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Achievement Emotions and Academic Performance: Longitudinal Models of Reciprocal Effects

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A reciprocal effects model linking emotion and achievement over time is proposed. The model was tested using five annual waves of the Project for the Analysis of Learning and Achievement in Mathematics (PALMA) longitudinal study, which investigated adolescents’ development in mathematics (Grades 5–9; N = 3,425 German students; mean starting age = 11.7 years; representative sample). Structural equation modeling showed that positive emotions (enjoyment, pride) positively predicted subsequent achievement (math end-of-the-year grades and test scores), and that achievement positively predicted these emotions, controlling for students’ gender, intelligence, and family socioeconomic status. Negative emotions (anger, anxiety, shame, boredom, hopelessness) negatively predicted achievement, and achievement negatively predicted these emotions. The findings were robust across waves, achievement indicators, and school tracks, highlighting the importance of emotions for students’ achievement and of achievement for the development of emotions.
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Running head: EMOTION AND ACHIEVEMENT
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Paper accepted for publication in: Child Development
Achievement Emotions and Academic Performance:
Longitudinal Models of Reciprocal Effects
Reinhard Pekrun
Stephanie Lichtenfeld
University of Munich
Herbert W. Marsh
Australian Catholic University and University of Oxford
Kou Murayama
University of Reading
Thomas Goetz
University of Konstanz and Thurgau University of Teacher Education
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Author Note
Reinhard Pekrun, Department of Psychology, University of Munich, Munich, Germany;
Stephanie Lichtenfeld, Department of Psychology, University of Munich, Munich, Germany;
Herbert W. Marsh, Institute for Positive Psychology and Education, Australian Catholic
University, Sydney, Australia, and Department of Education, University of Oxford, Oxford, UK;
Kou Murayama, Department of Psychology, University of Reading, Reading, UK; Thomas
Goetz, Department of Empirical Educational Research, University of Konstanz, Konstanz,
Germany, and Thurgau University of Teacher Education, Thurgau, Switzerland.
This research was supported by a LMU Research Chair grant awarded to R. Pekrun by
the University of Munich and four grants from the German Research Foundation (DFG) to R.
Pekrun (PE 320/11-1, PE 320/11-2, PE 320/11-3, PE 320/11-4). Parts of this paper were
presented at the annual meeting of the American Educational Research Association,
Philadelphia, PA, April 2014, and at the International Congress of Applied Psychology, France,
Paris, July 2014.
Correspondence concerning this article should be addressed to Reinhard Pekrun,
Department of Psychology, University of Munich, Leopoldstrasse 13, 80802 Munich, Germany.
E-mail: pekrun@lmu.de
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Abstract
A reciprocal effects model linking emotion and achievement over time is proposed. The model
was tested using five annual waves of the PALMA longitudinal study, which investigated
adolescentsdevelopment in mathematics (grades 5-9; N=3,425 German students; mean starting
age=11.7 years; representative sample). Structural equation modeling showed that positive
emotions (enjoyment, pride) positively predicted subsequent achievement (math end-of-the-year
grades and test scores), and that achievement positively predicted these emotions, controlling for
students’ gender, intelligence, and family socio-economic status. Negative emotions (anger,
anxiety, shame, boredom, hopelessness) negatively predicted achievement, and achievement
negatively predicted these emotions. The findings were robust across waves, achievement
indicators, and school tracks, highlighting the importance of emotions for students’ achievement
and of achievement for the development of emotions.
Keywords: achievement emotion, anxiety, academic achievement, mathematics
achievement, control-value theory
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Research has shown that children’s and adolescents’ emotions are linked to their academic
achievement. Typically, positive emotions such as enjoyment of learning show positive links
with achievement, and negative emotions such as test anxiety show negative links (for
overviews, see Goetz & Hall, 2013; Pekrun & Linnenbrink-Garcia, 2014; Zeidner, 1998).
However, most of the available studies were correlational and do now allow any inferences about
the causal ordering of emotion and achievement over time. As such, it remains unclear how the
observed links should be interpreted. It is open to question if students’ emotions impact their
learning, if success and failure at learning influence the development of their emotions, if other
variables cause the association, or if several of these possibilities are at work. Given the need to
acquire knowledge about the antecedents of both students’ achievement and their emotions, this
is an issue of considerable theoretical and practical importance. To address this issue, the present
investigation went beyond merely observing correlations at a single point in time and attempted
to disentangle the temporal ordering of these constructs across multiple waves of data collection
and a developmental time span of several school years.
The investigation is based on a reciprocal effects model of emotion and achievement which
posits that the two variables reciprocally influence each other over time. This stands in contrast
to traditional unidirectional perspectives, which suggest that the link between emotion and
achievement is simply due to effects of emotions on students’ learning and performance. For
example, correlations between test anxiety and students’ achievement were interpreted as
indicating that anxiety impacts achievement, and test anxiety theories put forward various
suggestions about mediating mechanisms (e.g., cognitive interference, motivation; Zeidner,
1998, 2014). In a similar vein, in studies on affect and performance more generally, researchers
have been interested in the impact of moods and emotions on cognitive performance and created
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various theories targeting this influence (Clore & Huntsinger, 2009).
Certainly an analysis of the effects of emotions is important as it can document the
functional relevance of emotions. However, what about the reverse causal direction, that is, what
about the impact of achievement on the development of emotions? In other words, what about
emotions as outcomes rather than causes of achievement? Herein we argue that this alternative
causal direction is no less important. Beyond their functions, emotions are developmental
outcomes that are in and of themselves important, because they are core components of identity,
well-being, and health. By implication, researchers and practitioners alike should attend to the
antecedents of students’ emotions, and academic achievement is certainly one promising
candidate---academic successes and failures possibly shape the development of emotions. As
such, we concur with traditional perspectives in assuming that emotions impact achievement, but
we also extend this notion and expect that achievement reciprocally influences emotion.
Empirical evidence on the causal ordering of students’ emotions and their achievement is
largely lacking, with a few exceptions pertaining to achievement-related anxiety. Specifically,
longitudinal investigations suggested that K-12 students’ test anxiety and academic achievement
reciprocally influence each other (Meece, Wigfield, & Eccles, 1990; Pekrun, 1992).
Furthermore, in a study of mathematics anxiety by Ma and Xu (2004), adolescents’ achievement
in mathematics had negative effects on their subsequent math anxiety, and anxiety had negative
effects on subsequent achievement for two of the five time intervals included. The failure to find
effects of anxiety on achievement for the other time intervals was likely due to the high stability
of the achievement variable across waves (autogressive ßs > .95). For children’s and adolescents’
achievement emotions other than anxiety, evidence on reciprocal links with academic
achievement is lacking.
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In the following sections, we use Pekrun’s (2006; Pekrun & Perry, 2014) control-value
theory of achievement emotions to derive a theoretical framework for the reciprocal causation of
emotion and achievement. This model expands upon previous models on the linkages of anxiety
and boredom with achievement (Meece, Wigfield, & Eccles, 1990; Pekrun, 1992; Pekrun, Hall,
Goetz, & Perry, 2014; Zeidner, 1998) by addressing not only negative emotions but positive
emotions as well. We tested this model using a longitudinal dataset that examined adolescents’
emotions and achievement in mathematics over a period of five school years.
