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Reciprocal relationships between math self-concept and math anxiety
Wondimu Ahmed ⁎, Alexander Minnaert, Hans Kuyper, Greetje van der Werf
University of Groningen, The Netherlands
abstractarticle info
Article history:
Received 26 April 2011
Received in revised form 28 November 2011
Accepted 2 December 2011
Keywords:
Math self-concept
Math anxiety
Reciprocal effects
Cross-lagged analysis
The present study examined the reciprocal relationships between self-concept and anxiety in mathematics. A
sample of 495 grade 7 students (51% girls) completed self-report measures assessing self-concept and anxi-
ety three times in a school year. Structural equation modeling was used to test a cross-lagged panel model of
reciprocal effects between math self-concept and math anxiety. The analysis showed a reciprocal relationship
between self-concept and anxiety in math (i.e., higher self-concept leads to lower anxiety, which in turn,
leads to higher self-concept). However, the magnitude of the path from anxiety to self-concept is almost
half of that from self-concept to anxiety. Overall, the results provide empirical support for the theoretical no-
tion that math self-concept and math anxiety are reciprocally related.
© 2011 Elsevier Inc. All rights reserved.
1. Introduction
Math anxiety is an unpleasant feeling associated with numerical
manipulations and math problem solving (Richardson & Suinn,
1972). It is widely documented that math anxiety is one of the
most significant factors that lead students to avoidance of math-
related educational tracks and career avenues (Ashcraft, 2002). Un-
fortunately, roughly 20% of students suffer from high math anxiety
(Ashcraft & Ridley, 2005). Historically, researchers have sought for
determinants of such a pervasive problem. Over the past decades,
studies have shown that students' competence beliefs are among
the most significant predictors of math anxiety. Particularly, re-
searchers have shown that domain specific self-appraisal of ability
such as self-concept and self-efficacy
1
substantially predict math
anxiety (e.g., Hembree, 1990; Meece, Wigfield, & Eccles, 1990;
Pajares & Miller, 1994). Much of the previous research on math anx-
iety is founded on cognitive models of test anxiety that postulate that
low self-perception of ability is a defining feature of anxiety. Howev-
er, an important issue that hinges on the role of self-concept of ability
in shaping an individual's math anxiety is that of ‘the chicken or the
egg’causality dilemma observed in most of the social sciences. Does a
low self-concept lead to anxiety or does anxiety lead to the develop-
ment of a low self-concept?
A dominant view is that a low self-concept of ability is a source of
high math anxiety. Because the self-concept involves self-evaluation
of one's knowledge and capability to deal with demands of the envi-
ronment, perceived incompetence signals that environmental threat
is imminent (see Bandura, 1997). Thus a low self-concept in math sig-
nifies that the individual is ill-equipped to cope with demands of
stressful situations involving math. This suggests that causality
flows from self-concept to anxiety. Another view is that because ex-
perience of anxiety is characterized by a distorted self-image, an indi-
vidual experiencing a high level of anxiety may judge his/her
capability in doing math as inadequate (see Beck & Clark, 1997). Fi-
nally, the contemporary view assumes that the relationship between
math self-concept and math anxiety is reciprocal (Bandura, 1997;
Pekrun, 2006; Zeidner, 1998). Although the three alternative assump-
tions appear to be equally convincing, much of the previous research
has been based on the hypothesis that a low self-concept precedes
math anxiety.
A considerable number of cross-sectional studies have provided
insight into the relationship between math self-efficacy or math
self-concept and math anxiety. In an earlier study, Hackett (1985)
found that math self-efficacy was a stronger predictor of math anxiety
than prior achievement in math. Recent studies have also reported
negative moderately significant cross-sectional correlations between
self-efficacy and math anxiety in both middle school (Jain &
Dowson, 2009) and university samples (Hoffman, 2010). With regard
to self-concept, Hembree's (1990) meta-analysis reported an average
correlation of −0.71 between math self-concept and math anxiety. A
handful of recent studies have also documented substantial concur-
rent relations between math self-concept and math anxiety (e.g.,
Frenzel, Pekrun, & Goetz, 2007; Goetz, Cronjaeger, Frenzel, Lüdtke,
Learning and Individual Differences 22 (2012) 385–389
⁎Corresponding author at: Institute for Educational Research, University of Groning-
en, Grote Rozenstraat 3, 9712TG, Groningen, The Netherlands. Tel.: +31 618198739;
fax: 31 50 363 6670.
