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The Relationship Between Working Memory and Mathematical Problem Solving in Children at Risk and Not at Risk for Serious Math Difficulties

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This study identified cognitive processes that underlie individual differences in working memory (WM) and mathematical problem-solution accuracy in elementary school children at risk and not at risk for serious math difficulties (SMD). A battery of tests was administered that assessed problem solving, achievement, and cognitive processing in children in first (N = 130), second (N = 92) and third grades (N = 131). The results were that (a) younger children and children at risk for SMD performed poorer on WM and problem-solving tasks, as well as measures of math calculation, reading, semantic processing, phonological processing, and inhibition, than older children and children not at risk for SMD and (b) WM predicted solution accuracy of word problems independent of measures of fluid intelligence, reading skill, math skill, knowledge of algorithms, phonological processing, semantic processing,'speed, shortterm memory, and inhibition. The results support the notion that the executive system is an important predictor of children's problem solving. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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The Relationship Between Working Memory and Mathematical Problem
Solving in Children at Risk and Not at Risk for Serious Math Difficulties
H. Lee Swanson
University of California, Riverside
Margaret Beebe-Frankenberger
University of Montana
This study identified cognitive processes that underlie individual differences in working memory (WM)
and mathematical problem-solution accuracy in elementary school children at risk and not at risk for
serious math difficulties (SMD). A battery of tests was administered that assessed problem solving,
achievement, and cognitive processing in children in first (N!130), second (N!92) and third grades
(N!131). The results were that (a) younger children and children at risk for SMD performed poorer on
WM and problem-solving tasks, as well as measures of math calculation, reading, semantic processing,
phonological processing, and inhibition, than older children and children not at risk for SMD and (b) WM
predicted solution accuracy of word problems independent of measures of fluid intelligence, reading
skill, math skill, knowledge of algorithms, phonological processing, semantic processing, speed, short-
term memory, and inhibition. The results support the notion that the executive system is an important
predictor of children’s problem solving.
Word problems constitute one of the most important mediums
through which students can potentially learn to select and apply
strategies necessary for coping with everyday problems. To com-
prehend and solve mathematical word, or story, problems, one
must be able to keep track of incoming information (see, e.g.,
Anderson, Reder, & Lebiere, 1996; Kail & Hall, 1999; Mayer &
Hegarty, 1996; Swanson & Sachse-Lee, 2001). This is necessary to
understand words, phrases, sentences, and propositions that, in
turn, are necessary to construct a coherent and meaningful inter-
pretation of word problems. Temporary storage of material that has
been read or heard is said to depend on working memory (WM;
see, e.g., Baddeley & Logie, 1999; Case, 1995), which takes into
account the storage of items for later retrieval and which is a
function of the individual’s level of text processing (see, e.g.,
Engle, Tuholski, Laughlin, & Conway, 1999; Ericsson & Kintsch,
1995). Previous studies have shown that a substantial proportion of
the variance related to solution accuracy in word problems is
related to WM (see, e.g., LeBlanc & Weber-Russell, 1996; Swan-
son, in press; Swanson & Sachse-Lee, 2001). For example, Le-
Blanc and Weber-Russell (1996) found, via computer simulation
and testing of children in Grade Levels K-3, that WM variables
accounted for a substantial proportion of variance (between 49%
and 57%) in children’s word-problem solutions.
One purpose of this study was to identify cognitive processes
and skills that underlie WM and word-problem-solving profi-
ciency in three age groups (first-, second-, and third-grade stu-
dents) in children at risk or not at risk for serious math difficulties
(SMD). Our framework for isolating components of WM that are
related to word-problem solving (as well as other domains) is
Baddeley’s (1986, 1996) multicomponent model. Baddeley (1986;
Baddeley & Logie, 1999) described WM as a limited-capacity
central executive system that interacts with a set of two passive
store systems used for temporary storage of different classes of
information: the speech-based phonological loop and the visual
sketchpad. The phonological loop is responsible for the temporary
storage of verbal information; items are held within a phonological
store of limited duration and are maintained within the store via the
process of articulation. The visual sketchpad is responsible for the
storage of visual–spatial information over brief periods and plays
a key role in the generation and manipulation of mental images.
Both storage systems are in direct contact with the central execu-
tive system. The central executive system is considered to be
primarily responsible for coordinating activity within the cognitive
system but also devotes some of its resources to increasing the
amount of information that can be held in the two subsystems
(Baddeley & Logie, 1999).
DISTINCTIONS BETWEEN WORKING MEMORY
AND SHORT-TERM MEMORY
The distinctions made between the central executive system and
specific memory storage systems (i.e., the phonological loop) in
some ways parallel the distinctions made between WM and short-
term memory (STM). WM is referred to as a processing resource
of limited capacity involved in the preservation of information
H. Lee Swanson, Educational Psychology, Graduate School of Educa-
tion, University of California, Riverside; Margaret Beebe-Frankenberger,
Department of Psychology, University of Montana.
This is the first-year preliminary report for a study funded by the U.S.
Department of Education’s Institute of Education Sciences, Cognition and
Student Learning Grant USDE R305H020055. We are indebted to Georgia
Doukas, Diana Dowds, and Rebecca Gregg for the data collection. Special
appreciation is given to the Colton School District and Tri City Christian
Schools. This article does not necessarily reflect the views of the U.S.
Department of Education or the school districts.
Correspondence concerning this article should be addressed to H. Lee
Swanson, Graduate School of Education, Area of Educational Psychology,
University of California, Riverside, CA 92521. E-mail: lee.swanson@
ucr.edu
Journal of Educational Psychology Copyright 2004 by the American Psychological Association
2004, Vol. 96, No. 3, 471–491 0022-0663/04/$12.00 DOI: 10.1037/0022-0663.96.3.471
471
while simultaneously processing the same or other information
(see, e.g., Baddeley, 1986; Baddeley & Logie, 1999; Engle et al.,
1999; Just & Carpenter, 1992). Individual differences in WM
capacity have been attributed to executive processing (see, e.g.,
Engle et al. 1999; Swanson, 2003), such as the ability to inhibit
irrelevant information (see, e.g., Chiappe, Hasher, & Siegel, 2000),
as well as to speed of processing (Salthouse, 1996) and knowledge
(see, e.g., Ericsson & Kintsch, 1995). In contrast, STM typically
involves situations where small amounts of material are held
passively (i.e., minimal resources from long-term memory [LTM]
are activated to interpret the task, e.g., digit or word span tasks)
and then reproduced in a sequential fashion. That is, participants
are asked to only reproduce the sequence of items in the order they
were presented (see, e.g., Daneman & Carpenter, 1980; Dempster,
1985; Klapp, Marshburn, & Lester, 1983).
1
Individual differences
on these STM measures have been primarily attributed to phono-
logical coding and rehearsal (see, e.g., Willis & Gathercole, 2001).
PHONOLOGICAL PROCESSING MODEL
How might WM mediate age-related and individual differences
in word-problem solving? We tested two competing models as an
explanation of the role of WM in age-related problem-solving
performance in children at risk or not at risk for SMD. The first
model hypothesizes that individual and age-related influence of
WM on childrens problem solving is primarily moderated by
processing efficiency at the phonological level. A simple version
of this hypothesis states that individuals at risk for SMD and
younger children are slower and/or less accurate at processing
verbal information (numbers, letters) than average-achieving chil-
dren or older children and that such reduced processing on the
participantspart underlies their poor WM and problem-solving
performance. This hypothesis is consistent with a number of
bottom-up models of reading (comprehension) that view the pri-
mary task of executive processing as one of relaying the results of
lower level linguistic analyses upward through the language sys-
tem (see, e.g., Shankweiler & Crain, 1986). Several studies have
suggested that the phonological system, via the phonological loop
(phonological store, subvocal rehearsal), influences verbatim
memory capacity, which in turn supports comprehension (see, e.g.,
Perfetti, 1985). Likewise, some studies have attributed individual
differences in mathematical problem solving to the phonological
system (see Furst & Hitch, 2000; Swanson & Sachse-Lee, 2001,
for reviews). This link occurs because mathematical word prob-
lems are a form of text, and the decoding and comprehension of
text draw on the phonological system (see Shankweiler & Crain,
1986, for a review). In summary, the hypothesis assumes that
phonologically analyzed information at word level or number level
is transferred to WM storage, which in turn is transferred (thus
freeing storage for the next chunk of phonological information)
upward through the processing system to promote online extrac-
tion of meaning. Consistent with this assumption, extraction of
meaning from text is compromised in children with SMD because
inefficient phonological analysis creates a bottleneck that con-
stricts information flow to higher levels of processing (see, e.g.,
Crain, Shankweiler, Macaruss, & Bar-Shalom, 1990).
What are potential measures of the phonological system? Sev-
eral studies have assumed that STM measures capture a subset of
WM performance, the utilization and/or operation of the phono-
logical loop (see Gathercole, 1998; Gathercole & Baddeley, 1993,
for comprehensive reviews). Some authors have suggested that the
phonological loop may be referred to as verbal STM (see, e.g.,
Baddeley, 1986; Dempster, 1985) because it involves two major
components discussed in the STM literature: a speech-based pho-
nological input store and a rehearsal process (see Baddeley, 1986,
for review).
Research to date suggests younger children rehearse less and
perform more poorly on tasks requiring the short-term retention of
order information than do older children (see, e.g., Ornstein, Naus,
& Liberty, 1975), signifying inefficient utilization of the phono-
logical rehearsal process (cf. Henry & Millar, 1993). Likewise,
children with SMD have been found to suffer deficits in short-term
retention when compared with children without SMD (see, e.g.,
Geary, Brown, & Samaranayake, 1991; Geary, Hoard, & Hamson,
1999; Siegel & Ryan, 1989).
There are clear expectations in the aforementioned model, spe-
cifically, that age-related and individual differences in childrens
problem solving are related to the phonological system. Thus,
problem-solving proficiency, as well as math and reading perfor-
mance, follows automatically with improvement in phonological
processing. Specifically, if individual and age-related differences
in WM and problem-solving performance are moderated by the
phonological system, then the relationship between problem solv-
ing and WM should be eliminated when measures of the phono-
logical system (e.g., STM, phonological awareness) are partialed
from a statistical analysis.
EXECUTIVE PROCESSING MODEL
In contrast to the above model, the second model views exec-
utive processes as providing resources to lower order (phonolog-
ical system) skills, as well as monitoring a general executive
system independent of those skills (see, e.g., Baddeley & Logie,
1999). Given that the phonological loop is partly controlled by the
central executive system, the development of problem solving may
be directly related to the controlling functions of the central
executive system itself. This model assumes there is variance that
is unique to particular systems of WM (executive processing,
phonological coding), as well as some shared variance with these
systems (see Swanson & Alexander, 1997, for further discussion).
Thus, in the context of WM development, the model suggests that
1
Everyday examples of WM tasks would thus include holding a per-
sons address in mind while listening to instructions about how to get there
or perhaps listening to the sequence of events in a story while trying to
understand what the story means. Everyday examples of STM tasks would
include recalling a series of digits, such as a telephone number, in order
immediately after their presentation. Although there is controversy con-
cerning the nature of STM and WM tasks (see Engle et al., 1999, for
review), there is some agreement that a transformation or active monitoring
(e.g., focusing on relevant information when competing information is
present) is required on WM tasks (see, e.g., Baddeley & Logie, 1999). For
the sake of parsimony, in the present study, we view WM tasks as those
that require some inference, transformation, and monitoring of relevant and
irrelevant information, whereas STM tasks require the storage of informa-
tion with minimal ongoing processing requirements that vary from initial
encoding. Thus, tasks in the present study were selected according to the
degree to which some monitoring of relevant and irrelevant information
would be required prior to output.
472 SWANSON AND BEEBE-FRANKENBERGER
both a general (executive) and specific (phonological) system
contribute significant variance to individual and age-related dif-
ferences in mathematical problem solving. In support of this
model, Swanson and Sachse-Lee (2001) showed that when math
knowledge and reading comprehension were controlled, phonolog-
ical processing and WM each contributed unique variance to
mathematical problem solving in math-disabled and non-math-
disabled children in Grades 5 and 6. The results also showed that
the entry of phonological processing first into a hierarchical re-
gression model did not partial out the influence of WM on solution
accuracy. These results suggest that fundamental processing prob-
lems in childrens WM at the executive level play an important
role in mediating accuracy in word-problem solving.
How might the executive system contribute to age-related and
individual differences in problem solving? One of the possibilities
we explored in this study relates to accurately accessing informa-
tion from LTM (see, e.g., Cantor & Engle, 1993). Baddeley and
Logie (1999) stated that a major role of WM is retrieval of stored
long-term knowledge relevant to the tasks at hand, the manipula-
tion and recombination of material allowing the interpretation of
novel stimuli, and the discovery of novel information or the
solution to problems(p. 31). They further stated, any increase in
total storage capacity beyond that of a given slave system is
achieved by accessing either long-term memory (LTM) or other
subsystems(Baddeley & Logie, 1999, p. 37). Thus, the influence
of WM performance on problem solving is related to ones ability
to accurately access information (e.g., appropriate algorithm) from
LTM to solve the problem. More specifically, a word problem
introduces information into WM. The contents of WM are then
compared with possible action sequences (e.g., associative links)
in LTM (Ericsson & Kintsch, 1995). When a match is found
(recognized), the contents of WM are updated and used to generate
a solution. This assumption is consistent with current models of
problem solving, which are based on recognizeact models of a
cognitive processor (Anderson et al., 1996; Ericsson & Kintsch,
1995).
