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Cognition and Emotion
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Worry and working memory influence each
other iteratively over time
Kelly Trezisea & Robert A. Reevea
a Melbourne School of Psychological Sciences, University of Melbourne,
Melbourne, VIC, Australia
Published online: 03 Feb 2015.
To cite this article: Kelly Trezise & Robert A. Reeve (2015): Worry and working memory influence each other
iteratively over time, Cognition and Emotion, DOI: 10.1080/02699931.2014.1002755
To link to this article: http://dx.doi.org/10.1080/02699931.2014.1002755
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Worry and working memory influence each other
iteratively over time
Kelly Trezise and Robert A. Reeve
Melbourne School of Psychological Sciences, University of Melbourne, Melbourne, VIC, Australia
(Received 22 August 2014; accepted 22 December 2014)
Little research has examined whether the relationship between working memory (WM) and anxiety/
worry remains stable or changes over time; and, if changes occur, the factor(s) influencing change.
Claims about influence are typically inferred from data collected at a single time point, and may
misrepresent the nature of influence. To investigate the iterative influence of WM and Worry and/or
vice versa, 133 fourteen-year-olds completed WM and Worry measures several times over the course
of a single day as they prepared for a math test. We used a bivariate latent difference score model to
analyse possible changes in WM–Worry relationships. The best fitting model indicated high Worry
predicts decreases in WM, and low or decreased WM predicts increases in Worry; high WM with
low Worry predicts accurate problem solving; low WM with high Worry predicts inaccurate problem
solving. Findings show relationships between WM and Worry varies considerably over a single day,
and initial disadvantages become worse over time.
Keywords:Working memory; Math anxiety/worry; Latent difference score model; Stability and
change; Cognition–emotion interactions.
INTRODUCTION
Most research that has studied the impact of
emotion on cognitive functions (or vice versa) has
examined the relationship at a single time point;
and little research has examined how these factors
might affect each other over time. Consider
studying for a math test. How do emotional states
(anxiety/worry) and cognitive functions [working
memory (WM)] change over a period of study? Do
they remain stable, or do they affect each other
iteratively over time? Practice may elicit anxiety/
worry, which in turn may impair cognitive func-
tions (e.g., WM); and impaired cognitive functions
may in turn increase anxiety, further diminishing the
effectiveness of cognitive functions. To investigate
the impact of emotion (worry) on cognition func-
tions (WM) (and vice versa), weinvestigate the ways
in which math worry and WM affect each other
iteratively over the course of single day as
14-year-olds study for a math test. We use a bivariate
latent difference score model (BLDSM; Grimm,
An, McArdle, Zonderman, & Resnick, 2012),
which allows examination of the interplay between
two constructs, to characterise the likely iterative
impact of WM and math worry on each other.
Research has shown associations between math
anxiety (MA), WM and math problem solving
Correspondence should be addressed to: Robert Reeve, Melbourne School of Psychological Sciences, University of Melbourne,
Melbourne, VIC, Australia. E-mail: r.reeve@unimelb.edu.au
© 2015 Taylor & Francis 1
COGNITION AND EMOTION, 2015
http://dx.doi.org/10.1080/02699931.2014.1002755
Downloaded by [Robert Reeve] at 12:13 05 February 2015
ability (Ashcraft & Kirk, 2001; Hoffman, 2010;
Mattarella-Micke, Mateo, Kozak, Foster, & Bei-
lock, 2011; Miller & Bichsel, 2004). However, the
direction of influence between MA, WM and
math ability is unclear. A commonly held view is
that MA increases cognitive demands, which
reduces WM capacity, which in turn, impairs
problem solving ability (Ashcraft & Kirk, 2001;
Derakshan & Eysenck, 2009; Faust, 1996; Miller
& Bichsel, 2004). It is of course possible that
deficits in math ability per se are responsible for
poor math problem solving and, in turn, are linked
to increases in MA (Ma & Xu, 2004; Maloney,
Ansari, & Fugelsang, 2011; Maloney, Risko,
Ansari, & Fugelsang, 2010).
Association between MA and WM has been
established by comparing individuals’WM and
self-reported MA (e.g., Ashcraft & Kirk, 2001;
Hoffman, 2010; Maloney et al., 2010; Miller &
Bichsel, 2004; Wu, Barth, Amin, Malcarne,
Menon, 2012). On the basis of current findings,
the direction of influence between MA and WM
cannot be determined. Ashcraft and Kirk (2001),
for example, found that, compared to students
with low MA, students with high MA had
reduced WM, were slower and less accurate
solving mental addition problems, and also
showed greater impairments solving difficult pro-
blems. Hoffman (2010) found that after control-
ling for self-efficacy and WM, MA was only
related to accuracy solving difficult arithmetic
problems; furthermore, after controlling for WM,
MA was not related to enumeration ability. These
findings suggest that both MA and WM are likely
related to math problem solving, but the precise
nature of the relationship is unclear. We suggest it
may be possible to better characterise the MA/
WM/problem solving relationship by examining
changes in MA and/or WM over time.
Cognitive functions (e.g., attention and WM)
may affect or be affected by MA. Findings
from non-math research show that cognition
(e.g., WM) may serve as a regulator or distracter
moderating emotion (e.g., anxiety; Hofmann,
Friese, Schmeichel, & Baddeley, 2011; Ochsner
& Gross, 2005; Van Dillen, Heslenfeld, & Koole,
2009). For example, Van Dillen and Koole (2007)
found self-reported mood was less negative when
WM load was high compared to when it was low.