E-mail address: w.ahmed@rug.nl (W. Ahmed).
1
Self-concept and self-efficacy beliefs may not be distinguished at a domain level of
specificity (Pajares, 1996) and even more so when only the cognitive aspect of self-
concept is considered (Pietsch, Walker, & Chapman, 2003). Moreover, according to
Schunk and Zimmerman (2006) both constructs refer to competence beliefs which de-
note ‘expectancies about one's capabilities to learn or perform actions’(p.350)
1041-6080/$ –see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.lindif.2011.12.004
Contents lists available at SciVerse ScienceDirect
Learning and Individual Differences
journal homepage: www.elsevier.com/locate/lindif
& Hall, 2010; Pajares & Miller, 1994). Among the cross sectional stud-
ies, Pajares and Miller's (1994) study stands out. The authors found
a strong correlation (r=−.87) between math anxiety and math
self-concept suggesting that about 76% variation in anxiety could be
explained by self-concept or vice versa. Although such a strong corre-
lation appears to suggest that the constructs may be conceptually
identical, one recent cross-cultural study of 41 countries using a
confirmatory factor analysis showed that math self-concept and
math anxiety are conceptually and empirically distinct constructs
(Lee, 2009).
The major limitation of the previous research on the relationship
between math self-concept and math anxiety is that they have been
cross-sectional by design. Only a few longitudinal studies have tested
the putative influence of self-concept on anxiety. For instance, Meece
et al. (1990) study of young adolescents found that self-concept
of ability measured at grade 7 predicted math anxiety at grade 9.
However, the authors did not test the reciprocal effect of anxiety on
self-concept.
Regardless of their design, previous studies tended to endorse the
prevalent view that self-concept precedes anxiety. However, as noted
earlier, most socio-cognitive models (Bandura, 1997; Pekrun, 2006;
Zeidner, 1998) suggest that the relationship between math self-
concept and math anxiety could be reciprocal. In particular,
Bandura's (1997) socio-cognitive theory hypothesizes that as indi-
viduals experience higher levels of anxiety, they also tend to report
lower levels of self-competence; however, as their self-efficacy
rises, individuals report a corresponding decrease in anxiety. In a
similar vein, Pekrun's (2006) control-value theory of achievement
emotions posits that self-appraisals of ability and emotions are recip-
rocally related. Pekrun argues that appraisals trigger emotions and
emotions act on appraisals by activating emotion-congruent memory
networks. Similarly, Zeidner's (1998) transactional model of anxiety
suggests that self-appraisal of capability and anxiety is reciprocally
related. Despite such a strong reciprocal causal assumption as well
as existing cross-sectional evidence on the relationship between
self-concept and anxiety in math, the temporal order remains
unclear. An understanding of the causal directions of self-concept
and anxiety helps in refining existing theories as well as in designing
interventions. Thus, on the basis of the theoretical models (Bandura,
1997; Pekrun, 2006; Zeidner, 1998) and existing empirical evidence
(e.g., Frenzel et al., 2007; Goetz, Pekrun, Hall, & Haag, 2006; Meece
et al., 1990), we expected self-concept to be reciprocally associated
with anxiety such that self-concept and anxiety would negatively
predict each other over time. More specifically, we examined the fol-
lowing research question: Does prior math self-concept predict subse-
quent math anxiety negatively and vice versa? In the previous
literature, researchers have used different definitions of anxiety and
self-concept in math. For the purpose of the current study, math anx-
iety is defined as students' anxious reactions to three important
achievement situations: attending classes, studying or doing home-
work and taking exams or tests based on Pekrun and colleagues' con-
ceptualization of achievement emotions (e.g., Pekrun, 2006; Pekrun,
Goetz, & Frenzel, 2005; Pekrun, Goetz, Frenzel, Barchfeld, & Perry,
2011; Pekrun, Goetz, Titz, & Perry, 2002). Similarly, math self-
concept is defined as student's self-perception of their ability and
their expectancy to perform well in math based on work of Eccles
and colleagues (e.g., Jacobs, Lanza, Osgood, Eccles, & Wigfield, 2002;
Wigfield & Eccles, 2000).