Although individual and age-related differences in problem-
solving accuracy are possibly related to the retrievability of con-
tents in LTM (e.g., knowledge of specific mathematical relations,
general problem-solving strategies) accessed in WM, some re-
search has suggested that other executive activities besides access-
ing information from LTM underlie the influence of WM solution
accuracy. Also, several cognitive activities have been assigned to
the central executive (e.g., see Miyake, Friedman, Emerson,
Witzki, & Howerter, 2000, for a review), such as control of
subsidiary memory systems, control of encoding and retrieval
strategies, attention switching during manipulation of material held
in the verbal and visualspatial systems, suppression of irrelevant
information, and so on, in addition to LTM knowledge retrieval
(see, e.g., Baddeley, 1996; Miyake et al., 2000; Oberauer, Sub,
Wilhelm, & Wittman, 2003). Recent studies have suggested that
specific activities of the central executive related to suppression of
irrelevant information and resource monitoring are deficient in
children with math and/or reading disabilities (Bull & Scerif, 2001;
Chiappe et al., 2000; Passolunghi, Cornoldi, & De Liberto, 1999;
Swanson, 1999). A review of these studies (see Swanson & Siegel,
2001a, 2001b, for reviews) indicates that children with SMD
and/or reading disabilities yield (a) poor performance on complex
divided attention tasks; (b) poor monitoring, such as an inability to
suppress (inhibit) irrelevant information; and (c) poor performance
across verbal and visualspatial tasks assumed to require both
storage and processing when compared with normally achieving
peers (also see Chiappe et al., 2000; De Beni, Palladino, Pazzaglia,
& Cornoldi, 1998; Swanson, 1993).
PURPOSE AND PREDICTIONS
In summary, the purpose of this study was to assess the contri-
bution of WM to individual and age-related performance in chil-
drens problem-solving performance. We considered two possible
models: (a) that the relationship between WM and problem solving
is primarily mediated by the phonological system or (b) that
executive processes operate independent of the phonological sys-
tem and therefore contribute unique variance to problem solving
beyond the phonological system. Measures of the phonological
system included tasks related to STM. Measures of the executive
system were modeled after Daneman and Carpenters (1980) WM
tasks. These tasks demand the coordination of both processing and
storage. Recent studies have suggested that these tasks capture at
least two factors of executive processing: susceptibility to inter-
ference and manipulation of capacity (see, e.g., Oberauer, 2002;
Whitney, Arnett, Driver, & Budd, 2001).
The major prediction in this study was that individual and
age-related differences in WM would be partially mediated by an
executive system that operates independent of the phonological
system. Thus, individual and age-related changes in WM would be
sustained when measures of phonological processing (e.g., STM,
phonological knowledge) were partialed from the analysis. Vari-
ables of interest besides WM were computation knowledge,
knowledge of processing operations, semantic processing, and
reading. Each of these variables has been suggested as important to
word-problem-solving accuracy (see Cooney & Swanson, 1990;
Swanson, Cooney, & Brock, 1993; Swanson & Sachse-Lee, 2001,
for reviews). We also investigated individual differences in chil-
drens recognition of the structural properties of word problems.
Analogous to the structural properties outlined by Mayer and
Hegarty (1996) and Cooney and Swanson (1990), four structural
properties were investigated in the present study, specifically, the
recognition of numerical, question, algorithmic knowledge, and
irrelevant propositions.
In summary, three research questions directed this study.
1. Does WM predict problem solving after various mea-
sures of phonological processing have been partialed
from the analysis?
2. Does the relationship between problem solving and WM
vary as a function of age and risk for SMD?
3. What cognitive processes (e.g., LTM, inhibition) and
skills (e.g., arithmetic, reading) mediate the relationship
between WM and problem solving?
METHOD
Participants
Three hundred and fifty-three children from a southern California public
school district and private school district participated in this study. Final
473
WORD PROBLEMS
Table 1
Means and Standard Deviations for Measures as a Function of Children at Risk for Serious Math Difficulties (SMD) and Not at Risk
(NSMD) as a Function of Age
Variable
Grade 1 (N!130) Grade 2 (N!92) Grade 3 (N!131)
SMD
(n!73)
NSMD
(n!57)
SMD
(n!34)
NSMD
(n!58)
SMD
(n!25)
NSMD
(n!106)
M SD M SD M SD M SD M SD M SD
Classification
Chronological age 6.19 0.49 6.23 0.46 7.41 0.50 7.29 0.50 8.52 0.51 8.20 0.49
Fluid intelligenceRaven
Standard score 102.77 14.19 112.72 13.47 106.18 14.00 111.76 16.65 96.00 9.73 108.58 13.08
Raw score 18.37 5.12 22.25 5.46 23.50 4.94 25.83 5.83 23.64 4.52 27.70 4.95
Mental computationWISCIII
Standard score 7.04 2.85 13.10 2.20 7.90 1.77 11.78 2.18 5.87 2.18 11.72 2.08
Raw score 8.56 2.75 12.63 1.10 11.82 1.76 14.28 1.35 11.43 3.34 15.75 1.57
Digit-naming speedCTOPP (standard score) 6.94 0.81 8.28 1.45 7.78 1.01 9.35 1.53 9.50 1.85 10.99 2.22
Criterion measuresWord problems
Word-problem-solving processes (raw score)
Question 1.46 0.91 2.12 0.80 1.68 0.87 1.97 0.86 1.80 1.04 2.42 0.80
Numbers 1.55 1.05 2.19 0.97 2.09 0.87 2.41 0.70 2.60 0.58 2.54 0.68
Goal 1.15 0.84 1.21 0.93 1.00 0.98 1.41 0.96 1.28 0.98 2.19 0.93
Operations 1.47 0.71 1.79 1.62 1.91 1.03 2.41 0.77 1.96 0.98 2.27 0.86
Algorithm 1.16 0.83 1.63 0.87 1.97 0.94 2.19 0.71 2.08 0.86 2.31 0.77
Irrelevant information 1.85 0.81 2.05 0.71 2.03 0.90 2.41 0.70 2.52 0.65 2.58 0.68
Total score 6.79 2.96 8.95 3.28 8.65 3.34 10.40 2.48 9.72 2.98 11.74 2.74
Word problemsSemantic structure varied
(raw score) 8.62 3.23 8.62 3.23 10.71 3.45 12.81 2.65 12.24 3.26 14.29 3.06
Criterion measuresMath
WRAT
Standard score 110.97 11.39 115.97 10.15 99.00 9.36 106.41 9.79 107.04 14.57 117.68 12.12
Raw score 18.71 1.61 19.70 1.51 21.94 1.72 23.34 1.87 28.36 3.60 30.13 2.90
WIAT
Standard score 102.63 13.18 110.91 13.59 100.79 11.28 109.36 10.64 108.28 13.46 116.25 9.97
Raw score 8.25 2.03 10.19 2.15 14.21 2.69 15.78 2.14 20.24 2.86 21.65 2.12
Computational fluency (raw score) 13.51 6.57 17.74 6.37 14.18 8.59 18.78 7.18 21.36 11.08 26.81 11.33
Prediction measuresReading
WRAT
Standard score 102.89 16.70 116.61 14.77 98.24 12.62 108.67 12.78 95.17 16.83 107.11 11.54
Raw score 21.64 4.72 25.54 4.16 27.33 4.20 30.78 4.27 29.91 5.44 33.38 4.11
ReadingTOWRE real words (raw score) 21.64 13.80 37.56 14.68 44.33 13.18 58.17 10.35 54.74 16.29 65.67 8.94
ComprehensiveWRMTR
Standard score 100.78 13.19 111.46 10.34 102.12 9.83 108.03 11.12 96.26 13.06 104.96 8.77
Raw score 10.92 7.81 19.37 7.76 24.39 6.37 29.88 5.92 28.74 8.64 33.91 4.42
Rapid namingLettersCTOPP
Standard (scale) score 6.83 0.84 8.16 1.28 7.81 0.94 9.17 1.45 9.39 1.88 10.17 1.80
Raw score 72.62 24.39 49.67 9.60 53.45 12.81 42.22 7.24 42.00 9.54 38.12 7.10
Phonological processing
PseudowordsTOWRE (raw score) 9.33 7.97 18.74 9.97 18.42 9.77 27.64 10.75 23.70 12.02 33.36 10.53
Elision (segmentation)CTOPP (raw score) 5.26 3.98 10.16 4.82 9.64 5.24 12.38 4.87 10.04 6.01 13.47 5.40
Phonological fluency (raw score) 5.58 3.19 6.56 3.53 7.00 2.90 7.07 3.09 7.13 2.85 8.26 3.40
Short-term memory (highest span scores)
Digits forwardWISCIII (span score) 2.44 1.00 2.82 0.97 3.06 0.75 3.19 0.89 2.52 0.85 3.33 1.06
Digits backwardWISCIII (span score) 1.00 0.60 1.37 0.62 1.30 0.59 1.49 0.57 1.30 0.63 1.66 0.77
Pseudoword span (span score) 3.38 0.91 4.32 2.49 4.58 1.85 5.16 2.84 3.87 2.10 4.93 2.20
Real-word span (span score) 3.33 0.91 3.75 0.74 3.67 0.99 4.09 0.68 3.70 0.70 3.98 0.72
474 SWANSON AND BEEBE-FRANKENBERGER
selection was related to parent approval for participation and achievement
scores. Of the 353 children selected, 169 were girls, and 184 were boys.
Gender representation was not significantly different among the three age
groups,
!
2
(2, N!353) !1.15, p".05. Ethnic representation of the
sample was 163 Anglo, 147 Hispanic, 25 African American, 14 Asian, and
4 other (e.g., Native American, Vietnamese). The mean socioeconomic
status (SES) of the sample was primarily middle class based on parent
education or occupation. However, the sample varied from low-middle
class to upper-middle class. Means and standard deviations for the selec-
tion variables used in this study are shown in Table 1.
Definition of Risk for Serious Math Difficulties
There are no generally agreed-upon criteria for defining children at risk
for SMD, especially in Grade 1, where instruction is only beginning to
address mathematical operations. Because first graders were used in our
sample, we attempted to control in our classification tasks demands placed
on reading and writing. In addition, our focus was on problem solving and
not arithmetic calculation, and therefore, we used different measures (i.e.,
mental computation of word problems vs. paper-and-pencil computation of
arithmetic problems) than studies that have defined math disabilities by
computation skill. In contrast to the literature on math disabilities or
reading disabilities, we assumed that children with SMD may not have skill
difficulties related to arithmetic calculation or reading but may neverthe-
less have difficulties in coordinating arithmetic and language processes to
solve a problem. Furthermore, because we were interested in the reasoning
processes related to problem solving, we focused on a reliable measure that
asked questions within a verbal context (e.g., If I have an apple and divide
it in half, how many pieces do I have?) rather than a computational context
(1 #1!?). Thus, we utilized the oral presentation of story problems as
a criterion measure of SMD.
In selecting children at risk, we also focused on a childs general fluency
with numbers. We assumed that number-processing speed underlies chil-
drens ability to automatically access arithmetic facts, knowledge of mean-
ing, signs, and procedures (Bull, Johnston, & Roy, 1999; Geary et al.,
1991). We assumed that children who have quicker access to numbers, that
is, faster number fluency, would be less at risk for mental computation
difficulties than children less fluent in number naming. This seemed
reasonable to us on the basis of Hitch and McAuleys (1991) finding that
children with math difficulties evidenced deficits in the speed of implicit
counting. Furthermore, speed of number naming has a parallel in the
reading literature, where both letter-naming speed and phonological knowl-
edge are assumed to underlie reading disabilities.
Thus, in this study, children at risk for SMD were defined as having
normal intelligence (standard score "85), but with performance below the
25th percentile (standard score of 90 or scaled score of 8) on standardized
measures related to (a) solving orally presented word problems and (b)
digit-naming fluency. The 25th-percentile cutoff score on standardized
achievement measures has been commonly used to identify children at risk
(see, e.g., Fletcher et al., 1989; Siegel & Ryan, 1989) and therefore was
used in this study. Classification of children at risk (SMD) and not at risk
(NSMD) was based on norm-referenced measures of computation on the
arithmetic subtest of the Wechsler Intelligence Scale for ChildrenThird
Edition (WISCIII; Wechsler, 1991) and digit-naming speed from the
Comprehensive Test of Phonological Processing (CTOPP; Wagner,
Torgesen, & Rashotte, 2000) described in the following section. Children
who yielded scaled scores at or less than 8 on both measures were
considered at risk for SMD. A scale score of 8 was equivalent to a standard
score of 90 or a percentile score of 25.
In the present sample, 132 children were classified at risk for SMD. As
expected, a larger proportion of children at risk for SMD were identified in
Grade 1 (52%) than Grades 2 (35%) and 3 (22%),
!
2
(2, N!353) !25.50,
p$.0001. No significant differences emerged between groups in terms of
ethnicity,
!
2
(5, N!353) !8.67, p".05, or gender,
!
2
(1, N!353) !