Moreover, WM has also been associated with
better emotion regulation ability: individuals with
greater WM capacity are better able to control
intrusive thoughts (Brewin & Beaton, 2002;
Brewin & Smart, 2005) and negative emotions
(Schmeichel & Demaree, 2010; Schmeichel,
Volokhov, & Demaree, 2008). Cognitive regula-
tion of emotion may have implications for indivi-
duals with low WM; poor regulation of emotion
may reflect high MA levels, impairing WM even
more and amplifying math problem solving
disadvantage.
One difficulty interpreting the significance of
MA–WM research findings is that the context
and/or purpose of the research activity is rarely
specified. Different outcomes tend to occur when
individuals complete a task as an end in itself,
compared to completing the task as part of a
meaningful goal (e.g., studying for an exam;
Brown, 1978,1992). Moreover, since MA–WM
relations are often examined at a single time point,
it is difficult to distinguish between trait and state
properties of MA. Test circumstance may interact
with WM and MA (and vice versa) and influence
outcomes. Indeed, there is evidence of short-term
changes in WM–MA relationships. In an adoles-
cent sample, Trezise and Reeve (2014a) found that
students initially belonging to a high WM-low
math Worry group remained in the same group
over a single day. Conversely, students who
initially belonged to a moderate WM or high
Worry group were likely to move to a lower WM
group. Trezise and Reeve demonstrated patterns
of stability/change in WM–MA relationships, but
they did not examine the change process. Conse-
quently, the Trezise and Reeve study was unable
to to characterise the effects between WM and
Worry (i.e., the direction of influence between
WM and Worry). It is possible that one factor
(WM or MA) may affect the other (MA or WM),
or there may be a mutual influence relationship
between WM and MA over time.
By early adolescence, students have experienced
solving arithmetic problems for many years, but are
likely to have only recently encountered algebra.
TREZISE AND REEVE
2COGNITION AND EMOTION, 2015
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Algebra is troublesome for many students (Knuth,
Alibali, McNeil, Weinberg, & Stephens, 2005):
compared to arithmetic, it requires abstract reason-
ing, and algebraic competence is claimed to require
formal reasoning ability (Piaget & Inhelder, 1969;
Tolar, Lederberg, & Fletcher, 2009). Surprisingly
little is known about the significance of MA and
WM in algebraic problem solving. We suggest a
better understanding of WM–MA interactions
will help characterise the associations between
algebra/math and WM, and between algebra/
math and MA.
The study
In the present study 14-year-olds completed sev-
eral WM and MA tests over the course of a single
day as they studied for an algebra test. The goal is
to examine the ways in which WM and MA affect
each other over the day, and how the change
pattern of influence affect algebraic problem solv-
ing. A dual span task (Conway et al., 2005)
comprised of alphanumeric symbols assessed
WM. An algebraic judgement task assessed task-
specific MA/worry. We focused on math Worry
since it has been identified as the cognitive
component of state anxiety (Eysenck, Derakshan,
Santos, & Calvo, 2007; Liebert & Morris, 1967),
and is thought to affect WM (Eysenck et al., 2007;
Hayes, Hirsch, & Mathews, 2008; Owens,
Stevenson, Hadwin, & Norgate, 2012; Trezise &
Reeve, 2014b). According to attentional control
theory (Eysenck et al., 2007), anxiety affects
cognitive processing because state anxiety increases
worry. Moreover, trait-worry is thought to be the
component of anxiety that impairs the ability to
filter threat-related information from WM (Stout,
Shackman, Johnson, & Larson, 2014). Problem
solving was assessed by mentally solving algebraic
problems.
We used a BLDSM (dual change score model-
ling) to examine changes in WM–worry relation-
ships. BLDSM is a form of structural equation
modelling that allows a test of influence between
two variables over time (Grimm et al., 2012;
McArdle, 2009). BLDSM tests whether change
(e.g., increase in Worry rating between two adjacent
time points) is predicted by previous levels (e.g.,
Worry rating at baseline). Furthermore, compar-
ison of models can be used to assess the nature of
the relationship within and between variables.
BLDSM is designed to model change parameters,
controlling for prior performance and change, and
in doing so we can examine changes in WM and/or
worry, and whether previous performance/change
predicts changes. Grimm et al. (2012) extended
the model to enable examination whether change
(e.g., increase in Worry rating between Time = 2
and Time = 3) is predicted by previous level
(e.g., Worry rating at Time = 2) and previous
changes (e.g., increase in Worry rating between
Time = 1 and Time = 2).
Our interest is whether a unidirectional model
(i.e., Worry influences WM, or vice versa, and
does not change outcomes) or a bidirectional
model best fits the data. Since previous research
has not examined bidirectional influence over
time, we cautiously suggest several outcomes. If
the model is consistent with the claim that high
Worry reduces WM capacity, then an immediately
preceding Worry score, or change in Worry,
would predict a change in subsequent WM
(Hypothesis 1a). If greater WM capacity reduces
Worry, then previous WM capacity, or change in
WM capacity, would predict subsequent change in
Worry (Hypothesis 1b). If both of the aforemen-
tioned relationships occur, it would suggest a
bidirectional relationship between Worry and
WM (Hypothesis 1c). Further, if changes (to
WM or Worry) are predicted only by previous
Worry score or WM capacity, it would suggest
that change occurs, but the nature of the change
relationship remains constant (Hypothesis 2a). In
other words, if WM or Worry changes are
predicted by previous changes, it would suggest
that the nature of the WM–Worry relationship
changes dynamically over time (Hypothesis 2b).
