Revealing Dynamic Relations Between Mathematics Self-Concept and Perceived Achievement From Lesson to Lesson: An Experience-Sampling Study

Abstract

Academic self-concept and achievement have been found to be reciprocally related across time. However, existing research has focused on self-concept and achievement scores that have been averaged over long time-periods. For the first time, the present study examined intraindividual (within-person) relations between momentary (state) self-concept and lesson-specific perceived achievement (i.e., self-reported comprehension) in students’ everyday school life in real time using intensive longitudinal data. We conducted an experience-sampling (e-diary) study with 372 German secondary school students in Grades 9 and 10 over a period of 3 weeks after each mathematics lesson. Multilevel confirmatory factor analyses confirmed a two-factor between-level and within-level structure of the state measures. We used dynamic structural equation modeling to specify a multilevel first-order vector autoregressive model to examine the dynamic relations between self-concept and perceived achievement. We found significant reciprocal effects between academic self-concept and perceived achievement on a lesson-to-lesson basis. Further, we found that these relations were independent of students’ gender, reasoning ability, or mathematics grades. We discuss implications for methodology, theory, and practice in self-concept research and educational psychology more generally.

Full text

Revealing Dynamic Relations Between Mathematics Self-Concept and Perceived
Achievement From Lesson to Lesson: An Experience-Sampling Study
Christoph Niepel
1
, Herbert W. Marsh
2
, Jiesi Guo
2
, Reinhard Pekrun
2, 3
, and Jens Möller
4
1
Department of Behavioural and Cognitive Sciences, University of Luxembourg
2
Institute for Positive Psychology and Education, Australian Catholic University
3
Department of Psychology, University of Essex
4
Institute for Psychology of Learning and Instruction, Kiel University
Academic self-concept and achievement have been found to be reciprocally related across time.
However, existing research has focused on self-concept and achievement scores that have been averaged
over long time-periods. For the first time, the present study examined intraindividual (within-person)
relations between momentary (state) self-concept and lesson-specific perceived achievement (i.e., self-
reported comprehension) in students’everyday school life in real time using intensive longitudinal data.
We conducted an experience-sampling (e-diary) study with 372 German secondary school students in
Grades 9 and 10 over a period of 3 weeks after each mathematics lesson. Multilevel confirmatory factor
analyses confirmed a two-factor between-level and within-level structure of the state measures. We used
dynamic structural equation modeling to specify a multilevel first-order vector autoregressive model to
examine the dynamic relations between self-concept and perceived achievement. We found significant
reciprocal effects between academic self-concept and perceived achievement on a lesson-to-lesson basis.
Further, we found that these relations were independent of students’gender, reasoning ability, or mathe-
matics grades. We discuss implications for methodology, theory, and practice in self-concept research
and educational psychology more generally.
Educational Impact and Implications Statement
This study suggests that students’momentary perception of their mathematics ability (i.e., their state
mathematics self-concept) directly influences their lesson-specific comprehension (i.e., perceived
achievement) from mathematics lesson to mathematics lesson. In turn, state mathematics self-con-
cept is itself influenced by students’previous perceived achievement (i.e., in showing reciprocal
relations). Therefore, our results indicate that students’state mathematics self-concept makes a sub-
stantial contribution to their academic development in their everyday life at school.
Keywords: state academic self-concept, mathematics self-concept, reciprocal relations, experience sam-
pling, intensive longitudinal data
Supplemental materials: https://doi.org/10.1037/edu0000716.supp
Academic self-concept (ASC)—the mental representation of
one’s own ability (Brunner et al., 2010;Marsh & Shavelson, 1985;
Shavelson et al., 1976)—is key for students’academic success and
well-being (e.g., Eccles, 2009;Marsh, 2006;Marsh et al., 2019).
Researchers have repeatedly found that ASC is related to a broad
range of outcomes, such as academic interests (Marsh et al., 2005;
Schurtz et al., 2014), academic emotions (Arens et al., 2017;Pek-
run et al., 2019), achievement goals (Dörendahl et al., 2021;
This article was published Online First November 11, 2021.
Christoph Niepel https://orcid.org/0000-0001-6376-7901
Herbert W. Marsh https://orcid.org/0000-0002-1078-9717
Jiesi Guo https://orcid.org/0000-0003-2102-803X
Reinhard Pekrun https://orcid.org/0000-0003-4489-3827
Jens Möller https://orcid.org/0000-0003-1767-5859
This research was funded by a grant from the Luxembourg National
Research Fund (FNR) to Christoph Niepel (C16/SC/11333571). All
statements expressed in this article are those of the authors and do not reflect
the official opinions or policies of the authors’host affiliations or any of the
supporting institutions. Individuals interested in the data can contact Christoph
Niepel by email (christoph.niepel@uni.lu). Apart from the commercially
available and copyrighted reasoning ability test items, all other items are
presented in the article or, for the original German-language wording, in the
online supplemental material. The Mplus code for the main analysis can also
be found in the online supplemental material.
