a deficit related to this
component, even when participants were 12 to 13 years old
(Grades 5 and 6).
2. It was expected that both groups with MD would continue to
show severe problems with arithmetic word problem solving in
Grades 5 and 6, as this type of task involves multiple knowledge
and skill components, not just problem-solving skill. However, it
was an open question whether the MD children would display a
specific deficit to the problem-solving skill component when the
conceptual and procedural components are controlled for.
3. One important question was whether children with MD-only
and MD-RD would continue to show a weakness in conceptual
knowledge (i.e., place value, calculation principles).
4. Another interest was to examine whether children with MD-
only and MD-RD would continue to show a weakness related to
procedural knowledge and skills in Grades 5– 6. That is, would
children with MD-only and MD-RD still perform lower than the
controls on written multidigit calculations when the influence of
the conceptual and factual knowledge components was controlled
for?
5. A specific interest was to examine whether reading difficul-
ties had a negative impact on arithmetic performance. Thus, would
children with MD-RD display poorer performance and poorer
development compared with MD-only children on tasks that can
be mediated by language, such as arithmetic story problems?
Furthermore, would children with RD-only display weaknesses
and slower progress on tasks that are mediated by language (De-
haene, 1992; Geary, 1993; Kulak, 1993)?
6. The present study sought to examine the time-telling skills
in children with MD. According to their chronological age, they
should manage to correctly identify most of the clocks used in
the study. However, as telling time is a skill that in many
respects resembles basic arithmetic, it is quite possible that
children with MD would have difficulties with time telling,
even in Grades 5 and 6.
7. As basic cognitive functions support the performance of
arithmetic, the present study sought to examine whether children
with MD-only and MD-RD displayed deficiencies related to cog-
nitive functions such as working memory, executive functions,
long-term memory, and general processing speed, and whether
these potential deficits persisted up to Grades 5 and 6 when they
were 12 to 13 years old. The assumption was that MD in children
to some extent is mediated by deficits in basic cognitive functions.
To address the research questions, the study performed by
Andersson (2008) was extended by including 67 additional par-
ticipants, primarily in the MD-RD and control groups, at Time 1
and adding two additional measure points. The selection of the
tasks was guided by previous research, which provided a theoret-
ically motivated set of tasks that have been commonly used to
assess arithmetic and cognitive skills in children.
Method
Participants
A total of 274 third and fourth graders were initially tested
during the first measure point. Out of these 274 children, 25
withdrew from the study. Thus, a total of 249 children attending 28
schools in the southern parts of Sweden completed all three mea-
sure points. Because this study is an extension of Andersson
(2008), only a condensed description of the tasks and overall
procedure are presented below. (For a more detailed description,
consult Andersson, 2008. An extended description is also available
in the online supplemental materials to this article.) Children who
were receiving special instruction in mathematics (MD-only),
reading (RD-only), or both mathematics and reading (MD-RD) at
the beginning of the study were included in the three groups with
difficulties. An additional selection criterion for being classified
into the MD-only group and the RD-only group was that the
child’s score on the screening tests of reading (MD-only) and
mathematics (RD-only) was at most 1.50 standard deviations be-
low the group mean of the normally achieving children of the same
age. The children in the control group were randomly selected
from the same classrooms as the children with learning difficulties
(i.e., MD-only, MD-RD, RD-only), with the criterion that the
children were not receiving any special instruction.
General Data-Collecting Procedure
During the three measure points, the children undertook nine
mathematics tasks, four to six cognitive tasks, and two measures of
IQ. The tasks were administered in a group test session and an
individual test session, with 1 to 4 weeks between the two sessions.
The same test order was used for all children, and all testing was
performed in the child’s school. All instructions regarding the
tasks were presented orally. The data collection was performed
from February to May each year by 12 female experimenters. To
avoid ceiling effects during Measure Points 2 and 3, more items
were added to some of the tests, without changing the structure of
the tests (cf. Aunola et al., 2004). The reported reliability estimates
have been calculated on the present sample of 249 participating
children from the first year assessments.
The following tasks were administered during the group session
(in order of presentation): screening test of mathematics and read-
118
ANDERSSON
ing (Malmquist, 1977), Raven’s Progressive Matrices test (Raven,
1976; Measure Point 1), place value task, word knowledge task
(Ja¨rpsten & Taube, 1997; Measure Point 1), and written multidigit
calculation. Three reading tasks were also administered but are not
reported here. In order of presentation, the following arithmetic
and cognitive tasks were administered individually to each child:
arithmetic fact retrieval, visual matrix span, calculation principles,
telling time, digit span (not administered at Measure Point 1),
approximate arithmetic, verbal fluency task, visual number match-
ing, one-step arithmetic problem solving, multistep word problem
solving, trail making, and listening span task (not administered at
Measure Point 1). The two test sessions took approximately 120
min each, divided into three 40-min sessions with 15-min breaks
between them.
Screening Tests and Tests of Verbal and Nonverbal IQ
Screening tests of mathematics and reading (Malmquist, 1977),
the word knowledge test (Ja¨rpsten & Taube, 1997), and Raven’s
Progressive Matrices test (Raven, 1976; sets B, C and D) were
administered in the group test session. The screening tasks were
included to establish that the groups of children classified as
having MD and/or RD indeed performed significantly worse than
the group of normally achieving children. A more detailed descrip-
tion of the screening tests and IQ tests is provided by Andersson
(2008).
Arithmetic Tasks and Procedure
Arithmetic fact retrieval. The material consisted of 12 addi-
tion, 12 subtraction, and 12 multiplication combinations (e.g., 5 ⫹
4;9–4;4⫻ 5). None of the 36 combinations were doubles. The
number combinations were presented in a column on three sepa-
rate sheets of paper, one for each operation. The addition condition
was administered first, followed by the subtraction and multipli-
cation conditions. The child was instructed to provide an answer
right away and encouraged to guess if the answer was not available
right away. A stopwatch was used to measure the total response
time and to register whether the response time was longer than 3 s
for each individual combination. The number of correctly solved
combinations with response times within 3 s and the total response
time for all 36 combinations were used as dependent measures.
Cronbach’s alpha coefficient calculated on all 36 combinations
(first-year assessment) was .87.
Written multidigit arithmetic calculation. During the first
two measure points, the child solved five addition and five sub-
traction problems (e.g., 4,203 ⫹ 5,825; 8,010 ⫺ 914), whereas
during Measure Point 3 the material included four additional
problems to avoid ceiling effects (e.g., 123.50 ⫹ 17.85; 30.7 ⫺
15.65). The problems were presented horizontally, and the children
responded in writing. They were allowed 8 min to perform the task
during all three measure points. The maximum scores were 10 and
14 (Measure Point 3), and Cronbach’s coefficient alpha was .82,
calculated on a subsample of 110 children.
One-step arithmetic word problems. The task was to solve
14 (Measure Point 1), 16 (Measure Point 2), and 20 (Measure
Point 3) word problems with paper and pencil during 15 min (e.g.,
“John had 65 crowns left when he had bought a book for 36
crowns. How much did he have to start with?”) The problems were
one to four sentences long and did not include any irrelevant
information. All but two problems included multidigit calcula-
tions. The task was designed so that the test items became succes-
sively more difficult. The experimenter read the individual prob-
lems while the child followed along on the paper. If requested by
the child, the experimenter reread the problem. The coefficient
alpha was .82, calculated on a subsample of 147 children, and was
based on the first 12 out of 14 items. None of the children
attempted to solve Items 13 and 14 during the first measure point.
Multistep arithmetic word problems. The task was to solve
7 (Measure Point 1), 10 (Measure Point 2), and 14 (Measure Point
3) word problems that required at least two calculation steps (e.g.,
“Mark weighs 38 kg. His dad weighs 35 kg more. How much do
they weigh together?”). All problems included multidigit calcula-
tions, were two to six sentences long, and did not include any
irrelevant information. Thus, the linguistic demand was low, as in
the one-step problems. The same procedure as for the one-step
word problems was used. The coefficient alpha was .72, calculated
on a subsample of 166 children.
Place value. The child’s understanding of the base-10 number
system and place value was tapped by three subtests. In Subtest 1,
the child was presented with five multidigit numbers and asked to
indicate, in writing, the value of a particular digit (e.g., “What is
the value of the digit 9 in 349?”). During Measure Point 3, two
additional problems were added to reduce ceiling effects. In
Subtest 2, the task was to answer, in writing, seven questions (nine
during Measure Point 3) about which number consists of a certain
number of thousands, hundreds, tens, and ones. The trials were
presented in writing and had the following format: Indicate in
writing the number that consists of 3 hundreds, 6 tens, and 3 ones
(i.e., 363). During Measure Point 3, two additional problems were
added. In Subtest 3, the child was presented with pairs of written
numbers (e.g., 799999 – 811111), and was required to indicate the
larger number of each pair by making a circle around the larger
number (see Russell & Ginsburg, 1984). The second number in the
pair was positioned beneath the first number in the pair. The
maximum scores were 17 and 21 (Measure Point 3). Cronbach’s
alpha calculated on the present sample (first-year assessment)
was .82.
