Dual-Task Studies of Working Memory and Arithmetic Performance: A Meta-Analysis

Abstract

We performed a meta-analysis of dual-task experiments to assess the robustness of the effects of conducting working memory secondary tasks on arithmetic performance. Four hundred effect sizes from 21 studies from 1,049 participants were analyzed across a variety of specifications. Results revealed that increases in working memory load resulted in slower (7% to 19% reduction) speed of solving of arithmetic problems. Of the potential moderators, working memory load type (i.e. central executive, phonological loop, and visuospatial sketchpad), arithmetic task type (e.g. addition verification, approximate addition, exact multiplication), and authors’ predictions for significance which served as a proxy for cross-talk were statistically significant across specifications, but participants’ age was not. Working memory load type was the most substantial moderator, with central executive tasks leading to the greatest slowing of performance, suggesting that the cognitive complexity of a working memory task may exert a larger influence on performance than the domain-specific overlapping processing demands of similar tasks. We discuss the apparent discrepancy between these findings and findings from correlational studies of the relation between arithmetic performance and working memory, which have reported similar correlations across working memory domains, on average.

Full text

Running Head: WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 1
Dual-Task Studies of Working Memory and Arithmetic Performance: A Meta-Analysis
Edward H. Chen, Drew H. Bailey
University of California, Irvine
Accepted at Journal of Experimental Psychology: Learning, Memory, & Cognition, 1/13/2020
Address inquiries to Edward Chen; cheneh1@uci.edu.
Author note: The authors thank Dave Geary, Susanne Jaeggi, Young-Suk Kim, Hal Pashler,
Lindsey Richland, and Jeff Rouder for comments on previous versions of this project. The
authors thank André Knops and Iro Xenidou-Dervou for sharing their summary data. Bailey is
funded by a Jacobs Fellowship.
©American Psychological Association, 2020. This paper is not the copy of record and may not
exactly replicate the authoritative document published in the APA journal. Please do not copy or
cite without author's permission.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 2
Abstract
We performed a meta-analysis of dual-task experiments to assess the robustness of the
effects of conducting working memory secondary tasks on arithmetic performance. Four hundred
effect sizes from 21 studies from 1,049 participants were analyzed across a variety of
specifications. Results revealed that increases in working memory load resulted in slower (7% to
19% reduction) speed of solving of arithmetic problems. Of the potential moderators, working
memory load type (i.e. central executive, phonological loop, and visuospatial sketchpad),
arithmetic task type (e.g. addition verification, approximate addition, exact multiplication), and
authors’ predictions for significance which served as a proxy for cross-talk were statistically
significant across specifications, but participants’ age was not. Working memory load type was
the most substantial moderator, with central executive tasks leading to the greatest slowing of
performance, suggesting that the cognitive complexity of a working memory task may exert a
larger influence on performance than the domain-specific overlapping processing demands of
similar tasks. We discuss the apparent discrepancy between these findings and findings from
correlational studies of the relation between arithmetic performance and working memory, which
have reported similar correlations across working memory domains, on average.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 3
Dual-Task Studies of Working Memory and Arithmetic Performance: A Meta-Analysis
It is widely agreed that various aspects of arithmetic performance are dependent on working
memory, but the strength of this relation and the degree to which different features of working
memory contribute to this performance is difficult to study. Arithmetic procedures involve the
temporary storage and manipulation of numerical elements across multiple steps (Hitch, 1978).
For example, individuals solving multi-digit arithmetic problems, such as 23 × 16 or 256 + 169
must encode the problem they are working with, perform a number of calculations, and maintain
these intermediate values in order to form a coherent solution to the arithmetic problem (for a
review, see Raghubar, Barnes, & Hecht, 2010). Arithmetical processing, therefore, appears to
rely heavily on working memory. Consistent with this theory, working memory has been found
to be reliably correlated with performance on mathematical tasks and has been found to
statistically predict children’s mathematics outcomes (for reviews, see Friso-van den Bos, van
der Ven, Kroesbergen, & van Luit, [2013] and Raghubar, Barnes, & Hecht, [2010]).
However, some specific questions about the nature of working memory resources influencing
arithmetic performance have been difficult to address. In particular, the specificity of these
effects to particular facets load types of working memory (e.g. differential impact of the
visuospatial system versus the phonological system on subtraction performance), types of
arithmetic, and interactions among them, is limited by the use of correlational designs. Two more
causally informative approaches are training studies, where participants learn to better utilize
working memory resources, and the dual-task experimental design, whereby participants perform
a primary cognitive task (e.g. multiplication) concurrently with another secondary task (e.g.
pressing a left key when hearing a low tone through a headset or the right key when hearing a
high tone). Dual-task experiments offer an alternative to correlational or training studies by

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 4
allowing the researcher to both experimentally manipulate working memory load and the overlap
between features of the working memory task and the arithmetic task to investigate task overlap
(Logie & Baddeley, 1987; Ashcraft, Donley, Halas, Vakali, 1992; for review, see Pashler, 1994).
Our goal is to investigate the specificity of working memory functions in arithmetic through such
a design and provide insight on the discrepancy between experimental and correlational findings.
Thus, we review the literature on the role of working memory in arithmetic processing and
provide meta-analytic evidence to characterize the causal relation between working memory and
arithmetic as studied in dual-task experiments.
Causal Effects of Working Memory on Arithmetic?
Working memory has been conceptualized in a variety of ways. Some models suggest
that it as distinct from executive functions while others argue that executive functions subsume
working memory functions. Research on the cognitive processes that underlie arithmetical
cognition has been largely influenced by the multicomponent model of working memory
conceptualized by Baddeley and Hitch (1974). According to this model, working memory is a
limited capacity system responsible for short-term storage and manipulation of elements within
cognitive processes (Diamond, 2013; Miyake & Shah, 1999). The model has been refined over
time but often separates working memory into three core subcomponents: the central executive
(CE), phonological loop (PL), and visuospatial sketchpad (VSSP). Miyake’s and colleagues’
(2000) theory of executive functions proposes three aspects of executive functions: updating,
shifting, inhibition. Updating involves the constantly monitoring and adding/deleting WM
contents, shifting involves switching between tasks and mental sets, and inhibition involves the
conscious overriding of predominant responses. Unlike the Baddeley model, there are no
subcomponent systems (i.e. the visuospatial sketchpad and phonological loop), and WM is

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 5
viewed separately as a passive-storage system that relies on executive functions. Diamond’s
model of EF is similar to Miyake’s in that it includes inhibition and views WM as a separate
construct, but it differs by including cognitive flexibility which involves task switching
(analogous to shifting). Dual-tasks have been thought to involve executive functions such as
those found in Miyake and Baddeley’s models (specifically that of shifting), but no consensus
has yet been found regarding the specificity of which executive function. Another perspective
involves EFs as part of WM, that is, WM capacity is the ability to use attention to maintain or
suppress information (Engle, 2002; Awh, Vogel, & Oh, 2006). Engle (2002) posits that a greater
WM capacity is indicative of greater ability to control attention rather than a larger memory
storage. While many alternative models to EF and WM have been proposed over the last few
decades, the focus of this meta-analysis is on Baddeley’s multicomponent model. The primary
reason for this is that dual-task research including arithmetic tasks is largely predicated on this
model, specifically predictions concerning its subcomponents and variations in arithmetic tasks.
Thus, arithmetic tasks are hypothesized to be subject to interference to the extent that they
overlap with specific arithmetic processing on Baddeley’s subcomponents. More general
processes, such as switching or inhibition, are likely required across most dual-task pairs,
although perhaps in different amounts, with tasks hypothesized to require CE resources
involving more switching and inhibition than tasks hypothesized to require only PL and VSSP.
Thus, results can be interpreted with respect to all of these theories, but hypothesized examples
of cross-talk pertain most directly to Baddeley’s.
The central executive is the most important component of Baddeley’s model. It acts in a
supervisory role between the other two subsystems by coordinating visual and verbal information
and between working memory and long-term memory. Compared to the PL and VSSP whose