Consistent with previous research, we expect that
both large WM capacity and low Worry would
predict good problem solving accuracy, and
that both small WM capacity and high Worry
would predict poor problem solving accuracy
(Hypothesis 3; Ashcraft & Kirk, 2001; Raghubar,
Barnes, & Hecht, 2010).
WORRY AND WM INFLUENCE OVER TIME
COGNITION AND EMOTION, 2015 3
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MATERIALS AND METHODS
Participants
One hundred forty-three 14-year-olds attending
mixed gender independent high schools in an
Australian city, participated in the study. Indivi-
duals who performed below chance, or showed
random responding at Time 1 were excluded
(n= 17), leaving a final sample of 133 participants
(M= 14 years 4 months, SD = 4 months),
comprising 97 boys and 40 girls. (The differences
in gender numbers reflected enrolments at the
schools in which the research was conducted.)
Common to Australian urban high schools, the
sample comprised students from diverse multi-
cultural and socio-economic backgrounds. Accord-
ing to school personnel, none of the participating
students had identified learning difficulties, and all
had normal or corrected to normal vision. The
research was approved by, and conducted in
accordance with the requirement of, the authors’
University’s Human Research Ethics Committee.
Approximately 95% of students invited to particip-
ate in the research did so.
Procedure
Students completed three algebraic tasks: (1)
Algebraic WM, (2) Algebraic judgement/Worry
(Worry) and (3) Algebraic Problem Solving. Tasks
were completed in two sessions in a single day (see
Figure 1 for test sequences). As WM is thought to
directly impact math reasoning, we deemed it
important to establish WM abilities at the begin-
ning of each test session, and immediately prior
to the problem solving task. In Session 1 the task
order was: (1) WM, (2) Worry, (3) WM, (4)
Worry and (5) WM. In Session 2 the order was:
(1) WM, (2) Worry, (3) WM and (4) algebraic
problem solving test. Students were informed they
would complete a sequence of activities over two
sessions, and there would be a test at the end of
the day. Students completed the initial session first
thing in morning, following which they attended
normal (non-math) classes; in the afternoon they
completed the second session. All tasks were
completed on 13′′ laptop computers running
Inquisit 3.0.6.0 software (2011). Each session
lasted 40 minutes.
Materials
The Algebraic WM was based on Turner and
Engle’s(1989) operation span task, modified to
use alphanumeric symbols and algebraic state-
ments. The task was designed to assess the ability
to process and remember alphanumeric symbols.
WM has been conceptualised as general abilities
(e.g., Lovett, Reder, & Lebiere, 1999) and as a set
of domain-specific constructs (Engle, 2010;
Miyake & Shah, 1999). Engle (among others),
for example, argues that WM involves “as many
domain-specific stores as there are different ways
of thinking”(p. S17, 2010). Indeed, evidence
suggests that math problem solving ability may
comprise a specific domain for which domain-
general measures may not capture ability differ-
ences (see LeFevre, DeStefano, Coleman, &
Shanahan, 2005; Raghubar et al., 2010). In the
present study we employed alphanumeric symbols
to assess domain-relevant WM. The task required
students to both appraise algebraic statements, and
remember alphanumeric symbols (see Figure 2).
Each trial consisted of an algebraic statement
appraisal, and the presentation of an algebraic
symbol to remember. Algebraic statement appraisal
Figure 1. Study design and task sequence for testing sessions one and two. Dark squares indicate task completed. Task sequence is left to
right.
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4COGNITION AND EMOTION, 2015
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required students to decide whether an algebraic
statement correct or incorrect (e.g., 3y + 2 = 20;
y = 2) by pressing a key on their computer (students
were given 15 seconds to respond). An alphanu-
meric symbol then appeared on the screen
(e.g., “4x”) for 1600ms. After n-trials, a 3 × 4
matrix consisting of 12 alphanumeric symbols
appeared on-screen (see Figure 2).
Students received instructions and training for
the appraisal and span components separately and
then completed two practice sets, prior to Session 1.
For the appraisal component, students were
instructed to judge the accuracy of a possible answer
to an algebraic equation, rather than solve an
algebraic equation. For the span component, stu-
dents were instructed to remember each alphanu-
meric symbol in the order they were presented.
Their task was to select the symbols from the matrix
that had appeared in the trials, in the order
presented. Students were instructed to do their
best on the appraisal and span components; they
were not informed about the researchers’interests.
Students completed two sequences of two-,
three-, four-, and five-trial sets (i.e., 2 × 2, 3, 4
and 5 trial sets; see Figure 2). Order of presentation
of set length was randomised. We were interested in
the proportion of alphanumeric symbols
correctly recognised. We used the partial scoring
procedure suggested by Redick et al., (2012) and
Conway et al. (2005), rather than ‘absolute’
maximum list span score typically used in develop-
mental research. Psychometric properties (e.g.,
test–retest and internal consistency) favour partial
scoring, rather than absolute scoring in complex
span task (see Redick et al., 2012).