Calculation principles. This task assessed the child’s under-
standing of the commutative principle (e.g., 26 ⫹ 32 ⫽ 58, so what
is 32 ⫹ 26?), the inversion principle (23 ⫹ 14 ⫽ 37, so what is
37 – 14?), and the double plus one principle (37 ⫹ 37 ⫽ 74, so
what is 37 ⫹ 38?). The answer to the first number combination
(e.g., 26 ⫹ 32 ⫽ 58) should be used to solve the second combi-
nation (32 ⫹ 26 ⫽ ?) The problems were presented in a vertical
format on separate sheets of paper. The experimenter read and
showed each problem to the child. A maximum response time of
5 s was employed to prevent the child from calculating (Hanich et
al., 2001). If the child did not respond within the maximum
response time, the answer was considered incorrect. The maximum
score was 10. Cronbach’s alpha calculated on the present sample
was .80.
Approximate arithmetic. The material consisted of seven or
eight (Measure Point 3) addition and subtraction combinations.
The combinations were presented in a vertical format on two
separate sheets of paper. All children started with the addition
combinations. Each item was accompanied by two proposed an-
swers (e.g., 72 or 60, 52 ⫹ 17 ⫽ ?). The task was to choose the
119
DEVELOPMENT IN ARITHMETIC
answer that was closest to the correct answer. The two proposed
answers were presented first, and then the arithmetic combination.
The experimenter displayed one combination at a time to the child
but did not read it to the child. The child was asked to provide an
answer right away and not to calculate an exact answer to the
combination. If the child did not respond within 5 s, the answer
was considered incorrect, and the child was instructed to respond
more quickly (Hanich et al., 2001). A stopwatch was used to
register whether the response time was longer than 5 s for each
individual problem. The maximum scores were 14 and 16 (Mea-
sure Point 3), and Cronbach’s alpha calculated on the present
sample was .81.
Telling time. This task consisted of four analog and four
digital clock faces presented on paper. The analog clocks were
supplemented with information regarding whether it was in the
morning, before or after noon, or in the evening or night. When
telling the time of the analog clocks, the child had to express the
time in words (e.g., “quarter past 10”) and in digital form. Thus the
child could receive 0, 1, or 2 points for each analog clock. When
telling the time of the four digital clocks, the child had to express
the time in words and also state whether it was in the morning,
before noon, afternoon, in the evening, or during the night. The
child could receive 0, 1, or 2 points for each of the four digital
clocks. The maximum score was 16, and the child responded in
writing. Cronbach’s coefficient alpha was .88.
Cognitive Tasks and Procedure
Visual number-matching task. This task measured general
processing speed. The child received a sheet of paper with 30 rows
of digits. Each row consisted of seven digits, with two of the digits
being identical. The task was to cross out the two identical digits
in each row as quickly and accurately as possible. Performance
was measured by the time taken to complete all 30 rows. Test–
retest reliability of .79 was established for this task by calculating
the correlation between the first and second measure points for the
controls and the RD-only children.
Trail-making task. This test assessed the ability to switch
between operations or retrieval strategies, which is assumed to be
an important executive function (Baddeley, 1996; Lee, Cheung,
Chan, & Chan, 2000; Miyake et al., 2000). The task included two
conditions. In the A condition, the material consisted of 25 circled
numbers on a sheet of paper. The task was to connect the 25 circles
in numerical order as quickly and accurately as possible. In the B
condition, half the circles had a number in the center (1–13) and
half had a letter (A–L). The task was to start at the number 1 and
make a trail with a pencil so that each number alternated with its
corresponding letter (i.e., 1-A-2-B-3-C ...12-L-13). The differ-
ence in solution time for the two conditions (i.e., B ⫺ A) was used
as dependent measure. Reported reliabilities for this task lie be-
tween .60 and .90 (Lezak, 1995).
Verbal fluency task. This task tapped semantic memory and
required the child to generate as many words as possible from two
semantic categories (animals, food) and two initial phoneme cat-
egories (/f/, /s/). Sixty seconds were allowed for each category.
The total numbers of words retrieved within 60 s for the four
categories were used as the dependent measure (Hodges & Patter-
son, 1995). The reported test–retest reliability for this task, with an
interval of 12 months between measures, is .71 (Lezak, 1995).
Visual-matrix span task. This task tapped visual working
memory (Swanson, 1992). The child was presented with a number
of dots in a matrix. The task was to remember the location of the
dots in the matrix. One matrix at a time was displayed on a sheet
of paper for 5 s. Then the matrix was removed, and the child was
asked a process question: “Were there any dots in the first col-
umn?” After answering the process question, the child was re-
quired to draw dots in the correct squares on an identical empty
matrix. The first matrix had nine squares and included two dots.
The next span size had nine squares and three dots. The complexity
of the matrices increased for each new span size, by increasing
either the size of the matrix or the number of dots. Testing stopped
when the child had failed both trials at any particular span length.
Visual working memory was measured as the most complex ma-
trix remembered correctly, plus .5 points if the participant man-
aged to replicate correctly both trials in the same span length. The
maximum score was 14.0. Andersson (2007) reported a test–retest
reliability of .60 for this task.
Listening span task. This task tapped verbal working mem-
ory (Daneman & Carpenter, 1980). The child was presented with
sequences of three-word sentences. The sentences were presented
orally, word by word, at a rate of approximately one word per
0.8 s. The child had to decide whether each presented sentence was
a normal or an absurd sentence. Half of the sentences were absurd
(e.g., “The fish drove the car”) and half were normal (e.g., “The
rabbit was fast”). The child had to answer “yes” (normal sentence)
or “no” (absurd sentence) before the next sentence was presented.
At the end of the sequence, the child had to recall in correct serial
order the first word in each sentence. Testing stopped when the
child failed all three trials of the same span length. Listening span
was measured as the longest sequences of sentences remembered
in correct serial order, plus .33 points for each trial the child
managed to replicate in the same span length. The maximum score
was 7.66. Test–retest reliability of .54 was established by calcu-
lating the correlation between the second and third measure points
for the controls.
Digit span task. This short-term memory task required the
child to recall and repeat sets of digits that had been spoken by the
experimenter in correct serial order. The digits were presented at a
rate of one digit per second. The first span size employed was
three. Testing stopped when the child failed both trials of the same
span length. The same scoring procedure as in the listening span
task was used. The maximum score was 9.5. The reported reliabil-
ity for 8- to 16-year-old children on this task is .85 (Wechsler,
1974).
Results
Background information and results on the Raven’s progressive
matrices test, the word knowledge test, and screening tasks for
mathematics and reading for each achievement group are pre-
sented in the upper part of Table 1. The lower part of Table 1
displays mean cognitive achievement scores by achievement group
and time. Mean arithmetic achievement scores by achievement
group and time are presented in Table 2 and Figures 1, 2, and 3.
Analyses of variance (ANOVAs) performed on measures of
nonverbal and verbal IQ, mathematics, and reading revealed sig-
nificant group effects on all four measures, F(3, 245) ⫽ 14.17, p ⬍
.05,
2
⫽ .15, MSE ⫽ 41.45 (Raven’s); F(3, 245) ⫽ 24.98, p ⬍
120
ANDERSSON
Table 1
Descriptive Information and Mean Cognitive Achievement Scores for Participants by Achievement Group and Time
Variable
MD-only MD-RD RD-only Controls
Time 1 Time 2 Time 3 Time 1 Time 2 Time 3 Time 1 Time 2 Time 3 Time 1 Time 2 Time 3
MSDMSDMSD M SDMSDMSDMSDMSDMSD M SDMSDMSD
Age in months 126 6.58 136 6.90 148 7.20 124 6.73 135 6.52 147 6.29 126 7.87 137 9.16 147 7.53 126 7.24 137 7.45 148 7.22
N (number of
boys) 39 (9) 80 (33) 36 (9) 94 (48)
Raven’s matrices 19.5 6.76 18.6 6.33 21.9 6.42 24.6 6.40
Word knowledge 13.8 3.01 12.0 3.56 11.6 3.79 16.2 3.68
Mathematics
screening task 10.8 5.04 10.4 5.63 17.0 6.51 19.9 7.01
Reading screening
task 11.2 3.11 6.2 3.44 7.3 3.27 12.6 3.20
Number matching 147 39 121 30 107 22 161 41 129 33 120 34 138 38 113 36 103 38 126 31 104 27 95 30
Trail-making 87 44 78 40 64 38 102 59 82 45 65 35 79 45 67 31 57 41 66 42 51 31 42 24
Verbal fluency 42 10 48 11 52 12 39 10 43 11 45 12 39 11 43 10 47 8 47 12 48 12 51 14
Visual matrix span 5.23 1.91 5.76 2.17 6.64 2.31 5.10 1.71 5.46 2.26 6.72 2.22 5.65 2.52 6.74 2.15 7.71 1.90 6.46 2.21 7.09 2.52 8.44 1.66
Listening span 3.48 0.65 3.62 0.68 3.31 0.68 3.54 0.72 3.30 0.74 3.55 0.52 3.68 0.70 3.81 0.58
Digit span 5.62 0.84 5.90 0.88 5.20 0.81 5.35 0.87 5.65 0.95 5.85 1.23 5.81 1.09 6.11 1.11
Note. MD ⫽ mathematics difficulty; RD ⫽ reading difficulty.
