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Journal of Experimental Child Psychology 99 (2008) 288–308

www.elsevier.com/locate/jecp

Working memory and arithmetic calculation

in children: The contributory roles of
processing speed, short-term memory, and reading
Derek H. Berg *
Faculty of Education, Mount Saint Vincent University, Halifax, Nova Scotia, Canada B3M 2J6

Received 4 July 2007; revised 10 December 2007

Available online 1 February 2008

Abstract

The cognitive underpinnings of arithmetic calculation in children are noted to involve working
memory; however, cognitive processes related to arithmetic calculation and working memory suggest
that this relationship is more complex than stated previously. The purpose of this investigation was
to examine the relative contributions of processing speed, short-term memory, working memory, and
reading to arithmetic calculation in children. Results suggested four important findings. First, pro-
cessing speed emerged as a significant contributor of arithmetic calculation only in relation to age-
related differences in the general sample. Second, processing speed and short-term memory did not
eliminate the contribution of working memory to arithmetic calculation. Third, individual working
memory components—verbal working memory and visual–spatial working memory—each contrib-
uted unique variance to arithmetic calculation in the presence of all other variables. Fourth, a full
model indicated that chronological age remained a significant contributor to arithmetic calculation
in the presence of significant contributions from all other variables. Results are discussed in terms of
directions for future research on working memory in arithmetic calculation.
Ó 2007 Elsevier Inc. All rights reserved.

Keywords: Mathematics/Numbers; Memory; Spatial ability/cognition; Education; Cognition

*
Fax: +1 902 457 4911.
E-mail address: derek.berg@msvu.ca

0022-0965/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jecp.2007.12.002
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 289

Introduction

Arithmetic calculation is one of the primary academic skills children learn in school.
Indeed, next to learning to read and write, the development of arithmetic calculation skills
takes a place near the top of the curricular pyramid. Given the value of these skills, diffi-
culties in these areas would pose substantial problems for children’s educational develop-
ment and for their daily experiences throughout life. For instance, the importance of
addition as a primary skill in schools rests in its place as a building block for the develop-
ment of increasingly advanced mathematical skills (Ashcraft, 1992; Jensen & Whang,
1994; National Council of Teachers of Mathematics, 2000). Indeed, the National Council
of Teachers of Mathematics (2006) recently underscored the importance of calculation
abilities early in school-based curriculum and instruction. Proficiency in simple addition
leads to the development of more complex addition skills (Geary & Brown, 1991), and
the development of multiplication skills stems from proficiency in addition (Cooney,
Swanson, & Ladd, 1988).
Across the field of mathematics, research has focused on describing factors that influ-
ence the development of arithmetic performance (e.g., problem-solving strategies [Fuchs &
Fuchs, 2002]), on isolating specific areas of arithmetic difficulty (e.g., level of problem dif-
ficulty [Ostad, 1998]), and on developing curricular and instructional interventions to
respond to poor arithmetic performance (Fleischner, Nuzum, & Marzola, 1987). However,
compared with the substantial body of research focused on identifying the cognitive pro-
cesses that undergird reading (e.g., phonological processing [Snow, Burns, & Griffin, 1998;
Stanovich, 1982]), little research has been directed at identifying the cognitive processes
that are involved in arithmetic calculation. Extant theoretical and empirical research sug-
gests that working memory, a cognitive system that is in part responsible for the construc-
tion of information and the transfer of this information into long-term memory, is likely to
play an integral role in arithmetic calculation (Baddeley, 1996; Baddeley, 2001).

Working memory

A generalized definition of working memory defines it as a limited capacity information

processing resource. It has two principal processes: the preservation of information and
the concurrent processing of the same or other information (e.g., Engle, Cantor, & Caru-
llo, 1992; Just & Carpenter, 1992). The latter element is important because it serves to dis-
tinguish working memory from other forms of memory such as short- and long-term
memory.
The most widely accepted and researched model of working memory was formulated by
Baddeley and Hitch (1974). Consistent with 30 years of research, this model supports a
three-component structure consisting of the central executive interacting with two subsys-
tems: the phonological loop and the visuospatial sketchpad. The central executive is
viewed as a limited capacity component responsible for directing information toward
the two subsystems, coordinating activity within the memory system, and retrieving infor-
mation from long-term memory (e.g., Baddeley, 1986; Baddeley, 1996). The phonological
loop (analogous to verbal working memory) is responsible for the temporary storage of
speech-based information and is vulnerable to rapid decay. Decay can be interrupted by
vocal and subvocal rehearsal through articulation (Vallar & Baddeley, 1984). The visuo-
290 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

spatial sketchpad (analogous to visual–spatial working memory) is responsible for the

temporary storage of visual–spatial information and is involved in the generation and
manipulation of mental images.

Arithmetic calculation and working memory

In the field of mathematics, the cognitive processes involved in performing arithmetic