The Algebraic Worry task was used to assess
worry students experienced while making algebraic
judgements. Students were shown pairs of algebra
equations of the form mx+c
1
=c
2
and judged
whether the value of the variable (x) in the two
equations had the same value (equivalent equa-
tions), or different values (non-equivalent equa-
tions; see Figure 2). The design of the judgement
procedure was based on Canobi, Reeve, and
Pattison’s(1998, 2003) arithmetic equation judge-
ment task which examined children’s ability to
“notice”the presence/absence of “commuted rela-
tionships”in two equations. Following each
judgement, the Faces Anxiety Scale (Bieri, Reeve,
Champion, Addicoat, & Ziegler, 1990) appeared
on the screen and students rated their Worry
experienced while making their judgement. The
Faces Anxiety Scale comprised six faces depicting
increasing Worry (neutral to extreme Worry) and
has been used to assess academic-related worry in
previous research (Punaro & Reeve, 2012; Trezise
& Reeve, 2014a).
Students completed training and practice in
both the judgement and worry components of the
task. For algebraic judgements, students were
Figure 2. Example stimuli for the (1) WM, and (2) Worry tasks. A two trial sequence on the algebraic WM task is depicted: two trials
each containing a processing (correct/incorrect) and memory (alphanumeric symbol) phase, and alphanumeric symbols on a recognition matrix.
Algebraic Worry task involves a judgement about the equivalence of two equations, followed by a worry rating.
WORRY AND WM INFLUENCE OVER TIME
COGNITION AND EMOTION, 2015 5
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instructed the task was to compare two equations
and indicate whether the value of xwas the same,
and they did not need to solve the equations.
The judgement component required an ability to
recognise commutative relationships. Students
received training on how to use the worry scale:
they were given non-math and math examples and
practice problems. A variety of situations that may
elicit different worry levels were described, and
students discussed which worry face they would
select to correspond with that worry level. Stu-
dents were instructed to evaluate if they felt calm,
a bit worried or extremely worried or anxious while
making their judgement. They were explicitly
instructed not to rate their confidence in their
answer, but to evaluate their worry/affect while
making their judgement. Students were not
informed of their judgement accuracy. In Session
1, 16 equations pairs were rated each testing time
point and in Session 2, 20 pairs were rated.
In the Algebra problem solving task students
solved linear algebra equations. The task com-
prised eight easy and eight hard equations pre-
sented in random order. The aim of the problem
solving task was to assess students’ability to solve
algebraic equations. The structure of the equations
was based on the format that students encounter
in their math classes. Difficulty was varied using
known properties of equivalence relationships
(see Humberstone & Reeve 2008; Trezise &
Reeve, 2014b). Students were not informed about
the form of the algebraic problem solving test prior
to attempting it. Easy equations comprised three-
terms problems with a variable, a coefficient and a
constant on the left side of the equivalence sign,
and a constant on the right (e.g., mx+c
1
=c
2
).
Hard equations comprised three-term problems,
with a constant on the left, and a variable, a
coefficient and a constant on the right side of the
equivalence sign (e.g., c
1
=m
1
x+c
2
) (see Alibali,
Knuth, Hattikudur, McNeil, & Stephens, 2007;
Humberstone & Reeve, 2008). Each hard equa-
tion also included negative integers. Solutions
ranged between −9 and +12. Students were given
up to 15 seconds to respond. Correctness was
recorded.
Analytic approach
We investigated the relationship between WM
and Worry using Grimm et al.’s(2012) BLDSM
approach. BLDSM is based on the premise of
identifying the difference between scores on the
same measure over two adjacent time points. In
BLDSM, scores at each time point are referred to
as levels and differences between scores are referred
to as changes. The models examined levels (the
WM capacity or Worry latent score at a single
testing time point), at each time point and change
(Δ) in levels between adjacent time points. For
example, a person who has a Worry value of 2 at
Time 1, and a Worry value of 4 at Time 2, their
ΔT
1
= +2. By separating out the change from
scores, we can examine predictors of change. In
these analyses, we examine how changes in WM
and Worry are predicted by previous levels and
change.
There are two stages in the analysis: first, a
number of models (of increasing complexity)
testing different WM–Worry relationships over
time were run, and the best fitting model of the
data were then identified; second, the best-fitting
model was examined. The aim of testing several
models was to evaluate different combinations of
relationships between WM and Worry, and to
identify the model that best describes the interac-
tions and influences between WM and Worry.
The model that best fitted the data was assessed
by comparing goodness of fit indices based on
likelihood ratio tests (-2LL). Information criteria
indices included the adjusted Bayesian Informa-
tion Criteria (aBIC), a comparative estimation of
best fit, that accounts for degree of parsimony in
the model by penalising the number of parameters
in the model; the standardised root mean square
residual (SRMR), which estimates the average
difference between the sample covariance and
estimated population (model) covariance; and a
chi-square test. For all indicators, a lower informa-
tion criterion indicates better fit; a value less than
.08 is considered a good fit for SRMR (Hu &
Bentler, 1999). We used the aBIC and SRMR
indices as Hu and Bentler (1999) recommend
reporting a comparative fit index and the SRMR
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6COGNITION AND EMOTION, 2015
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to evaluate the best fitting model. We also used
the Satorra-Bentler scaled chi-square test to com-
pare goodness-of-fit between nested models.