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 6
functions are more domain-specific and storage-based, the CE is amodal and facilitates
processing. Beyond a supervisory position, the CE’s other main functions include selective
attention, inhibiting or suppressing automatic responses, updating working memory with new
information, and shifting between tasks. Within Baddeley’s model, the CE would appear to play
a pivotal role across all types of single- and multi-digit arithmetic operations, because of the split
in attentional resources and the necessity to maintain intermediate results (for reviews, see
Raghubar, Barnes, & Hecht [2010] and DeStefano & LeFevre [2004]).
The phonological loop aids in temporarily storing and rehearsing verbal information. In
the context of arithmetic cognition, the phonological loop seems to be primarily involved in
verbally mediating calculation strategies, such as decomposing, transforming, and counting
up/down in multi-digit arithmetic (Furst & Hitch, 2000; Imbo & Vandierendonck, 2007a).
The visuospatial sketchpad is responsible for the storage and processing of visual and
spatial information of an element, such as its shape and position. The visuospatial sketchpad has
been viewed as especially important to the development of mental arithmetic in young children
whereby their use of the mental number line is reliant on visuospatial encoding (Hubbard et al.
2005; McKenzie, Bull, & Gray, 2003). While the role of the VSSP is less understood in mental
arithmetic, some have found evidence suggesting that it is involved in strategy use (though to a
much lesser extent than the PL and CE because these more sophisticated strategies take time to
develop) and more difficult arithmetic problems in both children and adults, such as those
requiring carrying operations or the encoding of intermediate results (Rasmussen & Bisanz,
2005; Xenidou-Dervou, van der Schoot & van Lieshot, 2015; Noël, Desert, Aubrun, & Seron,
2001; Logie, Gilhooly, & Wynn, 1994; Hubber, Gilmore, & Cragg, 2014).

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 7
Another component, the episodic buffer, was later added to the Baddeley model to
explain how the central executive interacts with the other subsystems; however, there is little
discussion or experimental manipulation of the episodic buffer within the dual-task literature for
arithmetic (Ketelsen & Welsh, 2010).
The relations between working memory and arithmetic performance may also differ by
participant age. The solving of simple arithmetic problems is often highly practiced in adults, but
children solve problems more slowly and often use less sophisticated strategies (e.g., counting
the smaller addend rather than retrieval) because they have yet to attain the same level of
expertise (Ashcraft, 1992; Anderson, 1987, Siegler, 1988). Children tend to use more efficient
strategies to solve arithmetic problems and rely less on working memory resources as they get
older (Imbo & Vandierendonck, 2007b; McKenzie, Bull, & Gray, 2003, likely due to the
acquisition of more efficient arithmetic strategies; Halford, Cowan, & Andrews, 2007).
Working memory’s role in arithmetic processing has been studied using several different
methodological approaches. We review these approaches, findings, and the costs and benefits
associated with each approach. Here, we will examine correlational studies and two experimental
approaches: working memory training and dual-task designs.
Correlations
Much of our understanding of the role of executive functions, specifically working
memory, in mathematical and arithmetic performance comes from correlational designs. In
general, all facets of working memory are known to be correlated with mathematical
performance (Friso-van den Bos, van der Ven, Kroesbergen, & van Luit, 2013; Bull & Lee,
2014), and arithmetic performance differences between children with and without mathematical
difficulties are smaller after statistically controlling for differences in working memory capacity

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 8
(Geary, Hoard, Byrd-Craven, & DeSoto, 2004; Geary, Hoard, Byrd-Craven, Nugent, & Numtee,
2007). However, while these correlational designs provide insight into the possible cognitive
processes influencing mathematical development, these designs have had inconsistent success at
demonstrating the specificity of working memory contributions. In one meta-analysis, the
average correlations for working memory and mathematical tasks were quite similar across the
phonological and visuospatial working memory tasks (r = .34 for visuospatial updating, r =.38
for verbal updating, r =.34 for VSSP, and r =.31 for PL; Friso-van den Bos, van der Ven,
Kroesbergen, & van Luit, 2013)
1
. Using a domain-specific model of working memory in which
the domains are divided into verbal (mathematics tasks with verbal components like word
problems), numerical (number related tasks like calculations), and visuospatial (mathematics
tasks with visuospatial components like geometry) working memory, another meta-analysis of
correlations between working memory and arithmetic tasks presented similar findings (r = .30
for verbal working memory, r =.34 for numerical working memory, and r =.31 for visuospatial;
Peng, Barnes, Namkung, & Sun, 2015). Thus, the correlational literature has not identified
substantial differences in the correlations between arithmetic performance and different working
memory task types.
These findings imply that if there is specificity in the effects of different working
memory components on arithmetic tasks, averaging correlations across studies does not reliably
1
Friso-van den Bos and colleagues (2013) used working memory components that were a
combination of those posited by Baddeley and Hitch (1974 – the central executive, phonological
loop, and visuospatial sketchpad) and Miyake et al. (2000 – updating, shifting, and inhibition). In
our meta-analysis, we primarily use Baddeley and Hitch’s model. We concluded this model was
more closely aligned with the dual task literature, because it more naturally makes a distinction
between domains (verbal and spatial), it makes a distinction between remembering and
manipulating information that applies to many of the working memory tasks used in the dual task
literature, and because almost all of the secondary tasks would be categorized as updating under
Miyake and colleagues’ model.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 9
differentiate among them. Statistically controlling for facets of working memory simultaneously
may not solve this problem, as the tasks used to measure each facet may differentially reflect
broader cognitive abilities (Schmidt, 2017). Taken together, this evidence is consistent with
different aspects of working memory contributing similarly to arithmetic performance, but this
method may also lack the required sensitivity to differentiate the contributions of different
aspects of working memory to arithmetic performance.
Interventions
In order to better understand the unique contributions of working memory facets to
arithmetic performance, we turn to evidence from experimental designs. One experimental
approach to estimating the effects of working memory on mathematical cognition is cognitive
training intervention. Typically, participants engage in an activity or game that targets either
general or specific cognitive skills and are later measured on both cognitive abilities and school-
related achievement tests (Diamond & Lee, 2011; Jaeggi, Buschkuehl, Jonides, & Shah, 2012;
Loosli, Buschkuehl, Perrig, & Jaeggi, 2012). For example, a recent study by Ramani et al. (2017)
trained three different groups of kindergarteners using a working memory game condition, math
game condition, and a no-contact control condition and found improvements in numerical
processing for both intervention groups. In this particular study, the working memory game
involved remembering the orientation and sequence of cartoon characters on a tablet screen. The
number of characters required of the children to remember would increase with successful
responses.
This approach has produced a number of positive effects, and its design appears to be a
straightforward approach for estimating causal links between working memory and arithmetic
performance. However, a meta-analysis of training interventions by Melby-Lervåg and Hulme