121
DEVELOPMENT IN ARITHMETIC
.05,
2
⫽ .23, MSE ⫽ 12.69 (word knowledge); F(3, 245) ⫽
41.04, p ⬍ .05,
2
⫽ .33, MSE ⫽ 38.89 (mathematics); F(3,
245) ⫽ 63.95, p ⬍ .01,
2
⫽ .44, MSE ⫽ 10.73 (reading). Post hoc
testing revealed that the MD-RD group performed poorer on all
four measures compared with the controls, whereas the scores of
the MD-only group were lower on three measures: Raven’s, word
knowledge, and mathematics. The RD-only group performed on a
par with the controls on Raven’s and the mathematics test, but
scored lower on the word knowledge test and the reading test. The
three groups with difficulties (MD-only, MD-RD, RD-only) per-
formed on a par with each other on Raven’s and the word knowl-
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
Grade 3/4 Grade 4/5 Grade 5/6
Test occasions
MD-only
MD-RD
RD-only
Controls
Figure 1. Mean performance time (in seconds) for the arithmetic facts
retrieval task, by achievement group and time. MD ⫽ mathematics diffi-
culty; RD ⫽ reading difficulty.
Table 2
Mean Arithmetic Achievement Scores for Participants by Achievement Group and Time
Measure
MD-only MD-RD RD-only Controls
Time 1 Time 2 Time 3 Time 1 Time 2 Time 3 Time 1 Time 2 Time 3 Time 1 Time 2 Time 3
M SD M SD M SD M SD M SD M SD M SD M SD M SD M SD M SD M SD
Fact retrieval
(score) 18 8 22 8 26 6 15 8 21 8 24 8 25 7 28 6 31 4 27 7 31 5 33 4
Fact retrieval
(in seconds) 243 87 194 76 136 44 269 124 202 110 169 87 170 67 136 56 111 53 135 53 106 36 89 32
Place value 13.6 1.99 14.0 1.78 16.4 2.29 12.2 3.20 13.5 2.29 15.8 2.53 14.0 1.95 14.4 1.71 16.7 3.06 14.3 1.46 15.1 1.10 18.5 2.23
Calculation
principles 6.13 2.41 7.54 2.02 7.51 2.53 5.29 2.60 7.00 2.43 7.35 2.37 7.89 2.40 8.56 1.61 8.83 1.34 8.20 2.10 9.09 1.25 9.37 1.26
Multidigit
calculation 2.90 1.67 4.05 1.45 5.77 1.93 2.89 2.19 4.29 2.56 5.86 2.99 4.56 2.37 5.78 2.17 6.78 2.61 5.71 2.35 7.35 2.08 8.78 3.05
One-step
problems 4.21 2.54 6.56 2.26 8.18 2.87 4.64 2.75 6.34 2.78 7.65 3.25 7.64 2.55 8.50 2.52 10.3 2.70 8.04 2.66 10.8 2.58 12.5 2.99
Multistep
problems 1.10 1.29 1.77 1.72 3.08 2.08 0.89 1.16 1.94 1.69 2.81 2.08 2.42 1.86 3.39 2.07 5.39 2.22 3.13 1.76 4.50 2.20 6.10 2.39
Approximation 5.56 3.40 6.38 3.38 8.46 4.38 5.03 3.16 6.38 3.27 8.84 3.99 7.19 3.51 9.17 3.22 10.8 4.31 7.49 3.88 9.40 3.14 12.0 3.16
Telling time 9.4 4.55 10.4 3.57 13.0 2.86 7.5 3.86 10.0 3.88 11.5 3.41 11.6 3.98 12.6 3.00 13.8 2.21 12.9 2.85 13.7 2.36 14.4 1.61
Note. MD ⫽ mathematics difficulty; RD ⫽ reading difficulty.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Grade 3/4 Grade 4/5 Grade 5/6
Test occasions
MD-only
MD-RD
RD-only
Controls
Figure 2. Mean number of correctly solved one-step arithmetic word
problems, by achievement group and time. The maximum score is 20.
MD ⫽ mathematics difficulty; RD ⫽ reading difficulty.
122
ANDERSSON
edge test (all ps ⬎ .05), but the RD-only group performed signif-
icantly better on the mathematics task compared with the two MD
groups. The MD-only group performed significantly better on the
reading test compared with the two RD groups.
To simplify the statistical analyses, the Raven’s test and the
word knowledge test were aggregated into a single IQ measure.
This was done by first z-transforming the two tasks and calculating
a mean from these two z variables. The correlation between the
word knowledge test and Raven’s calculated on the present sample
was r ⫽ .42, p ⬍ .05.
To address the research questions of the current study, analyses
of covariance (ANCOVAs) and effect sizes (partial
2
) were
computed to examine achievement group differences, effects of
time, and Group ⫻ Time interactions. The aggregated IQ measure
was used as a covariate in all ANCOVAs to control for group
differences in verbal and nonverbal IQ, which have been demon-
strated above. As the main effect of time was of minor interest in
the current study, this effect is reported but not discussed further.
An
2
of .09 or less is considered to be small, whereas an
2
between .10 to .24 is of medium size. An
2
of .25 or larger
represents a large effect (Cohen, 1988). ANCOVAs were first
performed on the basic cognitive tasks, followed by the arithmetic
tasks (see Table 3).
Cognitive Tasks
Visual number matching task. The ANCOVA performed on
the number-matching task demonstrated a medium-size achieve-
ment group effect, F(3, 244) ⫽ 7.02, p ⬍ .05,
2
⫽ .12, MSE ⫽
1700.56, and a large main effect of time, F(2, 488) ⫽ 199.36, p ⬍
.05,
2
⫽ .45, MSE ⫽ 370.39, but no Group ⫻ Time interaction
( p ⬎ .05). The Tukey-Kramer post hoc testing revealed that the
children in the control group performed significantly faster than
the MD-RD children, whereas the MD-only and RD-only children
performed on a par with the controls ( p ⬎ .05). The MD-RD group
was also outperformed by the RD-only group.
Trail-making task. A small achievement group effect was
found on the trail-making task, F(3, 244) ⫽ 5.59, p ⬍ .05,
2
⫽
.06, MSE ⫽ 2663.98. Post hoc testing revealed that the MD-RD
group performed the trail-making task slower than the controls.
The RD-only group and the MD-only group, on the other hand,
performed as fast as the controls ( p ⬎ .05). A medium-size time
effect was also obtained, F(2, 488) ⫽ 35.51, p ⬍ .05,
2
⫽ .13,
MSE ⫽ 1006.68, but no Group ⫻ Time interaction ( p ⬎ .05).
Verbal fluency task. The ANCOVA performed on the verbal
fluency task demonstrated only a medium-size effect of time, F(2,
488) ⫽ 54.54, p ⬍ .05,
2
⫽ .18, MSE ⫽ 43.42. Thus, no group
differences reached significance on this task.
Visual matrix span task. The ANCOVA on the visual matrix
span task yielded a small achievement group effect, F(3, 244) ⫽
6.35, p ⬍ .05,
2
⫽ .07, MSE ⫽ 6.87, and a significant main effect
of time, F(2, 488) ⫽ 50.99, p ⬍ .05,
2
⫽ .17, MSE ⫽ 3.09, but
no interaction effect ( p ⬎ .05). The Tukey-Kramer tests showed
that the control group scored significantly higher than the MD-
only group and the MD-RD group. No other group differences
reached significance on this task.
Listening span task. A significant main effect of time was
obtained, F(2, 488) ⫽ 12.08, p ⬍ .05,
2
⫽ .05, MSE ⫽ 0.29, but
no group effect or interaction effect ( p ⬎ .05).
Digit span task. The ANCOVA on the digit span task yielded
a significant main effect for group, F(3, 244) ⫽ 3.62, p ⬍ .05,
2
⫽ .04, MSE ⫽ 1.45, and time, F(2, 488) ⫽ 13.95, p ⬍ .05,
2
⫽
.05, MSE ⫽ 0.42, but not a significant interaction between group
and time ( p ⬎ .05). This significant group effect was due to the
lower scores of the MD-RD group compared with the control
group. No other group differences reached significance ( p ⬎ .05).
Arithmetic Tasks
Arithmetic fact retrieval. The ANCOVAs performed on the
arithmetic fact retrieval task demonstrated a significant and large
achievement group effect for the accuracy measure and a medium-
size group effect for the time measure (see Table 3). Significant
main effects of time were obtained for both measures (accuracy
and time). More important, the Group ⫻ Time interaction was
significant for both the accuracy measure and the time measure.
Post hoc testing of the main group effects revealed that both
groups with MD performed worse than the controls and RD-only
group on both accuracy and time, whereas the controls and the
RD-only group performed on a par.