An illustration of the predicted model is pre-
sented in Figure 3. Proportion of algebraic terms
remembered was used to indicate observed WM
score, and mean algebraic Worry was used to
indicate observed Worry. The model assumes true
scores of WM and Worry at the first test occasion
(baseline) are represented by latent intercept scores
M[0] and W[0], respectively. Change is represented
by a combination of constant univariate and
bivariate scores (see Figure 3). The constant scores
are: (a) constant latent slope score, estimate for both
WM and Worry, and represent a consistent change
over time (S
M
, WM slope, and S
W
, Worry slope),
and (b) constant additive parameter (α), with
assumed a value of one (Ghisletta & McArdle,
2012).
1
Univariate scores assess within factor
affects (i.e., effects of WM on WM, and Worry
on Worry), in particular (c) proportional change
(β), in which change is predicted by level at the
previous time point (e.g., ΔM
1
regressed on M
1
),
and (d) changes to changes (ϕ), where change is
predicted by changes at the previous time point
(e.g., ΔM
2
on ΔM
1
). ϕwas not estimated for
Worry, because there was only one change score
regression, resulting in a perfect linear dependency
between S
W
and ϕ
W
. Bivariate scores (termed as
coupling scores; Grimm et al., 2012) estimate the
effects of WM on Worry, and of Worry on WM,
and include (e) coupling proportional changes (γ),
where changes are predicted by the level of another
variable (e.g., ΔW
1
on M
1
), and (f) coupling
changes to changes (ξ) where changes are propor-
tional to changes of another variable (e.g., ΔW
1
on ΔM
1
).
A latent math problem solving true score (PS
μ
)
was represented by observed mean accuracy for
easy and hard algebraic problem solving (see
Figure 3). The problem solving true score was
regressed on final time point WM and Worry true
scores (ω). This allowed us to examine how WM
and Worry levels immediately before the problem
solving test, predicted problem solving accuracy.
There have been two suggestions about the
relationships between WM and (math) anxiety
previously: that WM helps control anxiety, and
that anxiety reduces WM (Eysenck et al., 2007;
Hofmann et al., 2011). For these reasons, we
examined models that examined no effect of WM
on Worry, or Worry on WM (Model 1), the effect
of Worry level on WM (Model 2), the effect of
WM capacity on Worry (Model 3) and the effects
of WM capacity on Worry, and Worry level on
WM (Model 4). However, few studies have
examined WM and Worry/anxiety relationships
over time: it is possible that changes in WM/
Worry predict subsequent change (e.g., changes in
WM may affect an individual’s ability to control
emotional responses). Therefore, we also examined
the effects of changes to WM/Worry on sub-
sequent changes. In addition to the effects of level
on change, we also examined models that exam-
ined only effects of WM change on WM, and
Worry change on Worry (Model 5), the effect of
changes to Worry on WM (Model 6), the effect of
changes to WM capacity to Worry (Model 7) and
the effects of WM changes on Worry and of
Worry changes on WM (Model 8).
We examined latent difference models of
increasing complexity using Mplus (Muthén &
Muthén, 2012). The models were constructed
iteratively and represented different WM–Worry
interactions. Eight models of increasing complex-
ity were tested. Figure 3 describes the models. The
four simplest models included change on level
regressions (βand γ), but no change on change
(ϕand ξwere not fitted). Model 1 was a baseline
model with “no coupling”, in which only within
variable (β) relationships (depicted by grey lines in
Figure 3) were tested; the model only examined
the effect of WM level on subsequent WM
change and Worry level on Worry change (i.e.,
the model that WM and Worry do not influence
each other). Model 2 (WM change on Worry)
assessed with within variable relationships in
Model 1 and Worry level predicting WM change
(γ
MW
: solid orange lines); the model tests the effect
of an individuals’Worry on their WM capacity
1
The letters a to f in parentheses represent the aspects of the parameterisation of the models.
WORRY AND WM INFLUENCE OVER TIME
COGNITION AND EMOTION, 2015 7
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Figure 3. Path diagram of a BLDSM with WM (M), Worry (W) and problem solving (PSμ; with easy, E and hard, H). Final levels of WM and Worry are regressed on algebraic
problem solving. There are five measurement occasions for WM, and three for Worry. Unlabeled paths are fixed equal to 1. Observed scores are represented by u; WM latent scores at time t
by M[t], Worry at time tby W[t]. M[0] and W[0] represent intercept (mean at t= 0), and S
M
and S
W
represent slope, for WM and Worry, respectively. Grey lines represent the base model
(Model 1), and are predicted in all models. Solid lines represent variations change on level (Models 2, 3 and 4), broken lines represent change on change (Models 5, 6, 7 and 8). Orange lines
represent the effects of Worry predicting WM. Purple lines represent the effects of WM predicting Worry. Blue lines represent the effects of WM on WM.
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(e.g., whether high Worry reduces WM). Model 3
(Worry change on WM) assessed within variable
relationships from Model 1 and WM level predict-
ing Worry change (γ
WM
: solid purple lines—the
model examines the effect of an individuals’WM
capacity on their Worry (e.g., whether low WM
capacity leads to increases in Worry). Model 4 is a
bivariate coupling model, which assessed within
variable, WM change on Worry, and Worry
change on WM change (all solid lines)—the model
examined whether Worry levels predicts a change
in WM, and whether WM capacity/level predicts a
change in Worry.