A Reciprocal Effects Model of Emotion and Achievement
The control-value theory (Pekrun, 2006; Pekrun & Perry, 2014) integrates propositions
from expectancy-value, attributional, and control approaches to achievement emotions (Folkman
& Lazarus, 1985; Pekrun, 1992; Turner & Schallert, 2001; Weiner, 1985). Achievement
emotions are defined as emotions related to achievement activities and their success and failure
outcomes. The theory posits that these emotions are aroused by cognitive appraisals of control
over, and the subjective value of, achievement activities and their outcomes. Control appraisals
consist of perceptions of one’s ability to successfully perform actions (i.e., academic self-
concepts and self-efficacy expectations) and to attain outcomes (outcome expectations). Value
appraisals pertain to the perceived importance of these activities and outcomes. Furthermore, the
theory posits that these emotions, in turn, influence achievement behavior and performance.
Since performance outcomes shape succeeding perceptions of control over performance, one
important implication is that emotions, their appraisal antecedents, and their performance
outcomes are linked by reciprocal causation. In terms of reciprocal causation, the theory is
consistent with reciprocal effects models for variables such as students’ self-concepts (Marsh &
Craven, 2006; Marsh, Trautwein, Lüdtke, Köller, & Baumert, 2005), achievement goals
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(Linnenbrink & Pintrich, 2002), and anxiety (Pekrun, 1992).
Effects of Emotion on Achievement
In the control-value theory, two dimensions describing human affect are used to distinguish
types of emotions, namely valence (positive vs. negative or pleasant vs. unpleasant) and
activation (activating vs. deactivating). Using these dimensions renders four groups of emotions:
positive activating (e.g., enjoyment, hope, pride), positive deactivating (e.g., relaxation, relief),
negative activating (e.g., anger, anxiety, shame), and negative deactivating (e.g., boredom,
hopelessness). The theory proposes that these emotions influence students’ cognitive resources,
motivation to learn, and use of learning strategies, thus impacting their achievement (for an in-
depth discussion, see Pekrun & Linnenbrink-Garcia, 2012).
Positive activating emotions (e.g., enjoyment of learning) are thought to preserve cognitive
resources and focus attention on the learning task, support interest and intrinsic motivation, and
facilitate deep learning. Accordingly, these emotions are expected to positively influence
students’ academic achievement under most task conditions. The opposite pattern of effects is
proposed for negative deactivating emotions (boredom, hopelessness). These emotions are
thought to reduce cognitive resources and task-related attention, to undermine both intrinsic and
extrinsic motivation, and to promote shallow information processing. Accordingly, negative
deactivating emotions are expected to negatively influence students’ achievement.
Achievement effects are posited to be more variable for the remaining two categories of
emotion. Deactivating positive emotions (relaxation, relief) are thought to reduce attention,
strategy use, and any immediate motivation to engage with learning tasks, but they can
strengthen long-term motivation to reengage with learning. Activating negative emotions (anger,
anxiety, shame) are thought to reduce cognitive resources by inducing irrelevant thinking, such
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as worries about failure in test anxiety, and to undermine intrinsic motivation. On the other hand,
these emotions can trigger extrinsic motivation to invest effort to avoid failure. Moreover, they
can facilitate the use of more rigid learning strategies, such as rote memorization. However,
notwithstanding individual differences regarding effects, we expect that the average overall
influence of positive deactivating emotions on achievement is positive, and that the average
overall influence of negative activating emotions is negative. For negative activating emotions
such as anxiety, this hypothesis is consistent with the available evidence, which indicates that the
correlations between these emotions and academic achievement are typically negative (Hembree,
1988; Zeidner, 1998, 2014).
Reverse Effects of Achievement on the Development of Emotion
Achievement reciprocally influences the appraisals that are considered to be proximal
antecedents of emotion. As implied by the control-value theory as well as other models of
achievement emotion (e.g., Folkman & Lazarus, 1985), positive emotions are thought to be
promoted when perceived competence and control over achievement activities are high. For
example, students should enjoy learning when they judge themselves competent to master the
learning task, provided they are interested in the material. Negative emotions should result when
perceived competence and control are low. For example, anxiety about an upcoming important
exam should be high if students judge themselves incompetent to pass it. One possible exception
is boredom, which could be promoted by high perceived competence if coupled with low task
demands (i.e., under-challenge); however, in an academic context, boredom also has been found
to be linked to students’ lack of perceived competence and control (e.g., Pekrun et al., 2010).
Competence and control are thought to influence both students’ momentary emotions within a
specific situation and their habitual, re-occurring emotions, which are based on re-occurring
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appraisals and related control-value beliefs (for summaries of empirical evidence, see Daniels &
Stupnisky, 2012; Pekrun & Perry, 2014).
Perceived competence and control depend on students’ individual achievement history,
with success strengthening control and failure undermining it. Hence, achievement is expected to
have positive effects on perceived control. Since achievement has positive effects on control, and
control has positive effects on positive emotions, it follows that students’ achievement should
have positive effects on the development of positive emotions. Similarly, since achievement has
positive effects on control, and control has negative effects on negative emotions, it follows that
achievement should have negative effects on the development of negative emotions.
Feedback Loops of Emotion and Achievement over Time
Because emotions are posited to influence achievement and achievement, in turn, to
influence emotion, the two constructs are thought to be linked by reciprocal causation over time.
Both effects are expected to be positive for positive emotions, amounting to positive feedback
loops, and both effects are expected to be negative for negative emotions, which also amounts to
positive feedback loops. We acknowledge that there may be negative feedback loops for negative
activating emotions in some students and under some conditions (e.g., failure on an exam
instigating a student’s anxiety, and anxiety eliciting effort to avoid failing the next exam; Pekrun,
1992). However, the existing evidence summarized above implies that negative activating
emotions typically are aroused by failure and contribute to subsequent failure, suggesting that
feedback loops should be positive for these emotions as well in the average student.
Overview of the Present Research
We tested the proposed reciprocal effects model using a longitudinal investigation of
adolescentsdevelopment in mathematics (Project for the Analysis of Learning and Achievement
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in Mathematics, PALMA; see Frenzel, Goetz, Lüdtke, Pekrun, & Sutton, 2009; Frenzel, Pekrun,
Dicke, & Goetz, 2012; Murayama, Pekrun, Lichtenfeld, & vom Hofe, 2013; Murayama, Pekrun,
Suzuki, Marsh, & Lichtenfeld, in press; Pekrun et al., 2007). To test models of reciprocal causal
linkages, designs are needed that assess both variables at multiple points in time (Little,
Preacher, Selig, & Card, 2007; McArdle, 2009; Rosel & Plewis, 2008). Although such designs
cannot fully rule out alternative causal explanations, they are better suited to test causal
propositions than cross-sectional designs or longitudinal designs that do not control for prior
levels of outcome variables. The PALMA study involved annual assessments of both emotions
and achievement, thus making it possible to conduct cross-lagged analyses examining reciprocal
causation. This study design made it possible to conduct multiple tests for the effects of emotion
on subsequent achievement, and of achievement on subsequent emotion, while controlling for
prior emotion and achievement levels.