calculations are embedded within the working memory system in that they require a com-
bination of temporary information storage while performing other mental operations. For
instance, to solve the problem 18 + 7, one must concurrently retain two or more pieces of
information (phonological codes representing the numbers 18 and 7) and then employ one
or more procedures (e.g., counting) to combine the numbers to produce an answer. Alter-
natively, employing carrying or regrouping involves maintaining recently processed infor-
mation while conducting a related operation. To solve 18 + 7, one must retain the 5 from
adding 8 + 7 while adding the 1 from the tens column of the 18 to the 1 from the tens col-
umn of the 15 produced from adding the 8 + 7.
Results from the limited number of studies conducted to date, largely with adults, suggest
that working memory resources are required to solve multistep problems such as 125 + 97
(Fürst & Hitch, 2000; Geary & Widaman, 1992; Logie, Gilhooly, & Wynn, 1994). In these
problems, working memory appears to be related to retaining intermediate values during cal-
culation and to carrying partial results across columns. Adams and Hitch (1997) explored
this relationship in 8- to 11-year-olds using simple mental addition (e.g., 8 + 1) and complex
mental addition (e.g., 231 + 16) problems. Results suggested that proficiency in solving prob-
lems presented verbally was associated with high working memory functioning, whereas pro-
ficiency in solving problems presented visually was related to low working memory
functioning. Findings were interpreted as indicating that problems presented visually require
more working memory resources than do problems presented verbally. In addition, level of
problem difficulty was related to working memory contributions, with higher working mem-
ory spans corresponding with more complex problems.
Studies have also indicated that individual working memory components have special-
ized roles in arithmetic calculation (e.g., Ashcraft, 1995; Geary, Hoard, Byrd-Craven,
Nugent, & Numtee, 2007). Verbal working memory has been associated with exact calcu-
lation in addition (Lemaire, Abdi, & Fayol, 1996), subtraction (Seyler, Kirk, & Ashcraft,
2003), and multiplication (Seitz & Schumann-Hengsteler, 2000). In addition, relationships
have been reported between verbal working memory and calculation procedures such as
counting (Logie & Baddeley, 1987), retaining problem information (Fürst & Hitch,
2000), and holding interim results during counting (Logie & Baddeley, 1987). Hecht
(2002) attempted to clarify the role of working memory by contrasting verbal working
memory with general working memory. In examining simple arithmetic calculation in
undergraduate students, Hecht found that verbal working memory was used to store inter-
mediate values, whereas general working memory resources were involved in coordinating
procedures to be used in calculations. Application of these results to describe the role of
working memory in arithmetic calculation should be made with caution given that Hecht’s
study focused on adults and did not include a measure for visual–spatial working memory.
The contribution of working memory to arithmetic calculation in children has not
received much attention. Since the early work of Hitch and colleagues (Adams & Hitch,
1997; Hitch, 1978), only a few studies have examined this relationship. Geary, Hoard,
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 291

Byrd-Craven, and DeSoto (2004) examined the involvement of working memory in the
strategy choice of elementary school children (Grades 1, 3, and 5) when solving simple
and complex addition problems. Results indicated that, similar to earlier studies, higher
working memory capacity was associated with higher accuracy in solving complex arith-
metic problems (Adams & Hitch, 1997). Geary and colleagues (2004), however, found that
this relationship was related to specific strategies. For instance, higher working memory
facilitated the use of more sophisticated strategies such as decomposition, whereas lower
working memory was related to less sophisticated strategies such as finger counting. In a
related study, Fuchs and colleagues (2006) investigated the cognitive predictors of chil-
dren’s arithmetic calculation. Their results suggested that working memory was not a sig-
nificant predictor in third graders’ performance on simple mental addition (e.g., 3 + 2),
complex mental addition (e.g., 35 + 29), or arithmetic word problems; rather, phonolog-
ical decoding, attention, and processing speed emerged as unique predictors. However, a
narrow selection of working memory measures might have overemphasized the contribu-
tion of verbal working memory. Similar to most studies on arithmetic calculation in chil-
dren, both Geary and colleagues (2004) and Fuchs and colleagues (2006) did not include
measures of visual–spatial working memory; rather, these studies used verbal working
memory tasks as the sole measure of working memory.
Largely absent from research on arithmetic calculation is consideration for the role of
visual–spatial working memory. Limited studies with adults suggest that the involvement
of visual–spatial working memory in arithmetic calculation is important during the initial
stages of arithmetic calculation for encoding problems presented visually (Heathcote,
1994; Logie et al., 1994). Heathcote (1994) examined the specific roles of verbal working
memory and visual–spatial working memory in pairs of three-digit addition problems pre-
sented visually and verbally. Whereas visual–spatial working memory was involved in
retaining carries, verbal working memory was involved in retaining interim results through
articulatory rehearsal. Furthermore, verbal working memory and visual–spatial working
memory both were involved during the initial stages of calculation through encoding of
the problem in working memory, with visual–spatial working memory involved in prob-
lems presented visually and verbal working memory involved in problems presented ver-
bally. Geary and colleagues’ (2007) study offered insight into the role of visual–spatial
working memory in arithmetic calculation in children. Specifically, Geary and colleagues
found that visual–spatial working memory was related to the use of counting procedures
(i.e., min strategy) when performing complex addition (e.g., 16 + 7) but not when perform-
ing simple addition (e.g., 4 + 3).
Although research suggests a relationship between working memory and arithmetic cal-
culation, with potential contributory influences of individual components related to verbal
and visual–spatial processing, the influence of other cognitive processes that inform work-
ing memory has received little attention. Specifically, arithmetic calculation proficiency is
supported by two cognitive processes that interact with working memory: processing speed
(Case, 1985) and short-term memory (Baddeley & Hitch, 1974).

Contributory processes: Processing speed and short-term memory

The speed at which information is introduced into working memory and used within
working memory is critical to the efficiency of the working memory system. Case (1985)
highlighted the importance of processing speed by implicating its role in increasing the
292 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

available short-term storage space. Based on research with children, Case and colleagues
found a linear relationship between processing speed and storage capacity of the working
memory (Case, Kurland, & Goldberg, 1982). In a study of 6- to 11-year-olds, they
reported that faster counting speed predicted higher counting spans. Case interpreted this
and similar findings as indicative of a speed–capacity relationship whereby the faster a
child processes relevant information, the more information the child can retain over a
short period of time. In essence, faster operations require less workspace, leaving more
available for information storage.
Although processing speed and short-term memory have been examined in relation to
various arithmetic activities such as problem solving (e.g., Swanson & Beebe-Frankenber-
ger, 2004) and to strategy use in arithmetic calculation (Geary & Brown, 1991; Jordan &
Montani, 1997), their relative importance in the complex of memory processes that have
been associated with arithmetic calculation is unclear. Bull and Johnston (1997) examined
the role of processing speed and short-term memory in children’s arithmetic calculation.
They reported that when a group of 7-year-olds were assessed on measures of short-term
memory, long-term memory, processing speed, and sequencing ability, the strongest pre-
dictor of arithmetic calculation was processing speed. Their study, however, used a set of
cognitive tasks that did not include measures of working memory.
The most telling evidence for the potential contribution of processing speed and short-
term memory to arithmetic calculation is available from recent studies by Swanson and col-
leagues (Swanson, 2006; Swanson & Beebe-Frankenberger, 2004). Swanson and Beebe-
Frankenberger (2004) examined the contributions of working memory to arithmetic prob-
lem solving and to arithmetic calculation in children in Grades 1 to 3. Results revealed that
processing speed contributed unique variance to each of these arithmetic areas in the pres-
ence of unique contributions of working memory, short-term memory, phonological pro-
cessing, and age. Although this study is notable in its inclusion of several cognitive
processing domains, it treated working memory as a general domain (creating a composite
score of all working memory tasks) and did not differentiate between verbal working memory
and visual–spatial working memory in predictive models. Swanson’s (2006) study investi-
gated the cognitive processes that contribute to arithmetic calculation in children with
advanced mathematical skills in Grades 1 to 3. This study differentiated between individual
components of working memory (verbal and visual–spatial) and included measures of the
executive system, processing speed, and inhibition (generation and fluency). In the presence
of significant contributions of age and reading, processing speed did not emerge as a signif-
icant contributor, yet the executive system and generation (numbers and letters) both proved
to be significant contributors. It should be noted, however, that the measures used to repre-
sent executive functioning in Swanson’s study (e.g., auditory digit sequence) have been
viewed as representative of verbal working memory in other studies (e.g., Swanson, 2004;
Swanson & Sachse-Lee, 2001). Thus, an alternative interpretation of the contribution of
executive functioning might be that verbal working memory was a significant contributor.