The final four models examined both level
(e.g., does WM capacity lead to a change in
Worry?) and change (e.g., does a change to WM
capacity lead to a change in Worry?) predicting
change: they include all change on level relation-
ships (βand γ) and all relationships depicted by
solid lines in Figure 3. Model 5 is a “no change
coupling”model; it includes the relationships
models in Model 4 and within variable change
(ϕ: broken blue lines) relationships (i.e., are
changes in WM predicted by previous changes in
WM, and are changes in Worry predicted by
previous changes in Worry?). In other words, the
no change coupling model assesses the effect of
levels on change (as in Models 4), and the effect of
within variable changes. Model 6 is a WM change
on Worry change model, assessing within variable
change (i.e., Model 5) and Worry change predict-
ing WM change (ξ
MW
: broken orange lines); it
examines the effect of changes to Worry on WM
(e.g., does a decrease in Worry lead to an increase
in WM?). Model 7 is a Worry change on WM
change model, assessing within variable change
and WM change predicting Worry change (ξ
WM
:
broken purple lines); it examines the effects of
changes to WM capacity on Worry (e.g., does an
increase in WM lead to a decrease in Worry?).
Finally, Model 8 is the full model, testing all
within and bivariate level and change relationships
(all solid and broken lines). This model examines
within variable effects of both level and change,
and all bivariate effects of both level and change
(i.e., WM level predicting Worry change, Worry
level predicting WM change, WM change pre-
dicting Worry change and Worry change predict-
ing WM change).
RESULTS
Means and standard deviations for each task, at
each time point are given in Table 1.
Table 2 shows the measures of fit for the
various models. We refer to the models by
the name and number presented in the method.
The top row assesses fit of traditional bivariate
models (Models 1–4), while the bottom row
assesses goodness of fit for the extensions pro-
posed by Grimm et al. (2012) (Models 5–8). We
expected the best fitting model would support
bivariate interactions (purple and orange lines in
Figure 3). The aBIC indicates the WM change on
Worry level (model 2), and the Worry change on
WM change (Model 7) are the best fitting models
Table 1. Means, standard deviations for WM, worry and algebraic problem solving
WM Judgement/Worry
Span Appraisal Accuracy Worry Problem solving
Time M(SD) M(SD) Time M(SD) M(SD) Diff M(SD)
1 .55 (.20) .80 (.16) 1 .69 (.20) 1.45 (1.15) E .53 (.38)
2 .47 (.24) .68 (.26) 2 .67 (.20) 1.54 (1.30) H .20 (.26)
3 .45 (.24) .56 (.34) 3 .66 (.20) 1.37 (1.14)
4 .50 (.25) .66 (.32)
5 .44 (.30) .57 (.36)
Time, Testing Time Point; Diff, Difficulty; E, Easy; H, Hard.
WORRY AND WM INFLUENCE OVER TIME
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(aBIC = –496). Chi-square and SRMR values
indicate the Worry change on WM change
(Model 7), and full bidirectional change coupling
(Model 8) models fit the data best (χ
2
= 75,
SRMR = .08). The Worry change on WM change
model (7) was deemed to be the model of best bit,
as it was supported by all indicators. We also
examined the full model (directional change
coupling, Model 8); parameter estimates were
very similar to that of the Worry change on WM
change model (7), indicating that the full model
did not add to the interpretation of the relation-
ship between WM and Worry.
Table 3 presents the parameter estimates and
standard error of the model. It shows an overall
mean increase in Worry (mean slope). The model
also shows that WM change is negatively pre-
dicted by Worry level (WM coupling change in
Table 3): the higher the Worry level, the larger the
decrease in WM; and Worry levels close to zero
predict stability in WM. Changes to Worry were
negatively predicted by WM level (Worry coup-
ling change in Table 3) and WM change (Worry
coupling change on change in Table 3). Thus the
model predicts the lower the WM level, the
larger the subsequent increase in Worry, whereas
higher WM predicts stability in worry levels.
Furthermore, if WM increases, then Worry will
decrease; and if WM decreases, Worry will
increase.
Problem solving was significantly affected by
both WM and Worry (see Table 3). Algebraic
problem solving was regressed on final scores
of WM and Worry (refer to Figure 3). High
algebraic problem solving accuracy was predicted
by high final WM score, and by low final Worry
score. The random effects, presented at the
bottom half of Table 3, were not statistically
significant, which indicates that random effects
do not affect model interpretation.
The findings are different from previous
research in two ways. First, the model shows that
WM capacity change (and level), predict changes
in Worry. Thus, if WM capacity is a value mat
time t, the changes to Worry will differ depending
on both m
t
and how different m
t
is from the
previous time point, m
t-1
. Second, these changes
to WM and Worry are iterative over time. It is
Table 2. Fit statisitics for traditional BLDSMs, and extensions of BLDSMs jointly fit to data
Levels →Change
12 3 4
Model No coupling Worry →WM change WM →Worry change Bidirectional coupling
aBIC −480.12 −495.91 −479.75 −495.29
χ
2
97.14 79.61 75.78 78.51
df 39 38 38 37
SRMR 0.11 0.10 0.12 0.09
Levels and change →Change (bivariate change coupling)
56 7 8
Model
No change
coupling
Worry change →WM
change
WM change →Worry
change
Bidirectional change
coupling
aBIC −494.91 −493.96 −495.92 −495.57
χ
2
77.16 76.38 74.42 74.04
df 36 35 35 34
SRMR 0.09 0.09 0.08 0.09
Note: Bold text indicates information criteria best fit.
aBIC, adjusted Bayesian Information Criterion; χ
2
= chi-square, SRMR, standardised Root Mean Square Residual.