Correspondence concerning this article should be addressed to Christoph
Niepel, Department of Behavioural and Cognitive Sciences, University of
Luxembourg, 11, Porte des Sciences, 4366 Esch-sur-Alzette, Luxembourg.
Email: christoph.niepel@uni.lu
1380
Journal of Educational Psychology
©2021 American Psychological Association 2022, Vol. 114, No. 6, 1380–1393
ISSN: 0022-0663 https://doi.org/10.1037/edu0000716
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Content may be shared at no cost, but any requests to reuse this content in part or whole must go through the American Psychological Association.

Niepel et al., 2014a), and career aspirations (Guo et al., 2017).
However, most research has focused on relations between ASC
and students’academic achievement (e.g., for an overview, see
Trautwein & Möller, 2016). At the core of this research is the find-
ing that ASC and achievement are reciprocally related across time
(Marsh & Craven, 2006;Marsh & Martin, 2011;Valentine et al.,
2004;Wu et al., 2021). This finding implies that students with
higher levels of achievement tend to develop higher ASC levels
over time, whereas, at the same time, students higher in ASC tend
to deliver higher levels of achievement in the long run.
However, despite the vast body of research on the relation
between ASC and achievement, it appears that no studies on the
longitudinal reciprocal relations between ASC and achievement
have applied intensive longitudinal methods on the situational
level to capture intraindividual dynamics in ASC and achievement
in real time in actual learning situations (through ambulatory
assessment, ecological momentary assessments, or experience
sampling; for an overview of these methods, see, e.g., Bolger &
Laurenceau, 2013;Hamaker & Wichers, 2017;Trull & Ebner-
Priemer, 2014;Zirkel et al., 2015). We know from previous
research that students’competence perceptions can be subject to
everyday variation (Malmberg & Martin, 2019;Tsai et al., 2008).
Nevertheless, due to the lack of intensive longitudinal studies on
the reciprocal relations between ASC and achievement, the mo-
mentary (state) intraindividual (within-person) dynamics between
ASC and achievement remain a black box. The existing longitudi-
nal research on their reciprocal relations does not allow inferences
to be made about within-person dynamics (see Murayama et al.,
2017). By implication (as will be described in more detail below),
the idea that ASC and achievement are mutually reinforcing has
yet to be verified when shifting toward an intraindividual, real-
time, and real-life perspective. To start filling this gap, we
employed experience sampling (i.e., e-diaries via smartphones) in
a sample of 372 German secondary school students. We tested the
reciprocal relations and temporal dynamics of lesson-specific
(state) ASCs and perceived achievement (i.e., self-reported com-
prehension) in the domain of mathematics in students’everyday
life at school across a period of 3 weeks.
Reciprocal Relations Between Academic Self-Concept
and Achievement
ASC is typically regarded as a highly domain-specific construct
(e.g., Brunner et al., 2010) with, for instance, mathematics self-
concept (MSC) representing a person’s mental representation of
their mathematics ability. There are a plethora of scientific articles
on the relation between students’ASC and achievement. Thereby,
achievement has been measured in various ways, such as standar-
dized achievement test scores, record-cards grades, teacher ratings,
or self-reports of achievement (see Valentine et al., 2004). Histori-
cally, three different main theoretical models that describe the two
constructs’causal ordering can be distinguished (Calsyn & Kenny,
1977;Marsh & Martin, 2011). First, the skill-development model
claims that students’previous achievement causes ASC (i.e., skill-
development effect), whereas students’ASC has no impact on
their later achievement. Second, the self-enhancement model
claims that students’ASC influences their achievement (i.e., self-
enhancement effect), whereas the latter is supposed to have no
impact on their later ASC. Third, the reciprocal effects model
claims that previous achievement affects ASC, and previous ASC
affects achievement (Marsh, 1990). According to the reciprocal
effects model, the relation between the constructs is characterized
by long-lasting, mutually reinforcing skill-development and self-
enhancement effects. Self-enhancement effects (ASC causes
achievement), which are claimed by both the self-enhancement
model and the reciprocal effects model, play a central role in ASC
theory and research. Their support implies that interventions that
are aimed at fostering ASC would also impact students’academic
achievement (Ehm et al., 2019).