The current study

In review, since early work by Hitch (1978), only a few studies have directly investigated
the role of working memory in children’s arithmetic calculation performance. Subsequent
research has been equivocal on the role of working memory. Three challenges in particular
underscore our current difficulty in reaching consensus on the role of working memory in
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 293

arithmetic calculation. First, many studies treat working memory as a unidimensional mem-
ory system (e.g., Hecht, 2002). Second, there has been a lack of consideration for the role that
processing speed plays in the complex of cognitive processes that are involved in children’s
arithmetic calculation. Third, the role of short-term memory in the complex of cognitive pro-
cesses that relate both to working memory and to arithmetic calculation has received little
attention. The current study was designed to address these issues with a central purpose of
examining the relative contributions of processing speed, short-term memory, verbal work-
ing memory, and visual–spatial working memory to arithmetic calculation in children.
Research suggests that, when examining the cognitive underpinnings of performance in
academic areas, it is important to consider similar contributory processes that relate to
each academic area. With respect to mathematical performance, reading has been associ-
ated with word problem solving (Swanson & Beebe-Frankenberger, 2004) and with arith-
metic calculation (Hecht, Torgesen, Wagner, & Rashotte, 2001). Ostensibly, although
reading and arithmetic use different knowledge structures (e.g., letter sound knowledge
vs. counting knowledge), performance in these areas draws on similar cognitive processes.
For instance, one will use his or her phonological processing ability when identifying
words through a sounding-out procedure (Bradley & Bryant, 1985). To solve arithmetic
calculations, implementing a counting-based procedure also engages the phonological sys-
tem (Buchner, Steffens, Irmen, & Wender, 1998; Geary, 1993). Consistent with these rela-
tionships, the current study sought to understand the influence of the target variables after
accounting for the interaction between reading and arithmetic calculation.

Method

Participants

A total of 90 children (44 boys and 46 girls) in Grades 3 to 6 from three schools in cen-
tral Canada participated in this study. Children’s ages ranged from 98 to 145 months. The
socioeconomic status of individual students was not assessed; however, each of the schools
that participated in the study was located in a predominantly middle-class neighborhood.
All children spoke English as their first language. No child had been identified as having a
neurological disorder (e.g., learning disability) or as having English language difficulties
that would have made it difficult for the child to complete any of the study activities.

Instruments

Children completed two test batteries. The first battery was administered to measure
children’s academic achievement in arithmetic calculation and reading. A second battery
measured four cognitive abilities: processing speed, short-term memory, verbal working
memory, and visual–spatial working memory.

Academic achievement

The Wide Range Achievement Test–Third Revision (WRAT3) (Jastak & Jastak, 1993)
was administered to measure children’s arithmetic and reading achievement. Raw scores
and standard scores (M = 100, SD = 15) based on age-appropriate norms were calculated
294 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

for each child. Cronbach’s alpha for the WRAT3-A and WRAT3-R measured .85 and .86,
respectively.

Cognitive processing

Processing speed
Digit naming A digit naming speed task was administered to assess each child’s speed at
retrieving phonological numerical representations from long-term memory. This task was a
modified version of a similar task used by Compton (2003). In the current study, the child was
required to read aloud sets of 9 randomly ordered Arabic digits as accurately and quickly as
possible. Digits 1 through 9 were arranged across three rows within three columns. Two trials
were administered separated by a 1-min rest. Each trial contained a different arrangement of
digits across rows and within columns. A stopwatch was used to measure the child’s naming
times. A child’s score for this task was his or her digit naming rate. Naming rate was calcu-
lated by dividing the number of digits read per trial (9, i.e., 3 digits in each of three rows) by
the mean time for the two trials. In the current study, Cronbach’s alpha for the digit naming
task measured .81.
Digit articulation A digit articulation task was administered to assess each child’s speed of
speech. This task was adapted from a similar task used by Kail (1997). The child was asked
to repeat a pair of single-syllable digits as quickly as possible five times. Four trials were
administered using the following digit pairs: 1–4, 5–8, 3–6, and 2–9. Each digit pair was pre-
sented orally by the researcher to the child. A practice trial was given using 9–8. A stop-
watch was used to measure the time to articulate each digit pair five times. A child’s
score for this task was his or her articulation rate. Articulation rate was calculated by divid-
ing the total number of digits articulated per trial (10, i.e., 2 numbers repeated five times) by
the mean time for the four trials. In the current study, Cronbach’s alpha for the digit artic-
ulation task measured .84.

Short-term memory
Digit span forward Digit span forward has been used widely as a test of short-term work-
ing memory (Passolunghi & Siegel, 2001). In the first part of the test, each child was asked to
listen to a series of single-digit numbers articulated by the researcher. In the second part, the
child was asked to repeat the number sequence in the order presented by the researcher. If the
child stated the number sequence correctly, another trial was administered. Successive trials
increased by one digit until the child failed two attempts within the same trial. The maximum
possible span was nine digits. A child’s score for this task was the highest number of digits
recalled in sequence correctly. In the current study, Cronbach’s alpha for the digit span for-
ward task measured .64.
Word span forward The word span forward task was similar to the forward digit span
task. Words were one-syllable frequently used words (Carroll & White, 1973). The maxi-
mum possible span was nine words. A child’s score for this task was the highest number of
words recalled in sequence correctly. In the current study, Cronbach’s alpha for the word
span forward task measured .58.