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possible that changes lessen over time; however,
our model suggests a multiplier effect between
WM and Worry over time.
In summary, the model shows that for an
individual with initial high WM and low Worry,
it is likely WM capacity will be maintained, Worry
be maintained/decrease, and algebraic problem
solving accuracy will be high. For an individual
with initial large WM and high Worry, WM will
decrease, Worry will remain stable and algebraic
problem solving accuracy will be moderately high.
For an individual with initial small WM and low
Worry, WM will decrease slightly, Worry increase
and algebraic problem solving accuracy will be low
to moderate. For an individual with small WM and
high Worry, the equations shows a steep decline in
WM, a positively accelerating increase in Worry
and low algebraic problem solving accuracy.
DISCUSSION
The present study investigated the iterative influ-
ences of Worry and WM on each other over a single
day, and their impact on algebraic problem solving.
The research provides evidence, for the first time, of
(1) the direction of influence between WM and
Worry, and (2) the changing nature of influences in
the WM–Worry relationship over time, and how
these changes affect problem solving. We have
shown high Worry scores predict decreases in
Table 3. Parameter estimates for model fit to WM, worry and problem solving data
WM Worry
PE SE PE SE
Fixed Effects
Mean intercept μ
y0
.53* .02 .29* .02
Mean slope μ
s
.01 .10 .52* .16
Proportional change β.06 .22 −.62 .51
Coupling change γ−.30* .07 −.36* .14
Change on change ϕ−.27 .24 ––
Coupling change on change ξ––−.65* .26
Final score on problem solving ω.79* .14 −.45* .16
Easy Hard
Problem solving loadings χ1 0 .61* .06
Random effects
Univariate PE SE PE SE
Intercept variance σ
2
y0
.03* .04 .04* .01
Slope variance σ
2s
.00*
a
.00
a
.04 .03
Covariant slope and level σ
y0,s
−.00
a
.01 .03 .02
Unique variance σ
2
u
.02* .00
a
.01* .00
a
Bivariate PE SE
WM intercept–Worry intercept variance σ
y0,y0
−.01 .00
a
WM intercept–Worry slope variance σ
y0,s
−.02 .01
WM slope–Worry intercept variance σ
s,y0
.01* .00
a
WM slope–Worry slope variance σ
s,s
.01 .01
Observed WM–Worry score variance σ
u,u
−.00
a
.00
a
Note: Bold text indicates the statistically significant dynamic parameters that describe the interplay between WM and Worry, and problem
solving.
PE, parameter estimate; SE, standard error of the parameter estimate.
a
indicates value <.01.
*p<.05.
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WM capacity, and small WM capacity, or decreases
to WM capacity predict increases in Worry (sup-
porting Hypothesis 1c). The findings demonstrate
dynamic changes in WM and Worry over a short
time period, supporting Hypothesis 2b. Consistent
with previous research and Hypothesis 3, the
findings also suggest math problem solving is
positively associated with WM, and negatively
associated with Worry scores.
The study shows that WM capacity and Worry
may change within a short time frame (e.g., a math
class or test). Our model suggests the effect of
Worry on WM (and problem solving) differs
depending on WM capacity; and the effect of
WM on Worry (and problem solving) differs
depending on Worry level. WM and Worry showed
changes that varied across time, rather than decreas-
ing/increasing linearly. Moreover, these changes to
WM and Worry affected problem solving abilities.
On the basis of the finalmodel, a student withinitial
higher WM and lower Worry will likely maintain
WM and Worry levels, and algebraic problem
solving accuracy would remain high. Conversely,
for a student with low WM and high Worry, WM
is likely to decrease, Worry increase and problem
solving would be impaired. Thus, the model
suggests that what begins as relatively small differ-
ences between individuals in WM and Worry,
through their mutual iterative influences, would
lead to much larger differences. Moreover, it appears
that individuals with low WM and/or high Worry
may be more vulnerable to changes, so that their
initial disadvantage may amplify.
The model fit indices suggest that high Worry
predicts WM decreases, supporting MA and general
anxiety-performance research (e.g., Ashcraft & Kirk,
2001; Brunyé et al., 2013; Eysenck & Derakshan,
2011) that anxiety increases cognitive demands,
resulting in reduced processing capacity available for
WM.Themodelalsosuggeststhatincreasesin
Worry ratings are predicted by low or decreases to
WM scores, and increased WM predicts subsequent
Worry decreases. The relationship may be a direct
influence, in which WM regulates Worry responses,
supporting emotion regulation research (Hofmann
et al., 2011). Alternatively, WM may affect Worry
indirectly: WM may predict problem solving, and
students’Worry may be affected by performance.
This indirect effect may support the notion that
deficits in cognitive abilities are responsible for poor
math problem solving and associated with increases in
MA/Worry (Jansen et al., 2013;Ma&Xu,2004;
Maloney et al., 2011).