For the present analysis, we used the grade 5 to 9 data from the PALMA study. As such,
the analysis involved five assessments for emotions and five assessments of achievement. These
assessments span the time from the beginning of secondary school (grade 5) to the end of
compulsory schooling in Germany (grade 9). At the start of secondary school, students are
selected into one of three tracks, including lower-track schools (Hauptschule), medium-track
schools (Realschule), and higher-track schools (Gymnasium), based on their elementary school
achievement. There is no additional school transition until the end of secondary school and
students usually remain in the same school. Whereas math teachers and the specific classroom
context can change, the broad academic context for students’ affective development remains
relatively stable across this time period. Specifically, contextual factors defining the emotional
salience of achievement, such as the visibility and frequency of feedback on achievement,
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remain stable during this period. The stability of context does not preclude changes in individual
levels of emotion (e.g., due to repeated success or failure and the influence of teachers and
peers). However, given the stability of context, we expected relations between students’ trait-like
emotions considered in this study and their achievement to be stable as well, with effects of these
emotions on achievement, and effects of achievement on emotions, showing equivalence (i.e.,
developmental equilibrium) across each of the one-year intervals included.
Seven distinct mathematics emotions were measured, including math-related enjoyment,
pride, anger, anxiety, shame, boredom, and hopelessness. These emotions were selected based on
their frequency and theoretical relevance (Pekrun et al., 2007). They were measured as trait-like
variables, that is, students’ habitual, re-occurring emotions in mathematics. Habitual emotions
can influence learning and achievement over a longer time span, in contrast to momentary
emotional episodes. In addition, we considered summary constructs of positive and negative
affect derived from integrating scores for positive and negative emotions, respectively. As
compared with multiple discrete emotions, these constructs render a more parsimonious
description of students’ affective development (Linnenbrink, 2007).
Achievement was assessed by students’ end-of-the-year grades in mathematics, which are
derived from multiple evaluations across the school year and represent students’ cumulative
performance. As such, these grades are suited to examining the impact of emotions on the long-
term development of achievement. In addition, test scores from the PALMA mathematical
achievement test (see Pekrun et al., 2007) were included to examine the generalizability of the
findings across different achievement outcomes. These scores reflect generic mathematical
competencies whereas grades represent students’ curriculum-related achievement in the
classroom, which should be more closely related to their emotions. Accordingly, we expected
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effects to be stronger for grades than for the test scores.
Structural equation modeling was used to test the reciprocal effects model. To ensure that
any observed relations were not mere artifacts of other plausible variables, we controlled for
students’ gender, intelligence, and family socio-economic status (SES) in the analysis. In
addition, we examined the equivalence of relations across school tracks. We expected the effects
linking emotion and achievement to be consistent over time and school tracks but modest in size
due to controlling for autoregressive effects, intelligence, and demographic variables.
Method
Participants and Design
The sample consisted of German adolescents who participated in the PALMA longitudinal
study (Pekrun et al., 2007). The study included annual assessments from grades 5 to 9 (2002-
2006). Sampling and the assessments were conducted by the Data Processing and Research
Center (DPC) of the International Association for the Evaluation of Educational Achievement
(IEA). Samples were drawn from schools within the state of Bavaria and were representative of
the student population of this state in terms of student characteristics such as gender, urban
versus rural location, and family background (SES; for details, see Pekrun et al., 2007). At each
grade level, the students answered the questionnaire towards the end of the school year. All
instruments were administered in the students’ classrooms by trained external test administrators.
At the first assessment (grade 5), the sample included 2,070 students from 42 schools
(49.6% female, mean age = 11.7 years). The sample comprised students from all three school
types within the Bavarian public secondary school system as described earlier, including lower-
track schools (Hauptschule, 37.2% ), intermediate-track schools (Realschule, 27.1%), and
higher-track schools (Gymnasium, 35.7%). These three school types differ in average student
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achievement due to the selection of students by entry-level achievement (see Murayama et al.,
2013). The distribution of students across tracks represents the distribution in the population. In
each subsequent year, the study not only tracked the students who had participated in the
previous assessment(s), but also incorporated those students who had not yet participated in the
study but had become members of PALMA classrooms at the time of the assessment (for more
details on sampling procedures, see Pekrun et al., 2007). This strategy resulted in the following
sample sizes for the subsequent years: 2,059 students in grade 6 (50.0% female, mean age = 12.7
years); 2,397 students at grade 7 (50.1% female, mean age = 13.7 years); 2,410 students at grade
8 (50.5% female, mean age = 14.8 years); 2,528 students at grade 9 (51.1% female, mean age =
15.6 years). Across all five assessments (i.e., grades 5 to 9), a total of 3,425 students (49.7%
female) took part in the study. 60.4% of the total sample participated in all five assessments, and
21.7%, 11.7%, 5.1%, and 1.1% completed four, three, two, or one assessment(s), respectively.
Measures
Emotions. Students’ emotions in mathematics were measured using the Achievement
Emotions Questionnaire-Mathematics (AEQ-M; Pekrun, Goetz, Frenzel, Barchfeld, & Perry,
2011). The instructions for the instrument ask respondents to describe how they typically feel
when attending class, doing homework, and taking tests and exams in mathematics; in this way,
the AEQ-M assesses students’ habitual, trait-like math-related emotions. The instrument
comprises seven scales measuring mathematics enjoyment (9 items, e.g., “I enjoy my math
class”), pride (8 items; e.g., “After a math test, I am proud of myself”), anger (8 items; e.g., “I
am annoyed during my math class”), anxiety (15 items; e.g., “I worry if the material is much too
difficult for me”), shame (8 items; e.g., “I am ashamed that I cannot answer my math teacher’s
questions well”), hopelessness (6 items; e.g., “During the math test, I feel hopeless”), and
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boredom (6 items; e.g., “My math homework bores me to death”). Participants responded on a 1
(strongly disagree) to 5 (strongly agree) scale, and the scores were summed to form the emotion
indexes (Alpha range .86 to .92 across all scales and measurement occasions; see Table 1). The
scores were also used to derive indexes for positive and negative affect factors combining
positive and negative emotions, respectively (see Data Analysis section).
Achievement. Students’ achievement was assessed by their end-of-the-year grades in
mathematics as retrieved from school documents and by standardized test scores.
End-of-the-year grades. These grades are summative scores based on multiple exams
within each school year; they represent students’ achievement in the math curriculum for the
respective year. Grades range from 1 (excellent) to 6 (poor). Grade scores were reversed prior to
the analysis to ease interpretation.
Test scores. The test scores were derived from the PALMA Mathematics Achievement
Test (Pekrun et al., 2007) which measures students’ competencies in arithmetics, algebra, and
geometry. The test includes different test forms for different grade levels and includes anchor
items to allow for the linkage of test forms across assessments. The obtained scores were scaled
using one-parameter logistic item-response theory (Rasch scaling; see Murayama et al., 2013).
Background variables. Demographic variables (gender and SES) and intelligence were
included as covariates in the analysis. Gender was coded 1=female, 2=male.
Socio-economic status. SES was assessed by parent report using the EGP classification
(Erikson, Goldthorpe, & Portocarero, 1979), which consists of six ordered categories of parental
occupational status. Higher values represent higher SES.
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Intelligence. Intelligence was measured at Time 1 (grade 5) using the 25-item nonverbal
reasoning subtest of the German adaptation of Thorndike’s Cognitive Abilities Test (Kognitiver
Fähigkeitstest [KFT 412 + R]; Heller & Perleth, 2000).
Strategy of Data Analysis
Structural equation modeling (SEM; Mplus, Version 7; Muthén & Muthén, 2012) was
used to evaluate the reciprocal effects model. We estimated two sets of models. The first set used
grades, and the second set used test scores as the achievement measure. In both sets, eight
different models were estimated, including seven separate models for the discrete emotions and
one integrative model combining all emotions into two second-order positive and negative affect
factors. There was substantial multicollinearity between the emotion variables in the dataset
(Table 1). As such, the present analysis combines two strategies to deal with multicollinearity,
namely, using single variables (separate discrete emotion models) and combining them by
constructing summary variables (integrative affect models). The separate discrete emotion
models also served to examine if the links between emotion and achievement were sufficiently
similar to combine emotions into the summary positive and negative affect constructs.