As can be seen in Table 2 and Figure 1, the interactions are a
result of the two MD groups progressing at a faster rate than the
controls. The slower progress of the controls on the accuracy
measure is probably due to a ceiling effect, but a ceiling effect
cannot account for their slower progress on the time measure (see
Figure 1). The interesting question was whether the MD-only or
MD-RD children would catch up with the controls during Measure
Points 2 and 3 or if they would continue to lag behind, even though
they progressed at a faster rate than the controls. To address this
question, three separate tests of simple effects ANCOVAs were
performed, one for each measure point, with IQ as a covariate.
0
1
2
3
4
5
6
7
8
Grade 3/4 Grade 4/5 Grade 5/6
Test occasions
MD-only
MD-RD
RD-only
Controls
Figure 3. Mean number of correctly solved multistep arithmetic word
problems, by achievement group and time. The maximum score is 14.
MD ⫽ mathematics difficulty; RD ⫽ reading difficulty.
123
DEVELOPMENT IN ARITHMETIC
The first measure point showed a large achievement group
effect for the accuracy measure, F(3, 732) ⫽ 39.71, p ⬍ .05,
2
⫽
.28, MSE ⫽ 42.38, and a medium-size effect for the time measure,
F(3, 732) ⫽ 32.12, p ⬍ .05,
2
⫽ .22, MSE ⫽ 5,621.88 The
second measure point revealed a large achievement group effect
for the accuracy measure, F(3, 732) ⫽ 25.92, p ⬍ .05,
2
⫽ .25,
MSE ⫽ 42.38, and a medium-size effect for the time measure, F(3,
732) ⫽ 18.46, p ⬍ .05,
2
⫽ .18, MSE ⫽ 5,621.88. The third
measure point showed medium-size achievement group effects for
the accuracy measure, F(3, 732) ⫽ 17.30, p ⬍ .05,
2
⫽ .22,
MSE ⫽ 42.38, and the time measure, F(3, 732) ⫽ 11.15, p ⬍ .05,
2
⫽ .18, MSE ⫽ 5,621.88.
Post hoc testing revealed the same pattern of results during all
three measure points for the accuracy measure. That is, both the
controls and the RD-only group obtained higher scores than the
MD-only group and the MD-RD group. No other group differences
reached significance ( ps ⬎ .05). For the time measure, both the
MD-only group and the MD-RD group performed significantly
worse than the controls and the RD group during Measure Points
1 and 2. On Measure Point 3, the MD-RD groups continued to
perform slower than both the controls and the RD-only group,
whereas the MD-only group only performed slower than the con-
trols.
Place value. On this task there was a small but significant
achievement group effect and a significant time effect, whereas the
interaction did not reach significance ( p ⬎ .05). Post hoc testing
showed that the MD-RD group was outperformed by the controls.
No other group differences reached significance.
Calculation principles. The ANCOVA yielded an achieve-
ment group effect and a significant time effect but no interaction
effect ( p ⬎ .05). Both MD groups were outperformed by controls
and the RD-only group, as indicated by the Tukey-Kramer post
hoc tests.
Written multidigit arithmetic calculation. A medium-size
achievement group effect was found on the multidigit arithmetic
calculation task, and a large time effect, but no interaction effect
( p ⬎ .05). Tukey-Kramer testing revealed that the controls scored
significantly higher than both MD groups. The RD-only group scored
higher than the MD-only group and the MD-RD group, whereas the
two MD groups performed equally on this task ( p ⬎ .05).
It is apparent that children with MD have problems with mul-
tidigit arithmetic calculation. To what extent are these problems
specifically related to the procedural knowledge and skills com-
ponent? To investigate this, the involvement of the conceptual
component and the factual component must be partialled out, as
multidigit calculation is supported by these two components (Ba-
roody & Wilkins, 1999; Geary et al., 2000; Geary, 2004; Jordan,
Hanich, & Kaplan, 2003; Mazzocco & Thompson, 2005; McCloskey
et al., 1991). An ANCOVA with the three tasks as covariates revealed
a small achievement group effect, F(3, 242) ⫽ 5.82, p ⬍ .05,
2
⫽
Table 3
Results of the Analysis of Covariance Performed on the Arithmetic Tasks
Task F df MSE
2
Fact retrieval, accuracy
Group effect 33.53
ⴱ
3,244 101.56 .29
Time effect 246.35
ⴱ
2,488 12.79 .50
Group ⫻ Time interaction 4.27
ⴱ
6,488 12.79 .05
Fact retrieval, time
Group effect 24.01
ⴱ
3,244 13,608.73 .23
Time effect 179.64
ⴱ
2,488 1,628.45 .42
Group ⫻ Time interaction 6.24
ⴱ
6,488 1,628.45 .07
Place value
Group effect 8.05
ⴱ
3,244 6.30 .09
Time effect 207.94
ⴱ
2,488 3.14 .46
Calculation principles
Group effect 25.42
ⴱ
3,244 5.93 .24
Time effect 35.27
ⴱ
2,488 3.12 .13
Arithmetic calculation
Group effect 17.76
ⴱ
3,244 10.66 .18
Time effect 165.13
ⴱ
2,488 2.50 .40
One-step word problems
Group effect 29.73
ⴱ
3,244 13.49 .27
Time effect 232.28
ⴱ
2,488 2.76 .49
Group ⫻ Time interaction 4.99
ⴱ
6,488 2.76 .06
Multistep word problems
Group effect 26.98
ⴱ
3,244 5.85 .25
Time effect 198.01
ⴱ
2,488 1.63 .45
Group ⫻ Time interaction 2.69
ⴱ
6,488 1.63 .03
Approximate arithmetic
Group effect 20.74
ⴱ
3,244 19.23 .20
Time effect 81.38
ⴱ
2,488 9.13 .25
Telling time
Group effect 18.39
ⴱ
3,244 20.82 .18
Time effect 105.96
ⴱ
2,488 3.82 .30
Group ⫻ Time interaction 5.16
ⴱ
6,488 3.82 .06
ⴱ
p ⬍ .05.
124
ANDERSSON
.07, MSE ⫽ 2.84; a medium-size contribution for the arithmetic
fact retrieval task, F(1, 242) ⫽ 33.42, p ⬍ .05,
2
⫽ .12, MSE ⫽
2.84; and a medium-size contribution for the place value task, F(1,
242) ⫽ 44.46, p ⬍ .05,
2
⫽ .16, MSE ⫽ 2.84, but not the
calculation principles task ( p ⬎ .05). This significant group effect
was due to the lower scores of the MD-only group and the MD-RD
group compared with the controls. No other group differences
reached significance ( p ⬎ .05).
One-step arithmetic word problems. The ANCOVA yielded
a large time effect and a large achievement group effect on
one-step word problems. The MD-RD group and the MD-only
group obtained significantly lower scores than the controls. The
MD-RD group and the MD-only group were also outperformed by
the RD-only group. The initial ANCOVA also revealed a small
interaction effect, indicating that the four groups progressed at
different rates (see Figure 2). The controls demonstrated an in-
crease of 4.45 points from Measure Points 1 to 3, whereas the
corresponding increases for the MD-only group, the MD-RD
group, and the RD-only group were 3.97 points, 3.01 points, and
2.69 points, respectively. Separate tests of simple effects
ANCOVAs were therefore performed to examine if the achieve-
ment group differences varied during the three measure points
while controlling for IQ.
Significant achievement group effects of medium size were
obtained on the first measure point, F(3, 732) ⫽ 19.05, p ⬍ .05,
2
⫽ .21, MSE ⫽ 6.33; the second measure point, F(3, 732) ⫽
20.03, p ⬍ .05,
2
⫽ .23, MSE ⫽ 6.33; and the third measure
point, F(3, 732) ⫽ 28.59, p ⬍ .05,
2
⫽ .21, MSE ⫽ 6.33. Post hoc
testing showed that during all three measure points, the two MD
groups scored significantly lower than the controls and the RD-
only group. No other group differences reached significance ( p ⬎
.05) during the three measure points.
Multistep arithmetic word problems. The ANCOVA per-
formed on the multistep word problem-solving task revealed a
large time effect, a large achievement group effect, and a small
interaction effect. Subsequent testing of the achievement group
effect demonstrated that both MD groups had significantly lower
scores compared with the controls. The scores of the MD-RD
group and the MD-only group were also significantly lower than
that of the RD-only group.
The observed interaction effect is a result of faster development
of the controls and the RD-only group compared with the two MD
groups (see Figure 3). The controls and the RD-only group dem-
onstrated an increase of 2.97 points from Measure Points 1 to 3,
whereas the corresponding increase for the MD-only group and the
MD-RD group were 1.98 points and 1.92 points, respectively. The
test of simple effects ANCOVAs performed on the three measure
points with IQ as a covariate and the follow-up comparisons
showed that the two MD groups performed worse than the controls
and the RD-only group on Measure Point 1, F(3, 732) ⫽ 11.01,
p ⬍ .05,
2
⫽ .18, MSE ⫽ 3.04; Measure Point 2, F(3, 732) ⫽
15.28, p ⬍ .05,
2
⫽ .15, MSE ⫽ 3.04; and Measure Point 3, F(3,
732) ⫽ 28.56, p ⬍ .05,
2
⫽ .21, MSE ⫽ 3.04.