Verbal working memory

Semantic categorization The semantic categorization task (Swanson, 1995) assessed
each child’s ability to recall related words within prearranged groups. The researcher
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 295

read aloud a set of words with a 2-s interval between words. Then the researcher asked
the child a process question, requested the child to choose a strategy depicted on a dis-
play card that would help him or her to remember the groups and words, and then
asked the child to recall each group name and each word within its respective group.
The strategies displayed were top-down subordinate organization, interitem discrimina-
tion, interitem associations, and subjective organization. To ensure standard adminis-
tration procedures, the Swanson–Cognitive Processing Test outlines specific
administration procedures for assessing participants’ strategy selection. The dialogue
was as follows:
Now, one way to help you remember these words is to think of words in their cat-
egory [point to Frame A]. For example, animals include dog and cat. Another way to
remember is to think of things that tie the words together [Frame B]. For example, I
know that dogs and cats have four legs and fur. Another way is to think of some-
thing to associate with each word [Frame C]. For example, for dog you can think
of a dog you know, and for cat you could think of a cat you petted once. Another
way to help you remember is for you to look at the difference between the words
[Frame D]. For example, dog and cat don’t sound the same, and they start with dif-
ferent letters. (Swanson, 1995, p. 69)

As an example of a single task administration, Set 1 was ‘‘flower—rose–daisy.” The

researcher asked the process question, ‘‘Which word, ‘rose’ or ‘violet,’ was presented?”
If the child answered incorrectly, the task was stopped. If the child answered correctly,
the child was asked to state the strategy he or she would use to remember each group
and the words within a group. Then the child was asked to recall the group and the
words within that group. If the child responded correctly, the next word set was
administered. Item set difficulty ranged from one group with two words to eight groups
with three words in each group. A child’s score was the number of sets recalled cor-
rectly. In the current study, Cronbach’s alpha for the semantic categorization task mea-
sured .61.
Auditory digit sequencing The auditory digit sequence task assessed each child’s ability
to recall numerical information contained within a short sentence. The researcher read
aloud a sentence containing a street address, asked the child a process question, asked
the child to select a strategy depicted on a display card that would help him or her
to remember the address, and then asked the child to recall part of the address. The
strategies were rehearsal, chunking, associating, and elaborating of information. The
standard Swanson–Cognitive Processing Test administration procedure for this task
was as follows:

Some ways that help you remember are: (1) saying the numbers over to yourself. For
example, if I say ‘‘2-4-6-3 Bader Street,” you would say to yourself ‘‘2-4-6-3” over
and over again [point to Frame A]. Or (2) you might say some numbers together
in pairs. For example, if I say ‘‘2-4-6-3,” you would say 24-63 [Frame B]. Or (3)
you may just want to remember that the numbers go with a particular street and
location. For example, if I say ‘‘2-4-6-3 Bader Street,” you would remember that
2-4-6-3 Bader Street go together [Frame C]. Or (4) you might think of other things
296 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

that go with the numbers. For example, if I say ‘‘2-4-6-3,” you might think ‘‘2-4-6-3 I
have to go climb a tree” [Frame D]. (Swanson, 1995, p. 38)
As an example of a single task administration, Set 1 was ‘‘Suppose somebody wanted to
have you drive them to the library at 2-9 Maple Street.” The researcher then asked the pro-
cess question, ‘‘What is the name of the street?” If the child answered incorrectly, the task
was stopped. If the child answered correctly, the child was asked to state the strategy he or
she used to remember the information. Then the child was asked to state the number
embedded within the address. If the child recalled the address number correctly, the next
sentence was administered. Sets ranged from two to nine sentences. A child’s score was the
number of sets recalled correctly. In the current study, Cronbach’s alpha for the auditory
digit sequence task measured .66.

Visual–spatial working memory

Visual matrix The visual matrix task (Swanson, 1995) assessed each child’s ability to
recall dots arranged within a matrix. The researcher presented the child with a matrix con-
taining a series of dots, gave the child 5 s to study the series of dots within the matrix, with-
drew the matrix from the child, and then asked the child a process question. The process
question was ‘‘Are there any dots in the first column?” If the child answered incorrectly,
the task was stopped. If the child answered correctly, he or she was then asked to repro-
duce the dot arrangement on a blank matrix of the same size. If the child reproduced the
original matrix correctly, the next matrix was administered. The items ranged in difficulty
from a matrix of 4 squares and 2 dots to a matrix of 45 squares and 12 dots. A child’s score
was the number of matrices recalled correctly. In the current study, Cronbach’s alpha for
the visual matrix task measured .67.
Corsi block task The Corsi block task (Corsi, 1972), as described by Milner (1971),
is one of the most widely used measures of visual–spatial working memory (Cornoldi &
Vecchi, 2003) and has been used in studies examining the arithmetic performance and
visual–spatial abilities of children with and without learning difficulties in arithmetic
(D’Amico and Guarnera, 2005; McLean & Hitch, 1999). The Corsi block task consists
of nine blocks arranged randomly on a wooden board (Milner, 1971). The researcher
pointed to a sequence of blocks at a rate of one per second. After the researcher com-
pleted tapping the sequence, the child was asked to replicate the sequence. If the child
recalled the sequence of blocks correctly, another trial was administered. Successive tri-
als increased by one block until the child failed two attempts within the same trial. The
maximum possible span was nine blocks. The score of this task was the highest num-
ber of blocks recalled in sequence correctly. In the current study, Cronbach’s alpha for
the Corsi block task measured .64.