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 10
(2013) found short-term, near-transfer improvements in working memory ability, but
inconsistent effects on substantively different cognitive tasks, including arithmetic. Subsequent
meta-analyses and a systematic review reported similar findings of strong near transfer effects of
cognitive training, with limited evidence that these improvements transfer to a variety of
cognitive tasks, including verbal ability, reading comprehension, and arithmetic (Sala & Gobet,
2017; Simons et al., 2016; Melby-Lervåg, Redick, & Hulme, 2016). Notably, an evaluation of a
school-based working memory training intervention in Australian first graders found persistent
impacts on some working memory tasks at 6 and 12 months after training, but no evidence of
transfer to math computation 12 or 24 months after training (Roberts et al., 2016).
Unfortunately, such findings are ambiguous with respect to the influence of working
memory on arithmetic performance, primarily because of debates about the breadth of the
constructs being trained. Though WM training is expected to transfer across WM tasks, this often
may not be the case (Colom et al. 2013). If training fails to generalize to even other WM tasks, it
is not clear if general cognitive processes like WM have been improved at all, and by extension,
the mechanism for transfer is not clear either. If training generalizes to arithmetic or other tasks,
the possibility of effects via mechanisms other than working memory improvement (especially in
studies including a passive control condition; Shipstead, Redick, & Engle, 2012) also calls into
question simple interpretations of such effects as evidence for effects of working memory on
arithmetic performance.
Dual-Task Studies
Dual-task studies were developed and primarily used to investigate the role of WM and
its components. In arithmetic cognition, dual-task experiments provide compelling evidence for
some of the cognitive processes involved in a task, such as mental arithmetic, because they can

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 11
be used to manipulate the roles of the different components of working memory across different
arithmetic tasks (Logie & Baddeley, 1987). A dual-task design involves the completion of a
primary cognitive task (in this case mental arithmetic) while simultaneously completing a
secondary distractor task (in this case working memory tasks). Participants’ accuracy and
reaction time in a single-task condition are then compared to performance in various dual-task
conditions.
Dual-task experiments vary on the types of tasks required of the participant (for review,
see Pashler, 1994). For example, a participant may be required to remember a string of letters (z,
h, d) while simultaneously completing an addition task. The interference effect of a concurrent
memory load on the speeded task is attributed to either a decrease in shared resource capacity
within WM or more likely, rehearsing the memory load causes interference in the stimulus-
response mapping or preparation of the speeded task delays the retrieval process (Logan, 1978;
1979). These designs generally produce small or null effects on performance across different task
modalities (Baddeley, 1986).
However, some studies have used secondary tasks with greater cognitive demands by
instructing participants to randomly generate letters (Vreugdenburg, Bryan, & Kemps, 2003;
Lemaire, 1996). As most commonly seen with central executive tasks, some designs have
participants perform perceptual judgments concurrently with an arithmetic task. For example, a
participant may perform an addition verification task (e.g., 5 + 6 = 11, indicating whether this is
correct or not) while completing a task designed to load the central executive, such as pressing
either a 1 or 2 key depending on whether they hear a high or low tone through a headset (e.g.
Imbo & Vandierendonck, 2007a; Tronsky, McManus, & Anderson, 2008). With regards to dual-
task interference in arithmetic, adult participants may rely on a small number of strategies when

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 12
performing arithmetic, with direct retrieval being the most common. Cognitive load induced by
dual-task experiments may interfere with the efficiency of using such strategies in calculations,
especially those that require more steps (e.g. decomposition) or that rely on specific resources
(Anderson, Reder, & Lebiere, 1996; Tronsky, 2005).
Two competing general theories have been posited to explain why these interference
effects occur in dual-task studies: serial processing and parallel processing. In serial processing,
people are believed to have some sort of structural limitation or a central bottleneck, whereby the
cognitive demands of the first cognitive process delay performance on the second (Ruthruff,
Pashler, & Klaasen, 2001). Ruthruff, Pashler, and Klaasen (2001) and Ruthruff, Pashler, and
Hazeltine (2003) provided experimental evidence that a structural limitation, rather than a
voluntary postponement, underlies slowed performance in dual-task studies. Assuming there is a
central bottleneck in processing, the memory load imposed by these secondary tasks (especially
difficult ones) are likely due to interference in the preparation of the arithmetic task rather than
the actual processing of arithmetic regardless of the modality.
Under the alternative graded capacity sharing model, cognitive processes are thought to
be performed in parallel, but interference effects occur due to capacity limitations rather than
with a bottleneck (Ruthruff, Pashler, & Hazeltine, 2003). A prediction specific to this model is
referred to as cross-talk, wherein if memory’s limited capacity relies on different cognitive
processes (e.g. visual and verbal processes), then completing two similar or within-modality
tasks leads to a greater decrement in performance than completing two dissimilar tasks (Lien &
Proctor, 2002; Miller, 2006; & Koch, 2009; Navon & Miller, 1987; Pashler, 1994). Cross-talk is
a recurrent hypothesis in dual-task studies of arithmetic and working memory. For example, Lee
and Kang (2002) hypothesized that arithmetic operations such as multiplication and subtraction

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 13
relied on two separate encoding processes – verbal for multiplication and visuospatial for
subtraction. Multiplication was more impaired by a verbal secondary task, while subtraction was
more impaired by a spatial secondary task, consistent with the hypothesis that separate facets of
working memory were required for different arithmetic tasks. While individuals may be able to
encode stimuli from similar tasks simultaneously, cross-talk designs would predict that the
processing of one of these tasks is harmful to the processing of the other (Treisman & Davies,
1973).
Although cross-talk is an influential hypothesis, specific findings have not always been
reliably replicated. For example, the differential interference of PL and VSSP secondary tasks on
multiplication and subtraction in the Lee and Kang (2002) study was partially replicated in
another study in a Chinese, but not a Canadian sample (Imbo & LeFevre, 2010). Some of the
apparently discrepant findings across studies may be related to confounding of working memory
task type with the cognitive demands of the working memory task: A recent study found no
selective interaction of working memory load type (PL and VSSP) with arithmetic conditions
(subtraction and multiplication) once the task demands for arithmetic and working memory were
matched on problem/set size and difficulty (Cavdaroglu & Knops, 2017). These findings do not
falsify the cross-talk hypothesis, but they suggest that any such effects may be difficult to
generalize across individuals and tasks. They also raise an alternative hypothesis: that the effects
of hypothesized specific overlapping task demands on delays in performance may be small,
especially in comparison to the general task demands inherent to the WM distractors themselves.
Altogether, it is unclear exactly how specific mental arithmetic is in recruiting working memory
resources among these dual-task designs.
Current Study