As arithmetic word problem-solving involves conceptual
knowledge and procedural knowledge and skill, not only problem-
solving skill, it is possible that the poor performance of the MD
groups on these tasks is due to their deficiencies with these two
components. The strong correlations between performance on the
one-step and the multistep problems (pooled over the three mea-
sure points) and multidigit calculation task (r ⫽ .84; r ⫽ .78, p ⬍
.05, respectively), the place value task (r ⫽ .68; r ⫽ .65, p ⬍ .05,
respectively), and the calculation principles task (r ⫽ .59; r ⫽ .57,
p ⬍ .05, respectively) support this assumption.
To isolate the problem-solving skill component, ANCOVAs
with three covariates were performed on the two arithmetic word-
problem solving tasks. On the one-step word problems a small
achievement group effect was found, F(3, 242) ⫽ 4.74, p ⬍ .05,
2
⫽ .06, MSE ⫽ 2.04; the calculation principles task provided a
small contribution, F(1, 242) ⫽ 14.55, p ⬍ .05,
2
⫽ .06, MSE ⫽
2.04; the place value task provided a medium-size contribution,
F(1, 242) ⫽ 31.50, p ⬍ .05,
2
⫽ .12, MSE ⫽ 2.04; and the
multidigit calculation task provided a large contribution, F(1,
242) ⫽ 193.81, p ⬍ .05,
2
⫽ .44, MSE ⫽ 2.04.
On the multistep word problems a small achievement group
effect emerged, F(3, 242) ⫽ 4.84, p ⬍ .05,
2
⫽ .06, MSE ⫽ 1.21;
the calculation principles task, F(1, 242) ⫽ 9.01, p ⬍ .04,
2
⫽
.06, MSE ⫽ 1.21, and the place value task, F(1, 242) ⫽ 23.74, p ⬍
.05,
2
⫽ .09, MSE ⫽ 1.21, provided small contributions; whereas
the multidigit calculation task provided a large contribution, F(1,
242) ⫽ 115.78, p ⬍ .05,
2
⫽ .32, MSE ⫽ 1.21. Follow-up
comparisons revealed that both groups with MD scored signifi-
cantly worse than the control group on both word-problem solving
tasks, indicating that they have problems with the problem-solving
skill component. In addition, both MD groups were outperformed
by the RD-only group on the multistep word problem-solving
tasks.
Approximate arithmetic. A large effect of time and a
medium-size achievement group effect were found on the approx-
imate arithmetic task, but no interaction effect ( p ⬎ .05). The
MD-RD group and the MD-only group performed worse than both
the controls and the RD-only group. No other group difference
reached significance ( p ⬎ .05).
Telling time. A medium-size achievement group effect and a
large time effect were found on the time-telling task. The control
group and the RD-only group scored equally but scored signifi-
cantly higher than the MD-RD group and the MD-only group. The
two MD groups reached the same achievement level ( p ⬎ .05). A
small interaction effect was also found. The performance increases
from Measure Points 1–3 for the MD-only group, the MD-RD
group, the RD-only group, and the controls were 3.57 points, 4.00
points, 2.19 points, and 1.54 points, respectively. Thus, controls
demonstrated a smaller increase compared with the two MD
groups; however, the slower progress of the controls is doubtless
a consequence of a ceiling effect, as the group mean score was
14.45 out of a maximum of 16 points during Measure Point 3. This
is expected, as children at the age of 8 to 10 years usually manage
to identify almost all analog and digital clocks used in the study
(Case et al., 1986; Friedman & Laycock, 1989; Siegler & McGilly,
1989; Vakali, 1991).
As the initial ANCOVAs already established that both MD
groups have problems with telling time, the interesting question is
whether children with MD continue to demonstrate problems with
telling time during Measure Points 2 and 3 when they are 11 to 13
years old. To address this question, tests of simple effects
ANCOVAs and post hoc tests were performed for each measure
point, with IQ as a covariate. On the first measure point, both MD
groups performed worse than the controls and the RD-only group,
F(3, 732) ⫽ 24.95, p ⬍ .05,
2
⫽ .19, MSE ⫽ 9.48; similar results
125
DEVELOPMENT IN ARITHMETIC
Table 4
Hierarchical Regression Analyses of Arithmetic Tasks
Variable Predictor  pr
2
Model 1: Fact retrieval time
Block 1: F(3, 209) ⫽ 35.71, R
2
⫽ .34
MD-RD vs. Con .52
ⴱ
.17
MD-only vs. Con .32
ⴱ
.08
IQ ⫺.14
ⴱ
.01
Block 2: F(7, 205) ⫽ 22.13, R
2
⫽ .43
MD-RD vs. Con .38
ⴱ
.08
MD-only vs. Con .26
ⴱ
.05
IQ ⫺.04 .00
Number matching .25
ⴱ
.04
Trail-making .04 .00
Visual-matrix span ⫺.02 .00
Digit span ⫺.16
ⴱ
.02
Model 2: Fact retrieval accuracy
Block 1: F(3, 209) ⫽ 50.44, R
2
⫽ .42
MD-RD vs. Con ⫺.56
ⴱ
.20
MD-only vs. Con ⫺.36
ⴱ
.10
IQ .18
ⴱ
.02
Block 2: F(7, 205) ⫽ 26.41, R
2
⫽ .47
MD-RD vs. Con ⫺.45
ⴱ
.11
MD-only vs. Con ⫺.32
ⴱ
.07
IQ .09 .00
Number matching ⫺.18
ⴱ
.02
Trail-making ⫺.00 .00
Visual-matrix span .06 .00
Digit span .13
ⴱ
.01
Model 3: Place value
Block 1: F(3, 209) ⫽ 50.06, R
2
⫽ .42
MD-RD vs. Con ⫺.30
ⴱ
.06
MD-only vs. Con ⫺.13
ⴱ
.01
IQ .46
ⴱ
.15
Block 2: F(7, 205) ⫽ 25.12, R
2
⫽ .46
MD-RD vs. Con ⫺.22
ⴱ
.03
MD-only vs. Con ⫺.07 .00
IQ .37
ⴱ
.09
Number matching ⫺.06 .00
Trail making ⫺.04 .00
Visual matrix span .20
ⴱ
.02
Digit span ⫺.01 .00
Model 4: Calculation principles
Block 1: F(3, 209) ⫽ 47.10, R
2
⫽ .40
MD-RD vs. Con ⫺.47
ⴱ
.15
MD-only vs. Con ⫺.30
ⴱ
.07
IQ .26
ⴱ
.05
Block 2: F(7, 205) ⫽ 22.50, R
2
⫽ .43
MD-RD vs. Con ⫺.40
ⴱ
.09
MD-only vs. Con ⫺.26
ⴱ
.05
IQ .19
ⴱ
.02
Number matching ⫺.07 .00
Trail making ⫺.03 .00
Visual matrix span .15
ⴱ
.01
Digit span .00 .00
Model 5: Arithmetic calculation
Block 1: F(3, 209) ⫽ 57.29, R
2
⫽ .45
MD-RD vs. Con ⫺.36
ⴱ
.09
MD-only vs. Con ⫺.36
ⴱ
.10
IQ .38
ⴱ
.11
Block 2: F(7, 205) ⫽ 33.17, R
2
⫽ .53
MD-RD vs. Con ⫺.25
ⴱ
.03
MD-only vs. Con ⫺.28
ⴱ
.06
IQ .29
ⴱ
.05
Number matching ⫺.16
ⴱ
.02
Trail making ⫺.17
ⴱ
.02
Visual matrix span .08 .00
Digit span ⫺.04 .00
Model 6: One-step word problems
Block 1: F(3, 209) ⫽ 87.06, R
2
⫽ .56
MD-RD vs. Con ⫺.42
ⴱ
.11
MD-only vs. Con ⫺.38
ⴱ
.11
IQ .41
ⴱ
.13
Block 2: F(7, 205) ⫽ 49.24, R
2
⫽ .63
MD-RD vs. Con ⫺.31
ⴱ
.05
MD-only vs. Con ⫺.30
ⴱ
.07
IQ .32
ⴱ
.06
Number matching ⫺.12
ⴱ
.01
Trail making ⫺.06 .00
(table continues)
126
ANDERSSON
were obtained on the second measure point, F(3, 732) ⫽ 13.06,
p ⬍ .05,
2
⫽ .13, MSE ⫽ 9.84. On the third measure point, the
ANCOVA and post hoc testing showed that the MD-RD group
performed worse than the controls and the RD-only group, F(3,
732) ⫽ 6.52, p ⬍ .05,
2
⫽ .11, MSE ⫽ 9.84. The MD-only group
performed on a par with the controls when IQ was considered.
However, an ANOVA performed on the third measure point, F(3,
245) ⫽ 19.89, p ⬍ .05,
2
⫽ .20, MSE ⫽ 6.71, revealed that the
MD-only group performed worse than the controls.