.06, p".05. As expected, the portion of children who scored below the
25th percentile in word recognition was significantly higher for children at
Table 1 (continued)
Variable
Grade 1 (N!130) Grade 2 (N!92) Grade 3 (N!131)
SMD
(n!73)
NSMD
(n!57)
SMD
(n!34)
NSMD
(n!58)
SMD
(n!25)
NSMD
(n!106)
M SD M SD M SD M SD M SD M SD
Working memory (highest span scores)
Listening sentence span 1.26 0.57 1.60 0.80 1.76 0.78 2.05 0.78 1.61 0.83 2.14 0.75
Digit/sequence spanS-CPT 1.56 0.42 1.70 0.45 1.70 0.56 1.86 0.63 1.80 0.56 2.19 0.84
Semantic associationS-CPT 1.66 0.28 1.79 0.15 1.73 0.26 1.78 0.15 1.79 0.10 1.86 0.32
Visual matrixS-CPT 2.52 1.19 2.72 1.16 2.82 1.27 3.28 1.23 3.40 1.29 3.44 1.29
Mapping/directionS-CPT 1.44 0.35 1.39 0.31 1.30 0.28 1.41 0.33 1.43 0.43 1.52 0.57
Inhibition and updating
Random-generation letters (computed score) 0.36 0.36 0.25 0.22 0.25 0.21 0.26 0.21 0.34 0.25 0.22 0.19
Random-generation numbers (computed score) 0.38 0.34 0.27 0.23 0.28 0.21 0.23 0.18 0.33 0.49 0.16 0.15
Updating (raw [proportion] score) 2.69 3.73 5.33 4.79 3.88 3.69 7.03 5.05 3.35 3.69 8.10 5.23
Semantic and vocabulary
Semantic fluency (raw score) 9.25 3.76 12.93 4.56 12.88 3.98 13.71 3.88 13.70 4.49 15.25 4.19
Vocabulary
Standard score 7.23 3.73 9.95 3.44 8.42 3.16 10.03 4.07 6.65 2.89 11.09 3.77
Raw score 10.29 4.98 13.93 4.56 16.18 4.79 18.29 6.39 17.13 4.71 23.87 6.63
Note. Raven !Raven Colored Progressive Matrices; WISCIII !Wechsler Intelligence Scale for ChildrenThird Edition; CTOPP !Comprehensive
Test of Phonological Processing; WRAT !Wide Range Achievement Test; WIAT !Wechsler Individual Achievement Test; TOWRE !Test of Word
Reading Efficiency; WRMTR!Woodcock Reading Mastery TestRevised; S-CPT !S-Cognitive Processing Test.
475
WORD PROBLEMS
risk for SMD (25% of the SMD sample) than children not at risk for SMD
(3%),
!
2
(1, N!353) !44.47, p$.0001.
Tasks and Materials
The battery of group- and individually administered tasks is described
below. Experimental tasks are described in more detail than published and
standardized tasks. Tasks were divided into classification, criterion, and
predictor variables. Cronbachs alpha reliability coefficients for the sample
were calculated for all measures and are provided.
Classification Measures
Fluid Intelligence
Fluid intelligence was assessed by the Raven Colored Progressive Ma-
trices (Raven, 1976). Strong correlations (e.g., r!.80) have been noted
between this measure and WM (see, e.g., Engle et al., 1999; Kyllonen &
Christal, 1990). Children were given a booklet with patterns displayed on
each page, each pattern revealing a missing piece. For each pattern, four
possible replacement pattern pieces were displayed. Children were required
to circle the replacement piece that best completed the patterns. After the
introduction of the first matrix, children completed their booklets at their
own pace. Patterns progressively increased in difficulty. The dependent
measure (range of 0 to 36) was the number of problems solved correctly,
which yielded a standardized score (M!100, SD !15). Cronbachs
coefficient alpha was .88.
Mental Computation of Word Problems
This task was taken from the arithmetic subtest of the WISCIII
(Wechsler, 1991). Each word problem was orally presented and was solved
without paper or pencil. Questions ranged from simple addition (e.g., If I
cut an apple in half, how many pieces will I have?) to more complex
calculations (e.g., If three children buy tickets to the show for $6.00 each,
how much change do they get back from $20.00?). The dependent measure
was the number of problems solved correctly, which yielded a scaled score
(M!10, SD !2). Cronbachs alpha for the WISCIII arithmetic subtest
was .66 in the present study.
Digit-Naming Speed
The administration procedures followed those specified in the manual of
the CTOPP (Wagner et al., 2000). For this task, the examiner presented
participants with an array of 36 digits. Participants were required to name
the digits as quickly as possible for each of two stimulus arrays containing
36 items, for a total of 72 items. The task administrator used a stopwatch
to time participants on speed of naming. The dependent measure was the
total time to name both arrays of numbers. The correlation between Array
Forms A and B was .91.
Criterion Variables
Word-Problem Solving and Components
Mathematical word-problem-solving processes. This experimental test
assessed the childs ability to retrieve processing components of word
problems (Swanson & Sachse-Lee, 2001). Two booklets were adapted
from Swanson et al. (1993) for students in Grades 2 and 3. A booklet was
developed for students in Grade 1 that was conceptually consistent with the
second- and third-grade experimental booklets but was grade-level appro-
priate in word and computational difficulty. Each booklet contained three
problems that included pages assessing the recall of text from the mathe-
matical word problems. The categories of mathematical word problems
were addition, subtraction, and multiplication (the last for third graders
only). Problems were four sentences in length and contained two assign-
ment propositions, one relation, one question, and an extraneous proposi-
tion related to the solution. To control for reading problems, the examiner
orally read (a) each problem and (b) all multiple-choice response options as
the students followed along. For example, a subtraction problem read as
follows: Darren found 15 pinecones [assignment]. He threw 5 pinecones
back [assignment]. Darren uses pinecones to make ornaments [extraneous].
How many pinecones did Darren keep [question]?(The combination of
Sentences 1 and 2 was the relation proposition.)
No titles were given to the problems except the titles Problem 1, Problem
2, and so on. Depending on the order of presentation, after the problem was
read, students were then instructed to turn to the next page on which the
following statement was written: Without looking back at the problem,
circle (from a choice of four options) the question the story problem was
asking on the last page.The multiple-choice questions for the sample
problem above were (a) How many pinecones did Darren have in all? (b)
How many pinecones did Darren start with? (c) How many pinecones did
Darren keep? and (d) How many pinecones did Darren throw back? This
page assessed student ability to correctly identify the question proposition
of each story problem.
On the next page for each problem, directions asking, Without looking
back at the problem, try to identify the numbers in the problemwere read.
The multiple-choice questions for the sample problem above were (a) 15
and 5, (b) 5 and 10, (c) 15 and 20, and (d) 5 and 20. This page assessed
student ability to correctly identify the numbers in the two assignment
propositions of each story problem.
Instructions on the next page were read as follows: Without looking
back at the problem, identify what the question wants you to find.The
multiple choice questions were (a) the total number of pinecones Darren
found all together, (b) what Darren plans to do with the pinecones, (c) the
total number of pinecones Darren had thrown away, and (d) the difference
between the pinecones Darren kept and the ones he threw back. This page
assessed the students ability to correctly identify the goals in the two
assignment propositions of each story problem.
Instructions for the final page were Without looking back at the
problem, identify whether addition, subtraction, or multiplication was
needed to solve the problem.Students were directed to choose one of the
two or three operations: (a) addition (b) subtraction, and (c) multiplication
(for third graders only). After choosing one of the two or three operations,
children were then asked to identify the number sentence they would use
to solve the problem: (a) 15 %5!, (b) 15 #10 !, (c) 15 &5!, or (d)
15 #5!. This page of the booklet assessed the students ability to
correctly identify the operation and algorithm, respectively.
At the end of each booklet, students were read a series of truefalse
questions. All statements were related to the extraneous propositions for
each story problem within the booklet. For example, the statement Darren
used pinecones to make ornamentswould be true, whereas the statement
Darren used pinecones to draw pictureswould be false. The total score
possible for propositions related to question, number, goal, operations,
algorithms, and truefalse questions was 12. Cronbachs alpha for the
experimental word-problem-solving booklet task was .77.
Word problems—Semantic structure varied. The purpose of this ex-
perimental measure was to assess mental problem solving as a function of
variations in the semantic structure of a word problem. Children were
orally presented the problem and asked to calculate the answer in their
head. The word problems were derived from the work of Riley, Greeno,
and Heller (1983); Kintsch and Greeno (1985); and Fayol, Abdi, and
Gombert (1987). There were four sets of questions. Eight questions within
each set were ordered by the difficulty of responses.
The first set of eight questions focused on change problems. The
solution difficulty of problems in this set involved sums of 9 or less. An
example change problem is Paul had 5 candies. His mother gave him 2
candies. How many candies does Paul now have?
476 SWANSON AND BEEBE-FRANKENBERGER
Although there were systematic variations in the presentation of each
change problem within the set, the computation difficulty of the problem
stayed the same. For example, the first question followed a standard format
in which a result was unknown. For example, Alan had 5 marbles. His
friend gave him 2 marbles. His sister then gave him 2 marbles. How many
marbles does Alan have?
The second question, when compared with the first problem above,
changed the location of the question. The question was presented in the
first sentence. For example, How many marbles does Alan have? Alan
had 5 marbles. His friend gave him 2 marbles. His sister then gave him 2
marbles.
The third question introduced a sequence of events. For example, This
morning Alan had 5 marbles in his pocket. At noon his friend gave him 2
marbles. Yesterday his sister gave him 2 marbles. How many marbles does
Alan have?
The fourth question changed both the location of the question and the
temporal sequence of the question. For example, How many marbles does
Alan have? This morning Alan had 5 marbles in his pocket. At noon his
friend gave him 2 marbles. Later in the day his sister gave him 2 marbles.
The next four questions followed the same format as the first four
questions (standard, location of question, time sequence, location of ques-
tion and time sequence), except the word now was introduced in either the
first or second sentence. This change in semantic structure differed from
the first four questions because the child had to determine the initial state
of a problem. For example, Alan has now 6 marbles. During morning
playtime at school, he won 2 marbles. At noon he won 2 marbles. How
many marbles did Alan have before he went to school this morning?The
introduction of the word now told the child the result. However, the child
had to determine the start of the question (initial state). The now in the first
sentence provided the child with the final state of the problem.
The second set of eight questions followed the same format of questions
in Set 1 except that the solution difficulties involved sums greater than 10
but less than 20.
The third set of questions focused on compare questions rather than
change questions. Compare questions are more difficult than change ques-
tions (Riley et al., 1983). Compare questions focus on differences in
quantity, for instance, how many more or how many less. For example,
Paul has 5 candies. Paul has 2 less candies than his sister. How many
candies does his sister have?Word problems followed the same format as
the other sets (standard, location of question, time sequence, location of
question and time sequence, initial state) and minuends of 9 or less. Set 4
included the same types of problems as Set 3, but problems had minuends
between 10 and 20.
Because the questions varied in difficulty, not all questions were admin-
istered. Questions in Set 1 were administered first. All eight questions were
administered unless three errors occurred. If two or fewer errors were
made, participants were administered Set 2 and so on. The dependent
measure was the number of problems solved correctly. The total possible
number of correct solutions was 32. Cronbachs alpha for the word-
problem-comprehension task was .82.
Arithmetic Calculation
Arithmetic computation. The arithmetic subtests from the Wide Range
Achievement Test (WRAT; Wilkinson, 1993) and the Wechsler Individual
Achievement Test (WIAT; Psychological Corporation, 1992) were admin-
istered. Both subtests required written computation for problems that
increased in difficulty. Problems began with simple calculations (2 #2!)
and moved up to algebra. The dependent measure was the number of
problems solved correctly, which yielded a standard score (M!100, SD !
15). Cronbachs alpha for the WRAT was .92 and for the WIAT was .93.
Computation fluency. This test was adapted from the Test of Compu-
tational Fluency (Fuchs, Fuchs, Eaton, Hamlett, & Karns, 2000). The
adaptations required students to write answers within 2 min to 25 basic
facts and algorithms for Grades 1, 2, and 3. The basic facts and algorithms
were problems matched to grade level. The dependent measure was the
number of problems solved correctly. Cronbachs alpha was .85.
Predictor Variables
Reading and Phonological Processing Measures
Because the phonological measures are commonly used and derived
from published standardized measures (i.e., Woodcock Reading Mastery
TestRevised [WRMTR; Woodcock, 1998], WISCIII, CTOPP [Wag-
ner et al., 2000]), we only briefly describe these tasks.
Real-word and pseudoword reading tasks. Two subtests were admin-
istered from the Test of Word Reading Efficiency (TOWRE; Wagner &
Torgesen, 1999). The two subtests required oral reading of a list of 120 real
words or pseudowords of increasing difficulty. Real-word reading effi-
ciency was assessed by the Sight Word Efficiency subtest. Students were
given 45 s to read aloud as many words as possible from a list of common
words. Pseudoword reading ability was assessed by the Nonword Effi-
ciency subtest. Students were given 45 s to read aloud as many words as
possible from a list of nonwords. The nonwords followed regular spelling
patterns, requiring students to quickly decipher pronunciations on the basis
of their existing knowledge of grammar. The dependent measures for both
subtests were the number of words read correctly in 45 s. Cronbachs
coefficient alpha for the Sight Word Efficiency subtest was .90 and for the
pseudoword subtest was .88.