Procedure

Children were assessed individually in two sessions within 1 week, with each session
lasting approximately 30 min. Each administration session corresponded with one of
the test batteries: academic achievement or cognitive processing. The order of admin-
istration of these two batteries was counterbalanced, with half of the participants
receiving the academic achievement battery first and half receiving the cognitive pro-
cessing battery first.
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 297

Results

Descriptive statistics

Means, standard deviations, and score ranges for all measures are reported in Table 1.
Due to participant selection criteria, large ranges in chronological age were expected. As a
result, large ranges were also expected between the academic achievement variables. Read-
ing raw scores ranged from 28 to 46, and arithmetic calculation raw scores ranged from 20
to 41. Reading standard scores ranged from 79 to 132 and arithmetic standard scores ran-
ged from 69 to 128.
Correlations among the measures are reported in Table 2. Inspection of the correlations
among cognitive tasks indicated that several constructs shared significant variance. That
processing speed measures were significantly related to both short-term memory and
working memory (with a stronger correlation with short-term memory) was interesting
but not surprising. Extant research suggests that processing speed underscores both
short-term memory and working memory functioning (Baddeley, 1986; Case et al.,
1982). The stronger relationships between processing speed and short-term memory likely
reside in the importance of speed in offsetting the influence information decay (Case, 1985;
Case et al., 1982).
Of particular interest, however, were the relationships between the independent vari-
ables (cognitive processing tasks) and participant descriptive variables (chronological
age and reading) and the dependent variable (arithmetic calculation). Correlations
between cognitive processing tasks and arithmetic calculation ranged from r(90) = .32,
p < .01, for auditory digit sequencing to r(90) = .57, p < .01, for digit span forward. Signif-
icant correlations between the cognitive processing tasks and chronological age ranged
from r(90) = .32, p < .01, for word span forward and semantic categorization to

Table 1
Descriptive statistics for chronological age, academic achievement, and cognitive processing
Variable M SD Range
Chronological age (months) 121.90 12.44 98–145
Academic achievement
Arithmetic calculation (raw) 28.78 4.39 20–41
Arithmetic calculation (standard) 95.64 10.63 69–128
Reading (raw) 37.54 4.51 28–46
Reading (standard) 109.21 11.76 79–132
Processing speed
Digit naming 2.70 0.51 1.51–3.81
Digit articulation 3.79 0.50 2.60–5.08
Short-term memory
Digit span forward 5.45 0.91 3.5–8.0
Word span forward 4.87 0.88 3–8
Verbal working memory
Auditory digit sequence 2.23 0.88 0–4
Semantic categorization 1.93 1.07 0–4
Visual–spatial working memory
Corsi block 5.59 0.89 3–8
Visual matrix 3.96 1.18 2–7
298 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

Table 2
Intercorrelations among chronological age, academic achievement measures, and individual cognitive processing
tasks
1 2 3 4 5 6 7 8 9 10 11
1. Chronological age —
2. Arithmetic calculation .65** —
3. Reading .38** .61** —
4. Digit naming .45** .44** .48** —
5. Digit articulation .38** .44** .30** .52** —
6. Digit span forward .38** .57** .55** .46** .45** —
7. Word span forward .32** .43** .48** .39** .39** .58** —
8. Auditory digit sequence .14** .32** .41** .19* .12* .29** .33** —
9. Semantic categorization .32** .52** .47** .26* .20* .43** .33** .35** —
10. Corsi block .40** .46** .33** .35** .34** .34** .22* .15* .27* —
11. Visual matrix .20** .44** .30** .41** .31** .54** .38** .09* .25* .33** —
*
p < .05.
**
p < .01.

r(90) = .45, p < .01, for digit naming. Both auditory digit sequencing, r(90) = .14, p = .19,
and visual matrix, r(90) = .20, p = .06, did not correlate significantly with chronological
age. Correlations between the cognitive processing tasks and reading ranged from
r(90) = .30, p < .01, for digit articulation and visual matrix to r(90) = .55, p < .01, for digit
span forward. Given the large range in chronological age among participants (98–145
months), and because of the strong significant correlation between chronological age
and arithmetic calculation, r(90) = .65, p < .01, chronological age was used as a control
variable in subsequent multiple regression analyses.

Predictive models of arithmetic calculation

The central purpose of this study was to examine the relative contributions of process-
ing speed, short-term memory, verbal working memory, visual–spatial working memory,
and reading to arithmetic calculation in children. Two specific objectives informed subse-
quent analyses. First, after accounting for the influence of reading on arithmetic calcula-
tion, do processing speed and short-term memory contribute to arithmetic calculation?
Second, in the presence of reading, processing speed, and short-term memory, do individ-
ual working memory components (i.e., verbal and visual–spatial) contribute unique vari-
ance to arithmetic calculation? To address these aims, specific associations among
cognitive processing variables and arithmetic calculation were examined by fixed-order
multiple regression analyses.
In light of the significant shared variance among the short-term memory and working
memory tasks (Table 2), examination of the individual constructs was deemed necessary
prior to investigating predictive models. Although agreement is increasing that short-term
memory and working memory represent related yet different constructs, literature in the
area has not reached consensus (Engle, Tuholski, Laughlin, & Conway, 1999; Heitz, Uns-
worth, & Engle, 2005). Inspection of intercorrelations among short-term memory and
working memory tasks (Table 2) indicated significant shared variance among these con-
structs. For example, whereas semantic categorization correlated with auditory digit
sequence, r = .35, p < .01, semantic categorization also correlated significantly with digit
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 299