2. Method
2.1. Participants and procedure
The participants were 522 seventh grade students (mean
age=12.7, at T1) in The Netherlands. The participants were recruited
from twenty-one classrooms in two secondary schools located in two
predominantly middle class suburban communities. The average class
size was 23. The participants provided data at three occasions: at the
beginning (T1), in the middle (T2) and at the end (T3) of a school
year. Twenty seven students did not complete data at two occasions
and were therefore not used in the current analysis. The final sample
was 495 students (51%, girls) of which 98.8%, 97.5% and 96% provided
data at T1, T2 and T3 respectively.
2.2. Measures
2
2.2.1. Math self-concept
Math self-concept was assessed using four items adapted from
Wigfield and Eccles (2000).The measure assessed among others
how good the participants thought they were at math, how well
they expected to do in the future in math, and how good they thought
they would be at learning something new in math. Example items in-
clude: “How good at math are you?”,“Compared with other students
how good in math are you?”(1 = not at all good,5=very good). The
measure demonstrated good internal consistency (see Table 1).
2.2.2. Math anxiety
Math anxiety was assessed using items adapted from the math
anxiety subscale of the Academic Emotions Questionnaire —Mathemat-
ics (Pekrun et al., 2005). The subscale assessed the participants' anx-
ious experiences when, in math class (e.g., I feel anxious in math
class), studying or doing homework (e.g., “I get tense when studying
math”) and taking exams or test in mathematics (e.g., “I feel nervous
during taking math tests”). Students responded to 8 items assessing
their level of math anxiety. Participants responded on a 1 (not at
all)to5(very much) scale and the scores were averaged to form the
math anxiety index. The measure demonstrated good internal consis-
tency (see Table 1).
2.2.3. Prior achievement
Students' scores on a national test at the elementary school (grade
6, USA) were taken as a proxy for prior cognitive ability which is like-
ly to determine the initial levels of anxiety and self-concept in math-
ematics. As students start obtaining feedback on their performance in
the secondary school context, this feedback is more likely to be a
source of information for cognitive and affective self-evaluations
than their prior academic achievement. The students' scores on this
test ranged from 513 to 550 (M =538, SD =6.6). The possible range
is between 501 and 550.
3. Results
Table 1 presents the means, standard deviations, and inter-
correlations between measured variables. The correlations between
math anxiety and math self-concept at the three measurement occa-
sions were all significant and were all in the expected direction. More
importantly, the two cross-temporal correlations between self-
concept and anxiety were all significant suggesting the need to exam-
ine reciprocal effects.
To examine the reciprocal relationships between math self-
concept and math anxiety, we used structural equation modeling.
The cross-lagged models were tested with LISREL 8.80 for Windows
(Jöreskog & Sörbom, 2006), using full information maximum likeli-
hood estimation. To control for error variance, we randomly assigned
items into two parcels for both math anxiety and math self-concept.
We used parcels because they produce relatively more reliable esti-
mates of latent variables than individual items (Little, Cunningham,
2
The measures were translated from English into Dutch by a senior researcher with
an excellent command of both languages and were used in our previous publications
(e.g., Ahmed, Minnaert, van der Werf & Kuyper, 2010a; Ahmed, Minnaert, van der Werf
& Kuyper, 2010b).
386 W. Ahmed et al. / Learning and Individual Differences 22 (2012) 385–389
Shahar, & Widaman, 2002). The use of parcels also results in an in-
crease in estimates of structural parameters and a decrease in residual
variances compared to aggregate scores (Coffman & MacCallum,
2005). Moreover, parcels provide distributions that more closely ap-
proximate normality (Bagozzi & Heatherton, 1994). To assess the fit
of the models, we incorporated four fit indices: root mean square re-
sidual (RMSEA .08 or less), the Tucker–Lewis index (TLI, .90 or great-
er), comparative fit index (CFI, .90 or greater) and chi-square divided
by degree of freedom (χ
2
/df,b2)(Hu & Bentler, 1999).