All of the models represent a cross-lagged format, with emotion at each assessment
influencing subsequent achievement one year later, and achievement at each assessment
influencing subsequent emotion one year later (Figure 1). As such, the discrete emotion models
include four paths from emotion to achievement and four paths from achievement to emotion. In
the affect models, there were eight paths from positive and negative affect to achievement, eight
paths from achievement to positive and negative affect, as well as four paths from positive to
negative affect and four paths from negative to positive affect (Figure 1). The emotion variables
were modeled as latent constructs. The achievement measure and the three background measures
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(gender, SES, and intelligence) were evaluated as manifest variables. The background variables
were included as covariates; for each of these variables, directional paths to all of the emotion
variables and to all of the achievement variables were included.
We estimated two versions for all of the 16 models. In the first version, autoregressive
coefficients, cross-paths, and factor residual variances were freely estimated. In the second
version, all three parameters were constrained to be invariant across time intervals
(developmental equilibrium; e.g., the effects of Time n emotion on Time n+1 achievement were
constrained to be the same from each wave to the next).
Measurement models for latent variables. The emotion scale items were used as
indicators for each of the latent emotion variables. Following recommendations by Pekrun et al.
(2011), a correlated uniqueness approach was used by including correlations between residuals
for items representing the same setting (attending class, doing homework, and taking tests and
exams in mathematics). In addition, correlations between residuals for identical emotion items
across measurement occasions were included to control for systematic measurement error.
The latent affect factors were constructed in a two-step procedure. We first conducted
separate confirmatory factor analyses for each of the seven emotions across the five assessments
and derived emotion factor scores from these analyses (it was not possible to conduct a
confirmatory factor analysis with all emotion items across all assessments, i.e., 60 x 5 = 300
items, due to computational limitations). We then used these factor scores to construct one
integrative affect measurement model. For this model, factor scores for the positive emotions
served as indicators for positive affect, and factor scores for the negative emotions served as
indicators for negative affect. As such, the two affect constructs represent second-order factors.
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Measurement equivalence across waves and school tracks. Prior to the main SEM
analyses, we sought to establish measurement equivalence of the latent emotion and affect
constructs over time and schools tracks. For each of the emotion and affect variables, we
sequentially evaluated models of configural, metric, scalar, and residual invariance (Meredith,
1993). Configural invariance is defined by equal patterns of factor loadings. Metric invariance
additionally requires equal factor loadings, scalar invariance requires equal factor loading and
intercepts, and residual invariance requires equal factor loadings, intercepts, and residual
variances. To establish equivalence of constructs for analyzing correlations and path coefficients,
metric invariance is the minimum needed (Chen, 2007; Steenkamp & Baumgartner, 1998). To
compare model fit, we followed recommendations by Chen (2007). Provided adequate sample
size, for testing metric invariance, a change of > -.010 in CFI, supplemented by a change of >
.015 in RMSEA or a change of > .030 in SRMR would indicate noninvariance; for testing scalar
or residual invariance, a change of > -.010 in CFI, supplemented by a change of > .015 in
RMSEA or a change of > .010 in SRMR would indicate noninvariance. As recommended, we
did not use the difference test because it is overly sensitive to sample size (Marsh, Balla, &
McDonald, 1988).
Hierarchical data structure, estimator used, and missing values. As students were
nested in schools, we corrected for the clustering of the data using the “type=complex” option
implemented in Mplus (Muthén & Muthén, 2012). As noted, schools in the German public
secondary school system differ in average student achievement due to the between-schools
tracking based on achievement, indicating that nestedness within schools needs to be considered.
The <type=complex> corrects standard errors for nestedness while preserving use of the
covariance matrix from the full sample to calculate parameters.
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18
To estimate the model parameters, the robust maximum likelihood estimator (MLR) was
employed which is robust to nonnormality of the observed variables. To make full use of the data
from students with missing data, we applied the full information likelihood method (FIML;
Enders, 2010). FIML has been found to result in trustworthy, unbiased estimates for missing
values even in the case of large numbers of missing values (Enders, 2010) and to be an adequate
method to manage missing data in longitudinal studies (Jeličič, Phelps, & Lerner, 2009). To
examine the robustness of the analysis, we replicated the cross-lagged analyses for emotion and
achievement with the subsample of students who participated in the study from the beginning (N
= 2,070). As compared to the models using the full sample, there were no substantial differences
in model fit ( CFI < .007, RMSEA < .006, and SRMR < .007 for all of the models), and the
substantive results were essentially the same (see Supplemental Material, Tables S6 and S7).
Goodness-of-fit indexes to evaluate model fit. We applied both absolute and
incremental fit indices to evaluate the fit of the models, including the comparative fit index
(CFI), the Tucker-Lewis index (TLI), the root-mean-square-error of approximation (RMSEA),
and the standardized-root-mean residual (SRMR). Traditionally, values of CFI and TLI higher
than .90 and close to .95, values of RMSEA lower than .06, and values of SRMR lower than .08
were interpreted as indicating good fit (Browne & Cudeck, 1993; Hu & Bentler, 1999). We
report these fit indexes to make the present analysis comparable with previous research.
However, it should be noted that the recommended cutoff values are often not met with datasets
derived from more complex studies, suggesting that they should be used with caution (Heene,
Hilbert, Draxler, Ziegler, & Bühner, 2011; Marsh, Hau, & Wen, 2004).
Results
Preliminary Analysis
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Alpha coefficients for the emotion scales and manifest correlations for the emotions and
achievement are outlined in Table 1 (for information about distributions, see Table S1).
Correlations between the emotion measures indicated that enjoyment and pride were positively
related, as were anger, anxiety, shame, hopelessness, and boredom. The correlations between
positive and negative emotions were negative. Overall, this pattern of relations is consistent with
previous evidence on the structures of students’ academic emotions (e.g., Pekrun et al., 2011).
Enjoyment and pride correlated positively with mathematics achievement in each year, whereas
anger, anxiety, shame, hopelessness, and boredom correlated negatively with achievement.
Confirmatory Factor Analysis (CFA) for the Emotion Constructs
To further examine the relations between emotions, item-based CFA models including
the seven emotions were estimated. This was done separately for the five measurement
occasions. The models showed a good fit to the data (Supplemental Material, Table S2),
supporting the measurement quality of the emotion variables. The latent correlations between the
emotion variables showed the same pattern as the manifest correlations (Table 1). These
correlations are corrected for measurement error and indicate that the latent emotion variables
are closely related but nevertheless distinct (for similar findings with university students, see
Pekrun et al., 2011). This is also true for emotions that might be presumed to constitute opposite
ends of a bipolar continuum, such as enjoyment and boredom, which showed moderately
negative relationships. The strongest correlations were found for neighboring, like-valenced
emotions such as enjoyment and pride, and anxiety, shame, and hopelessness. In interpreting
these correlations, it is important to note that the present study used the AEQ-M to assess
students’ trait-like emotions. As noted by Pekrun et al. (2011), like-valenced trait emotions are
known to be strongly correlated, in contrast to state emotions which show more divergence.