Arithmetic Performance of the RD-Only Group
One interest of the present study was to examine whether children
with RD-only display weaknesses and slower progress on arithmetic
tasks that are mediated by language. The previously performed anal-
yses have not revealed any such weaknesses when controlling for the
RD-only children’s poorer verbal IQ. However, it might be possible to
detect weaknesses and slower progress on certain tasks (e.g., one-step
word problems) if verbal IQ is not considered in the analyses. To
examine this question further, additional ANOVAs were performed
on the one-step and multistep word problem-solving tasks and the
written multidigit calculation task. The ANOVAs and the follow-up
comparisons of the main group effects showed that the RD-only
group performed worse than the controls on one-step word problems,
F(3, 245) ⫽ 54.20, p ⬍ .05,
2
⫽ .40, MSE ⫽ 17.37; multistep word
problems, F(3, 245) ⫽ 49.07, p ⬍ .05,
2
⫽ .38, MSE ⫽ 7.85; and
the multidigit calculation task, F(3, 245) ⫽ 37.12, p ⬍ .05,
2
⫽ .31,
MSE ⫽ 12.49. Small interaction effects were also obtained on one-
step word problems, F(6, 490) ⫽ 4.99, p ⬍ .05,
2
⫽ .06, MSE ⫽
2.76, and on multistep word problems, F(6, 490) ⫽ 4.05, p ⬍ .05,
2
⫽ .04, MSE ⫽ 1.64. Tests of simple effects ANOVAs and post hoc
testing revealed that the RD-group obtained significantly lower scores
on one-step problems than the controls on Measure Point 2, F(3,
735) ⫽ 59.08, p ⬍ .05, MSE ⫽ 5.68, and Measure Point 3, F(3,
735) ⫽ 67.84, p ⬍ .05, MSE ⫽ 5.68. On the multistep problems, the
RD-only group obtained lower scores on Measure Point 2, F(3,
735) ⫽ 32.09, p ⬍ .05, MSE ⫽ 3.71, but not Measure Points 1 and 3.
The Contribution of Basic Cognitive Functions to
MD in Children
The above results show that children with MD have deficits
related to all examined areas of arithmetic. Can these deficits to
some extent be accounted for by the cognitive weaknesses dis-
Table 4 (continued)
Variable Predictor  pr
2
Visual matrix span .22
ⴱ
.03
Digit span ⫺.02 .00
Model 7: Multi-step word problems
Block 1: F(3, 209) ⫽ 88.12, R
2
⫽ .56
MD-RD vs. Con ⫺.39
ⴱ
.10
MD-only vs. Con ⫺.35
ⴱ
.10
IQ .45
ⴱ
.15
Block 2: F(7, 205) ⫽ 47.75, R
2
⫽ .62
MD-RD vs. Con ⫺.28
ⴱ
.05
MD-only vs. Con ⫺.28
ⴱ
.06
IQ .34
ⴱ
.07
Number matching ⫺.08 .00
Trail making ⫺.07 .00
Visual matrix span .21
ⴱ
.03
Digit span .02 .00
Model 8: Approximate arithmetic
Block 1: F(3, 209) ⫽ 23.78, R
2
⫽ .25
MD-RD vs. Con ⫺.52
ⴱ
.17
MD-only vs. Con ⫺.40
ⴱ
.13
IQ ⫺.05 .00
Block 2: F(7, 205) ⫽ 15.28, R
2
⫽ .34
MD-RD vs. Con ⫺.38
ⴱ
.08
MD-only vs. Con ⫺.32
ⴱ
.08
IQ ⫺.14
ⴱ
.01
Number matching ⫺.23
ⴱ
.03
Trail making ⫺.08 .00
Visual matrix span .12 .00
Digit span .00 .00
Model 9: Telling time
Block 1: F(3, 209) ⫽ 40.83, R
2
⫽ .37
MD-RD vs. Con ⫺.45
ⴱ
.13
MD-only vs. Con ⫺.24
ⴱ
.05
IQ .27
ⴱ
.05
Block 2: F(7, 205) ⫽ 21.12, R
2
⫽ .42
MD-RD vs. Con ⫺.35
ⴱ
.07
MD-only vs. Con ⫺.18
ⴱ
.02
IQ .17
ⴱ
.02
Number matching ⫺.07 .00
Trail making ⫺.07 .00
Visual matrix span .18
ⴱ
.02
Digit span .02 .00
Note. pr
2
⫽ squared part correlations, representing the proportion of unique variance explained by each variable; MD ⫽ mathematics difficulty; RD ⫽
reading difficulty; Con ⫽ control.
ⴱ
p ⬍ .05.
127
DEVELOPMENT IN ARITHMETIC
played by the MD children? To address this question, hierarchical
regression analyses, with two blocks, were performed on the
pooled mean scores calculated over the three measure points for all
arithmetic tasks. Because the aim was to determine if the observed
group differences could be attributed to problems with basic cog-
nitive functions, the first block included the group contrast vari-
ables, whereas the second block included all cognitive tasks on
which the MD children performed worse than the controls. Two
group contrast variables were created via dummy coding (0 and 1).
One contrast variable captured the difference between the control
group and the MD-RD group (Con vs. MD-RD), and the second
captured the difference between the control group and the MD-
only group (Con vs. MD-only). The control group was used as
reference group and therefore always coded as 0, whereas the
contrast group in each contrast variable was coded as 1. The
remaining group in each contrast variable was also coded as 0. All
variables in the blocks were entered simultaneously. The RD-only
group was not included in the regression analyses. As can be seen
in Table 4, the number-matching task emerged as a significant
predictor of both measures of fact retrieval (Models 1 and 2),
written arithmetic calculation (Model 5), one-step word problems
(Model 6), and approximate arithmetic (Model 8). The visual
matrix span task turned out to be a significant predictor of place
value (Model 3), calculation principles (Model 4), one-step and
multistep word problems (Models 6 and 7), and telling time
(Model 9). The trail-making task provided significant contribu-
tions to the two measures of fact retrieval (Models 1 and 2) and
written arithmetic calculation (Model 5), whereas the digit span
task contributed to the two measures of fact retrieval (Models 1
and 2).
However, the important aspect of the hierarchical regression
analyses is whether the contribution of the group contrast variables
was eliminated or reduced when the cognitive tasks were included
in the model (i.e., Block 2). The second blocks in all nine models
show that the MD-RD vs. Con contrast variable continued to
account for significant amounts of variance, estimated by the
squared part correlations, on all arithmetic tasks. Similar results
were obtained for the MD-only vs. Con contrast variable, with the
exception of the place value task (Model 3). When the cognitive
tasks, especially the visual matrix span task, were included in
Model 3, the effect of the MD-only vs. Con contrast variable was
completely eliminated. Although the group contrast variables con-
tinued to be significant predictors even when controlling for group
differences on the cognitive tasks, the amount of variance ac-
counted for by the group contrast variables was reduced by
3%–9% for the MD-RD vs. Con variable and by 1%–5% for the
MD-only vs. Con variable, compared with the first blocks. Thus,
deficiencies in cognitive function cannot fully account for MD in
children, but they do account for it to some extent.
Discussion
The present study was theoretically based on Geary’s (2004;
Geary & Hoard, 2005) conceptual framework, stating that any
mathematical domain is supported by domain-specific competen-
cies (conceptual and procedural) and a domain-general cognitive
system. These two aspects were examined in children with differ-
ent types of learning difficulties from a longitudinal developmental
perspective. Before discussing the results in relation to the specific
research questions in more detail, an overall conclusion is dis-
cussed.
The overall picture regarding development is that group differ-
ences in development were found on very few arithmetic tasks and
none of the cognitive tasks. Furthermore, the different rates of
development that were observed were small and did not change the
achievement group differences during the three measure points.
The conclusion that can be drawn is that all four groups developed
at similar rates in all domain-specific knowledge and skill com-
ponents as well as basic cognitive functions (Jordan & Hanich,
2003; Jordan, Hanich, & Kaplan, 2003).
Factual Knowledge Component and MD
Consistent with the prediction, the children with MD displayed
severe problems when performing the arithmetic fact retrieval task
(cf. Geary et al., 1999; Jordan & Hanich, 2003; Jordan, Hanich, &
Kaplan, 2003; Ostad, 1997, 1998; Russel & Ginsberg, 1984).
Although the MD children enhanced their ability to solve simple
arithmetic problems by means of fast and accurate retrieval more
than the controls, they still displayed considerable problems during
Measure Point 3. Thus, the present study provides further support
to the statement that a defective factual knowledge component is a
cardinal characteristic of children with MD-only and MD-RD.
This deficit is among the first to be observed in children with MD;
it appears to persist during the entire childhood and maybe into
adolescence, but it does not seem to increase with age (cf. Ostad,
1997).
Conceptual Knowledge Component and MD
During all three measure points and independent of the influ-
ence of IQ, the MD-RD group performed worse than the controls
on both the place value and the calculation principles tasks used to
assess conceptual knowledge. A similar finding was obtained for
the MD-only group on the calculation principles task. Although
most controls reached the ceiling on the calculation principles task
at Measure Point 2, the two MD groups did not catch up with the
controls but continued to display lower achievement levels. These
observations extend previous research by demonstrating that
weaknesses in conceptual knowledge regarding basic concepts
such as the base-10 number system and place value and under-
standing of the relationships within and between arithmetic oper-
ations is not only present in younger children with MD (7 to 10
years old), but continues to be a weakness in older children with
MD (12 to 13 years old; cf. Hanich et al., 2001; Jordan & Hanich,
2000; Jordan & Hanich, 2003; Jordan, Hanich, & Kaplan, 2003).