Word recognition. Word recognition was assessed by the reading
subtest of the WRAT. The task provided a list of words of increasing
difficulty. The childs task was to read the words until 10 errors occurred.
The dependent measure was the number of words read correctly. Cron-
bachs alpha for the word-recognition task was .89.
Reading comprehension. Reading comprehension was assessed by the
Passage Comprehension subtest from the WRMTR (Woodcock, 1998).
The purpose of this task was to assess the childs comprehension of topic
or subject meaning during reading activities. Comprehension questions
were drawn from the reading of short paragraphs. The dependent measure
was the number of questions answered correctly. Cronbachs coefficient
alpha was .90.
Letter-naming speed. The administration procedures followed those
specified in CTOPP (Wagner et al., 2000), including the presentation of
practice trials. The manual reported correlations between parallel forms
ranging from .80 to .93. For this task, the examiner presented participants
with an array of 36 letters. Participants were required to name the letters as
quickly as possible for each of two stimulus arrays containing 36 letters, for
a total of 72 letters. The task administrator used a stopwatch to time
participants on speed of naming. The dependent measure was the total time
to name both arrays of letters. The correlation between Array Forms A and
B was .90.
Phonological deletion. The Elision subtest from the CTOPP (Wagner
et al., 2000) was administered. The Elision subtest measures the ability to
parse and synthesize phonemes. The child was asked to say a word and to
say what word is left if part of the word is deleted. For example, Say heat.
Now say the word if I said heat without saying the /t/.There were four
practice items and 15 test items. The dependent measure was the number
of items said correctly. Cronbachs coefficient alpha was .94.
Phonological fluency. This experimental measure was adapted from
Harrison, Buxton, Husain, and Wise (2000). Children were given 60 s to
generate as many words as possible beginning with the letter B. Children
were told, I want to see how many words you can say that begin with a
certain letter. Do not say proper nouns or numbers or the same word with
different endings, and try not to repeat yourself. Keep naming words that
start with the letter until I say, Stop.Speak clearly and loud enough so
that I can hear the word you are saying. Do you understand? The letter is
B, begin.Repetitions, proper name errors, and contravention of the stem
repetition were deleted from the analysis. The dependent measure was the
477
WORD PROBLEMS
number of words correctly stated in 60 s. Cronbachs coefficient alpha
was .88.
Short-Term Memory Measures
Four measures of STM were administered: forward digit span and
backward digit span, word span, and pseudoword span. The digit subtest
from the WISCIII was administered. The forward and backward parts of
this subtest were maintained as separate variables rather than combining
them to create a composite STM span. The forward digit span task required
participants to recall and repeat in order sets of digits that had been spoken
by the examiner and that increased in number. The technical manual
reported a testretest reliability of .91. The backward digit span task
required participants to recall in reverse order sets of digits administered in
the same manner as the forward digit span task. The reliability reported for
this task was .76. Dependent measures for both tasks were the highest set
of items recalled in order (range of 0 to 8 for digits forward; range of 0 to
7 for digits backward). It was assumed that forward spans presumably
involved a subsidiary memory system (the phonological loop). However,
the backward span task was assumed to involve a subsidiary system
(phonological), as well as some resources from the executive system. It
was reasoned that if a backward span test represented additive effects of a
subsidiary system plus attention demands or control processes representa-
tive of a central executive system, then it should load with the WM tasks
in the factor analysis (to be discussed below). However, if the backward
digit span task primarily reflected a subsidiary system, it would load in the
factor analysis with the STM measures. Cronbachs alpha for both the
forward and backward tasks was .84.
The word span and pseudoword span tasks were presented in the same
manner as the forward digit span measure. The word span task was
previously used by Swanson, Ashbaker, and Lee (1996). The word stimuli
were one- or two-syllable high-frequency words. Students were read lists
of common but unrelated nouns and then were asked to recall the words.
Word lists gradually increased in set size from a minimum of two words to
a maximum of eight. The phonetic memory task (pseudoword span task;
Swanson & Berninger, 1995) used strings of nonsense words (one syllable
long), which were presented one at a time in sets of two to six nonwords
(e.g., DES, SEEG, SEG, GEEZ, DEEZ, DEZ). The dependent measure for
all STM measures was the highest set of items retrieved in the correct serial
order (range of 0 to 7). Cronbachs alpha was .62 for the word span task
and .82 for the phonetic memory task.
Working Memory Measures
The WM tasks in this study required children to hold increasingly
complex information in memory while responding to a question about the
task. The questions served as distractors to item recall because they
reflected the recognition of targeted and closely related nontargeted items.
A question was asked for each set of items, and the tasks were discontinued
if the question was answered incorrectly or if all items within a set could
not be remembered. Thus, WM span reflected a balance between item
storage and correct responses to questions. Consistent with a number of
previous studies, our WM tasks required the maintenance of some infor-
mation during the processing of other information. For example, consistent
with Daneman and Carpenters (1980) seminal WM measure, the process-
ing of information was assessed by asking participants simple questions
about the to-be-remembered material (storage plus processing demands),
whereas storage was assessed by accuracy of item retrieval (storage de-
mands only). The question required a simple recognition of new and old
information and was analogous to the yesno response feature of Daneman
and Carpenters task. It is important to note, however, that in our tasks, the
difficulty of the processing question remained constant within task condi-
tions, thereby allowing the source of individual differences to reflect
increased storage demands. Furthermore, the questions focused on the
discrimination of items (old and new information) rather than deeper levels
of processing such as mathematical computations (see, e.g., Towse, Hitch,
& Hutton, 1998). A previous study with a different sample had established
the reliability and the construct validity of the measures with the Daneman
and Carpenter measure (Swanson, 1996). For this study, four WM tasks
were divided into those requiring the recall of verbal (sentence/digit task,
semantic association task) and visualspatial information (e.g., visual ma-
trix task, mapping/direction task) and were selected from a standardized
battery of 11 WM tasks because of their high construct validity and
reliability (see Swanson, 1992). The complete description of administration
and scoring of the tasks is reported in Swanson (1995). A childrens
adaptation of the Daneman and Carpenter measure (Swanson, 1992) was
also administered. Task descriptions follow.
Listening sentence span. The childrens adaptation (Swanson, 1992) of
Daneman and Carpenters (1980) sentence span task was administered. The
construction of and pattern of results associated with the two measures are
comparable. The only difference was that each sentence was read to the
child with a 5-s pause that indicated the end of a sentence. The original
sentence span measure was used with university students, whereas the
current measure used a simpler sentence structure and reading vocabulary.
As a common measure of WM (see Daneman & Carpenter, 1980; Just &
Carpenter, 1992), this task required the presentation of groups of sentences,
read aloud, for which children simultaneously tried to understand the
passage and remember the last word of each sentence. The number of
sentences in the group gradually increased. After each group, the partici-
pant answered a question about a sentence and then recalled the last word
of the sentence. WM capacity was defined as the largest group of ending
words recalled. The mean sentence-reading level was approximately 3.8.
The dependent measure was the highest set recalled correctly (range of 0
to 8) in which the process question was answered correctly. Cronbachs
coefficient alpha was .79.
Semantic association task. The purpose of this task was to assess the
participants ability to organize sequences of words into abstract categories
(Swanson, 1992, 1995). The participant was presented a set of words (one
every 2 s), asked a discrimination question, and then asked to recall the
words that go together.For example, a set might include the following
words: shirt, saw, pants, hammer, shoes, nails. Participants were directed
to retrieve the words that went together (i.e., shirt, pants, and shoes;saw,
hammer, and nails). The discrimination question was Which word, saw or
level, was said in the list of words?Thus, the task required participants to
transform information encoded serially into categories during the retrieval
phase. The range of set difficulty was from two categories of two words to
five categories of four words. The dependent measure was the highest set
recalled correctly (range of 0 to 8) in which the process question was
answered correctly. Cronbachs coefficient alpha was .85.
Digit/sentence span. This task assessed the childs ability to remember
numerical information embedded in a short sentence (Swanson, 1992,
1995). Before stimulus presentation, the child was shown a card depicting
four strategies for encoding numerical information to be recalled. The
pictures portrayed the strategies of rehearsal, chunking, association, and
elaboration. The experimenter described each strategy to the child before
administration of targeted items. After all strategies had been explained,
the child was presented numbers in a sentence context. For example, Item
3 stated, Now suppose somebody wanted to have you take them to the
supermarket at 8 6 5 1 Elm Street?The numbers were presented at 2-s
intervals, followed by a process question, for instance, What was the name
of the street?Then, the child was asked to select a strategy from an array
of four strategies that represented the best approximation of how he or she
planned to practice the information for recall. Finally, the examiner
prompted the child to recall the numbers from the sentence in order. No
further information about the strategies was provided. Students were al-
lowed 30 s to remember the information. Recall difficulty for this task
ranged from 3 digits to 14 digits; the dependent measure was the highest set
478 SWANSON AND BEEBE-FRANKENBERGER
correctly recalled (range of 0 to 9) in which the process question was
answered correctly. Cronbachs coefficient alpha was .79.
Visual matrix task. The purpose of this task was to assess the ability of
participants to remember visual sequences within a matrix (Swanson, 1992,
1995). In contrast to the standardization procedures (Swanson, 1995), the
visual matrix task was administered in small groups. An overhead projector
was used to display stimuli to groups of children instead of individually by
use of the examiners manual. This change in format required students to
circle their answer to the process question, rather than verbally responding.
Otherwise, the task was administered as per the manual instructions.
Participants were presented a series of dots in a matrix and were allowed
5 s to study the matrix. The matrix was then removed, and participants
were asked, Are there any dots in the first column?To ensure the
understanding of columns prior to test, participants were shown the first
columns location and practiced finding it on blank matrices. In addition,
for each test item, the experimenter pointed to the first column on a blank
matrix (a grid with no dots) as a reminder of the first columns location.
After answering the discriminating question (by circling yfor yes or nfor
no), students were asked to draw the dots they remembered seeing in the
corresponding boxes of their blank matrix response booklets. The task
difficulty ranged from a matrix of 4 squares and 2 dots to a matrix of 45
squares and 12 dots. The dependent measure was the highest set recalled
correctly (range of 0 to 11) in which the process question was answered
correctly. Cronbachs alpha was .42.
Mapping and directions. This task required the child to remember a
sequence of directions on a map (Swanson, 1992, 1995). The experimenter
presented a street map with dots connected by lines; arrows illustrated the
direction a bicycle would go to follow this route through the city. The dots
represented stoplights, and lines and arrows mapped the route through the
city. The child was allowed 10 s to study the map. After the map was
removed, the child was asked a process question, for example, Were there
any stoplights on the first street (column)?The child was then presented
a blank matrix on which to draw the street directions (lines and arrows) and
stoplights (dots). Difficulty ranged on this subtest from 4 dots to 19 dots.
The dependent measure was the highest set of a correctly drawn map
(range of 0 to 9) for which the process question was answered correctly.
Cronbachs alpha was .94.
Inhibition and Updating Measures
Random generation of letters and numbers. The random-generation
task has been well articulated in the literature (e.g., Baddeley, 1996;
Towse, 1998). The task was assumed to measure inhibition because par-
ticipants were required to actively monitor candidate responses and sup-
press responses that would lead to well-learned sequences, such as 12
34 or abcd (Baddeley, 1996). Because this task has been primarily
used with adult samples who have quicker access to letters and numbers,
it was modified for the age groups in this study. Each child was asked to
write as quickly as possible numbers (or letters) first in sequential order to
establish a baseline. Children were then asked to quickly write numbers (or
letters) in a random, nonsystematic order. For example, for the number
section, students were first asked to write numbers from 0 to 9 in order (i.e.,
1, 2, 3, 4, 5, 6, 7, 8, 9) as quickly as possible in a 30-s period. They were
then asked to write numbers as quickly as possible out of order in a 30-s
period. Scoring included an index for randomness, information redun-
dancy, and percentage of paired responses to assess the tendency of
participants to suppress response repetitions. The measure of inhibition was
calculated as the number of sequential letters or numbers minus the number
of correctly unordered numbers or letters divided by the number of se-
quential letters or numbers plus the number of unordered letters or num-
bers. Cronbachs alpha for the random number-generation task in the
current sample was .89 and for random letter generation was .91.
Updating. The experimental updating task was adapted from Morris
and Jones (1990). A series of one-digit numbers was presented that varied
in set lengths of 9, 7, 5, and 3. No digit appeared twice in the same set. The
examiner told the child that the length of each list of numbers might be 3,
5, 7, or 9 digits. Participants were then told that they should recall only the
last three numbers presented. Each digit was presented at approximately
1-s intervals. After the last digit was presented, the participant was asked
to name the last three digits in order. It was stressed that some of the lists
of digits would be only three digits long so the participants should not
ignore any items. In contrast to the aforementioned WM measures, which
involved a dual-task situation where participants answered questions about
the task while retaining information (words or spatial location of dots), the
current task involved the active manipulation of information such that the
order of new information was added to or replaced the order of old
information. That is, to recall the last three digits in an unknown (N!3,
5, 7, 9) series of digits, the order of old information (previously presented
digits) had to be kept available along with the order of newly presented
digits. Thus, task performance reflected the activity of both the phonolog-
ical system and the executive system. The dependent measure was the total
number of digits correctly repeated (range of 0 to 16). Cronbachs alpha for
the current sample was .94.