span forward, r = .43, p < .01, and word span forward, r = .33, p < .01. A possible expla-
nation for the presence of such strong shared variances among short-term memory and
working memory measures might be the stimulus similarities between tasks. For instance,
the auditory digit sequence task involves processing both numeric and nonnumeric infor-
mation, whereas the two short-term memory span tasks require processing numeric infor-
mation (digit span forward) and nonnumeric (word span forward).
To determine whether a priori assumptions that short-term memory measures and
working memory measures reflected different but related constructs, a confirmatory factor
analysis was conducted. The latent construct short-term memory was measured by digit
span forward and word span forward. In light of literature differentiating between verbal
working memory and visual–spatial working memory (e.g., Baddeley, 2001; Baddeley &
Hitch, 1974), and because these working memory components were examined for individ-
ual and cumulative influence on arithmetic calculation in subsequent multiple regression
models, each of these working memory components was examined as a separate latent
construct. Verbal working memory was measured by auditory digit sequencing and
semantic categorization. Visual–spatial working memory was measured by the Corsi block
test and the visual matrix test.
Analysis supported a three-factor model representing the hypothesized latent con-
structs. The chi-square test, v2 = 5.51, df = 6, p = .480, suggested a good fit for the overall
model. In addition, the goodness of fit index (GFI) was .98, the root mean square error of
approximation (RMSEA) was less than .01, and the comparative fit index (CFI) was .96,
indicating that the model was a good fit for the data. The standardized path coefficients for
short-term memory were .88 and .66 (digit span forward and word span forward, respec-
tively), for verbal working memory were .51 and .69 (auditory digit sequencing and seman-
tic categorization, respectively), and for visual–spatial working memory were .45 and .72
(Corsi block and visual matrix, respectively). Results indicating the differentiation between
short-term memory and working memory corresponded to studies using similar measures
within child populations (e.g., Swanson & Beebe-Frankenberger, 2004). Furthermore, the
three-factor model presented suggested that working memory can be reliably separated
into related but different factors related to verbal and visual–spatial constructs.
Supported by the confirmatory factor analysis, measures related to processing speed,
short-term verbal working memory, and visual–spatial working memory were aggregated
corresponding to theoretical assumptions into composite scores by summing z scores for
each appropriate task based on the total sample. The mean intercorrelations among com-
posite scores are reported in Table 3.
Using the composite z scores for each cognitive construct, fixed-order multiple regres-
sion analyses were conducted to examine relative contributions of the cognitive constructs
to arithmetic calculation. Because the order of entry influences the outcome of a regression
equation, a series of models was created that considered the individual contributions of
each cognitive processing domain, the cumulative variance accounted for by combined
processes, and the unique variance accounted for by individual cognitive processing
domains. A total of four models were created and are reported in Table 4.
Model 1 reports the individual contributions of each cognitive processing domain and
reading to arithmetic calculation after accounting for the influence of chronological age.
Two findings are of particular importance. First, each cognitive processing domain con-
tributed significant individual variance to arithmetic calculation after accounting for the
influence of chronological age. Second, after controlling for chronological age, reading
300 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

Table 3
Intercorrelations among chronological age, academic achievement measures, and composite cognitive processing
tasks
1 2 3 4 5 6 7
1. Chronological age —
2. Arithmetic calculation .65* —
3. Reading .38* .61* —
4. Processing speed .48* .51* .45* —
5. Short-term memory .39* .56* .58* .54* —
6. Verbal working memory .28* .51* .53* .27* .47* —
7. Visual–spatial working memory .37* .55* .39* .50* .51* .28* —
*
p < .01.

Table 4
Summary of multiple regression analyses: Predictive models of arithmetic calculation
Order of entry in equation R2 Change in R2 df F ratio p
Model 1: Contributions of cognitive tasks and reading after controlling for age
1. Chronological age .42 — 88 63.07 <.001
2. Speed .46 .05 87 8.21 .005
2. STM .53 .11 87 20.96 <.001
2. VSWM .54 .12 87 22.23 <.001
2. VWM .54 .12 87 23.59 <.001
2. Reading .57 .15 87 31.99 <.001
Model 2: Contributory roles of VSWM and VWM after controlling for age and reading
3. VSWM .63 .06 86 13.48 <.001
4. VWM .66 .03 85 6.93 .010
3. VWM .61 .04 86 7.52 .007
4. VSWM .66 .05 85 12.79 .001
Model 3: Contributory roles of speed, VSWM, and VWM after controlling for age and reading
3. Speed .58 .01 86 2.06 .155
4. VSWM .63 .05 85 11.13 .001
5. VWM .66 .03 84 6.91 .010
4. VWM .62 .04 85 7.61 .007
5. VSWM .66 .04 84 10.36 .002
Model 4: Contributory roles of STM, VSWM, and VWM after controlling for age and reading
3. STM .60 .03 86 5.33 .023
4. VSWM .64 .04 85 9.04 .003
5. VWM .66 .02 84 5.91 .017
4. VWM .62 .02 85 5.35 .023
5. VSWM .66 .04 84 9.58 .003
Note. Speed, processing speed; STM, short-term memory; VSWM, visual–spatial working memory; VWM, verbal
working memory.

was the strongest contributor to arithmetic calculation, accounting for an additional 15%
of the variance. In light of these results, follow-up models were constructed to examine the
interaction among cognitive processing domains and reading on arithmetic calculation.
However, given the substantial contribution of chronological age, accounting for 42%
of the variance in arithmetic calculation, this effect needed to be attended to so as to
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 301

increase the interpretability of successive models. Thus, subsequent models were created to
examine whether working memory contributed significant variance to arithmetic calcula-
tion after accounting for the contributions of reading, processing speed, and short-term
memory. Specific attention was directed at whether individual working memory compo-
nents contributed individual, cumulative, and unique variance to arithmetic calculation
when controlling for all other variables.
To address limitations of previous studies that did not examine the potential individual
contributions of verbal working memory and visual–spatial working memory, each of
these variables was entered last into subsequent models (Swanson & Beebe-Frankenber-
ger, 2004). Model 2 examined the individual contributions of verbal working memory
and visual–spatial working memory and the unique contributions of each after the influ-
ence of the other was controlled. Both verbal working memory and visual–spatial working
memory contributed significantly to arithmetic calculation (4 and 6%, respectively). In
addition, in the presence of the other component, verbal working memory and visual–spa-
tial working memory both contributed unique variance to arithmetic calculation (3 and
5%, respectively).
Models 3 and 4 were designed to examine the individual contributions of processing
speed and short-term memory to arithmetic calculation after controlling for chronological
age and reading and to investigate whether working memory contributed significant var-
iance to arithmetic calculation after controlling for the influence of processing speed and
short-term memory. Inspection of Model 3 indicated that processing speed did not con-
tribute significantly to children’s arithmetic calculation above the contribution of chrono-
logical age and reading. Because it failed to contribute significant individual variance to
arithmetic calculation, processing speed was not included in subsequent models. Model
4 indicated that short-term memory contributed significant variance to arithmetic calcula-
tion (3%). Moreover, in the presence of short-term memory and the other working mem-
ory components, both verbal working memory and visual–spatial working memory
retained unique contributions to arithmetic calculation (2 and 4%, respectively).
In sum, a full regression model predicting arithmetic calculation accounted for 66% of
the variance, F(5, 84) = 32.81, p < .001. Within the complete model, after controlling for
chronological age and reading ability, individual working memory components (verbal
and visual–spatial) contributed significant variance to arithmetic calculation in the pres-
ence of processing speed and short-term memory. Said differently, verbal working memory
and visual–spatial working memory both were related to individual differences in arithme-
tic calculation independent of chronological age, reading ability, processing speed, and
short-term memory.
Given the large range in age among participants and the high significant correlations
between chronological age and the achievement and cognitive measures, a series of fol-
low-up regression models was calculated to investigate which cognitive processes
accounted for the age-related differences in arithmetic calculation. As noted above, read-
ing has been found to be a significant contributor to arithmetic calculation; thus, it was
entered first into all subsequent regression models.
Model 1 indicated that after accounting for the influence of reading, chronological age
accounted for an additional 20% of the variance in arithmetic calculation. Four separate
regression equations were calculated (Model 2) to investigate the individual contributions
of processing speed, short-term memory, verbal working memory, and visual–spatial
working memory on arithmetic calculation after accounting for reading and to investigate
302 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