Running head: EMOTION AND ACHIEVEMENT
20
For positive and negative affect based on the emotion factor scores, we conducted an
integrative CFA including both constructs across all five measurement occasions. The fit for this
CFA model was good (Supplemental Material, Table S3, configural invariance model). Latent
correlations between the positive and negative affect factors were r = -.19, -.23, -.25, -.23, and -
.21 (all ps < .01) for Time 1, 2, 3, 4, and 5, respectively, showing that the two affect constructs
were sufficiently distinct.
Measurement Invariance of the Emotion Constructs over Time and School Tracks
Measurement invariance across waves was tested separately for the seven emotions and
for positive and negative affect. The configural invariance models showed a good fit to the data,
with CFI > .93, RMSEA < .03, and SRMR < .05 for all seven discrete emotion constructs
(Supplemental Material, Table S3). As compared with these models, the loss of fit for the metric
invariance models was CFI < -.004, RMSEA < .001, and SRMR < .006 for all models,
indicating clear support for metric invariance for all of the emotions. The loss of fit for the scalar
invariance models was CFI < -.007,RMSEA < .004, and SRMR < .007 for all of the
emotions, documenting that scalar invariance was supported as well. The loss of fit for the
residual invariance models was CFI < -.010 for all emotions except shame, CFI = -.010, as
well as RMSEA < .003 and SRMR < .008 for all emotions, indicating support for residual
invariance. For positive and negative affect, the loss of fit was CFI < .008, RMSEA < .004,
and SRMR < .005 for the metric, intercept, and residual invariance models, demonstrating
support for invariance for these second-order constructs as well. In sum, the findings show that
the latent emotion and affect variables showed strong measurement equivalence over time, thus
meeting the requirements to be included in longitudinal analysis. Furthermore, in supplemental
analyses using multi-group analysis, the emotion constructs also showed strong measurement
Running head: EMOTION AND ACHIEVEMENT
21
equivalence across the three school tracks (see Supporting Information, Table S8).
Reciprocal Effects Models of Emotions and Achievement
The fit indexes provided support for the cross-lagged structural equation models for all
seven emotions as well as positive and negative affect and across both measures of achievement.
For all of the models freely estimating autoregressive effects, cross-lagged effects, and factor
residual invariances, CFI was > .92, TLI > .90, RSMEA < .06, and SRMR < .08 (Table 2 and
Supplemental Material, Table S4). When constraining autoregressive effects, cross-lagged
effects, and factor residual variances to be equal across time intervals, the loss of fit was CFI <
.003, RMSEA < .001, and SRMR < .003 for all of the models. These findings support the
invariance of these parameters, suggesting developmental equilibrium in autoregressive stability
and in the links of emotion and achievement across time. Accordingly, we adopted the
constrained models for further interpretation, which have the additional advantage of providing
more robust and precise parameter estimates (note that these constraints equalize unstandardized
coefficients; to ease interpretation, we report standardized coefficients which can still differ due
to the standardization procedure).
Emotions and grades. Factor loadings, path coefficients, and residual variances for the
reciprocal effects models including grades are displayed in Table 3. In the enjoyment and pride
models, both the emotion variables and students’ achievement showed considerable stability over
time, as indicated by the autoregressive effects for these variables. Furthermore, there were
significant relations between the positive emotions and achievement at grade 5 in these models,
latent rs = .26 and .26, ps < .001, for enjoyment and pride, respectively. Over and above these
pre-existing relations, and despite autoregressive stability, results showed enjoyment and pride to
positively predict each subsequent achievement outcome (ß range .11 to .13, ps < .001) while
Running head: EMOTION AND ACHIEVEMENT
22
controlling for gender, SES, and intelligence. In addition, positive paths emerged from each
achievement outcome to the subsequent enjoyment and pride variables (all βs = .11, ps < .001).
In the negative emotion models, there were substantial initial links between anger, anxiety,
shame, boredom, and hopelessness at grade 5, latent rs = -.31, -.39, -.32, -.16, and -.37,
respectively, ps < .001. Despite these links and the considerable stability of the emotion and
achievement variables over time, anger, anxiety, shame, boredom, and hopelessness negatively
predicted each subsequent achievement outcome (ß range -.08 to -.14, all ps < .001) while
controlling for gender, SES, and intelligence. The effects were especially pronounced for anxiety
and hopelessness (all ßs > -.11). In addition, negative paths from each achievement outcome to
subsequent anger, anxiety, shame, boredom, and hopelessness were observed (ß range -.06 to
-.14; all ps < .001).
These effects were similar across the two positive emotions, and similar across the five
negative emotions, thus justifying their combination into positive and negative affect constructs.
In the reciprocal effects model for positive and negative affect, the initial links with achievement
were rs = .26 and -.33 for positive and negative affect, respectively, ps < .001. Despite these
links and strong autoregressive coefficients for both positive and negative affect as well as
achievement, positive affect positively predicted achievement, and negative affect negatively
predicted achievement. Because both types of affect were included in the analysis, these findings
indicate that positive and negative affect had independent predictive effects on achievement.
Achievement, in turn, had positive predictive effects on positive affect and negative predictive
effects on negative affect. Regarding cross-paths between positive and negative affect, we had
not expected any effects of this type and none of the paths were significant.
Emotions and test scores. The findings for emotions and test scores replicated the results
Running head: EMOTION AND ACHIEVEMENT
23
for grades, demonstrating generalizability across different achievement measures (Supplemental
Material, Table S5). As expected, however, the effects were weaker than for grades. Positive
emotions were positive predictors of test scores, ß range = .04 to .05, and negative emotions were
negative predictors, ß range = -.03 to -.08, all ps < .001. Test scores were a positive predictor of
positive emotions, ß range = .05 to .07, and a negative predictor of negative emotions, ß range =
-.04 to -.11, all ps < .001. In the positive and negative affect model, positive affect was not a
significant predictor of test scores (all ßs = .01, ns), whereas negative affect predicted test scores,
ß range = -.06 to -.07, ps < .001. Test scores, in turn, were a positive predictor of positive affect,
ßs = .03, ps < .01, and a negative predictor of negative affect, ß range = -.04 to -.05, ps < .001.
Effects of the covariates. Intelligence had positive effects on grades and test scores as
well as negative effects on students’ anger, anxiety, shame, and hopelessness (Tables 3 and S5).
SES also had positive, albeit weaker, effects on math achievement. Gender had significant
effects on all of the emotions except anger, indicating that girls reported lower enjoyment, pride,
and boredom, and higher anxiety, shame, and hopelessness in mathematics than boys.
Equivalence of effects across school tracks. In supplemental analyses, we used multi-
group analysis to examine the equivalence of cross-paths, autoregressive effects, and effects of
covariates across the three school tracks. Comparing models constraining versus not constraining
these coefficients to be invariant (using Chen’s, 2007, criteria outlined in the Data Analysis
section), the findings provide robust support for invariance across tracks for all of the emotion
and affect constructs included and both math grades and test scores (see Tables S9, S10).
Discussion
The findings of this study provide robust evidence for the proposed reciprocal effects
model of emotion and achievement. As indicated by longitudinal SEM, adolescentsmath-
Running head: EMOTION AND ACHIEVEMENT
24
related positive emotions (enjoyment and pride) positively predicted their subsequent end-of-the-
year math grades, and grades, in turn, positively predicted the development of positive emotions.