Consistent with general practice of structural equation modeling,
a two step procedure was used. First, we examined the measure-
ment model of the variables. For this purpose, a confirmatory factor
analysis (CFA) was used. To rule out the assumption that math anx-
iety and math self-concept are indistinguishable constructs, we test-
ed a one-factor and a two-factor CFA models by freely estimating
the factor loadings for the six latent variables measuring self-
concept and anxiety at T1, T2 and T3. Whereas a one-factor CFA
demonstrated a poor fit to the data (χ
2
/df=2.8; RMSEA=.09;
CFI=.89; TLI=.86), a two-factor CFA model fitted the data well
(χ
2
/df=1.6; RMSEA=.05; CFI= .99; TLI= .98) suggesting that
math anxiety and math self-concept are conceptually and statistical-
ly distinct constructs.
Second, we examined several structural models. We tested each of
the models by allowing autoregressive effects between adjacent
times to control for stability effects. We also allowed the concurrent
disturbances to correlate to account for variances due to measure-
ment occasion. To control for initial differences in self-concept and
anxiety, we estimated the effect of prior academic achievement on
both constructs at T1. In the course of testing the structural models,
four models were compared. To compare the competing models, we
used chi-square difference test (Δχ
2
). In the first model (model A)
we constrained all cross-lagged effects to zero. This model served as
the baseline model against which subsequent models were evaluated.
This model did not fit the data well (χ
2
/df=2.2; RMSEA=.09;
CFI=.88; TLI=.89). In the second model (model B), we partially re-
laxed the constraints imposed in the first model by freeing the lagged
effects of self-concept on anxiety. This model helped us to test the hy-
pothesis that low self-concept leads to anxiety. The model replicated
the data very well (χ
2
/df=1.4; RMSEA=.04; CFI= .99; TLI= .98)
and fitted better than model A (Δχ
2
(2)=10, pb.05). In the third
model (model C), we lifted the constraints on the cross-lagged effects
of anxiety on self-concept and constrained the cross-lagged effects of
self-concept on anxiety to zero. This model was used to test the alter-
native view that high levels of anxiety may lead to appraisal of one's
ability as low. This model also fitted the data very well (χ
2
/df=1.5;
RMSEA=.03; CFI=.99; TLI=.99) and fitted better than model A
(Δχ
2
(2)=6.2, pb.05). In the final model (model D), we relaxed
the constraints in the third model and tested the reciprocal effects
of anxiety and self-concept. The reciprocal effects model fitted
the data very well (χ
2
/df=1.3; RMSEA=.03; CFI= .99; TLI= .99)
and better than model A (Δχ
2
diff (4)=18, pb.01) model B (Δχ
2
diff
(2)=8, pb.05) and model C (Δχ
2
diff (2)=13, pb.01). This suggests
that the proposed cross-lagged model replicates the data and is better
than the baseline or the alternative models.
3
The standardized structural coefficients are presented in Fig. 1.
Prior achievement significantly predicted Time 1 self-concept but
not Time 1 anxiety. The three disturbance correlations were signifi-
cant. The autoregressive effects were also substantially significant
suggesting the relative stability of the constructs. Above all else, the
cross-lagged coefficients are all significant suggesting that math
self-concept and math anxiety are reciprocally related. However,
one can easily discern from the figure that the effect of self-concept
on anxiety is twice as large as the effect of anxiety on self-concept.
This suggests that self-concept has a stronger effect on anxiety than
vice versa.
4. Discussion
The main goal of the current study was to investigate the recipro-
cal relationships between self-concept and anxiety in math. On the
basis of theory and previous research, we hypothesized that students'
math self-concept would be reciprocally associated with their math
anxiety. Consistent with the assumptions of cognitive mediational
models (Bandura, 1997; Pekrun, 2006; Zeidner, 1998), the three-
wave cross-lagged longitudinal analyses revealed significant recipro-
cal effects. The analysis revealed that lower levels of math self-
concept predicted subsequent higher levels of anxiety, controlling
for prior levels of anxiety. Simultaneously, higher levels of math anx-
iety predicted subsequent lower levels of self-concept, controlling for
prior levels of self-concept.