Procedural Knowledge and Skills and MD
Similar to studies performed on second, third, and fourth graders
with MD, the present results show that written multidigit arith-
metic calculation is an area of substantial weakness for children
with MD ages 12–13 years old in Grades 5 and 6 (Hanich et al.,
2001; Jordan & Hanich, 2000; Jordan et al., 2002; Jordan, Hanich,
& Kaplan, 2003; Russell & Ginsburg, 1984). Children in the
MD-only group and the MD-RD group both scored worse than the
controls, even when controlling for measures tapping the factual
knowledge component and the conceptual component, demonstrat-
128
ANDERSSON
ing that they have a deficit specifically related to procedural
knowledge and skills. However, this weakness in knowledge of
arithmetic calculation strategies and procedures and the skill to
apply them in a flexible and accurate manner does not seem to
increase with age, as suggested by the lack of group differences in
skill development (cf. Jordan & Hanich, 2000; Jordan, Hanich, &
Kaplan, 2003).
The ANCOVA also revealed that the place value task and the
arithmetic fact retrieval task accounted for 12% variance each in
the performance on the written multidigit calculation task and also
reduced the achievement group effect from
2
⫽ .18 to
2
⫽ .07.
These findings are consistent with Geary’s conceptual framework,
as they suggest that conceptual knowledge concerning the under-
standing of the base-10 number system (i.e., place value) and
factual knowledge are important during the performance of com-
plex multidigit arithmetic calculation tasks (Geary, 2004; see also
McCloskey et al., 1985). More specifically, factual knowledge
provides the automation that is necessary to effectively solve
multidigit arithmetic tasks; to understand and perform trading
operations, the child must possess conceptual knowledge about the
base-10 number system (Ashcraft, 1992; Baroody & Wilkins,
1999; Geary, 1994). The MD children’s problems with these two
components contribute to their problems in performing multidigit
arithmetic calculation tasks as well as their procedural weaknesses.
Problem-Solving Skill and MD
As expected, both groups with MD displayed severe problems with
the arithmetic word problem-solving tasks during all three measure
points (Fuchs & Fuchs, 2002; Jordan & Hanich, 2000; Parmar et al.,
1996; Russell & Ginsburg, 1984). As a matter of fact, their perfor-
mance on these two tasks developed at a slightly slower rate com-
pared with the controls. A more important finding, though, was that
the MD children obtained lower scores than the controls on the two
arithmetic word problem-solving tasks when controlling for their
weaknesses in the conceptual and procedural components. Thus, the
problem-solving skill component appears to be defective in children
with both MD-only and MD-RD. Theoretically, this deficit might be
connected to problems with identifying or formulating the problem,
constructing a problem representation, or developing a solution plan
(Geary, 1994; Kilpatrick et al., 2001; Kintsch & Greeno, 1985; Mayer
& Hegarty, 1996). However, because most arithmetic word problems
presented in school as well as those used in the present study are
clearly specified, a weakness with problem identification is less likely.
Thus, a reasonable interpretation of the present results is that children
with MD have problems with integrating the different propositions
(i.e., relation, number, and question propositions) contained in the
arithmetic problem into a problem representation. Children with MD
might also have problems with developing a solution plan based on
the problem representation—that is, deciding the arithmetic calcula-
tions required to solve the problem and the order in which the
different calculations should be executed.
In line with Geary’s framework (2004; Geary & Hoard, 2005),
the present study shows that conceptual and procedural knowledge
and skills are vital during arithmetic word problem solving, as
these two components accounted for significant amounts of the
variation on the one-step word problem-solving task and the com-
plex multistep word problem-solving task. More specifically, an
adequate conceptual understanding of the relationships within and
between arithmetic operations (i.e., calculation principles) and the
base-10 number system and place value and adequate procedural
knowledge about arithmetic calculation strategies and procedures
and the skill to apply them in a flexible and accurate manner
contribute to children’s skill in arithmetic word problem solving
(cf. Baroody & Dowker, 2003; Delazer, 2003; Dowker, 2005;
Goldman & Hasselbring, 1997; Kilpatrick et al., 2001).
Approximate Arithmetic, Time Telling, and MD
Approximate arithmetic. The present study replicates and
extends previous findings (Hanich et al., 2001; Jordan, Hanich, &
Kaplan, 2003) by showing that approximate arithmetic also is an
area that is problematic for children with MD who are 9 to 13 years
of age. Thus, children with MD may well have a deficit related to
the visual-spatial number lines used to manipulate and estimate
quantities, as suggested by Jordan and colleagues (Hanich et al.,
2001; Jordan, Hanich, & Kaplan, 2003). Further support for a
deficit related to visual-spatial processing in children with MD is
provided by their poorer performance on the visual-spatial work-
ing memory task used in the present study (see below).
Time telling. The important result related to time telling is
that both groups with MD continued to show lower scores at the
second and third measure points, although most controls reached
the ceiling on this task and thereby seemed to develop slower than
the children with MD. These findings, in combination with previ-
ous studies showing that most children are able to identify almost
all analog and digital times at the age of 8 to 10 years, indicate that
telling time is a considerable weakness in most children with MD
that seems to persist up to 12 (Grade 5) and 13 (Grade 6) years of
age, and probably beyond (cf. Case et al., 1986; Friedman &
Laycock, 1989; Siegler & McGilly, 1989; Vakali, 1991).
Poor time-telling skills in children with MD is theoretically
reasonable; this skill in many respects resembles basic arithmetic,
as whole-hour and half-hour analog times are usually solved by
retrieving preexisting representations of the associations between
particular configurations and the time names, whereas counting
procedures are used to tell analog times, such as 07:12 (Friedman
& Laycock, 1989; Siegler & McGilly, 1989). Telling the time of
digital clocks (e.g., 16:35) is, on the other hand, quite similar to
decoding and retrieving number names of regular one- and two-
digit numbers (6 or 19), as the hour and minute values are pre-
sented as numbers (Friedman & Laycock, 1989). Thus, basic
arithmetic skills that are crucial for successful time telling are well
established as core deficits (i.e., fact retrieval and counting proce-
dures) in children with MD.
Basic Cognitive Functions and MD
The present results provide further empirical evidence that chil-
dren with MD, and especially children with MD-RD, have small
weaknesses in some basic cognitive functions. Contrary to findings
reported by Andersson (2008), both the MD-only group and the
MD-RD group performed worse than the controls on the visual
matrix span task; thus, their capacity to simultaneously process and
store visual-spatial information in working memory is low (cf.
Andersson & Lyxell, 2007). This finding, combined with the
results on the approximate arithmetic task, suggests that children
with MD might have a general weakness in their ability to process
129
DEVELOPMENT IN ARITHMETIC
and represent visual-spatial information. The MD-RD children’s
lower scores on the digit span task indicate that they have some
problems with storing numeric information in verbal short-term
memory (cf. Geary et al., 1991, 1999; McLean & Hitch, 1999;
Passolunghi & Siegel, 2001,2004; Siegel & Ryan, 1989; Swanson,
1994; Swanson & Beebe-Frankenberger, 2004). In addition to the
working memory and short-term memory problems, the MD-RD
children appear to have problems with executive functions (i.e.,
shifting ability) and general processing speed, as indicated by their
poor performance on the trail-making task and the number-
matching task (Andersson & Lyxell, 2007; Bull et al., 1999; Bull
& Johnston, 1997; Bull & Scerif, 2001; McLean & Hitch, 1999;
Passolunghi & Siegel, 2001, 2004). The MD-RD group in the
Andersson (2008) study did not show problems with executive
functions, as they performed on a par with the controls on the
trail-making task.
In contrast to previous studies, both groups with MD performed
on a par with the controls on the listening span task and the verbal
fluency task, suggesting that their verbal working memory capac-
ity and their ability to retrieve semantic and phonological infor-
mation from long-term memory is intact (Andersson & Lyxell,
2007; Bull & Johnston, 1997; Landerl et al., 2004; Swanson &
Beebe-Frankenberger, 2004; Temple & Sherwood, 2002). Further-
more, the RD-only children did not display any weaknesses related
to the basic cognitive functions assessed in the present study.
As basic cognitive functions support the performance and de-
velopment of arithmetic in children (Geary, 2004; Geary & Hoard,
2005), hierarchical regression analyses were performed to test the
assumption that MD in children is mediated to some extent by
deficits in basic cognitive functions. The results revealed that
controlling for group differences in basic cognitive functions did
not eliminate the lower arithmetic performance of the two MD
groups; it was only reduced by 3%–9% for the MD-RD group and
by 1%–5% for the MD-only group. This finding provides support
for the assumption that MD in children is mediated to some extent
by deficits in basic cognitive functions. However, as MD in
children was not fully accounted for by their basic cognitive
deficiencies, other explanations of MD must be considered.