The vast majority of empirical findings have supported the re-
ciprocal effects model (e.g., Arens et al., 2017;Guay et al., 2003;
Marsh & O’Mara, 2008;Marsh et al., 2018;Retelsdorf et al.,
2014; but see Ehm et al., 2019). These findings include meta-ana-
lytical evidence (Valentine & DuBois, 2005;Valentine et al.,
2004;Wu et al., 2021; cf. Huang, 2011). For example, Wu et al.
(2021) reported average effect sizes of .08 for self-enhancement
and somewhat stronger effect sizes of .16 for skill-development
effects. Moreover, the reciprocal effects model pattern has been
found to generalize across gender as well as different cultures and
ability levels (e.g., Gorges et al., 2018;Marsh & Martin, 2011;
Marsh et al., 2005;Seaton et al., 2015;Valentine et al., 2004; cf.
Wu et al., 2021).
Lesson-to-Lesson Dynamics Between Academic
Self-Concept and Perceived Achievement: Shifting
Toward a Within-Person and Short-Term Perspective
Despite the extensive empirical support for reciprocal relations
between ASC and achievement, there appears to be no research
that has used (a) a within-person, intraindividual approach com-
bined with (b) a short-term, state-based approach in drawing on in-
tensive longitudinal data obtained through experience sampling.
Instead, previous research has typically taken a between-person
approach and focused solely on long-term relations in drawing on
only a few assessments of ASC and achievement, bridging time
spans of several months to years (Ehm et al., 2019;Huang, 2011;
Valentine et al., 2004;Wu et al., 2021).
Most research has employed cross-lagged panel models to
examine the long-term relations between ASC and achievement
longitudinally. Therefore, such research has focused on students’
relative rank-order position in their self-concept and their relation
to their relative rank-order position in their future achievement
(and vice versa). Such cross-lagged panel models do not allow
researchers to properly distinguish intraindividual (within-person)
processes from stable interindividual (between-person) differences
(Ehm et al., 2019;Hamaker et al., 2015). This is problematic
because major theoretical models on the longitudinal relations
between ASC and achievement (e.g., the reciprocal effects model)
clearly focus on self-enhancement and skill-development effects
and thus on motivational within-person processes.
Indeed, in their recent attempt to disentangle within- and
between-person processes in drawing on four measurement occa-
sions bridging 3.5 years in German primary school children using
a random intercept cross-lagged panel model, Ehm et al. (2019)
found no evidence of self-enhancement effects, contradicting
assumptions of the reciprocal effects model. However, the authors
acknowledged the need for future research. Notably, the lack of
self-enhancement effects might be due to the relatively long lags
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between their measurement occasions. In order to detect cross-
lagged effects, optimal time lags should be rather short so that
within-person processes can be examined while controlling for
interindividual differences (Dormann & Griffin, 2015; see Ehm et
al., 2019).
To this end, researchers need to shift toward a short-term perspec-
tive. Studying longitudinal short-term relations is typically associated
with intensive longitudinal (or microlongitudinal) data such as those
obtained with experience sampling to study students’experiences in
their everyday life at school (e.g., Hamaker & Wichers, 2017;Trull
& Ebner-Priemer, 2014;Zirkel et al., 2015). Intensive longitudinal
data is characterized by its focus on within-person regulatory mecha-
nisms and associated dynamics as well as its short-term perspective
in typically drawing on multiple measurement time points that are
close together (McNeish & Hamaker, 2020).
Shifting toward within-person and short-term perspectives in
studying dynamics between ASC and achievement with experi-
ence sampling is crucial for several reasons. Most importantly,
between-person and within-person relations between variables are
most likely to be statistically independent unless the (unlikely)
assumption of ergodicity holds (Molenaar, 2004;Murayama et al.,
2017). Importantly, this implies that research findings on recipro-
cal effects between ASC and achievement may actually not hold
when shifting toward an everyday, within-person perspective (see
Ehm et al., 2019). Also, in collecting intensive longitudinal data,
the researcher obtains a relatively large number of repeated meas-
urements of interest variables in real time and real life. As such,
they shift their perspective from merely a trait-based perspective
toward a state-based perspective.
Specifically, in the present study, we applied an experience-sam-
pling, e-diary methodology to obtain lesson-specific—or state—
measures of MSC and perceived achievement (i.e., self-reported
comprehension) for every single lesson in mathematics. We suggest
that this represents repeated snapshots (Hamaker & Wichers, 2017)
of students’ASC and perceived achievement over time (described
in the Method section). Notably, state assessments capture a
broader range of momentarily perceived situations and may be less
biased than conventional trait-based self-reports (Trull & Ebner-
Priemer, 2014). When research on the reciprocal relations between
ASC and achievement exclusively relies on only a few assessment
occasions that bridge longer time spans, such an approach is argu-
ably insufficient for capturing the bandwidth of the dynamics that
students actually perceive in their everyday life at school. For
instance, students who generally perform well in mathematics may
experience difficulty understanding the material in particular les-
sons or may fail to complete some tasks from time to time. Con-
versely, students who generally struggle in mathematics may feel
that they were able to follow a particular lesson well or may some-
times feel that they understood the material the teacher went over.