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 14
The purpose of the current study is to test theories of arithmetic cognition and dual-task
performance by meta-analyzing a body of research that has used the dual-task paradigm to study
the underlying cognitive processes involved in arithmetic. The aim of this meta-analysis is
threefold. The first aim is to address the robustness of arithmetic performance’s reliance on
working memory resources as predicted by dual-task studies by means of meta-analysis. The
second aim is to address how the effects of working memory load on arithmetic performance
might depend on factors typically manipulated in these dual-task studies: overlapping features
characterized by the types of working memory load and arithmetic operations, task complexity
characterized by the type of working memory task, level of expertise in arithmetic, as
approximated by the age of participants, and author’s prediction about significant effects of WM
load types as a proxy for the cross-talk hypothesis. The last aim is to in some way reconcile the
discrepancy between previously reported correlational and experimental findings.
Methods
Inclusion criteria and screening
A flow-chart of the identification and screening process can be found in Figure 1.
Keyword searches were used in Google Scholar and ProQuest (databases used were ERIC (1966-
Current) and PsychINFO (1806-Current)) to obtain the sample of studies to be screened for this
meta-analysis. The following search terms used included: ("working memory" OR WM OR
"executive function" OR cognition OR visuospatial OR visu* OR spatial OR "phonological
loop" OR "verbal" OR "slave system") AND (mathematics OR math OR arithmetic) AND ("dual
task" OR paradigm OR interference OR suppression). In total, 1071 results were found from
ProQuest and 5000 results were searched through Google Scholar. After removing duplicates, we
were left with a total of n=4119 records. We then proceeded with pre-screening of titles and

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 15
abstracts for relevance, which brought the number of records down to n=850 after title screening
and then n=55 after abstract screening.
Studies were considered eligible for this meta-analysis if they met the following criteria.
First, studies had to be of a dual-task design such as those described in Pashler (1994) and
Ashcraft et al. (1992). Second, a primary arithmetic task must have been performed concurrently
with a secondary working memory or cognitive load task such that accuracy or reaction time
(RT) was measured. Third, the working memory or cognitive load task needed to be
experimentally manipulated in either a within or randomized between subjects design, such that
the same or comparable individuals also completed the math task under no (or different)
cognitive load. Experimental manipulation of load allows for comparisons on arithmetic tasks
with either participants’ baseline performance or the performance of a randomized control
condition. Fourth, to be included in the database, studies had to report sufficient statistical
information to enable the computation of an unstandardized effect size (i.e., group RT means).
When information was not directly presented in a published manuscript, we asked authors whose
papers were published within the last ten years for the data (n=5) – of which we received data
from 2. Fifth, studies must only include participants who are children above the age of 5 or
adults below the age of 65. Dual-task literature generally excludes pre-school-aged children as
well as adults over the age of 65, because of the difficulty in obtaining reliable estimates within
these age groups (most data are collected from undergraduate or middle-adult samples). Lastly,
we excluded studies that used only participants who had been identified as having a learning
disability or special need prior to participating in the study.
The literature search yielded 55 records after abstract screening. Of these, 20 were excluded
during the full text screening, because cognitive load was not experimentally manipulated.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 16
During final assessment for eligibility, 5 were excluded because studies included participants
with either a math learning disability or autism spectrum disorder. Three studies were excluded
because the primary task was a working memory task rather than an arithmetic task. Six studies
were excluded due to lack of available reaction time data. In total, 21 eligible papers containing
400 effect sizes from 51 unique samples, obtained from 1,049 individuals, were included in these
analyses.
Coding
The following variables were coded for the dataset. Study characteristics included: (a)
whether the study design was within or between subjects, (b) a unique identifier for the
experiment number, because some studies included multiple experiments with different samples
or more than two conditions within the same experiment, (c) descriptions of the arithmetic and
working memory tasks, (d) the arithmetic task type (i.e. exact addition, approximate addition,
exact subtraction, exact multiplication, addition verification, and multiplication verification), (e)
the number of items per condition, (f) the two conditions being compared (e.g. central executive
cognitive load vs. control), (g) whether authors made a prediction about the significance of a
certain condition, and (h) whether strategies were reported. Sample characteristics included (i)
the mean age of the sample and (j) the type of sample (participants between the ages of 4 and 17
were considered children and participants between the ages of 18 and 65 were considered
adults). Coding for the WM distractor tasks can be found in the online supplemental materials,
Table S3. Statistical characteristics included: (l) control and experimental sample sizes and (m)
RT means and standard deviations to calculate effect sizes. While accuracy data were coded,
they were not included in effect size analyses for 2 reasons: (1) mean accuracies were very high
(around .90, as reported in Table 1) and (2) some studies (n=4) did not report accuracy data by

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 17
condition, but every study included RT means. Analyses lacked sufficient statistical power to
detect statistical interactions between every arithmetic type and every WM load type, so we used
author’s predictions as a proxy to measure the predicted non-additive effects of WM distractor
and arithmetic types. Authors’ predictions were obtained by coding each of the 21 studies’
introduction and methods sections for statements hypothesizing the effect of WM distractor on
arithmetic performance. For example, “Given these arguments, we predict that the CRT-R task
will interfere more with both non-retrieval-based and retrieval-based subtraction problem solving
than the SRT-R task because of its response selection component” (Tronsky, McManus, &
Anderson, 2008, p. 194). These predictions were then dichotomously coded as having
hypothesized effect of WM load or not. The passages justifying each coded prediction are
included in the online supplementary materials, Table S2.
Data Imputation and Transformations
Reaction time is a commonly used dependent variable used to draw inferences about
cognitive processing, but studies often do not account for residual processes involved with
completing RT tasks, such as encoding the item and producing a response (Rouder, 2005). For
example, Geary, Widaman, and Little (1986) estimated that encoding single digits in complex
addition and multiplication problems required approximately 170 ms. To try to account for such
residual processes, 200 ms was subtracted from all RT means prior to analyses. Missing standard
deviations for reaction times that were in studies older than ten years or those which we could
not obtain from the authors were imputed by regressing RT standard deviations on RT means
and the square of RT means. Imputing the missing standard deviations was deemed appropriate
because of the high predictability of standard deviations from a two predictor model:
SDt = b0 + b1Meant + b2Meant2 + et

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 18
SDc = b0 + b1Meanc + b2Meanc2 + ec
In these models, the subscript t indicates values from the experimental conditions across study,
and the subscript c indicates values from the control conditions across studies. The regression
equations for standard deviation used a second order polynomial term to account for the non-
linear association between means and standard deviations across effects. The models explained
the vast majority of variance in both the control and experimental conditions (control: r2 = .92;
experimental: r2 = .84). Scatterplots with best fitting lines and polynomials appear in the online
supplementary materials, Figure S1. In all, 168 (84 in the experimental conditions and 84 in the
control conditions) standard deviations (or 21%) were missing from the data. Only those
standard deviations with an available mean RT (n = 164 standard deviations: 82 in the
experimental conditions and 82 in the control conditions) were imputed, while the remaining 4
standard deviations were left missing. Imputation led to a small number of predicted negative
values of experimental standard deviations (n = 6). We set these values to the minimum observed
(i.e., non-imputed) standard deviation of experimental conditions, which was 58.19 ms. As
explained below, the meta-analysis was re-run with non-imputed values only to test the
robustness of our main findings to these data processing decisions. Initial histograms revealed
the RT data to be positively skewed (Figure S2). Further, there was a substantial relation
between RT means and RT experimental effects (b = .090, se = .023, p < .001; Figure S5, left
panel). In other words, secondary tasks resulted in more absolute slowing for arithmetic tasks
that took longer to complete. In the dual-task arithmetic literature, we found little discussion of
the functional relation between arithmetic speed and the presence of a secondary task.
Specifically, did the secondary task slow each trial by a short but constant amount of time for
participants to encode and rehearse the stimulus, or did the secondary task slow the rate of