Semantic Processing and Vocabulary
Semantic fluency. The experimental measure was adapted from Harri-
son et al. (2000). Children were given 60 s to generate as many names of
animals as possible. Children were told, I want to see how many animals
you can name. Try not to repeat yourself. Dont say pet names. Keep going
until I tell you to stop. Ready, begin.Repetitions were deleted from the
analysis. The dependent measure was the number of words correctly stated
within 60 s. The coefficient alpha for the experimental semantic fluency
task was .91.
Vocabulary. A word-knowledge score was obtained from the WISC
III vocabulary subtest. Children were read a word by the examiner and
were asked to provide the meaning of the word. Words increased in
complexity. Scoring followed the WISCIII manual. The dependent mea-
sure was based on the quality of word definitions. The coefficient alpha for
the current sample was .88.
Procedures
Three doctoral-level graduate students trained in test administration
tested all participants in their schools. Two sessions of approximately 45 to
60 min each were required for small-group test administration and one
session of 45 to 60 min for individual administration. During the group-
testing session, data were obtained from the Raven Colored Progressive
Matrices, WIAT, WRAT, mathematical word-problem-solving processes
task (WPS-P), visual matrix test, and arithmetic calculation fluency. The
remaining tasks were administered individually. Test administration was
counterbalanced to control for order effects. Task order was random across
participants within each test administrator.
RESULTS
The means and standard deviations for intelligence, accuracy in
recognizing problem-solving components, problem-solving solu-
tion accuracy, arithmetic calculation, phonological processing,
reading, STM, WM, inhibition/updating, and semantic processing
and vocabulary are shown in Table 1. The analyses and results
were divided into two sections. The first section focused on age
and ability group differences. SMD children and NSMD children
were compared across three grade levels (first, second, and third
grades). This approach lent itself to comparing age and ability
group but had the disadvantage of ignoring information about the
variability of participants in each group. The second section fo-
cused on correlations between WM and problem solving in the
479
WORD PROBLEMS
complete sample. The approach allowed us to study the entire
range of scores in WM and problem solving as well to focus on
common and unique variance. Regression models were computed
to isolate unique processes that underlie word-problem-solving
performance.
Age and Ability Group Comparisons
Preliminary Analysis
Prior to the analysis, we compared the age and risk groups on
measures related to fluid intelligence (IQ). A 3 (age) %2 (group
risk: SMD vs. NSMD) analysis of variance (ANOVA) was com-
puted on standard scores from the Raven Colored Progressive
Matrices. A shown in Table 2, a significant main effect emerged
for ability group. Thus, standard scores from the Raven Colored
Progressive Matrices test were used as a covariate in the subse-
quent analysis. ANOVAs were also computed on the classification
measures related to the arithmetic subtest of the WISCIII and the
rapid number naming subtest from the CTOPP. As shown in Table
2, high effect sizes emerged between the two ability groups on
these measures. Table 2 shows the univariates and effect sizes (
"
2
)
for each comparison. An
"
2
of .13, .05, or .02 corresponded to a
Cohensdof .80, .50, or .20, respectively.
A series of 3 (age) %2 (group risk: SMD vs. NSMD) factorial
multivariate analyses of covariance (MANCOVAs) was conducted
to examine differences in means for the following processes: (a)
word-problem components, (b) arithmetic calculation, (c) reading,
(d) phonological processing, (e) STM, (f) WM, (g) inhibition/
updating, and (h) semantic processing and vocabulary. Results
related to the univariate analyses of covariance (ANCOVAs) are
shown in Table 2. Because of the number of comparisons, a
significant alpha level of p$.001 was adopted to control for
Table 2
Analysis of Covariance (ANCOVA) Word-Problem Solving, Calculation, Reading, and Cognitive Measures
Variable
Age Group
Variable
Age Group
Fratio
"
2
Fratio
"
2
Fratio
"
2
Fratio
"
2
Classification
Chronological age 571.53.61 3.29 .009
Fluid IntelligenceRaven
Standard score 1.90 .005 31.97.08
Raw score 60.18.15 30.88.08
Arithmetic subtestWISCIII
Standard score 6.16 .017 399.43.53
Raw score 185.4.34 251.1.41
Digit-naming speedCTOPP
Standard score 124.77.26 56.06.14
Raw score 134.43.27 95.63.21
CriterionWord-problem solving
Word-problem solving processes
(raw score)
Question 6.20 .01 13.98.04
Numbers 20.09.05 4.59 .01
Goal 14.05.03 7.95 .02
Operations 11.20.03 3.39 .01
Algorithm 35.10.09 2.95 .01
Irrelevant information 17.60.05 3.22 .01
Word problemsSemantic structure
varied
43.64.11 15.01.04
Math calculation
WRAT 519.30.59 13.79.04
WIAT 785.90.69 19.90.05
Computational fluency 23.03.06 13.12.04
Predictor: Reading
Word recognitionWRAT 115.42.24 28.62.08
Real-word fluencyTOWRE 200.42.36 54.65.13
Note. ANCOVAs were computed for raw scores on all but the four Classification tasks. Effect sizes are based on Fratios with influence of Raven scores
partialed out. Raven !Raven Colored Progressive Matrices; WISCIII !Wechsler Intelligence Scale for ChildrenThird Edition; CTOPP !
Comprehensive Test of Phonological Processing; WRAT !Wide Range Achievement Test; WIAT !Wechsler Individual Achievement Test; TOWRE
!Test of Word Reading Efficiency; WRMTR!Woodcock Reading Mastery TestRevised; S-CPT !S-Cognitive Processing Test.
*** p$.001. p$.0001.
Predictor: Reading (continued)
ComprehensionWRMTR 222.25.38 35.91.09
Letter rapid namingCTOPP 62.45.14 50.13.12
Phonological
PseudowordsTOWRE 64.11.15 37.93.09
ElisionCTOPP 24.13.06 21.68.06
Phonological fluency 7.01*** .02 1.79 .005
Short-term memory
Digits forwardWISCIII 6.95 .02 15.04*** .04
Digits backwardWISCIII 5.97 .02 11.37.03
Pseudoword span 4.99 .01 5.61 .02
Real-Word span 6.33 .02 9.77*** .03
Working memory
Listening sentence span 15.81.04 8.90 .03
Digit/sequence spanS-CPT 9.37.02 4.94 .02
Semantic associationS-CPT 3.99 .01 4.78 .02
Visual matrixS-CPT 11.20.03 .77 .002
Mapping/directionsS-CPT 2.40 .006 0.11 .001
Inhibition and updating
Random-generation letters .75 .002 4.32 .01
Random-generation numbers 8.23.02 5.31 .02
Updating 5.42 .02 25.44.08
Semantic and vocabulary
Semantic fluency 18.71.05 11.85.03
Vocabulary 70.81.16 18.21.05
480 SWANSON AND BEEBE-FRANKENBERGER
experimentwise alpha inflation and Type I error. None of the
Age %Group interactions for the MANCOVAs met the .001 alpha
level, and therefore, the Fratios related to the interactions are not
reported.
Word-Problem-Solving Components and Accuracy
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted examining the six components of word-problem solv-
ing, identifying (a) the question, (b) the numerical information, (c)
the goal of the problem, (d) the arithmetical operation, (e) the
algorithm, and (f) irrelevant information. These measures were
taken from the WPS-P. A significant multivariate main effect
emerged for age, Wilkss'!.72, F(12, 682) !10.07, p$.0001,
"
2
!.28, and for group, Wilkss'!.94, F(6, 341) !6.56, p$
.0001,
"
2
!.06. The overall pattern for the MANCOVA, which
tests for the linear combination of the variables (determined by
variable intercorrelations), is that students in third grade scored
higher on all WPS-P components than those in second and first
grades and that children not at risk scored higher than those at risk
for SMD. The results of the ANCOVAs showed that five WPS-P
components (identifying number, goal, operations, algorithms, and
irrelevant information) differed by age. The largest effect size as a
function of age was knowledge of algorithms. Only one ANCOVA
(identify question) was significantly related to ability group. Over-
all, effect sizes related to age and ability group were in the low to
moderate range.
An ANCOVA was computed on the solution-accuracy scores
for the word-problem-solving measure that varied the semantic
structure of the sentences. As shown in Table 2, the ANCOVA was
significant for age and ability group. The results showed that
students in Grade 3 scored higher than those in the earlier grades
and that children not at risk scored higher on measures related to
word-problem accuracy than those at risk for SMD.
Arithmetic
A 3 (age) %2 (group risk: SMD vs. NSMD) factorial
MANCOVA was conducted on accuracy scores for arithmetic
measures (WRATmath, WIATmath, computation fluency). A
significant multivariate main effect emerged for age, Wilkss'!
.13, F(6, 688) !194.44, p$.0001,
"
2
!.87, and for group,
Wilkss'!.89, F(3, 344) !12.88, p$.0001,
"
2
!.11. As
expected, the general pattern of results is that students in Grade 3
scored higher than those in Grades 1 and 2 and that children not at
risk scored higher on arithmetic measures than those at risk for
SMD. It is important to note, however, that although a clear
advantage was found for children not at risk, standard scores for
the children at risk were in the normal range, suggesting that
mathematical problem solving (as reflected by low standard scores
on the arithmetic subtest of the WISCIII) shares some indepen-
dence from calculation skill.
Reading Measures
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted on accuracy scores for reading (WRATreading,
WRMTRreading comprehension, TOWREreal words, letter-
naming speed). A significant multivariate effect emerged for age,
Wilkss'!.40, F(8, 678) !49.56, p$.0001,
"
2
!.60, and for
group, Wilkss'!.82, F(4, 339) !19.92, p$.0001,
"
2
!.18.
The general pattern of the results is that students in Grade 3 scored
higher than those in Grades 1 and 2 and that children not at risk
scored higher on reading and phonological processes measures
than those at risk for SMD.
Phonological Measures
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted on accuracy scores for phonological measures
(TOWREpseudowords, elision, phonological fluency). A signif-
icant multivariate effect emerged for age, Wilkss'!.71, F(6,
682) !21.00, p$.0001,
"
2
!.29, and for group, Wilkss'!
.89, F(3, 341) !14.92, p$.0001,
"
2
!.11. The general pattern
of the results is that students in Grade 3 scored higher than those
in Grades 1 and 2 and that children not at risk scored higher on
reading and phonological processes measures than those at risk for
SMD.
Short-Term Memory
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted on measures related to memory using span scores for
STM (forward digit, backward digit, pseudoword span, real-word
span). A significant multivariate main effect emerged for age,
Wilkss'!.90, F(8, 678) !4.25, p$.0001,
"
2
!.10, and for
group, Wilkss'!.92, F(4, 339) !7.70, p$.0001,
"
2
!.08.
Students in third grade scored higher on memory measures than
those in second and first grades, and children not at risk scored
higher than those at risk for SMD. The ANCOVAs show that
significant age effects emerged for only the digit forward task. In
contrast, significant ANCOVAs as a function of ability group
emerged on the digit forward, digit backward, and real-word span
tasks. Effect sizes for the significant ability group ANCOVAs
were in the low to moderate range.
Working Memory
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted on measures related to span scores for WM (listening
sentence span, digit/sentence, semantic association, visual matrix,
mapping/directions). A significant multivariate main effect
emerged for age, Wilkss'!.82, F(10, 687) !6.62, p$.0001,
"
2
!.18, and for group, Wilkss'!.95, F(5, 339) !3.10, p$
.001,
"
2
!.05. Students in third grade scored higher on memory
measures than those in second and first grades, and children not at
risk scored higher than those at risk for SMD. The ANCOVAs
showed age effects on all measures except for the semantic asso-
ciation and the mapping/directions tasks (all effect sizes were in
the moderate range). ANCOVAs for isolated tasks as a function of
ability group were nonsignificant at the .001 level. Furthermore,
effect sizes as a function of ability group were in the low to
moderate range.
Random Generation and Updating
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted on measures of random generation and updating (ran-
dom generation of letters and random generation of numbers,
481
WORD PROBLEMS
updating task). Rapid naming of numbers was not included be-
cause it served as a classification measure. A significant multivar-
iate main effect emerged for age, Wilkss'!.92, F(6, 678) !
4.35, p$.0001,
"
2
!.08, and for group, Wilkss'!.91, F(3,
339) !10.64, p$.0001,
"
2
!.09. As shown in Table 1, students
in third grade scored higher than those in second and first grades,
and students not at risk for SMD scored higher than students at risk
for SMD. Significant ANCOVAs as a function of age emerged on
the random generation of numbers task and as a function of ability
group on the updating task.
Semantic Fluency and Vocabulary
A 3 (age) %2 (group risk: SMD vs. NSMD) MANCOVA was
conducted on vocabulary and semantic fluency measures (WISC
III Vocabulary subtest and semantic fluency). A significant mul-
tivariate main effect emerged for age, Wilkss'!.69, F(4,
684) !34.23, p$.0001,
"
2
!.31, and for group, Wilkss'!
.93, F(2, 342) !21.39, p$.0001,
"
2
!.07. Students in third
grade scored higher than those in second and first grades, and
students not at risk for SMD scored higher than students at risk for
SMD. All ANCOVAs were significant at the .001 level.