According to the defective number module hypothesis, MD in
children is due to deficits related to a basic and innate specialized
capacity to understand and manipulate numerosities, not to defi-
ciencies in general cognitive functions (Butterworth, 2005; Land-
erl et al., 2004). This innate and specialized capacity constitutes
the foundation for developing an understanding of numbers (i.e.,
Arabic numerals) and arithmetic (Butterworth, 2005). However, in
a recent study, Rousselle and Noel (2007) found support for an
alternative hypothesis—the access deficit hypothesis—stating that
MD is not due to difficulties with processing numerosity per se but
rather in accessing the internal magnitude representations associ-
ated with Arabic numerals. As the ability to access and process the
underlying magnitude of Arabic numerals is central to mathemat-
ical achievement, the access deficit hypothesis might explain the
present findings that both MD groups failed to catch up with the
controls on the arithmetic tasks, all of which included the use of
Arabic numerals. More specifically, tasks such as multidigit cal-
culation, fact retrieval, word problem solving, approximate arith-
metic, comprehension of place value, and calculation principles
involve connecting numbers and mathematical symbols to their
underlying semantic representations (Gersten & Chard, 1999; Ger-
sten, Jordan, & Flojo, 2005). A deficit in this access ability should,
according to Rousselle and Noel (2007), generate an increasing
amount of difficulties that are characteristic for children with
MD—for example, problems with decomposing a complex prob-
lem into simpler ones, employing developmentally immature
counting procedures, learning arithmetic facts, and detecting gross
calculation errors (Butterworth, 2005; Geary, 1993, 2004). Thus,
future research should focus on basic processing of numerosities
and internal representations of nonsymbolic and symbolic (i.e.,
Arabic numerals) magnitude to identify underlying causes of MD
in children (Geary, Hoard, Nugent, & Byrd-Craven, 2008; Rubin-
sten & Henik, 2005, 2006).
The Impact of Reading Difficulties on Arithmetic
Ability in Children
In contrast to previous research, the MD-RD group and MD-
only group performed at the same achievement levels and devel-
oped at similar rates on almost all arithmetic tasks, except for the
place value task (Measure Point 3) and the time-telling task
(pooled scores; Hanich et al., 2001; Jordan et al., 2002; Jordan,
Hanich, & Kaplan, 2003). However, the lower performance of the
MD-RD group was eliminated when controlling for IQ. Overall,
these results provide no support to previous research indicating
that the reading difficulties of MD-RD children have a negative
influence on their skill or development in arithmetic (Fuchs &
Fuchs, 2002; Hanich et al., 2001; Jordan, Hanich, & Kaplan, 2003;
Jordan & Montani, 1997). One interpretation of this result is that
the present study examined third to sixth graders with MD-RD,
whereas second and third graders have been studied in previous
studies. Thus, although the reading skills of the MD-RD children
remain inferior to the MD-only children when in Grades 3 to 6,
their reading skills appear to have reached such a high level that it
no longer constitutes an additional difficulty when performing
arithmetic tasks that are mediated by language, such as arithmetic
word problems. Furthermore, the arithmetic word problem-solving
tasks used in the present study were designed to impose low
demands on the child’s linguistic skills but high demands on the
child’s arithmetic skills (see Method section). Thus, the main
challenge for the MD-RD children in solving the arithmetic word
problem tasks is related to the arithmetic demands, which is
something they share with the MD-only children.
An important result concerning the influence of RD on arith-
metic is that the RD-only children’s performance on the multidigit
calculation task and the two arithmetic word problem-solving tasks
was worse than the controls’ (cf. Geary et al., 2000; Hanich et al.,
2001; Jordan & Hanich, 2003; Jordan, Hanich, & Kaplan, 2003).
However, in Andersson (2008) the RD-only group performed on a
par with the controls on all arithmetic tasks. Another salient result
is that the RD-only children’s ability to solve one-step arithmetic
word problems developed more slowly than the controls (cf. Jor-
dan et al., 2002; Jordan & Hanich, 2003). These findings suggest
that reading difficulties might have a negative effect on children’s
arithmetic skill and its development and give further support to the
claim that children with specific RD are at risk of encountering
arithmetic difficulties later in elementary school, especially on
tasks that are mediated via language (e.g., word problem solving),
due to their poor reading and language skills. The fact that the
RD-only group did not score lower than the controls on any
130
ANDERSSON
arithmetic task during the three measure points when adjusting for
the RD-only children’s lower scores on the word knowledge test
does not contradict such a conclusion, because the RD-only chil-
dren’s lower word knowledge skills might very well be an effect of
their reading difficulties. Reading contributes to the growth of
vocabulary; because children with RD usually spend less time in
reading activities than good readers, they have fewer opportunities
to develop sufficient language and vocabulary skills (see Stanov-
ich, 1986, for a review).
In theory, reading and language comprehension can influence
growth and skills in arithmetic in a number of ways. For example,
almost all mathematics learning activities in the classroom require
language and reading comprehension skills—when listening to the
teacher’s instructions and perhaps at the same time taking notes,
for instance, which is another language skill (i.e., writing). Fur-
thermore, as the child moves up in grades the subject of mathe-
matics becomes more complex; more of the problems are pre-
sented in a linguistic context (word problems) and new, complex
concepts are presented that require longer and linguistically more
complex explanations. In relation to arithmetic word problem
solving, reading and language comprehension processes are most
important to establish a correct problem representation—that is,
comprehending the problem (see above; Hegarty, Mayer, & Monk,
1995; Kail & Hall, 1999; Kintsch & Greeno, 1985; Mayer &
Hegarty, 1996). When constructing a representation of the prob-
lem, the child must read and translate each statement in the word
problem and integrate it with the other statements; thus, children
with RD should have a disadvantage compared to children with
good reading abilities during this process (Hegarty et al., 1995;
Hegarty, Mayer, & Green, 1992; Mayer & Hegarty, 1996).
Educational Implications
The present findings show that the MD children have deficits
related to all four domain-specific components of knowledge and
skills important to the development of adequate arithmetic skills.
Thus, to promote arithmetic skills in children with MD, special
instruction should focus on all four components. This is clear when
considering arithmetic word problem solving, because this com-
plex task requires the involvement of both conceptual and proce-
dural skill components besides problem-solving skill. A focus on
all four components might be the most efficient way to improve
the arithmetic skills in children with MD because there seems to be
a reciprocal interdependence between them. The development of
conceptual knowledge appears to be facilitated by procedural
knowledge and skills and vice versa (Baroody & Dowker, 2003;
Byrnes & Wasik, 1991; Geary, 1994; Kilpatrick et al., 2001;
Rittle-Johnson & Alibali, 1999). Thus, to promote the MD chil-
dren’s poor conceptual knowledge, they should engage in learning
activities that directly focus on conceptual aspects of arithmetic
but also on procedural knowledge promoting activities.
As the MD children, especially the MD-RD children, also
display weaknesses in supporting basic cognitive functions, one
way to help these children improve their mathematical skill devel-
opment might be to reduce the demands on their working memory
system while they perform learning activities (see Gathercole &
Alloway, 2004). This can be accomplished by providing external
memory aids and giving short, simple instructions (possibly in
writing; see Gathercole & Alloway, 2004; Gathercole, Alloway,
Willis, & Adams, 2006).
Assuming that MD in children is attributed to deficits in access-
ing numerical magnitude representations or less accurate (numer-
ical) magnitude representations, teaching during the early school
years should include elements that focus on the connection be-
tween numerical and mathematical symbols and their underlying
semantic representations—for example by using concrete objects
to represent magnitudes or simple arithmetic problems. This might
prevent or reduce the cardinal deficit in children with MD: poor
factual knowledge (Rousselle & Noel, 2007).
Conclusions
The present findings demonstrate clearly that children identified
as having MD-only or MD-RD when they are 9 to 10 years old do
not catch up with their normally achieving peers in the later
elementary school grades, when they are 12 to 13 years old.
However, caution is advised when interpreting the present results
in relation to boys with MD-only or RD-only, because these two
groups included only nine boys each. In view of Geary’s frame-
work (2004; Geary & Hoard, 2005), the present longitudinal
developmental data show that children with MD continue to lag
behind their peers in all domain-specific components of arithmetic.
They also continue to lag behind their peers in the domain-general
cognitive system; however this weakness is small compared to
weaknesses observed in the domain-specific components. More
specifically, a defective factual knowledge component is a cardinal
characteristic of children with MD, and knowledge and skills
related to the conceptual and procedural components still remain a
weakness in children with MD when they are 12 to 13 years old.
The domain-specific shortcomings (factual, procedural, concep-
tual) contribute to some extent to the MD children’s problems with
multidigit calculation and arithmetic word problem solving. Fur-
thermore, the problem-solving skill component appears to be de-
fective both in children with MD-only and MD-RD. Approximate
arithmetic and telling time are two other aspects of arithmetic that
are problematic for children with MD. Children with MD ages 9 to
10 years display small weaknesses in some basic cognitive func-
tions that persist up to the age of 12–13. These cognitive weak-
nesses mediate to some extent the difficulties MD children have
with arithmetic.
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Received September 25, 2008
Revision received June 16, 2009
Accepted June 22, 2009 䡲
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