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 19
mental arithmetic proportionally, such that more difficult problems would be more slowed than
easier problems? In the former case, one parameter of interest is a constant, which represents the
average amount of time that each trial is slowed. However, if arithmetic slowing is hypothesized
to be proportional to the amount of time the arithmetic task takes to perform in the absence of a
secondary task, a more useful parameter of interest is the percentage by which the average trial is
slowed in the presence of a secondary task. Based on our reading of the literature, authors seem
to have the latter case in mind, whereby keeping information from the secondary task in working
memory slows performance on the arithmetic task. Therefore, we used a logarithmic
transformation in our main specification to better capture the underlying cognitive effects of
secondary tasks. In addition, we also report an analysis of untransformed RT means in an
alternative specification (Table S1, No Log). We performed logarithmic transformations of the
RT data using Method 1 from Higgins, White, and Azures-Cabrera (2008). This method assumes
log-normal distributions with different standard deviations. The approximate transformations
were then converted to log base 10 for interpretability in our analyses by dividing these means
and SD by the constant, ln(10). Following the transformation, we again checked the distributions
of RT means, which were substantially less skewed (Figure S4). Average RTs and effect sizes
were no longer substantially associated (b = .020, se = .019, p = .30; Figure S5, right panel).
Effect Size Calculation
Following Method 1 from Higgins and colleagues (2008), we computed the raw and log
mean differences in the experimental and control conditions. The equation for calculating the
standard error of these differences is shown below for between-subjects designs:
Between: SEdiff =
√(SD
c2/nc + SDt2/nt)

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 20
However, 17 out of the 21 eligible studies were conducted within participants, requiring a
measure of covariance between control and experimental scores to compute the standard error of
these effects. Because this information was not directly reported in most studies, we computed
within-subjects standard errors under three different assumptions of correlated performance
across conditions: r = .2, .5, and .8. These correlations were then each multiplied with the control
and experimental RT standard deviations to create 3 measures of covariance, using the following
equation:
Within: SEdiff =
√(SD
c2/n + SDt2/n – 2*covtc/n)
Analyses
The metafor package in R was used to conduct this meta-analysis (Viechtbauer, 2010). A
multilevel random-effects meta-analysis of these data was used to estimate and account for the
amount of heterogeneity between papers and between different samples included in the same
paper. Effect sizes were modeled as nested within samples, which were nested within papers
across all of these specifications. We performed several sensitivity analyses, including an
alternative estimation strategy of robust variance estimation adjustment of the standard errors
using the robumeta package in R (Park & Beretvas, 2018; Fisher & Tipton, 2015). All
specifications included some of the same characteristics: Nine specifications (1 main, 3
alternatives, 1 PEESE (precision-effect estimate with standard errors) adjustment, 4 robustness
checks) were used in this analyses and are as follows: (1) Main=assumed within subject
correlation=.5, (2) Alt 1=assumed within subject correlation=.2, (3) Alt 2=assumed within
subject correlation=.8, (4) Alt 3=assumed within subject correlation=.5 and RT data restricted to
≤ 5000 ms. Four other alternative specifications used for sensitivity analyses included: (6)
Nonimputed=analyses without SD imputations, (7) RVE=standard errors adjusted using robust

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 21
variance estimation with small sample correction, (8) Adult=analyses included only studies with
adult participants, (9) No Log=same as Main model but without log transformations, and (10)
PEESE=PEESE adjusted model, assumed within subject correlation=.5.. Specifications Main,
Alt 1, and Alt 2 reflected the three assumed correlations used to calculate the standard errors for
within-subject studies. Alt 3 used a subset of the RT data, such that only RTs ≤ 5000 ms were
included for analyses, because several child effect sizes had much larger RT means (Figure S4),
and because of the possibility that cognitive processing lasting longer than 5000 ms could
plausibly be differentially affected by a secondary task. We ran the latter 4 alternatives
specifications using the assumed correlation structure of the Main model as robustness checks.
The Nonimputed models used data without the SD imputations to examine any potential bias
from these estimates. Robust variance estimation (RVE) models were used as an alternative to
multilevel modeling to account for the non-independence of observations within samples and
papers. Finally, Adult models that only contained adult participants were included, because of
the small sample of child effect sizes (n=36) and some effect sizes with unreported participant
age (n=7).
To address the second aim of this meta-analysis, 5 potential moderators of the overall
effect were tested to determine differences between subgroups of samples. The first moderator
examined was participant age (adult vs. child), as a proxy for expertise in arithmetic. The second
moderator was working memory load type: central executive, verbal, visuospatial, or spatial. The
third moderator was arithmetic problem type: addition verification, exact addition, approximate
addition, exact multiplication, exact subtraction, or multiplication verification. The last
moderator was the authors’ predictions of whether load had significant effects on arithmetic
performance (given the very large number of possible interactions between arithmetic problem