The findings highlight the importance of
accounting for the relationship between WM and
worry/MA, in examining the effect of WM or MA
on math problem solving. Not accounting for the
reciprocal relationship between MA and WM
reduces the interpretability of findings. Accounting
for WM–MA relationships in more traditional
single time point studies could be achieved by
examining individual profiles of WM/MA relation-
ships. As noted by Trezise and Reeve (2014b), in
single time point studies analytical techniques that
group individuals based on their response patterns
for WM and MA (e.g., latent class analysis) will
likely lead to a more complete characterisation of the
impact of WM or MA on math performance. Our
findings also highlight the importance of assessing
emotion–cognition relationships over time to exam-
ine changes within MA/WM relationships over
time. Assessing WM and/or MA at multiple time
points and using analytic methods that allow a
characterisation of the stability and/or change in the
interaction among factors (e.g., WM and MA) in
individuals may provide a more complete account of
the dynamic relationship between cognition and
emotion.
Our online worry assessment procedure shows
that Worry changes within a test session. Most
measures of MA are questionnaire based (for
example, the Abbreviated Math Anxiety Scale;
Hopko, Mahadevan, Bare, & Hunt, 2003), and
scores tend to be related to math problem solving
or WM. Questionnaire measures are not designed
to assess within task movements. Assessing the
worry/anxiety students exhibit over time allowed
us to examine whether their worry/anxiety rela-
tionships remained stable or changed (i.e., and
ipso facto whether MA has trait or state proper-
ties). The findings show MA may fluctuate over
short time periods (e.g., a math class), consistent
with a state interpretation of MA. Distinguishing
between trait and state MA, and their respective
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effects on math problem solving is an important
issue for further research. Understanding the
temporary (or stable) nature of MA may provide
useful information for those interested in the
prevention and/or treatment of MA.
In our study, context remained constant through-
out the study: all stimuli were algebraic, the level of
difficulty for the WM and Worry tasks remained
constant and students received no feedback. How-
ever, the WM–Worry relationship may change with
age, or in response to context, such as feedback,
pressure or domain. The extent of change in WM
and/or Worry may be smaller in an adults compared
to the adolescent sample used in our research (see
Steinberg, 2005). Providing students with feedback
while progressing (for example, about the correctness
of their responses) may change Worry ratings: Worry
has been shown to increase after negative feedback,
and decrease after positive feedback (Daniels &
Larson, 2001; Morris & Fulmer, 1976; Morris &
Liebert, 1973). Pressure (e.g., test situations) may
also affect the WM–worry relationship. We assessed
algebraic problem solving at the end of the day. If
students had also completed an initial algebraic
problem solving test, it may have affected the initial
WM/Worry scores. A change in context (e.g., a
change in domain) may also weaken the WM–worry
relationship. Research has shown that changing
domains can alter cognitive/emotional states
(Ilkowska & Engle, 2010;Punaro&Reeve,2012).
We therefore worried that changing domains
between tasks (e.g., a language WM task and an
algebraic worry task) might have affected the WM–
Worry relationship. Even changes between math
domains may affect WM–Worry relationships.
Algebra is a relatively new math domain for early
high school students and requires formal reasoning
ability (Tolar et al., 2009). Conversely, arithmetic is a
well-learned domain and involves more fact-retrieval
which places less demand on WM (Imbo &
Vandierendonck, 2007, Raghubar et al., 2010).
Further research is required to better understand the
WM–Worry interactions for variations in context.
We used a WM task with alphanumeric sym‐
bols in order to assess domain-specific WM, and
its relationship with algebraic problem solving.
Changes in algebraic WM span likely reflect
changes in individual’s ability to access and process
(i.e., WM capacity) alphanumeric symbols, rather
than fluctuations in algebraic expertise over a
single day. WM span showed an overall decrease
across time; however, there was a slight increase in
WM span between Set 3 and Set 4. The change
may reflect distributed practice effects: learning is
enhanced when a time lag separates study episodes
(spaced practice), compared to when items are
learned in a single episode (massed practice; see
Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006).
The changes may also be associated with perform-
ance decrements that have been observed in time-
on-task fatigue effects research (see Hockey, 2013;
Matthews, Davies, Westerman, & Stammers,
2000): whether individual differences in WM
and Worry are similarly associated with fatigue
onset is an issue for further research.
Bivariate latent difference modelling has advan-
tages over other methods of addressing hypotheses
about dynamic relationships among variables. The
model found a large regression weight of WM
change on Worry change, but also large variability.
This finding suggests that there may be latent
categorical individual differences the effects of
WM on Worry, which were unable to be fitted in
the model. Future research should examine possible
categorical differences in WM-Worry influences.
Conclusions
In sum, the present study demonstrates that WM
and Worry influence each other iteratively over
time. Research examining these relationships has
often focused on one direction of the influence and
assumed stability. Our findings reveal more com-
plex relationships: there is a bidirectional influence
between students’WM and Worry, and levels of
WM and Worry change considerably within one
day. These changes ultimately affect math prob-
lem solving. Of significance, individuals with low
WM and/or high Worry may be more vulnerable
to changes: their initial disadvantage may become
amplified over time. Moreover, the study has
demonstrated considerable changes in both cogni-
tion and emotion. It is important to further char‐
acterise these changes by exploring how the
WORRY AND WM INFLUENCE OVER TIME
COGNITION AND EMOTION, 2015 13
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cognition–emotion relationships vary in response to
external factors, such as feedback or domain.
Disclosure statement
No potential conflict of interest was reported by the
authors.
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