whether chronological age remained a significant contributor after accounting for the
influence of each cognitive process. Results indicated that each cognitive process contrib-
uted to arithmetic calculation: processing speed (7%), short-term memory (7%), verbal
working memory (5%), and visual–spatial working memory (12%). Furthermore, in the
presence of each cognitive process, chronological age accounted for additional variance
in arithmetic calculation.
Then a series of models was constructed to examine whether the relationship between
chronological age and arithmetic calculation was due to the influence of processing speed
and short-term memory (Model 3), of verbal working memory and visual–spatial working
memory (Model 4), of processing speed and working memory (Model 5), and of short-
term memory and working memory (Model 6). Inspection of each of these models sug-
gested that each cognitive process was influential in accounting for age-related differences
in arithmetic calculation, yet chronological age remained a significant contributor. A final
model (Model 7) was constructed to examine the degree to which all cognitive processes
accounted for age-related differences in arithmetic calculation. All variables emerged as
significant contributors; together reducing the contribution of chronological age to arith-
metic calculation from 20 to 11% (Table 5).

Discussion

Results of the current study provide further evidence of the role of working memory
and related cognitive processes in arithmetic calculation (e.g., Adams & Hitch, 1997;
Swanson & Beebe-Frankenberger, 2004). Building on a dearth of research on arithmetic
calculation in children, this study sought to examine the relative contributions of process-
ing speed, short-term memory, and working memory in children’s arithmetic calculation.
Results suggested four important findings. First, processing speed emerged as a significant
contributor of arithmetic calculation only in relation to age-related differences in the gen-
eral sample. Second, processing speed and short-term memory did not eliminate the con-
tribution of working memory to arithmetic calculation. Third, individual working memory
components (i.e., verbal working memory and visual–spatial working memory) each con-
tributed unique variance to arithmetic calculation in the presence of all other variables.
Fourth, a full model indicated that chronological age remained a significant contributor
to arithmetic calculation in the presence of significant contributions from all other
variables.
Unexpectedly, and in contrast to previous studies, the current results did not find that
processing speed contributed to arithmetic calculation in the general sample. Although
several studies have highlighted a relationship between processing speed and arithmetic
calculation (Bull & Johnston, 1997; Swanson & Beebe-Frankenberger, 2004), the absence
of corresponding results might be due, in part, to participant selection. Participants in Bull
and Johnston’s (1997) study had a mean age of 89 months, whereas participants in the cur-
rent study were approximately 122 months of age. Similarly, the children in Swanson and
Beebe-Frankenberger’s (2004) study were from Grades 1 to 3, whereas participants in the
current study were from Grades 3 to 6. Subsequent models within the current study that
addressed age-related difference within the general sample corresponded with previous
research, suggesting that processing speed does indeed contribute significant variance to
arithmetic calculation. Again, these confounding results underscore the work of Kail
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 303

Table 5
Summary of multiple regression analyses: Predictive models of age-related differences in arithmetic calculation
Order of entry in equation R2 Change in R2 df F ratio p
Model 1: Contribution of age after controlling for reading
1. Reading .37 – 88 52.55 <.001
2. Chronological age .57 .20 87 40.90 <.001
Model 2: Contributory role of each cognitive process after controlling for reading
2. Processing speed .44 .07 87 10.42 .002
3. Chronological age .58 .14 86 29.66 <.001
2. STM .44 .07 87 10.06 .002
3. Chronological age .60 .16 86 34.27 <.001
2. VWM .42 .05 87 7.43 .008
3. Chronological age .61 .19 86 39.90 <.001
2. VSWM .49 .12 87 19.97 <.001
3. Chronological age .63 .14 86 31.97 <.001
Model 3: Contributory roles of speed and STM after controlling for reading
2. Speed .44 .07 87 10.42 .002
3. STM .47 .03 86 4.52 .036
2. STM .44 .07 87 10.06 .002
3. Speed .47 .03 86 4.85 .030
4. Chronological age .60 .13 85 28.25 <.001
Model 4: Contributory roles of VWM and VSWM after controlling for reading
2. VWM .42 .05 87 7.43 .008
3. VSWM .53 .11 86 18.86 <.001
2. VSWM .49 .12 87 19.97 <.001
3. VWM .53 .04 86 6.57 .012
4. Chronological age .66 .13 85 31.55 <.001
Model 5: Contributory roles of speed, VWM, and VSWM after controlling for reading
2. Speed .44 .07 87 10.42 .002
3. VWM .49 .05 86 7.55 .007
4. VSWM .55 .06 85 11.07 .001
3. VSWM .51 .07 86 11.95 .001
4. VWM .55 .04 85 6.75 .011
5. Chronological age .66 .11 84 26.99 <.001
Model 6: Contributory roles of STM, VWM, and VSWM after controlling for reading
2. STM .44 .07 87 10.06 .002
3. VWM .47 .03 86 4.49 .037
4. VSWM .54 .07 85 12.82 .001
3. VSWM .51 .07 86 12.39 .001
4. VWM .54 .03 85 4.96 .029
5. Chronological age .66 .12 84 26.69 <.001
Model 7: Contributory role of all cognitive processes after controlling for reading
2. STM .44 .07 87 10.06 .002
3. Speed .47 .03 86 4.85 .030
4. VWM .50 .03 85 5.36 .023
5. VSWM .55 .05 84 9.06 .003
4. VSWM .52 .05 85 8.99 .004
5. VWM .55 .03 84 5.47 .022
6. Chronological age .66 .11 83 26.41 <.001
Note. STM, short-term memory; VWM, verbal working memory; VSWM, visual–spatial working memory;
Speed, processing speed.
304 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308