In summary, the results clearly show that all MANCOVAs were
significant. Older children and children not at risk for SMD out-
performed younger children and children at risk for SMD across
aggregate measures of problem solving, arithmetic calculation,
word-problem-solving processes, reading, phonological pro-
cesses, STM, WM, random generation and updating, and vo-
cabulary and semantic fluency. All effect sizes related to the
MANCOVAs met or exceeded a Cohensdof .50. Except for
the calculation and reading measures, the effect sizes for the
ANCOVAs as a function of ability group were low to moderate,
suggesting that ability group differences are more reliable at the
composite level.
Correlations
The next analyses examined the relationship between math
problem solving and WM in the total sample. We predicted that if
the WM system played an important role in accounting for age-
related and individual differences in problem solving independent
of the phonological system, then WM measures would predict
problem-solving performance after various measures of the pho-
nological system had been partialed from the analysis. We exam-
ined this hypothesis through a series of regression analyses in
which phonological processes (i.e., STM, phonological knowl-
edge) and WM were the independent variables and problem solv-
ing was the dependent measure. We also considered whether
reading and calculation skill, semantic processing, speed, inhibi-
tion, and fluid intelligence mediated performance. Prior to our
regression analysis, however, the intercorrelations between WM
and problem-solving measures were examined.
Because all the MANCOVAs were significant, several of the
above measures were aggregated into composite scores. This was
done to cluster variables along theoretical lines, as well as for data
reduction purposes. The composite scores were created by com-
puting zscores for each task based on the total sample. The z
scores for the appropriate tasks were then summed to create the
composite scores. Two composite scores served as criterion mea-
sures: word problems (word-problem solvingsemantic structure
varied, WISCIII Arithmetic subtest, mean intercorrelation !.45),
and mathematical computation (WRATarithmetic, WIAT
arithmetic, computation fluency, mean intercorrelation !.70).
Three composite scores served as predictor variables that yielded
one value for reading (WRATreading, WRTcomprehension,
TOWREreal words, mean intercorrelation !.88), phonological
processing (elision, phonological fluency, TOWREpseudoword
reading, mean intercorrelation !.45), and semantic processing
(vocabulary, semantic fluency, r!.47). Because of the impor-
tance of speed in discussions of STM and WM (Kail & Hall, 2001;
Salthouse, 1996), a composite measure of speed was also created
(rapid naming of digits, rapid naming of letters, r!.88). In
addition, because the correlation between the random-generation
measures was weak (r!.30), the random generation of numbers
task was used as the primary measure of inhibition.
2
In addition,
the raw scores from the Raven Colored Progressive Matrices test
served as a measure of fluid intelligence.
Although the composite scores fit a logical structure, we were
uncertain about the structure of the memory measures. This was
because several recent studies have argued that there are no pure
measures of STM or WM and therefore that latent scores must be
used (see, e.g., Conway, Cowan, Bunting, Therriault, & Minkoff,
2002; Engle et al., 1999). Thus, to reduce the data set, we submit-
ted the memory measures to a principal factor analysis. Also
included in the analysis was the updating measure. Because of the
broad selection of measures and in consideration of our sample
size, we relied on factor scores to keep the number of estimated
parameters within a reasonable range. In addition, variance across
the tasks was wanted variance, and task-specific variance was of
less interest. As shown in Table 3, a two-factor model emerged
(eigenvalues of 1.40 and 1.10, respectively). To interpret Table 3,
we used a varimax rotation (orthogonal solution) and considered
factor loadings of .30 as meaningful. We used the common factor
analysis and varimax rotation because scores on each measure had
a reasonable degree of reliability and shared common variance
with scores on other measures. We were also interested in the
independent contribution of each dimension in explaining the
covariation of individual differences on the measures. Therefore,
an orthogonal solution was used to retain the independent dimen-
sions. As shown in Table 3, all STM measures (except backwards
digit) and the updating task loaded on Factor 1. All WM tasks
loaded on Factor 2.
To determine if the two-factor structure was an adequate ex-
traction of the matrix, we obtained maximum-likelihood estimates
(Jo¨reskog & So¨rbom, 1984) for the two-factor model. The likeli-
hood ratio chi-square test yielded
!
2
(26, N!353) !32.84, p!
2
Measures of fluency are associated with inhibition in adult samples
(see, e.g., Conway & Engle, 1994). However, the correlation between
semantic fluency and phonological fluency, with age partialed out, in the
present sample was weak (r!.28). A weak partial correlation also
emerged between semantic fluency and random number generation (r!
&.16) and between semantic fluency and random letter generation (r!
&.09). Likewise, a weak partial correlation emerged between phonological
fluency and random number generation (r!&.08) and between phono-
logical fluency and random letter generation (r!&.13). Thus, the fluency
measures did not appear to capture inhibition in the age groups represented
here.
482 SWANSON AND BEEBE-FRANKENBERGER
.16. Nonsignificance was considered one criterion for model ac-
ceptance discussed by Bentler and Bonett (1980). The goodness-
of-fit index (Bentler & Bonett, 1980) was computed from the null
model (which hypothesizes that the variables are uncorrelated) in
the population,
!
2
(45, N!353) !442.95, p$.001, and the
current two-factor model as (442.95 32.84/442.95) !.926. Thus,
the model was 93% of the way to a perfect fit. Further testing of
this model included an analysis of the
!
2
/df ratio, the root-square
residual, and the Tucker-Lewis Index (TLI). The
!
2
/df ratio pro-
vided information on the relative efficiency of the alternative
model in accounting for the data (Marsh, Balla, & McDonald,
1988). Values of 2.0 or less were interpreted as representing an
adequate fit. The present two-factor model was 1.25. The root-
mean-square residual (RMSR) measured average residual correla-
tion (Jo¨reskog & So¨rbom, 1984). Smaller values (e.g., .10 or less)
reflected a better fit. The RMSR for the two-factor structure was
.003. The TLI roughly scaled the chi-square from 0 to 1, with 0
representing the fit of the null model (Bentler & Bonett, 1980),
which assumed that the variables were uncorrelated, and 1 repre-
sented the fit of a perfectly fitting model. Values less than .90
would suggest that the model could be improved substantially (see
Marsh et al., 1988, p. 292, for discussion), whereas values close to
1.0 would indicate a better fit. This measure, when compared with
the other indices, was relatively independent of sample size. The
TLI in the present study was .97.
As shown in Table 4, the two factor scores (STM and WM)
based on the common factor analysis were correlated with math,
reading, fluid intelligence, and cognitive measures. Also included
in the correlation analysis were chronological age and the
problem-solving component score related to knowledge of algo-
rithms. This latter measure was a subcomponent of the word-
problem-processing task and was selected because it was found to
eliminate the significant contribution of WM for samples in the
higher grades (Swanson & Sachse-Lee, 2001). In addition, a factor
analysis of component scores of the present data showed that the
knowledge of algorithms loaded highest on the first factor. Thus,
for simplicity, only the knowledge of algorithms component score
was analyzed. As shown in Table 4, several significant correlations
emerged. Because of the number of comparisons, alpha was set to
.001 (N!353, rs".18, p$.001).
Table 3
Factor Structure of Short-Term Memory, Working Memory, and Updating Tasks
Task 1 2 3 4 5 6 7 8 9 10
Factors
(varimax rotation)
I II
Short-term memory
1. Digit forward .51 &.001
2. Digit backward .08 .24 .36
3. Pseudoword span .30 .13 .53 .17
4. Real-word span .30 .23 .38 .55 .16
Working memory
5. Listening span .11 .22 .19 .26 .24 .40
6. Digit/sentence .13 .27 .21 .19 .29 .22 .54
7. Semantic association .15 .11 .17 .15 .15 .27 .17 .36
8. Visual matrix .08 .17 .09 .03 .19 .24 .10 .06 .37
9. Mapping/directions &.05 .07 .10 .08 .15 .24 .22 .14 &.03 .42
Updating
10. Updating .35 .24 .37 .35 .19 .26 .14 .12 .01 .57 .18
Note. Boldfaced numbers in the factor columns indicate factor loadings ".35.
Table 4
Intercorrelations Among Mathematics, Reading, and Cognitive-Processing Variables
Variable 1 2 3 4 5 6 7 8 9 10 11 12
1. Age
2. Word-problem accuracy .50
3. Arithmetic calculation .79 .65
4. Reading skills .60 .72 .80
5. Phonological knowledge .41 .63 .62 .79
6. Processing speed &.49 &.58 &.63 &.72 &.60
7. Semantic processing .49 .62 .63 .67 .62 &.51
8. Inhibition &.19 &.31 &.25 &.34 &.28 .30 &.31
9. Short-term memory .25 .47 .40 .50 .48 &.44 .43 &.26
10. Working memory .37 .54 .51 .52 .49 &.37 .53 &.22 .26
11. Algorithm knowledge .36 .62 .42 .48 .41 &.33 .44 &.20 .32 .35
12. Fluid intelligence .43 .58 .55 .62 .51 &.42 .51 &.27 .37 .41 .42
Note. rs".21, p$.0001.
483
WORD PROBLEMS
An inspection of Table 4 shows four important findings (to
interpret the results, we considered rs".50 as substantial corre-
lations). First, problem-solving accuracy was significantly related
to all variables except inhibition ( p".0001). Second, phonolog-
ical processes (phonological knowledge and STM) were signifi-
cantly related to the majority of measures. Substantial correlations
occurred between phonological processing and composite scores
of word-problem-solving accuracy, arithmetic calculation, pro-
cessing speed, and semantic processing. Third, WM was substan-
tially correlated with arithmetic calculation, semantic processing,
and fluid intelligence.
Finally, the correlation between WM and word-problem solving
was substantial, r(351) !.54, p$.0001. To examine this rela-
tionship further, we partialed from the correlation analysis vari-
ables assumed to underlie individual differences in WM. The
correlation between WM and problem solving was r!.46 when
partialed for age, r!.33 when partialed for age plus arithmetic
calculation, and r!.30 when partialed for age plus reading. When
processing variables were considered, the partial correlation was
r!.32 when partialed for age plus phonological processing, r!
.40 when partialed for age plus speed, r!.31 when partialed for
age plus semantic processing, r!.44 when partialed for age plus
inhibition, r!.42 when partialed for age plus STM, r!.41 when
partialed for age plus knowledge of algorithms, and r!.37 when
partialed for age plus fluid intelligence. Thus, the smallest coeffi-
cient related to WM and problem solving partialed out the influ-
ence of age and reading. However, all the partial coefficients were
significant (all ps$.0001), and none of the partial coefficients
that included age differed significantly from the other partialed
coefficients in magnitude (via Fisher z-score transformation, all
ps".001).
We analyzed the correlations between problem solving and WM
within grades. A significant correlation between problem solving
and WM was found in Grade 1, r(128) !.57, p$.0001; Grade 2,
r(88) !.42, p$.0001; and Grade 3, r(127) !.34, p$.0001. We
also partialed the influence of phonological processing, inhibition,
and speed from these coefficients. The partial coefficients were
r!.46, p$.0001; r!.28, ns; and r!.25, p$.001, for Grades
1, 2, and 3, respectively. These findings show that the magnitude
of the zero-order correlations and partial correlations was greater
in Grade 1 than in Grades 2 and 3. However, the magnitude of the
coefficients was reduced when measures of processing efficiency
(phonological processing, inhibition, and naming speed) were par-
tialed from the analysis. It is important to note that the magnitude
of the partial coefficients did not differ significantly between
grades ( ps".001).
Predictions of Problem Solving
The next analysis determined those variables that significantly
predicted word-problem solving when the effects of all other
competing variables were partialed from the analysis. Subsequent
regression analyses assessed the variables that contributed unique
variance to problem-solving performance. We investigated
whether the relationship between problem solving and WM was
maintained when blocks of variables related to age, phonological
processing, semantic processing, inhibition, and reading skill were
entered into the analysis. We also addressed the same question
when comparing the relationship between math calculation and
WM. In summary, the regression analysis was done (a) to deter-
mine if the influence of WM was partialed out when variables
related to phonological and related processes were entered into the
analysis and (b) to determine if blocks of variables related to
reading, math, inhibition, and speed increased the predictability of
word-problem solving when compared with a model that included
only WM.
The criterion and predictor variables were the same as those
shown in Table 4. Criterion measures were zscores (converted
from raw scores based on the total sample) from the problem-
solving and math calculation measures. Predictor variables were z
scores related to WM, STM phonological and semantic processing,
reading, inhibition, chronological age, and the component score
related to knowledge of algorithms.
For our first set of analyses, we determined the amount of
variance in problem-solving performance that was accounted for
by WM alone (Model 1). As shown in Table 5, WM contributed
approximately 30% of the variance to problem solving and 26% of
the variance to arithmetic calculation. For each subsequent model,
variables were entered simultaneously such that the beta values
reflected unique variance (the influence of all other variables
partialed out). In Model 2, we determined the contribution of the
factor scores related to STM, WM, and age when entered in the
model. As shown, WM contributed significant variance to both
math calculation and word-problem solving even when STM and
age were partialed from the analysis. In addition, both STM and
age contributed unique variance. The predictor variables in Model
2 contributed approximately 50% of the variance in word-problem
solving and 62% of the variance in math calculation. When com-
pared with Model 1, Model 2 significantly improved the predic-
tions for math calculation, F
inc
(2, 345) !163.42, p$.001, and
problem-solving accuracy, F
inc
(2, 345) !69.03, p$.001.