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 22
type and working memory load type, this was the method we chose for examining the strength of
evidence for cross-talk). Although hypothesized cases of cross-talk make more complex
predictions, such as interactions among secondary task modality (e.g., visual vs. verbal), type of
arithmetic (e.g., multiplication vs. subtraction), and presentation format (e.g., vertical vs.
horizontal), a complex set of such instances was predicted by authors corresponding to a large
number of parameters to test. Thus, we chose a broader definition of cross-talk captured through
the authors’ predictions instead. The moderators were examined across the nine specifications.
Results
Publication Bias
A funnel plot of the distribution of effect sizes was used as a visual aid to detect
publication bias (Figure 2) (Egger et al. 1997). Effect estimates from studies are plotted against a
precision measure from those studies (e.g. standard error). Estimates of effects from smaller
studies are more variable than those from larger studies leading to larger amount of scatter
towards the base of the plot. In the absence of bias, a symmetrical funnel shape is observed.
However, asymmetrical distribution of points around the average RT effect indicate possible
publication bias. The main specification (assuming a within subject correlation= .5) was used.
The effect estimates (log difference in RT means) were plotted along the x-axis while the
standard errors of the effect estimates were plotted along the y-axis. The vertical line in the
middle of Figure 2 represents the location of the estimated effect of a working memory task on
performance (b = .074). Examination of the dispersion of effect sizes in the funnel plot revealed
some asymmetry – specifically, for smaller studies – suggesting the possibility of some
publication bias.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 23
The PEESE test is a meta-analytic approach to detecting publication bias using
metaregression. PEESE (precision effect estimate of standard error) uses a weighted-least-
squares regression model where the variance (squared standard errors) of each sample is used to
predict the distribution of effect sizes (Stanley & Doucouliagos, 2014). Assuming a true effect,
publication bias is stronger for studies when the standard error is large and weaker when the
standard error is small. The PEESE model revealed a smaller, non-significant effect of a working
memory task on performance (b1 = .031 se = .017, p = .070), and a moderate degree of
publication bias (QM = 24.11, p < .001) (Table S1, PEESE).
Summary Effect Size
The main model specification (assumed within subject correlation=.5) produced a
summary effect of b = .074, z = 5.21, p < .001 across 400 comparisons, which suggests a .074
difference in log RT. An effect of this size corresponds to an 18.7%
2
slowing of performance for
participants in experimental conditions where they are performing dual-tasks compared to
control conditions where arithmetic was tested by itself. This effect remained stable across the
six other specifications including the Nonimputed, RVE, and Adult models (Table 2). However,
as described above the PEESE adjustment decreased the estimated effect to b = .031, z = 1.81, p
= .070, which equates to a non-significant 7.3% decrease in speed. However, this model obscures
important heterogeneity in the estimates, which was explored in the following moderation
analyses.
Moderators
2
18.7% and subsequent percentage effects were calculated by taking 10b where b (e.g. .074) is
the coefficient estimate.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 24
Participant age, working memory task type, arithmetic task type, and authors’ prediction
of whether load would be significant were entered separately as moderators in different models
(Table 3). Results of sensitivity analyses for moderators can be found in the online
supplementary materials (see Table S1). Across most specifications, working memory task type,
arithmetic task type, and authors’ predictions statistically moderated differences in log reaction
times, suggesting that differences in RT performance are dependent on the type of working
memory load (main specification: Q(2) = 796.04, p < .001), the type of arithmetic problem (main
specification: Q(5) = 12.11, p = .033), and authors’ predictions (main specification: Q(1) =
265.14, p < .001). Compared to other moderators, working memory load type explained the most
heterogeneity (22% of the total heterogeneity) across effect sizes. This finding was largely robust
across specifications.
Furthermore, dual-tasks involving the central executive appear to have the most impact
out of all load types (CE: b = .146, z = 12.35, p < .001, CI [0.123, 0.169]) (Table 3, Column 1,
WM load type). Indeed, across all specifications working memory load reflected a strong effect
of the central executive (a 40% decrease in speed) with much smaller effects for the other load
types (Table 3, Column 1, WM load type). For example, verbal tasks generated an effect of b = -
.115, z = -27.64, p < .001, CI [-0.123, -0.107] in relation to the intercept (CE); meaning an effect
of verbal distractors on performance of .031 or a 7.4% decrease in speed for dual-tasks with
verbal distractors (Table 3, Column 1, WM load type). The strong effect of CE was robust,
showing similar estimates across the sensitivity analyses.
Of the various arithmetic problem types, addition and multiplication verification tasks
had consistent effects on arithmetic performance across most specifications (reference task,
addition verification: b = .077, z = 2.53, p = .011, CI [0.018, 0.137], with more slowing for

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 25
multiplication verification: b = .046, z = 2.99, p = .003, CI[0.016, 0.076]) (Table 3, Column 1,
Arithmetic problem type)
3
. On average, participants performing multiplication verification tasks
were slowed 13.3% more than participants performing addition verification tasks. Although
these differences were smaller than the differences among working memory load types, the effect
of arithmetic problem type was robust to the inclusion of controls for working memory load type
(addition verification: b = .131, z = 5.50, p < .001, CI [0.084, 0.178], multiplication verification:
b = .044, z = 2.99, p = .003, CI [0.015, 0.074]; Table 3, Column 1, WM load + arithtype).
Another metaregression model was analyzed to test whether differences in RT
performance were driven by authors’ predictions for significance, which measured the predicted
non-additive effects of WM distractor and arithmetic types. This analysis revealed a highly
significant effect of authors’ predictions across most specifications (b = .072, z = 16.28, p < .001,
CI [.063, .081], Q(1) = 265.14; Table 3, Column 1, sig predict). On average, authors’ predictions
predicted an 18.0% decrease in performance between experimental and control conditions in
dual-task studies. To further test whether predicted effects were driven by main effects of WM
load and arithmetic task types, another metaregression model was estimated using authors’
predictions while controlling for WM load and arithmetic task type. These analyses revealed a
smaller but usually still significant effect of authors’ prediction of significance controlling for
WM load type and arithmetic task type (b = .023, z = 4.58, p < .001, Q(8) = 820.41, p < .001;
Table 3, Column 1, sig predict + WM + arithtype). Expressed differently, the authors’
predictions were associated with a 18.0% decrease in RT performance, but this dropped to 5.3%
after controlling for WM and arithmetic task types, suggesting that authors’ predictions about the
3
However, it should be noted that both verification tasks were statistically significant only after
they were log-transformed (Table S1, Column 6, Arithmetic problem type).

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 26
effect of load likely contribute somewhat to slower arithmetic performance, but this is somewhat
confounded with the type of working memory and arithmetic tasks. Importantly, the effect of
unpredicted effects of CE distractors on addition verification tasks (b = .107, z = 4.33, p < .001,
CI [.058, .156]) was still significant and larger than the remaining effect of authors’ predictions
with a 27.9% slowing in arithmetic performance (Table 3, Column 1, sig predict + WM+ arith
type).
Discussion
We conducted a meta-analysis for the purpose of assessing how robust the speed of
solving arithmetic problems is affected by changes in WM resources as predicted by dual-task
studies. Consistent with our predictions, we found strong evidence for the influence of
performing a secondary WM task on arithmetic performance. The main effect of WM distractors
on arithmetic performance in dual-task studies was robust across all 9 of our specifications.
These findings suggest that arithmetic performance relies heavily on WM, specifically central
executive resources.
Among the moderators, working memory load type was the most substantial moderator
of performance decrements, followed by the type of arithmetic operation, and authors’
predictions. Tasks taxing the central executive specifically incurred greater decrements in
performance than any other WM load type, indicating the importance of considering the general
cognitive complexity of the secondary task when predicting its influence on primary task
performance. This finding was consistent with previous literature citing the overall importance of
the central executive/executive functions to working memory (Engle, 2002; Engle & Kane,
2004). Assuming the central bottleneck theory to be correct, these more difficult cognitive loads
may not be competing for shared WM resources but rather there is delay in preparation or switch

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 27
to arithmetic processing. In comparison, the PL and VSSP tasks showed much smaller impacts
than CE tasks. Consequently, the parallel processing theory is not entirely ruled out, as the
impact of PL and VSSP tasks would imply processing interference within-modality. Of course,
the larger effects of CE tasks may also be indicative of a cognitive bottleneck. The null effect of
age was somewhat surprising given its prevalence in the literature. However, it should be noted
that the number of child participants was quite small compared to adults (see Table 1).
Furthermore, both distractor and arithmetic tasks did vary in the number of observations across
our models which led some categories to have higher standard errors (see Table 3), so these
estimates are less precise. Variability in precision across estimates would imply that greater
scrutiny is required for the categories with fewer observations. For example, VSSP secondary
tasks had been used in fewer studies, and therefore estimates are less precise for this category.
However, despite a higher standard error in our analyses, the VSSP load estimate was still 2
standard errors below the estimate for CE load (Table 3, Column 1, WM load type).
Importantly, this meta-analysis’s finding that CE tasks have a much greater impact on
arithmetic processing than other WM load types appears to contrast with findings from prior
meta-analyses based on correlations between working memory and math tasks. These prior meta-
analyses reported very similar correlations across working memory facets (Friso-van den Bos,
van der Ven, Kroesbergen, & van Luit, 2013; Peng, Barnes, Namkung, & Sun, 2015).A possible
explanation for this discrepancy is that factors common to working memory tasks, such as
maintaining a constant memory of a single element (number or letter) or an element’s position in
space (carrying values or grids) inflate the correlations between arithmetic performance and
visuospatial and phonological working memory. It may also be possible that the specific
encoding behind similar modality tasks, especially subcomponent WM tasks, are being