and colleagues on the developmental nature of processing speed, and of researchers in the
field of cognitive processing and arithmetic calculation, indicating that the relationship
between processing speed and arithmetic calculation is more pronounced during the early
stages of arithmetic development and decreases as children become more proficient. As
children get older, there is less variability in processing speed (Salthouse & Kail, 1983).
That is, age-related differences are more pronounced in younger children, with fewer sig-
nificant differences between older children and adolescents and between adolescents and
adults. Furthermore, processing speed has been linked to age-related differences in arith-
metic calculation. In their study examining phonological processing relationships to math-
ematical computation, Hecht and colleagues (2001) found that speed of processing
phonological-based information was a significant predictor in mathematical computa-
tional differences between Grades 2 and 3 but did not surface as a significant predictor
in differences between Grades 2 and 5, Grades 3 and 4, and Grades 4 and 5. Processing
speed might hold particular importance during the early stages of arithmetic calculation
development (e.g., development of automaticity of number representations in long-term
memory), with its influence decreasing over time corresponding to increases in the influ-
ence of other cognitive factors (short-term memory and working memory) and in the
sophistication of arithmetic development (simple arithmetic progressing to more complex
arithmetic).
A second notable finding of the current study was that the contribution of working
memory in general or its individual components to arithmetic calculation was not elimi-
nated by the presence of processing speed and short-term memory. Although both of these
latter processes are argued to be implicit to the working memory system (Case, 1985; Case
et al., 1982) and to arithmetic calculation (Baddeley & Hitch, 1974; Swanson & Beebe-
Frankenberger, 2004), the question arises as to what other processes account for the con-
tribution of working memory to arithmetic calculation. Recent theoretical perspectives
suggest that attentional resources within working memory (e.g., executive functions
related to mental coordination and flexibility) are likely avenues for interpretation of
the current results and future studies (Baddeley, 2001).
Baddeley (2001) offered a revised model of his earlier working memory model (Baddeley
& Hitch, 1974). The newer model proposed that the central executive has no capacity
(storage) capabilities; rather, it is primarily a processing component directed toward atten-
tion. The reconceptualized functions of the central executive focus on differentiated atten-
tion processes related to selective, sustained, and divided attention. One or a combination
of these attention processes might interact during arithmetic calculations. Furthermore,
the presence of significant associations between short-term memory and arithmetic calcu-
lation in this sample is worthy of note and is consistent with theoretical explanations
(Baddeley, 2001). It is possible that the variance accounted for by individual working
memory components, after short-term memory was partialled from the equation, captured
aspects of the attention-related processes of the central executive.
The third important finding was the emergence of individual working memory compo-
nents as significant contributors to arithmetic calculation. Although both verbal working
memory and visual–spatial working memory contributed to arithmetic calculation, each
also emerged as a unique contributor in the presence of the other working memory com-
ponent. Moreover, these contributions were independent of reading ability, chronological
age, processing speed, and short-term memory. These findings provide support for one of
the central limitations of current research, specifically, the need to address potential
 D.H. Berg / Journal of Experimental Child Psychology 99 (2008) 288–308 305

differentiated roles of verbal working memory and visual–spatial working memory in

arithmetic calculation, particularly in children. The precise role of the each working mem-
ory component remains unclear. Although evidence from adult populations indicates that
the verbal and visual–spatial components of working memory have specialized roles in
solving arithmetic calculation—verbal working memory is involved in retaining parts of
the solution, and visual–spatial working memory is involved in encoding the problem
addends when the problem is presented visually (Logie et al., 1994)—the current study
offers no insight into such specialized roles. Indeed, this study assessed general arithmetic
calculation performance and did not differentiate between operations (e.g., addition and
subtraction) or among levels of problem complexity. Research and practice would benefit
from understanding such differentiated functions with respect to these factors. For
instance, educators would be able to develop curricular and instructional approaches to
arithmetic calculation that take advantage of the memory processes that children are likely
to use to solve arithmetic problems.
Fourth, full regression models indicated that chronological age remained a significant
contributor to arithmetic calculation in the presence of significant contributions from
all other variables. From these models, two important findings are particularly notewor-
thy. First, the emergence of each cognitive process as a contributor to arithmetic calcula-
tion might reside in particular relationships to the range of arithmetic skills embedded
within the arithmetic calculation measure administered. For instance, the WRAT3
includes questions of addition, subtraction, multiplication, and division, each of which
is presented in simple and complex forms. It is possible that the range of cognitive pro-
cesses that surfaced as contributors to general arithmetic on the WRAT3 are at least par-
tially attributable to some arithmetic skills (e.g., simple addition) and not other arithmetic
skills (e.g., complex multiplication). A growing body of work is suggesting that problem-
specific characteristics (e.g., simple vs. complex problems) are differentially related to dif-
ferent cognitive processes (e.g., Geary et al., 2007; Heathcote, 1994). Second, chronolog-
ical age still contributed unique variance to arithmetic calculation after controlling for the
effects of reading and all other cognitive variables. That the cognitive measures adminis-
tered in the current study did not capture all variance in arithmetic calculation due to
age is a limitation of this study and the field of mathematical cognition in general. This
study focused on a discrete set of cognitive processes related to processing speed and dif-
ferent memory constructs; absent were other cognitive measures such as intelligence and
attention. In addition, compared with other fields of study (e.g., reading), research in
mathematics, in particular arithmetic calculation, is in its infancy. The field awaits consen-
sus on the cognitive processes that best predict the development of specific mathematical
skills and on the complex of cognitive processes that undergird the range of mathematical
abilities.

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