In Model 3, we assessed the contribution of WM to problem
solving as well as when composite scores related to phonological
processing, speed, and inhibition were added to Model 2. As
shown, Model 3 accounted for 58% of the variance in word-
problem solving and 70% of the variance in math calculation. All
variables contributed significant variance to word-problem-solving
accuracy. In contrast, all variables except STM and inhibition
contributed unique variance to math calculation. Model 3 im-
proved the prediction for calculation and problem solving when
compared with Model 2, F
inc
(3, 342) !30.57, p$.001, and
F
inc
(3, 342) !22.16, p$.001, respectively.
In Model 4, we determined whether the variables that contrib-
uted significant variance in Model 3 were eliminated when the
reading and the semantic-processing composite scores were en-
tered into the model. Model 4 captured approximately 61% of the
variance in word-problem solving and 73% of the variance in math
calculation. The important finding related to Model 4 is that the
significant influence of phonological processing and processing
speed was eliminated in the prediction of word-problem-solving
accuracy. The only variables that contributed unique variance to
solution accuracy were reading, semantic processing, age, STM,
and WM. Likewise, Model 4 showed that reading, age, and WM
contributed unique variance to math calculation. Model 4 im-
proved the prediction for calculation and problem solving when
compared with Model 3, F
inc
(2, 340) !37.97, p$.001, and
F
inc
(2, 340) !13.08, p$.001, respectively.
484 SWANSON AND BEEBE-FRANKENBERGER
In Model 5, we determined whether the variables that contrib-
uted significant variance in Model 4 were eliminated when mea-
sures of fluid intelligence were entered into the model. The im-
portant finding related to Model 5 is that the significant influence
of age was eliminated in the prediction of word-problem-solving
accuracy. The only variables that contributed unique variance to
solution accuracy were fluid intelligence, reading, STM, and WM.
In contrast, Model 5 provided a comparable pattern of results to
Model 4 in predicting math calculation. The results showed that
reading, age, and WM were the only variables that contributed
unique variance to math calculation. Model 5 did not improve the
prediction for calculation, F$1, but did improve the prediction of
problem-solving accuracy when compared with Model 4, F
inc
(2,
339) !4.46, p$.01.
A final regression was computed related to word-problem solv-
ing. For this model, we entered the composite score related to math
calculation and the component score related to knowledge of
algorithms into the regression model. The results are shown in
Table 6. When comparing Model 5 in Table 5 with Model 6 in
Table 6, one finds that R
2
increased by 7%. More importantly, the
results show that WM remained a significant predictor of word-
problem solving. The results also show that algorithmic knowledge
and processing speed contributed unique variance to solution ac-
curacy. Model 6 significantly improved the prediction for problem
solving when compared with Model 5, F
inc
(3, 337) !25.32, p$
.001.
In summary, there are two important findings related to the
hierarchical regression analysis. First, in the complete model (see
Table 6), only WM, STM, fluid intelligence, processing speed, and
knowledge of algorithms contributed unique variance to problem-
solving accuracy. Second, partialing out phonological processing,
STM, speed, and inhibition did not eliminate the significant role
Table 5
Hierarchical Analysis of Math Calculation and Word-Problem Solving
Model & variable
Math calculation Word-problem solving
B SE
#
tratio B SE
#
tratio
Model 1
WM 1.89 .17 .51 11.04*** 1.80 .14 .54 12.19***
Model 2
Age 1.55 .09 .58 16.23*** 0.73 .09 .31 7.50***
STM 0.64 .12 .19 5.34*** 0.90 .12 .29 7.39***
WM 0.96 .12 .26 7.27** 1.18 .13 .36 8.72***
Model 3
Inhibition &0.13 .09 &.04 &1.42 &0.21 .09 &.08 &2.19*
Speed &0.17 .05 &.12 &3.13*** &0.19 .05 &.15 &3.29**
Phonological processing 0.30 .04 .26 6.44*** 0.24 .04 .24 4.88***
Age 1.25 .09 .46 13.29*** 0.44 .09 .18 4.50***
STM 0.15 .13 .04 1.29 0.27 .14 .15 3.61***
WM 0.54 .12 .14 4.22** 0.77 .15 .25 6.00***
Model 4
Reading 0.35 .05 .37 6.18*** 0.26 .06 .31 4.28***
Semantic processing 0.10 .06 .06 1.57 0.15 .06 .11 2.29*
Inhibition &0.07 .08 &.02 &0.81 &0.15 .09 &.06 &1.67
Speed &0.03 .05 &.02 &0.55 &0.08 .06 &.06 &1.33
Phonological processing 0.09 .05 .08 1.70 0.06 .06 .06 1.13
Age 0.98 .09 .37 10.12*** 0.21 .10 .09 2.09*
STM 0.02 .11 .01 0.20 0.33 .12 .11 2.71**
WM 0.37 .12 .10 2.94** 0.63 .19 .19 4.66***
Model 5
Fluid intelligence 0.03 .09 .01 0.38 0.36 .10 .15 3.55**
Algorithm knowledge 0.26 .03 .39 7.93*** 0.26 .06 .31 4.28***
Reading 0.34 .05 .36 5.86*** 0.20 .06 .24 3.27**
Semantic processing 0.09 .06 .06 1.51 0.13 .06 .09 1.93
Inhibition &0.06 .08 &.02 &0.77 &0.13 .09 &.05 &1.43
Speed &0.03 .05 &.02 &0.58 &0.09 .06 &.08 &1.65
Phonological processing 0.09 .05 .08 1.61 0.06 .05 .06 1.13
Age 0.98 .09 .36 10.04*** 0.18 .10 .07 1.79
STM 0.02 .11 &.01 0.17 0.30 .12 .05 2.48*
WM 0.37 .12 .07 2.89** 0.59 .13 .18 4.39***
Note. Model 1: Math calculation: F(1, 347) !121.32, p$.001, R
2
!.26; Word-problem solving: F(1, 347) !
148.65, p$.001, R
2
!.30; Model 2: Math calculation: F(3, 345) !191.24, p$.001, R
2
!.62; Word-problem
solving: F(3, 345) !113.87, p$.001, R
2
!.50; Model 3: Math calculation: F(6, 342) !131.31, p$.001,
R
2
!.70; Word-problem solving: F(6, 342) !77.45, p$.001, R
2
!.58; Model 4: Math calculation: F(8,
340) !116.26, p$.001, R
2
!.73; Word-problem solving: F(8, 340) !65.70, p$.001, R
2
!.61; Model 5:
Math calculation: F(9, 339) !103.10, p$.001, R
2
!.73; Word-problem solving: F(9, 339) !61.79, p$.001,
R
2
!.62. WM !working memory; STM !short-term memory.
*p$.05. ** p$.01. *** p$.001.
485
WORD PROBLEMS
that WM plays in predicting problem solving. In fact, no signifi-
cant variance could be attributed to phonological processing and
inhibition when partialed for the influence of other variables.
DISCUSSION
The purpose of this study was to determine the mechanisms that
mediate the relationship between WM and problem-solving
accuracy in elementary school children. Two models were
tested. Before discussing the results related to these models, how-
ever, we briefly summarize age-related and individual differences
across the various problem-solving, achievement, and cognitive
measures.
In terms of age-related differences, the results yielded the ex-
pected finding that older children outperform younger children on
WM and problem-solving tasks. Although increases in WM and
problem solving are age related, age accounted for only a small
proportion of the variance in predicting problem-solving accuracy,
suggesting that developmental or school-based processes other
than increases in age mediated the relationship. That is, the cor-
relation between WM and problem solving was moderate at r!
.54 and was reduced to only r!.46 when partialed for chrono-
logical age. Furthermore, there was no significant difference in the
magnitude of the correlation between WM and problem solving
across the three grades. These results suggest that the influence of
WM across grades is stable. In addition, in the full regression
model, we found that chronological age did not contribute unique
variance to problem solving.
In terms of ability group findings, performance of children at
risk for SMD was below that of children not at risk on aggregate
measures related to problem solving, calculation, reading, phono-
logical processing, random generation/updating, STM, WM, and
semantic processing/vocabulary. Of particular interest to our study
was identifying those cognitive processes that, when partialed out
of the analysis, would mitigate the relationship between individual
differences in WM and problem-solving accuracy. The results
show a significant relationship between WM and problem solving
even when the influence of phonological processing, inhibition,
speed, and math calculation and reading skill was partialed from
the analysis. However, it is important to note in our study that none
of the individual WM tasks separated the two ability groups at the
.001 alpha level. This may be because WM and fluid intelligence
share important variance (Engle et al., 1999; Kyllonen & Christal,
1990), and therefore, partialing out fluid intelligence in the anal-
ysis may have removed important variance related to ability group
differences on individual tasks. Regardless, ability group differ-
ences in WM did emerge on aggregate WM scores, suggesting that
latent measures that cut across the WM task better discriminate
between ability groups than isolated tasks. This latter finding has
emerged in other studies (e.g., Wilson & Swanson, 2001).
The important findings, however, relate to the two models of
WM and its influence on mathematical word-problem solving. One
model tests whether phonological processes (e.g., STM, phono-
logical knowledge) play a major role in predicting performance in
problem solving and whether the phonological system mediates
the influence of executive processing (WM) on problem solving.
Phonological processes in this study were related to factor scores
of STM and composite scores of phonological knowledge (mea-
sures of elision, pseudoword reading, phonological fluency). The
model follows logically from the reading literature that links
phonological skills to new-word learning (see, e.g., Baddeley,
Gathercole, & Papagno, 1998), comprehension (Perfetti, 1985),
and mental calculation (see, e.g., Logie, Gilhooly, & Wynn, 1994).
The model assumes that low-order processing, such as phonolog-
ical coding, provides a more parsimonious explanation of ability
group differences in problem solving than measures of WM do.
Recall that in this model, WM regulates the flow of information
from a phonological store. Thus, the model suggests that poor
problem solvers have deficits in the processing of phonological
information, creating a bottleneck in the flow of information to
higher levels of processing.
The second model suggests that problem-solving performance
relates to executive processing, independent of the influence of the
phonological system. This assumption follows logically from the
problem-solving literature suggesting that abstract thinking, such
as comprehension and reasoning, requires the coordination of
several basic processes (see, e.g., Engle et al., 1999; Just, Carpen-
ter, & Keller, 1996; Kyllonen & Christal, 1990). Measures of
executive processing in this study were related to a factor score of
WM and measures assumed to reflect activities on the executive
system, for example, inhibition (random generation of numbers)
and activation of LTM (composite measures of reading, arithmetic
calculation, knowledge of algorithms). The findings on these two
models are as follows.
First, WM contributes unique variance to problem solving be-
yond what phonological processes (e.g., STM, phonological
knowledge), as well as reading skill, calculation, inhibition, pro-
cessing speed, and semantic processing, contribute. The results
show that WM contributed approximately 30% variance to
problem-solving accuracy when entered by itself in the regression
analysis. Furthermore, although age-related and ability group
related differences emerged on measures of phonological process-
ing, these measures did not partial out the influence of individual
differences in WM on problem solving. There is clear evidence
that the executive system of WM does contribute important vari-
ance to problem-solving performance beyond processes that relate
to the phonological system. Thus, the results do not support the
first model. There is weak support for the assumption that
Table 6
Hierarchical Analysis for Cognitive Variables and Math
Calculation Skill Predicting Word-Problem Solving
Variable B SE
#
tratio
Model 6
Fluid intelligence 0.25 .09 .11 2.62**
Algorithm knowledge 0.70 .08 .29 8.20***
Math calculation 0.09 .05 .11 1.79
Reading 0.12 .05 .14 2.03*
Semantic processing 0.05 .06 .04 0.91
Inhibition &0.13 .08 &.08 &1.47
Speed &0.11 .05 &.09 &2.05*
Phonological processing 0.05 .05 .05 1.03
Age &0.01 .12 &.005 &0.01
STM 0.29 .11 .09 2.61**
WM 0.54 .13 .17 4.32***
Note. Model 6: R
2
!.69, F(11, 337) !67.18, p$.0001. STM !
short-term memory; WM !working memory.
*p$.05. ** p$.01. ** p$.001.
486 SWANSON AND BEEBE-FRANKENBERGER
bottom-up processes (i.e., the phonological system) mediate indi-
vidual differences in WM performance and its influence on solu-
tion accuracy. A refinement of this model may consider phono-
logical processing as important to problem solving, but no more
important than other processes.
No doubt, a theoretical problem emerges, however, when one
considers how to reconcile specific phonological deficits (e.g.,
problems related to the phonological system) recently attributed to
some individuals with SMD (see, e.g., Hecht, Torgesen, Wagner,
& Rashotte, 2001) with the notion that an executive system may
override such processes. One means of reconciling this issue with
the current study would be to test (a) whether problems in specific
activities of the central executive system can exist in individuals
with SMD independent of their problems in phonological process-
ing (Swanson, 1993) and/or (b) whether a general manipulation of
processing and storage demands may indirectly account for low-
order processing deficits (especially on language-related tasks).