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 28
suppressed by their relation to the central executive. For example, performance on a subtraction
task following a VSSP matrix task may be impacted by need to switch between tasks in addition
to requiring similar resources. Dual-task measures can be viewed as forms of inhibition or
shifting tasks, which may indicate that the CE is being used in addition to the PL or VSSP.
Future work might apply methods that attempt to make the magnitudes of effects from
correlational and experimental studies more directly comparable by using regression-adjusted
estimates and intervention effect sizes (e.g., Bailey, Duncan, Watts, Clements, & Sarama, 2018)
to studies of working memory and arithmetic to test this prediction directly.
We attempted to estimate cross-talk or the non-additive effects of WM distractors on
arithmetic performance by coding authors’ predictions of the effect of specific WM secondary
tasks on arithmetic performance. The effects of these predictions were nonzero, but after
accounting for the main effects of WM and arithmetic load type, the author-predicted effects
were not substantially larger than the non-predicted effects. Even these may be somewhat over-
estimated, because almost all studies were conducted before study preregistration was
encouraged in psychology. Thus, it is possible that these predictions could have been changed
over the course of data-collection or through pilot testing, inflating the estimated effect of cross-
talk when operationalized as authors’ predictions. We realize that research teams with more
specific hypotheses about cross-talk may prefer to code predicted effects in a different way. Our
hope is that, by publishing the meta-analytic database, this will allow others to test these
hypotheses in future work. Taken together, while having to process similar modalities in mental
arithmetic may explain some of the underlying processes behind mental arithmetic, general
structural limitations brought on by the general demands of each distracting task may deserve
additional scrutiny, relative to the cross-talk effects often predicted in this literature.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 29
Notably, our preliminary analyses indicated that the detrimental effects of secondary WM
tasks on performance were strongly associated with the average RT on the arithmetic task. In
other words, RT in arithmetic tasks increased more under more difficult secondary WM tasks.
This finding mirrors Salthouse’s (1988) findings that aging-related performance deficits are
higher on more complex cognitive tasks. Our findings also provide convergent evidence for
Salthouse’s theory that changes in general cognitive resources exert the largest effects on
complex cognitive tasks. Specifically, in this meta-analysis, for participants of the same age
taking the same tests on the same day, the proportionality of WM demands is larger for more
complex arithmetic tasks when using an experimental research design.
Limitations & Future Directions
One of the key limitations was that our analyses lacked sufficient statistical power to
detect smaller effects of the interactions between arithmetic and WM load types, prompting us to
use authors’ predictions as a proxy for cross-talk. This may hide important heterogeneity in
cross-talk effects, with some being substantial and others null. Further, as noted, a moderate
degree of publication bias was present in the eligible studies; this may have inflated the apparent
effect of authors’ predictions.
As stated previously, accuracy data were not always available and mean accuracy scores
were consistently high leading us to solely use reaction time data in our analyses. We recognize
that there is a speed-accuracy trade-off meaning participants will sacrifice time in order to
correctly answer questions or vice-versa. For example, Kalaman & LeFevre (2007) reported
more errors in two-digit plus two-digit addition with carrying vs. no carrying but found no
significant differences in speed. These findings suggest that such speed-accuracy trade-offs in

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 30
dual-task studies could potentially reduce an estimated effect of working memory on mental
arithmetic based on RT data alone.
We recognize that Baddeley’s multicomponent model of WM is one of several models
used to describe the relations between memory processes. This model is commonly used in the
arithmetic dual-task literature; thus, we chose to use similar terminology as it more closely
aligned with those designs. However, more recent research (Friso-van den Bos, van der Ven,
Kroesbergen, & van Luit, 2013; Christophel et al., 2017) as well as our own findings suggest that
the use of this model in the dual-task literature lacks appropriate discussion of the central
executive’s role interacting with the more specialized domains. Moreover, other models of WM
have proposed promising alternative perspectives on the role of WM and executive functioning
(see Miyake et al., 2000; Diamond & Lee, 2011; Engle, 2002). These models were introduced
previously and most if not all would point to the strong influence of general cognitive ability
within EF or central executive functions driving dual-task performance rather than specific
modalities highlighted in Baddeley’s model. Indeed, most would argue that dual-tasks involve
quickly switching between mental tasks. Results are also consistent with Engle’s (2002) model
which posits attention rather than capacity as the limiting factor in performance. This model
would predict dual-task performance to be reliant on more domain-general processes rather than
domain-specific components. Altogether, many other perspectives of EF and working memory
are in line with our conclusions about the importance of central executive functions in mental
arithmetic.
Most of the studies in our sample did not report data on participant strategy use.
Aggregating reaction time data across different arithmetic strategies obscures information about
the cognitive processes underlying performance (Siegler, 1987). Because both the frequency and

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 31
efficiency of arithmetic strategy use are correlated with working memory capacity (Bailey,
Littlefield, & Geary, 2012; Geary et al., 2004, 2007), understanding whether working memory
distractors influence performance via strategy changes or slowing within strategies would be
theoretically useful.
Because identical tasks were generally not used across age groups, our analyses may have
been unable to detect a moderating effect of age, despite the strong support for this prediction in
the literature. Larger dual-task studies with child participants will be required to test for
hypothesized developmental differences in the nature of working memory effects on arithmetic
performance (Meyer, Salimpoor, Wu, Geary, & Menon, 2010; for review, see Anderson, 1987).
Because cognitive load was not matched across primary arithmetic or secondary WM
tasks types, the moderating effects of these task types may reflect some combination of the kinds
of demands and the magnitudes of demands of different tasks. Based on descriptions of the
different WM tasks, we suspect the CE tasks differed in their magnitudes of general cognitive
demands, but future work that experimentally manipulates the magnitude of demands in the
secondary tasks would be useful for testing the importance of this construct directly. For
example, a recently used approach of systematically equating demands of WM and arithmetic
tasks through the use of an adaptive psychophysical staircase to determine appropriate span sizes
per individual (Cavdaroglu & Knops, 2017) is a useful model for separating the effects of cross-
talk from general task demands.
Finally, like the previous correlational work on arithmetic and WM, the exact constructs
being measured or manipulated in the dual-task literature are not wholly clear and warrant
further attention. For example, it is not clear whether variations in working memory task type or
complexity within individuals are qualitatively similar approximations of between-individual