Math-related negative emotions (anger, anxiety, shame, hopelessness, and boredom) were
negative predictors of subsequent math grades, and grades, in turn, were a negative predictor for
the development of negative emotions. Similar predictive effects were obtained for the
integrative constructs of positive and negative affect, respectively, and for test scores as a
measure of achievement. The findings were consistent across models for the seven discrete
emotions, the combined positive and negative affect model, four time intervals, two different
measures of achievement (grades, test scores), and the three school tracks while controlling for
students’ gender, SES, and intelligence. All of the effects were significant with the single
exception of the effects of positive affect on test scores.
Because prior links between emotion and achievement as well as intelligence and
demographic background variables were controlled, the path coefficients are likely to represent
effects of emotion on achievement, and vice versa, rather than simply the influence of prior
emotion, prior achievement, gender, intelligence, or socio-economic status. As expected, the size
of these coefficients was modest. However, it is important to note that the coefficients represent
incremental predictive effects due to prior emotion and achievement being controlled. Thus, the
coefficients represent effects of each variable on change in the other from one assessment to the
next, rather than effects on the absolute levels of these variables. Furthermore, both emotion and
achievement showed considerable stability over time, leaving little variance to be explained and
making it difficult to detect the effects of additional variables. From this perspective, the
consistency of effects lends credibility to the notion that emotion and achievement are indeed
linked by reciprocal causation over time.
Running head: EMOTION AND ACHIEVEMENT
25
Reciprocal Effects Linking Emotion and Achievement
The findings are congruent with previous evidence showing that emotions and academic
achievement are correlated (Goetz & Hall, 2013; Pekrun & Linnenbrink-Garcia, 2014; Zeidner,
1998). However, they go beyond correlational evidence by disentangling the directional effects
underlying the emotion-achievement link. Specifically, the findings suggest that emotions indeed
have an influence on adolescents’ achievement, over and above the effects of general cognitive
ability and prior accomplishments. These effects are in line with Pekrun’s (2006) control-value
theory which posits that emotions influence learning and achievement outcomes.
Of specific importance is the finding that adolescents’ positive emotions in mathematics
had positive predictive effects on their math grades over time. Previous research has produced
mixed findings on the relation between students’ positive affect and their learning, with most
studies reporting positive relations (see Linnenbrink, 2007) but some others null findings (e.g.,
Pekrun, Elliot, & Maier, 2009). The present analysis suggests that positive emotions can have
positive effects, in line with theory and the views of educational practitioners. However, the
effects were weaker for positive emotion than for the negative emotion constructs, and did not
reach significance for the predictive effect of positive affect on test scores. Future research
should examine possible reasons why negative emotion is a stronger predictor of students’
academic achievement than positive emotion. This difference may relate to general asymmetries
in the impact of negative versus positive states and events on human memory and action (see
e.g., Baumeister, Bratslavsky, Finkenauer, & Vohs, 2001).
The results also contribute to our understanding of the developmental origins of students’
emotions. The findings suggest that achievement impacts the development of emotions. More
specifically, it appears that doing well in school can strengthen students’ positive emotions and
Running head: EMOTION AND ACHIEVEMENT
26
reduce their negative emotions over time, whereas doing poorly in school undermines positive
emotions and exacerbates negative emotions. These effects are likely mediated by students’
perceptions of competence and control over achievement, with high control promoting
enjoyment and pride and low control leading to negative emotions (e.g., Pekrun et al., 2010).
Taken together, these effects amount to positive developmental feedback loops linking
emotions and achievement. As noted, a few longitudinal studies have found that students’ test
anxiety and their achievement were linked by positive feedback loops (Meece, Wigfield, &
Eccles, 1990; Pekrun, 1992). The present research adds to this literature by showing that
emotions other than anxiety share similar links with achievement. As such, it would appear that
unidirectional models are unable to adequately capture the complex reality of students’ emotions.
Rather, systems-oriented perspectives are needed that take more complex patterns of causal links
into account, including feedback loops between emotions, their antecedents, and their effects.
Discrete Emotions versus General Affect
It is noteworthy that the cross-paths were similar across different discrete emotions. For
effects of achievement on emotion, this is to be expected, as success and failure are thought to
impact the development of different positive and negative emotions in similar ways. As outlined
in our reciprocal effects model, success is expected to generally increase perceived control, thus
enhancing positive emotions, and failure is expected to decrease control, leading to negative
emotions. However, regarding effects of emotion on achievement, emotion theories such as the
control-value theory (Pekrun, 2006) imply that the effects of some emotions (e.g., deactivating
negative emotions such as boredom) may be more consistent than the effects of other emotions
(e.g., activating negative emotions such as anxiety). Instead, the findings clearly indicate that the
predictive effects of emotions on students’ long-term achievement were also similar across
Running head: EMOTION AND ACHIEVEMENT
27
different emotions. Accordingly, whereas constructs of discrete emotions are needed to explain
the impact of emotions on functional mechanisms and different types of cognitive performance,
parsimonious summary constructs of positive and negative affect may be sufficient to explain
their relations with overall academic achievement. This possibility is underscored by the robust
findings for positive and negative affect documented in the present analysis.
Effects of Gender, Intelligence, and SES
The findings on gender differences are consistent with previous evidence showing that
girls report less enjoyment and more anxiety and shame in mathematics even if they perform as
well as boys. Lower competence beliefs and perceived values in mathematics may be possible
explanations (Goetz, Bieg, Lüdtke, Pekrun, & Hall, 2013). However, girls reported less boredom
than boys, in line with previous evidence (Pekrun et al., 2010). As such, the findings suggest that
girls exhibit a more maladaptive profile of math emotions, except for boredom.
As expected, intelligence had substantial predictive effects on the achievement variables.
Furthermore, intelligence had negative effects on math-related anger, anxiety, shame, and
hopelessness. Given that students’ mathematics achievement was included in the analysis, this
finding suggests that higher general cognitive ability can help to reduce negative mathematics
emotions, above and beyond any effects of students’ academic success in mathematics. Finally,
SES also had positive, albeit weaker, effects on math achievement, suggesting that the family
exerts an influence on students’ achievement, over and above any effects of cognitive ability.
Limitations, Suggestions for Future Research, and Implications for Practice
The present study represents a significant advancement over previous research, because it
documents reciprocal effects of emotion and achievement over time while controlling for general
cognitive ability and critical demographic background variables. Nevertheless, several
Running head: EMOTION AND ACHIEVEMENT
28
limitations should be considered when interpreting the study findings and can be used to suggest
directions for future research.
Methodological considerations. As compared with experimental studies, the power of
non-experimental field studies to derive causal conclusions is limited. As such, although the
present analysis used multi-wave longitudinal structural equation modeling and controlled for
related variables and autoregressive effects, the possibility still exists that our findings are
attributable to other variables that were not included in the study. On the other hand, field studies
may be more ecologically valid than experimental emotion studies, which are limited in terms of
situational representativeness and ethical concerns about experimentally manipulating emotions.
Furthermore, statistical power is higher in field studies such as the present one due to large
sample size. To balance the benefits and drawbacks of different methodologies and make
headway in this avenue of research, future studies should further pursue the approach taken
herein while complementing this approach with experimental studies.