This issue can be put within the context of Baddeleys (1986,
1996; Baddeley & Logie, 1999) WM model. In Baddeleys (1986)
model, the central executive system is an undifferentiated generic
system that is used to support low-order systems. However, if the
executive system is overtaxed, it cannot contribute resources to
low-order processing. This is because the phonological loop is
controlled by the central executive (Baddeley, 1996), and there-
fore, any deficits in phonological functioning may partially reflect
deficiencies in the controlling functions of the central executive
itself (see Baddeley, 1996; Gathercole & Baddeley, 1993).
Second, WM captures unique variance in predicting solution
accuracy beyond measures of LTM and inhibition. Measures of
LTM in this study were related to reading and calculation ability,
as well as knowledge of algorithms. It has recently been argued
that the executive system functions to access information from
LTM (see, e.g., Baddeley & Logie, 1999). We found that the
contribution of knowledge of algorithms to the regression equation
contributed unique variance to solution accuracy. However, addi-
tional variables that uniquely predicted problem solving in the
complete model were speed, STM, reading, and WM. This finding
clearly supports the notion that components of executive process-
ing (other than LTM, processing speed, and inhibition) contribute
unique variance to solution accuracy. Thus, it seems there are
activities besides the aforementioned variables that contribute
unique variance to problem-solving ability. We posit that moni-
toring of processing and storage demands may be one aspect of
this unique variance.
The study also addressed two hypotheses discussed in the liter-
ature as playing a major role in accounting for individual differ-
ences in WM. One hypothesis relates to the speed of processing. A
simple version of this hypothesis states that individuals with SMD
are slower at processing language information than are average-
achieving children and that such reduced processing on the SMD
participantspart underlies their poor WM performance. Several
models of WM assume that operations related to language are time
consuming (see, e.g., Salthouse, 1996). Therefore, speed of pro-
cessing may underlie the general pattern of WM difficulties noted
in the present study. Furthermore, Kail (1993) has argued that a
common pool of cognitive resources related to processing speed is
used to perform a variety of tasks, with the pool increasing across
ages in children. Clearly, our findings show a significant relation-
ship between processing speed and measures of problem solving,
semantic processing, inhibition, STM, and WM, as well as reading
and calculation (rs range from &.33 to &.72; see Table 4). How-
ever, when measures of fluid intelligence and reading were entered
into the regression analysis in predicting problem solving and math
calculation, speed did not contribute significant variance (see
Model 5, Table 5). Although speed does play an important role in
the complete model, it was only one of several variables that
contributed unique variance. More importantly, the correlation
coefficient between WM and problem solving was significant
when partialing out the influence of speed and age.
A second hypothesis considers whether individuals with SMD
are less resistant to interference (see Baddeley, 1996; Brainerd &
Reyna, 1993; Rosen & Engle, 1997; Towse, 1998, for further
discussion of this model). Such a hypothesis assumes that an
inhibition deficit limits SMD participantsability to prevent irrel-
evant information from entering WM during the processing of
targeted information (see Passolunghi & Siegel, 2001, for discus-
sion of this model). An activity related to the central executive that
has been implicated as a deficit in children with reading disabilities
is their ability to suppress irrelevant information under high pro-
cessing demand conditions (see, e.g., Chiappe et al., 2000; De Beni
et al., 1998; Swanson & Cochran, 1991). Earlier studies showed
that children with reading disabilities vary from controls in their
ability to recall targeted (relevant) and nontargeted (incidental)
information (see, e.g., Swanson & Cochran, 1991). Likewise,
individuals with SMD may have difficulty preventing unnecessary
information from entering WM and, therefore, would be more
likely to consider alternative interpretations of material (such as
asked for in the processing questions) that are not central to the
task when compared with average achievers. This interpretation
fits within several recent models that explain individual differ-
ences in memory performance as related to inhibitory mechanisms
(see, e.g., Cantor & Engle, 1993; Conway & Engle, 1994) without
positing some form of a capacity deficit.
Our results show that children with SMD experienced difficul-
ties on the number random-generation tasks when compared with
children not at risk for SMD. However, partialing out performance
on the inhibition measure did not eliminate the significant rela-
tionship between WM and problem solving. One could argue,
however, that the process question in our WM tasks put demands
on the participants ability to suppress competing information and
therefore that individual differences in inhibition mediated WM
performance. Clearly, the process questions for the current WM
tasks constituted a temporary competing condition with storage.
As a consequence, children at risk for SMD may have had diffi-
culty preventing unnecessary information from entering WM and,
therefore, considered alternative interpretations of material (such
as those asked for in the process questions) that were not central to
the task. Although we see the above model as a viable alternative
to the results, we have three reservations. First, only the span levels
of participants who answered the process question correctly were
analyzed. If a process question was missed, the participantsrecall
of previously stored information was not scored. This procedure is
different from previous studies (e.g., Daneman & Carpenter, 1980)
that have allowed dissociation between the process question (i.e.,
it is not necessary for participants to answer the process question
correctly) and the retrieval question in the analyses. Our procedure
removed from the analysis irrelevant responses that emerged be-
tween the processing of the distractor question and the retrieval
487
WORD PROBLEMS
question. Second, if children at risk for SMD suffer more inter-
ference (i.e., diminished inhibition in that a large number of traces
are simultaneously active) than children not at risk for SMD, then
one would expect the effects of WM to be nonsignificant in
predicting both word-problem solving and calculation when mea-
sures of inhibition are partialed from the analysis. Such was not the
case in this study. That said, we do assume that inhibition effi-
ciency may be a consequence of capacity constraints (see Cantor &
Engle, 1993, for discussion). Children at risk for SMD may use
more WM capacity than children not at risk for SMD to inhibit or
resist potential interference from irrelevant items (see Chiappe et
al., 2000, for discussion).
Overall, these findings give partial support to Model 2. We say
partial because the role of the executive system has not been
clearly delineated. We argue that because partialing out speed,
inhibition, and general knowledge (e.g., arithmetic calculation,
knowledge of algorithms) did not eliminate the significant effects
of WM on problem solving, individual differences in WM are
related to constraints in regulating and/or manipulating storage
capacity. Of course, this study provides no direct measure of
mentally coordinating resources across the WM tasks, and there-
fore, one could argue that the results merely reflect the fact that
WM tasks draw on finite resources from a phonological storage
system. We argue, however, that the executive system coordinates
the distribution of finite resources of the verbal and visualspatial
system. Support for this common system is found in the factor
analysis in that both verbal and visualspatial WM loaded on the
same factor (see Table 3), whereas STM tasks loaded on a separate
factor.
Implications
There are three implications of our findings for current litera-
ture. First, bottom-up processes (e.g., the phonological system) are
not the primary mediators between age-related and individual
differences in WM and problem solving. Of course, these results
apply only to the age and ability groups represented in this sample.
However, similar results have occurred with older children (Swan-
son, in press; Swanson & Sachse-Lee, 2001). Our findings further
suggest that although skills associated with phonological processes
(i.e., naming speed and STM) are important to age-related changes
in children in calculation and problem solving, they are no more
important than WM. Such a finding qualifies bottom-up models of
problem solving of children by suggesting that if low-order pro-
cesses, such as phonological processes (e.g., STM), moderate the
influence of executive processing (WM) on problem-solving per-
formance, their effects may be indirect or minimal for children
who have perhaps met a minimum threshold in mathematics and
reading skills.
The second implication relates to the independence of WM and
STM. We argue that STM and WM may make independent con-
tributions to problem solving because STM measures draw on
phonological codes (Salame & Baddeley, 1982), whereas WM
measures draw on resources from the executive system (see, e.g.,
Engle et al., 1999). Our results are consistent with those of others
who have argued that STM tasks and WM tasks are inherently
different (Engle et al., 1999). We found separate loadings for STM
and WM factors. That is, although phonological coding might be
important to recall in STM, it may not be a critical factor in WM
tasks. This finding is important because a common opinion is that
STM tasks are a proper subset of processes of which WM is
capable. As a qualification to this view, however, the present
analysis shows that components of WM operate independent of
STM. This finding is consistent with other experimental work with
adults (see, e.g., Engle et al., 1999) and poor readers (Swanson et
al., 1996). The implication of this finding is that problems in WM
may co-occur with STM but also maintain some independence
from the development of STM.
Some comment is necessary as to why children at risk for SMD
suffer deficits on aggregate scores related to both STM and WM
tasks. The above research suggests that children with SMD per-
form poorly on tasks that require accurate recognition/recall of
letter and number strings or real words and pseudowords. Tasks
such as these, which have a read in and read outquality to them
(i.e., place few demands on LTM to infer or transform the infor-
mation), reflect STM. One common link among these tasks is the
ability to store and/or access the sound structure of language
(phonological processing). There is evidence that participants with
SMD suffer deficits in STM, a substrate of the phonological
system. However, some children with SMD also do poorly on
tasks that place demands on attentional capacity, a characteristic of
WM tasks. The findings on STM tasks make sense to us because
there are many mnemonic situations in which a stimulus in mem-
ory is attended to and the other stimuli exist as a backgroundthat
is, they are not the center of current awareness. These situations, in
our opinion, do not challenge monitoring. We argue that WM tasks
require the active monitoring of events and that these events are
distinguishable from simple attention to stimuli held in STM.
Monitoring within WM implies attention to the stimulus that is
currently under consideration together with active consideration
(i.e., attention) of several other stimuli whose current status is
essential for the decision to be made. Results from our lab have
suggested that the tasks differ in subtle ways. Although a substrate
of STM may contribute to problems in verbal WM, children with
SMD can suffer problems in a substrate of WM that are indepen-
dent of problems in verbal STM. Problems in verbal WM have
been found to persist in children with average IQ and learning
problems even after partialing out the influence of verbal STM
(Swanson et al., 1996), verbal articulation speed (Swanson &
Ashbaker, 2000), reading comprehension (Swanson, 1999), or
fluid intelligence (Swanson & Sachse-Lee, 2001).
Finally, although the influence of individual differences in WM
on problem-solving performance is robust, this does not mean that
its influence cannot be compensated for. As previously stated,
increased performance on measures related to speed and knowl-
edge of algorithms can reduce the influence of individual differ-
ences in WM on problem solving. What remains to be studied is
the influence of instruction and age-related development on these
processes.
Conclusion
In summary, our findings converge with studies on individual
differences that suggest WM plays a critical role in integrating
information during problem solving. We argue that WM plays a
major role because (a) it holds recently processed information to
make connections to the latest input and (b) it maintains the gist of
information for the construction of an overall representation of the
488 SWANSON AND BEEBE-FRANKENBERGER
problem. Yet WM is not the exclusive contributor to variance in
problem-solving ability. This study also supports previous research
about the importance of reading skill, processing speed, and ac-
cessing information from LTM (knowledge of algorithms) in so-
lution accuracy. Moreover, our findings are consistent with models
of high-order processing suggesting that WM resources activate
relevant knowledge from LTM (Baddeley & Logie, 1999; Ericsson
& Kintsch, 1995) but also suggesting that a subsystem that controls
and regulates the cognitive system plays a major role (Baddeley,
1986). Thus, we think one of the core problems children face in
solving mathematical word problems relates to operations ascribed
to a central executive.
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... Studies that mixed elementary school children with preschool or middle school students were excluded (e.g., Holochwost et al., 2017); (3) in cases where the same data were reported in multiple publications, only one source was selected. Specifically, journal articles were preferred over dissertations (e.g., Niu et al., 2018); (4) they included studies featuring both typically developed children and children with developmental disability, but presented the results separately for each group (e.g., Swanson & Beebe-Frankenberger, 2004); (5) they included data from participants who did not receive any interventions (i.e., only baseline and control group data from intervention studies were used); (6) they reported at least one cross-sectional correlation coefficient between inhibitory control and mathematical ability. While longitudinal studies provide valuable insights into causal relationships, the limited number of longitudinal studies in this field restricts our ability to investigate longitudinal relationships. ...
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... Working memory impacts all learning processes (Alloway, 2006) by facilitating the integration of incoming information with existing knowledge, as well as the transformation of this information into new knowledge (Swanson & Saez, 2003;Swanson & Beebe-Frankenberger, 2004). In this sense, working memory is seen as a highly influential structure on learning as a whole (Savage, Lavers, & Pillay, 2007) and plays an important role in academic skills such as reading and mathematics (Gathercole & Alloway, 2004). ...
... Working memory is often described as a unique cognitive resource serving both short-term maintenance and processing [1]. It is thus central in problem solving [23]. Working memory is known to be mediated by the prefrontal cortex [7,13,20], and it is clearly linked to intelligence [9,11], particularly when complex span tasks are used [26]. ...
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The Raven Progressive Matrices (RPM) tests measure “general cognitive ability” or, better, eductive, or “meaning making,” ability (Raven, Raven, & Court, 1998a,2000). The term “eductive” comes from the Latin root educere, which means, “to draw out.” The basic version of the test, known as the Standard Progressive Matrices (or SPM), consists of five sets of items of the kind shown in Figures 11.1 and 11.2. Within each set, the items become progressively more difficult. At the beginning of each set, the items, although easy again, follow a different logic. The sets in turn become progressively more difficult. The five sets offer those taking the test five opportunities to become familiar with the method of thought required to solve the problems. In addition to the Standard series, there is the Coloured Progressive Matrices (CPM), which is designed to spread the scores of children and less able adults and the Advanced Progressive Matrices (APM), developed to spread the scores of the top 20% of the population.