Bandura (1997) argued that while lower self-appraisal of capability
to exercise control over stressors might lead to anxiety, higher levels of
anxiety might also lead to lower self-efficacy judgments. Similarly, a
number of test anxiety researchers have suggested a dynamic recipro-
cal relationship between anxiety and self-related cognitions (Krampen,
1988; Sarason, 1988;Zeidner, 1998). The current finding is consistent
with such assumptions and shows a dynamic association between
self-concept and anxiety. The significant effect of prior self-concept
on subsequent anxiety found in the current study gives credence to
the generalized assumption that higher efficacy judgments lead to low-
ered anxiety and is consistent with Meece et al. (1990) study in which
prior self-concept of ability predicted subsequent math anxiety. On the
other hand, the significant effect of prior anxiety on subsequent levels
of self-concept tends to support the processing efficiency theory of
anxiety (Eysenck & Calvo, 1992) that hypothesizes that anxiety pro-
duces negative self-evaluations. On a broader level, this finding is also
consistent with experimental research on the influence of affect on
self-related cognitions that generally shows that negative emotions
lead to negative self-evaluative judgments (Sedikides, 1992). Although
we are not aware of studies that investigated the reciprocal relation-
ship between self-concept and anxiety in math, one previous study
(Krampen, 1988) that investigated the reciprocal effects of math self-
3
We tested a multivariate cross-lagged multilevel model using MLwIN 2.23 (Ras-
bash, Browne, Healy, Cameron, & Charlton, 2011) to check whether the cross-lagged
effects vary across classrooms. First, we fitted two unconditional multilevel models
to partition the variance in the dependent variables (i.e., anxiety and self-concept) into
component parts. The results showed that the proportion of variance that lies between
classes (i.e., intra-class correlation coefficient, ICC) was 0.044 for anxiety and 0.068 for
self-concept. To check if the cross-lag effects of anxiety on self-concept (or vice versa)
vary across classrooms, we tested a multivariate cross-lagged multilevel model with
random slopes in which T3 anxiety and T3 self-concept were each simultaneously
regressed on T2 self-concept and T2 anxiety; and, T2 anxiety and T2 self-concept were
each simultaneously regressed on T1 self-concept and T1 anxiety, controlling for auto-
regressive effects. The patterns of the results were almost identical to those reported.
However, although the effect of self-concept on anxiety did not vary across classrooms,
the effect of anxiety on self-concept at the second lag i.e., T2 to T3 (but not at the first
lag) showed small but significant variability across classrooms [Estimate= 0.058
(SE=0.026)].
Table 1
Descriptive statistics and inter-correlations between measured variables.
123456
1. MS1
2. MS2 .60
3. MS3 .56 .71
4. MA 1 −.43 −.34 −.32
5. MA2 −.38 −.48 −.35 .56
6. MA3 −.29 −.33 −.38 .40 .53
M 3.43 3.32 3.16 2.00 2.03 2.06
SD .60 .70 .77 .55 .57 .64
α.84 .86 .87 .86 .89 .82
All correlations are significant at p= .01; MS= math self-concept; MA= math anxiety;
Numeric suffixes denote measurement occasions.
387W. Ahmed et al. / Learning and Individual Differences 22 (2012) 385–389
concept and test anxiety found support for the prospective effect of
math self-concept on test anxiety but not vice versa. The current
study provided support for a reciprocal effects model. The study also
showed that the effect of self-concept on anxiety is twice the effect of
anxiety on self-concept. This may suggest that because of previous neg-
ative experience with math, individuals might have developed dys-
functional self-schemas that might have precipitated biased appraisal
of their ability to do math, which in turn led to a higher level of anxiety
(see Beck & Clark, 1997). A large body of theoretical and empirical
work regarding anxiety has demonstrated that the perceived inability
to personally influence actions and outcomes in one's environment
are the major determinants of anxiety (Pekrun, 2006; Zeidner, 1998).