Achievement was assessed by students’ end-of-year grades and test scores. By using
grades, we sought to employ an ecologically valid measure of student achievement (for a similar
procedure, see Pekrun et al., 2014). As is typical for grades, more detailed information about
reliability was not available; as such, it was not possible to disattenuate the link between
emotions and grades for potential unreliability of this achievement measure. However, in
German secondary schools, end-of-the-year grades are summative scores based on multiple
exams within each school year, which may boost their reliability in comparison to grades on
single exams. In the present study, the stability of grades across years (all ßs > .50) could be
considered as a lower bound to reliability. Furthermore, from the perspective of grades as
sources of students’ emotional development, they could be seen as having almost perfect
Running head: EMOTION AND ACHIEVEMENT
29
reliability---grades, rather than objective achievement, provide the feedback that shapes students’
perceptions of success and failure and any development based on these perceptions. In addition,
an advantage of grades is that they represent achievement in terms of the math curriculum taught
in students’ classes. They represent the specific contents learned by students and may be superior
to alternative measures in terms of curricular validity. Finally, the findings based on grades
proved to be generalizable, as the results were essentially the same for test scores.
Substantive issues. The present research examined academic emotions as experienced by
adolescents in the domain of mathematics. It is open to question whether the present findings
would generalize to other age groups, such as elementary school children or post-secondary
students. Furthermore, it is possible that there is individual variation in the link between
emotions and achievement. To examine such variation, within-person analyses of the relations
between emotion and achievement over time are needed (e.g., by using experience sampling
methodology; Goetz, Sticca, Pekrun, Murayama, & Elliot, 2016). Because the present research
involved samples of German adolescents, it also remains an open question as to whether the
findings would generalize to students in other cultures. Additionally, future research should
explore if these findings generalize to emotions in achievement domains other than mathematics,
The study considered a broad range of important mathematics emotions but did not
include an exhaustive list of emotions. It is open to question whether the observed reciprocal
effects would also occur for emotions not assessed herein. Specifically, the study did not include
students’ deactivating positive emotions, such as relief and relaxation. Future studies could
explore how these emotions are linked to students’ academic achievement. Furthermore, the
present study examined students’ trait-like emotions which are known to be highly correlated
(Pekrun et al., 2011), which makes it difficult to determine unique variance in achievement
Running head: EMOTION AND ACHIEVEMENT
30
attributable to different emotions. Future research should examine the unique impact of multiple
state emotions, which are less correlated (Goetz et al., 2016), on students’ learning.
Finally, the study addressed the overall developmental relations between emotion and
achievement but did not examine the mechanisms that mediate the observed links. In the
proposed model of reciprocal effects, it is posited that effects of emotion on achievement are due
to the influence of emotions on cognitive resources, motivation, and strategy use. The effects of
achievement outcomes on the development of emotion are thought to be mediated by perceptions
of competence and control over performance, and could additionally be mediated by value
appraisals. More research on the link between emotion and achievement as mediated by these
cognitive and motivational mechanisms is needed to better understand students’ emotions and
their relations with important school outcomes.
Implications for educational practice. Two important messages follow from the present
research. First, the results suggest that emotions have effects on adolescent students’ academic
achievement, and that these effects are not merely an epiphenomenon of prior performance---
more likely, they represent a true causal influence of students’ emotion experiences. By
implication, the findings suggest that educators, administrators, and parents alike should consider
intensifying efforts that strengthen adolescentspositive emotions and minimize their negative
emotions. Second, the results imply that achievement outcomes reciprocally influence students’
emotions, suggesting that successful performance attainment and positive achievement feedback
can facilitate the development of positive emotions, and failure experiences can contribute to the
development of negative emotions. Accordingly, providing students with opportunities to
experience success (e.g., using intrapersonal standards to evaluate achievement; emphasizing
mastery over competition goals) may help to promote positive emotions and prevent negative
Running head: EMOTION AND ACHIEVEMENT
31
emotions (also see Pekrun, Cusack, Murayama, Elliot, & Thomas, 2014). By documenting the
influence of achievement outcomes on students’ emotions, the present findings elucidate one
important factor that can be targeted by educators to reduce students’ negative affect and
facilitate the development of emotional well-being.
Running head: EMOTION AND ACHIEVEMENT
32
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Running head: EMOTION AND ACHIEVEMENT
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Running head: EMOTION AND ACHIEVEMENT
38
Table 1
Alpha Coefficients and Pearson Product-Moment Correlations for Emotions and Achievement
Enjoyment
Pride
Anxiety
Shame
Boredom
Hopelessness
Enjoyment
(.87) a
.83
-.53
-.36
-.60
-.48
(.87)
.84
-.51
-.33
-.63
-.51
(.88)
.86
-.48
-.30
-.62
-.49
(.85)
.86
-.46
-.30
-.57
-.49
(.89)
.88
-.42
-.23
-.50
-.46
Pride
.73
(.87)
-.37
-.25
-.39
-.38
.74
(.88)
-.42
-.27
-.50
-.44
.75
(.88)
-.40
-.26
-.47
-.43
.76
(.89)
-.37
-.25
-.47
-.43
.78
(.89)
-.35
-.18
-.43
-.39
Anger
-.55
-.35
.88
.76
.84
.93
-.55
-.40
.86
.73
.82
.82
-.56
-.39
.86
.69
.79
.83
-.53
-.39
.86
.68
.72
.85
-.49
-.37
.87
.68
.75
.84
Anxiety
-.41
-.29
(.90)
.92
.67
.90
-.39
-.31
(.90)
.92
.60
.91
-.35
-.29
(.91)
.87
.53
.92
-.33
-.26
(.91)
.88
.51
.92
-.32
-.26
(.92)
.87
.55
.91
Shame
-.27
-.19
.78
(.86)
.55
.82
-.23
-.18
.77
(.88)
.48
.79
-.20
-.16
.74
(.87)
.37
.78
-.19
-.16
.75
(.87)
.36
.78
-.14
-.09
.74
(.89)
.42
.78
Boredom
-.51
-.27
.44
.37
(.86)
.63
-.53
-.35
.39
.31
(.89)
.60
-.52
-.33
.33
.25
(.90)
.54
-.48
-.32
.29
.23
(.90)
.56
-.41
-.29
.32
.28
(.90)
.57
Hopelessness
-.41
-.34
.83
.74
.43
(.86)
-.43
-.38
.86
.73
.42
(.88)
-.42
-.37
.86
.71
.37
(.88)
-.43
-.37
.86
.70
.37
(.87)
-.43
-.37
.86
.68
.38
(.83)
Achievement
.20
.18
-.37
-.33
-.37
-.12
(end-of-year
.25
.22
-.38
-.34
-.37
-.09
grades)
.34
.29
-.37
-.29
-.39
-.15
.41
.36
-.37
-.29
-.39
-.15
.45
.38
-.40
-.29
-.45
-.22
Note. a 1st, 2nd, 3rd, 4th, 5th coefficient in each column: Grade 5, 6, 7, 8, and 9, respectively. Coefficients
below main diagonal are manifest correlations. Coefficients above main diagonal are latent correlations
based on confirmatory factor analyses for each wave. Coefficients in parentheses are Cronbach’s Alphas.
p < .01 for all coefficients.