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 32
differences in working memory capacity. Fully reconciling these two literatures would require
more precise models of the processes and parameters underlying both the correlations and dual-
task effects.
Conclusion
Our meta-analysis indicates that the dual-task literature provides strong evidence that
mental arithmetic relies on working memory resources. We hope that further work will attempt
to build on these findings by both attempting to quantify the underlying domain-specific and
domain-general effects of working memory on arithmetic and by building quantitative models
that will reconcile the discrepancy between correlational and experimental literature.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 33
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WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 44
Table 1
Descriptive Statistics of Analysis Variables
N
Count
Age
400
Adult
359
Child
36
NA1
7
Within vs. Between Subjects
400
Within
328
Between
72
Working Memory task type
400
CE
169
Verbal
178
VSSP
53
Arithmetic task type
400
Add verification
83
Approximate addition
8
Exact addition
211
Exact multiplication
22
Exact subtraction
22
Mult verification
48
NA
6
mean
sd
Sample Size
20.8
13.1
Accuracy
Experimental
.89
.10
Control
.94
.05
RT
Experimental
3014
2894
Control
2360
2657
RT SD
Experimental
2038
2261
Control
1263
1577
Log RT
Experimental
3.28
0.32
Control
3.19
0.31
Log RT SD
Experimental
0.19
0.08
Control
0.17
0.06
Number of observations
N=400
Note. N is number of effects. Frequencies calculated from non-
missing RT data. 1 One study did not report age of participants

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 45
Table 2
Main effects by specifications
k
b
10b
Q
QE
QM
Main
400
.074***
1.187
3847.37***
(.014)
Nonimputed
318
.073***
1.183
3586.29***
(.016)
RVE
400
.091***
1.232
(.006)
Adult
357
.077***
1.193
3425.04***
(.014)
Alt1
400
.074***
1.187
2544.01***
(.014)
Alt2
400
.074***
1.185
8667.91***
(.015)
Alt3(subset)
339
.077***
1.193
2809.28***
(0.013)
No Log
400
318.609***
2909.20***
(74.451)
PEESE
400
.031
1.073
3606.48***
24.11***
(0.017)
Note. SE in parentheses. QE = test for residual heterogeneity. QM = test of
moderator. Main: assumes within subject correlation = .5, Nonimputed: Without SD
imputations, assumes within subject correlation= .5, RVE: Robust Variance
Estimation for small samples (used with Main specification), Adult: subset of data
that excludes child and no age data, Alt1: assumes within subject correlation = .2,
Alt2: assumes within subject correlation = .8, Alt3: Subset of data where only RT’s
<=5000 ms are kept, PEESE: corrects for publication bias, assumes within subject
correlation = 0.5. No Log: Analyses conducted without log transformation, assumes
within subject correlation =.5 * p<.05 ** p<.01 *** p<.001

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 46
Table 3 Moderator Analyses of Working Memory and Arithmetic Performance
Main model
Alt 1
Alt 2
Alt 3
k/
Qm
b
k/
Qm
b
k/
Qm
b
k/
Qm
b
Age
393
393
393
339
Intercept(adult)
0.38
.076***
0.34
.076***
0.51
.077***
2.89
.071***
(.016)
(.016)
(.016)
(.012)
child
-.027
-.026
-.031
.085
(.044)
(.044)
(.048)
(.050)
WM load type
400
400
400
339
Intercept(ce)
796.04***
.146***
528.81***
.146***
1775.14***
.145***
586.04***
.140***
(.012)
(.012)
(.012)
(.012)
verbal
-.115***
-.116***
-.113***
-.102***
(.004)
(.005)
(.003)
(.004)
vssp
-.128***
-.129***
-.127***
-.122***
(.007)
(0.008)
(.004)
(.007)
Arithmetic problem type
394
394
394
333
Intercept(add verification)
12.11*
.077*
11.31*
.077*
14.61*
.077*
13.18*
.074**
(.031)
(.030)
(.031)
(.023)
approx addition
-.041
-.038
-.049
N/A
(.080)
(.078)
(.086)
N/A
exact addition
-.001
-.001
-.002
-.012
(.037)
(.036)
(.037)
(.029)
exact multiplication
-.027
-.028
-.026
-.038
(.043)
(.043)
(.044)
(.034)
exact subtraction
-.010
-.010
-.011
-.022
(.043)
(.042)
(.044)
(.034)
multiplication verification
.046**
.046***
.046***
.045**
(.015)
(.015)
(.015)
(.015)
WM+arith type
394
394
394
333
Intercept(ce+add ver)
802.31***
.131***
533.95***
.131***
1783.45***
.130***
592.49***
.125***
(.024)
(.024)
(.024)
(.023)
verbal
-.115***
-.116***
-.113***
-.102***
(.004)
(.005)
(.003)
(.004)
vssp
-.129***
-.129***
-.128***
-.125***
(.007)
(.009)
(.005)
(.008)
approx addition
.013
.016
.005
N/A
(.053)
(.051)
(.059)
N/A
exact addition
.024
.026
.022
.029
(.029)
(.029)
(.029)
(.029)
exact multiplication
-.010
-.013
-.007
-.012
(.033)
(.033)
(.033)
(.032)
exact subtraction
.006
.004
.008
.004

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 47
(.032)
(.032)
(.033)
(.031)
multiplication verification
.044**
.044**
.046**
.044**
(.015)
(.015)
(.015)
(.015)
Sig Predict
400
400
400
339
Intercept(no)
265.14***
.021
170.10***
.021
611.36***
.022
152.63***
.013*
(.014)
(.014)
(.014)
(.013)
yes
.072***
.072***
.071***
.061***
(.004)
(.006)
(.003)
(.005)
Sig + WM + arith type
394
394
394
333
Intercept (no+ce+add ver)
820.41***
.107***
542.70***
.108***
1838.28***
.104***
596.10***
.112***
(.025)
(.025)
(.025)
(.024)
sig(yes)
.023***
.021***
.024***
.012*
(.005)
(.006)
(.003)
(.005)
verbal
-.106***
-.108***
-.104***
-.097***
(.005)
(.006)
(.003)
(.005)
vssp
-.117***
-.118***
-.114***
-.118***
(.008)
(.009)
(.005)
(.008)
approx addition
.013
.017
.006
N/A
(.056)
(.053)
(.062)
N/A
exact addition
.026
.028
.024
.030
(.029)
(.029)
(.030)
(.029)
exact multiplication
-.007
-.010
-.004
-.010
(.033)
(.033)
(.034)
(.032)
exact subtraction
.009
.007
.011
.005
(.033)
(.033)
(.034)
(.031)
multiplication verification
.045**
.044**
.046**
.044**
(.015)
(.015)
(.015)
(.015)
Note. Model names are bolded. The variable names following each model name are the moderators included in that model. The first row of each model represents the
estimated effect at the reference group, which is given in parentheses after Intercept, for each model. Standard errors are in parentheses. k = number of effect sizes included in
the model; Qm = Q test for heterogeneity explained by the predictors (total heterogeneity for each specification appears in Table 2), ce=central executive, vssp=visuospatial
sketchpad, approx=approximate, add ver=addition verification, sig(yes)=author made a prediction on significance. Main: assumes within subject correlation = 0.5; Alt1:
assumes within subject correlation = 0.2; Alt2: assumes within subject correlation = 0.8; Alt3: Subset of data where only RT’s <=5000ms are kept, assumes within subject
correlation=.5

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 48
Figure 1. Flowchart of Literature Search and Screening Process.

WORKING MEMORY AND ARITHMETIC IN DUAL TASK STUDIES 49
Figure 2. Funnel Plot of Log RT Differences.
Note: Standard errors were calculated using the assumptions from the main specification, assuming a within subject correlation=.5.