A low self-concept signals a sense of dimensioned control. Prior expe-
rience of lack of control may put individuals at an ultimate risk to expe-
rience anxiety through the development of a generalized belief that
manipulation of numbers and math problem solving is not within
one's control (see Schwarzer, 1986). Experiencing weakened control
over math during learning may establish that math is fraught with
threat which subsequently leads to experience of frustration and anxi-
ety. Thus, early experience of failure in math and associated self-
concept of inability to succeed in the domain may be thought of as a
crucial pathway to the development of anxiety. This is because such ex-
perience may promote an increased probability to process math related
challenges as not within one's control. However, our findings also sug-
gest that, to some extent, experiences of high math anxiety produces
lower self-evaluation of ability do math. This suggests that in spite of
magnitude differences, self-concept and anxiety in math are reciprocal-
ly related.
One strength of the current study is that the reciprocal effects
model was tested using a prospective, longitudinal design involving
three measurement occasions. The availability of three measurement
points provided us with the opportunity to test if the hypothesized
reciprocal process was evident across more than one time period.
The other strength of this study is that it evaluated the cross-lagged
relations controlling for concurrent disturbance correlations and
autoregressive effects. This conservative modeling approach helps to
reduce the concern that associations among variables are simply an
artifact of unmeasured third variables.
In spite of these methodological strengths, the present study also
has some limitations. First, the correlational design of the research
makes it difficult to draw a definitive causal conclusion. However, the
fact that the analyses controlled for temporal ordering of variables is
an improvement over cross-sectional research designs. Second, the
present study considered a relatively short time span. Although this pe-
riod covers an important critical transition period worth of investiga-
tion, a longer time span might help to paint a clearer picture of the
relations between self-concept and anxiety over time. Third, the
study used only self-reports for both self-concept and anxiety risking
a common method variance. In the current study, however, the cross-
lagged effects are unlikely to account for such bias because the tests
controlled for autoregressive (stability) effects as well as correlated dis-
turbances. Fourth, although we found an evidence for reciprocal effects
model of self-concept and anxiety, the effect sizes of the cross-lagged
paths are generally small. However, the small cross-lagged paths partly
reflect the fact that the model was tested under a very conservative
condition, reflecting a design effect and thus should not be viewed as
having no meaning or importance (see Prentice & Miller, 1992). Finally,
the current study focused only on the domain of mathematics in an
early adolescent sample. Although the focus on the domain and age
group is of particular relevance to researchers and practitioners, it is
worth noting that the relations between the two constructs could be
different in other school domains. Previous research has demonstrated
that both self-concept and emotions are domain specific(e.g.,Goetz,
Frenzel, Pekrun, Hall, & Lüdtke, 2007; Marsh, 1990) and that the rela-
tions between the two may depend on age (Goetz et al., 2010). Thus,
future research should test the viability of the current reciprocal
model in other school subjects and using younger or older samples.
These limitations notwithstanding, the findings of the current
study have important implications for theory and practice. The results
suggest that the dynamic relations between self-concept and anxiety
are theoretically consistent with cognitive mediational models. More-
over, the findings suggest that the prevalent view that low self-
concept leads to anxiety is stronger than the vice versa. Thus, the
findings inform emerging theoretical models of emotions in educa-
tion by highlighting the fact that despite the existences of the recipro-
cal effects, magnitude issues are of concern. With regard to
implications for educational intervention, the reciprocal effects
found in the current study point to the possibility that enhancing stu-
dents' self-concept is beneficial for reducing math anxiety and vice
versa. Treatments for math anxiety have generally involved helping
students to regulate their emotions. Although such programs are likely
to reduce anxiety, the current findings suggest that enhancing students'
self-concept through either skill development or self-enhancement
strategies (see Marsh & Craven, 2006) may be more effective than anx-
iety focused training of self-regulatory strategies. In conclusion, al-
though the hypothesis that self-concept and anxiety are reciprocally
associated is empirically supported, only future research will discern
the mechanisms involved in such processes.
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