Running head: EMOTION AND ACHIEVEMENT
39
Table 2
Reciprocal Effects Models for Emotion and Grades: Fit Indexes
2
df
CFI
TLI
RMSEA
SRMR
Model
Cross-paths, autoregressive effects, and residual variances
freely estimated
Enjoyment
4125.280**
1147
.940
.928
.027
.052
Pride
2729.201**
722
.940
.928
.028
.048
Anger
3238.875**
918
.941
.927
.027
.049
Anxiety
9091.434**
2992
.920
.909
.024
.050
Shame
2168.850**
907
.965
.957
.020
.044
Boredom
1384.409**
532
.974
.966
.021
.038
Hopelessness
2018.158**
562
.959
.949
.027
.055
Positive and negative
affect
6837.618**
685
.947
.930
.051
.075
Cross-paths, autoregressive effects, and residual variances
invariant across waves
Enjoyment
4210.435**
1165
.938
.927
.027
.053
Pride
2794.131**
740
.942
.930
.028
.049
Anger
3285.829**
936
.940
.928
.027
.050
Anxiety
9148.887**
3010
.920
.909
.024
.050
Shame
2244.200**
925
.964
.956
.020
.045
Boredom
1500.094**
550
.971
.963
.022
.041
Hopelessness
2058.064**
580
.959
.950
.027
.055
Positive and negative
affect
6976.520**
721
.946
.933
.050
.078
** p < .01.
Running head: EMOTION AND ACHIEVEMENT
40
Table 3
Reciprocal Effects Models for Emotion and Grades: Standardized Factor Loadings, Path Coefficients, and Residual Variances
Enjoyment model
Pride model
Anger model
Anxiety model
Shame model
Enjoyment
Grades
Pride
Grades
Anger
Grades
Anxiety
Grades
Shame
Grades
Factor loadings
.37-.81a
.55-.77 a
.58-.77 a
.44-.77 a
.48-.78 a
Autoregressive effects
T1 T2
.67***
.57***
.62***
.57***
.58***
.57***
.60***
.56***
.62***
.58***
T2 T3
.66***
.59***
.64***
.59***
.61***
.59***
.64***
.58***
.61***
.60***
T3 T4
.66***
.61***
.65***
.61***
.62***
.60***
.66***
.60***
.60***
.62***
T4 T5
.65***
.59***
.65***
.59***
.62***
.58***
.68***
.58***
.60***
.60***
Cross-lagged effects
Grades
Enjoyment
Enjoyment
Grades
Grades
Pride
Pride
Grades
Anger
Grades
Grades
Anger
Grades
Anxiety
Anxiety
Grades
Grades
Shame
Shame
Grades
T1 T2
.11***
.13***
.11***
.11***
-.12***
-.10***
-.08***
-.11***
-.06***
-.09***
T2 T3
.11***
.13***
.11***
.12***
-.13***
-.10***
-.08***
-.13***
-.06***
-.09***
T3 T4
.11***
.13***
.11***
.12***
-.14***
-.10***
-.07***
-.14***
-.06***
-.09***
T4 T5
.11***
.12***
.11***
.12***
-.13***
-.10***
-.07***
-.14***
-.06***
-.08***
Effects of Covariates at T1
Gender
.14***
.02
.17***
.02
-.03
.02
-.16***
.02
-.09**
.02
SES
-.05**
.09***
.05*
.09***
.03
.09***
-.04
.09***
-.03
-.09***
Intelligence
-.02
.40***
-.00
.40***
-.12***
.40***
-.18***
.40***
-.17***
.40***
Residual Variances
T1
.98
.82
.97
.82
.98
.82
.94
.82
.96
.82
T2
.50
.57
.57
.58
.62
.57
.59
.57
.55
.58
T3
.51
.56
.54
.56
.59
.56
.53
.56
.58
.56
T4
.52
.58
.53
.58
.57
.57
.50
.58
.60
.58
T5
.52
.56
.52
.56
.57
.55
.50
.56
.61
.56
Running head: EMOTION AND ACHIEVEMENT
41
Table 3 (continued)
Boredom model
Hopelessness model
Positive and negative affect model
Boredom
Grades
Hopelessn.
Grades
Pos. affect b
Neg. affect b
Grades
Factor loadings
.56-.77 a
.63-.85 a
.77-.96 a
.41-.93 a
Autoregressive effects
T1 T2
.63***
.59***
.53***
.56***
.80***
.74***
.54***
T2 T3
.65***
.61***
.57***
.59***
.81***
.76***
.56***
T3 T4
.66***
.63***
.58***
.60***
.82***
.78***
.57***
T4 T5
.66***
.61***
.59***
.58***
.82***
.79***
.56***
Cross-lagged effects
Grades
Boredom
Boredom
Grades
Grades
Hopelessn.
Hopelessn.
Grades
Grades
Pos. affect
Grades
Neg. affect
Pos. affect
Grades
Neg. affect
Grades
T1 T2
-.06***
-.08***
-.11***
-.11***
.05***
-.04***
.10***
-.08***
T2 T3
-.06***
-.08***
-.12***
-.12***
.05***
-.04***
.10***
-.08***
T3 T4
-.06***
-.09***
-.12***
-.13***
.05***
-.04***
.10***
-.09***
T4 T5
-.06***
-.09***
-.11***
-.13***
.05***
-.04***
.10***
-.09***
Effects of Covariates at T1
Gender
.09**
.02
-.16***
.02
.15***
-.13***
.02
SES
-.03
.09***
-.04
.09***
-.05**
-.03
.09***
Intelligence
.00
.40***
-.13***
.40***
-.02
-.15***
.40***
Residual Variances
T1
.99
.82
.95
.82
.97
.96
.82
T2
.59
.58
.66
.58
.34
.41
.58
T3
.56
.56
.61
.58
.33
.36
.57
T4
.54
.57
.60
.56
.32
.35
.59
T5
.53
.55
.59
.56
.32
.33
.57
Note. a Range of factor loadings. p < .001 for all loadings. b Cross-paths between positive and negative affect were not significant (all ps > .05).
* p < .05. ** p < .01. *** p < .001.
Running head: EMOTION AND ACHIEVEMENT
42
Figure 1. Basic structure of cross-lagged reciprocal effects models. Upper part: emotion and
achievement. Lower part: positive affect, negative affect, and achievement. The models include
cross-lagged effects, autoregressive effects, and directional paths from the covariates to emotion
or affect and achievement at all waves. Correlations between the covariates and between
residuals are not displayed.
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Goodness-of-fit (GOF) indexes provide "rules of thumb"—recommended cutoff values for assessing fit in structural equation modeling. Hu and Bentler (1999) proposed a more rigorous approach to evaluating decision rules based on GOF indexes and, on this basis, proposed new and more stringent cutoff values for many indexes. This article discusses potential problems underlying the hypothesis-testing rationale of their research, which is more appropriate to testing statistical significance than evaluating GOF. Many of their misspecified models resulted in a fit that should have been deemed acceptable according to even their new, more demanding criteria. Hence, rejection of these acceptable-misspecified models should have constituted a Type 1 error (incorrect rejection of an "acceptable" model), leading to the seemingly paradoxical results whereby the probability of correctly rejecting misspecified models decreased substantially with increasing N. In contrast to the application of cutoff values to evaluate each solution in isolation, all the GOF indexes were more effective at identifying differences in misspecification based on nested models. Whereas Hu and Bentler (1999) offered cautions about the use of GOF indexes, current practice seems to have incorporated their new guidelines without sufficient attention to the limitations noted by Hu and Bentler (1999).
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The International Handbooks of Teacher Education cover major issues in the field through chapters that offer detailed literature reviews designed to help readers to understand the history, issues and research developments across those topics most relevant to the field of teacher education from an international perspective. This volume is divided into two sections: The organisation and structure of teacher education; and, knowledge and practice of teacher education. The first section explores the complexities of teacher education, including the critical components of preparing teachers for teaching, and various aspects of teaching and teacher education that create tensions and strains. The second examines the knowledge and practice of teacher education, including the critical components of teachers’ professional knowledge, the pedagogy of teacher education, and their interrelationships, and delves into what we know and why it matters in teacher education.
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