In addition, feedback from teachers and peers, which students can
use to infer their current performance, can also vary across lessons.
Such dynamics arguably remain undetected in long-term studies in
which longitudinal assessments are separated by months or years.
We know from previous research that even allegedly well-estab-
lished notions had to be revised in the field after a state-based per-
spective was adopted (see, e.g., Goetz et al., 2013; research on state
vs. trait mathematics anxiety).
Notably, the intraindividual, within-person approaches used in
the present study also capture differences between persons. As
such, they enable researchers to potentially reveal interindividual
differences across the observed within-person relations, or stated
differently, the heterogeneity of the functional within-person rela-
tions between the variables of interest (see, e.g., Pekrun et al.,
2002).
The Present Study
In the present experience-sampling study, we drew on e-diary data
collected in German secondary schools across a time-period of 3
weeks to uncover real-life and real-time dynamics between students’
ASC and their lesson-specific perceived achievement in the domain
of mathematics. This study is the first to revisit the reciprocal rela-
tions between ASC and (perceived) achievement in everyday life in
shifting toward a state-based, within-person perspective. The over-
arching aim of this study was to examine the existence and signifi-
cance of self-enhancement and skill-development effects when
studying students in every single lesson in a given domain at school.
In focusing on the mathematics domain, we studied everyday rela-
tions between ASC and perceived achievement within the arguably
most frequently analyzed domain in ASC research (Marsh, 2006),
thus enabling us to better compare and embed our results into the
existing between-person research.
As mentioned earlier, previous between-student research on the re-
ciprocal relations between ASC and achievement has deployed differ-
ent indicators and proxies for measuring achievement (Huang, 2011;
Valentine et al., 2004;Wu et al., 2021). Standardized achievement tests
or report-card grades may be the indicators of choice when drawing on
panel designs spanning longer time intervals (Marsh & Martin, 2011).
However, when researchers examine intraindividual skill-development
and self-enhancement effects in real time and real life, students’
achievement should be assessed in an ecologically valid way on a les-
son-to-lesson basis. In comparison with conventional panel designs,
e-diary designs allow researchers to collect longitudinal data in a natu-
ral, spontaneous context (Reis, 2014)inafarlessintrusivemannerwith
fewer barriers (McNeish & Hamaker, 2020). To measure students’cog-
nitive learning outcomes, previous e-diary studies have widely
employed self-reports of learning or perceived achievement (see, e.g.,
Giannakos et al., 2020;Peterson & Miller, 2004;Shernoff, Sannella, et
al., 2017) instead of more objective (but arguably also more intrusive)
daily measures of achievement, such as standardized tests. The use of
self-reports of learning or perceived achievement has a long tradition
(e.g., Richmond et al., 1987). Previous research has shown that students
are able to accurately assess their own learning (e.g., Brown et al.,
2015;Chesebro & McCroskey, 2000;Ross, 2006), and measures of
students’perceived achievement have been widely used as a valid way
to measure students’cognitive learning outcomes (e.g., Kurucay &
Inan, 2017;Rovai et al., 2009;Shin, 2003;Yoon et al., 2020). In the
present study, we asked students to indicate their comprehension and
their learning progress for every single mathematics lesson. Students
thus reported in real time how well they understood the material that
they had just gone through in class. This understanding indicates stu-
dents’lesson-specific perceived achievement with respect to what they
were supposed to learn (see the Method section below).
However, it is important to note that although we built on existing
research on the reciprocal effects model (Marsh, 1990;Marsh & Cra-
ven, 2006;Wu et al., 2021), the present study should not be under-
stood as a direct replication of the classic reciprocal effects model at
the within-person and short-term levels. Standardized achievement
1382 NIEPEL, MARSH, GUO, PEKRUN, AND MÖLLER
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test scores or report card grades have been recommended to test the
reciprocal effects model (e.g., Marsh & Martin, 2011). In contrast,
perceived achievement measures were used in the present study.
Specifically, our overarching aim resulted in two focal research
questions, which are both located on the within-person, intraindi-
vidual level:
RQ 1: Is there a positive and significant path from previous per-
ceived achievement to subsequent MSC (i.e., the skill-develop-
ment effect) on a lesson-to-lesson basis?
RQ 2: Is there a positive and significant path from previous
MSC to subsequent perceived achievement (i.e., the self-
enhancement effect) on a lesson-to-lesson basis?
Both research questions are critically important for ASC theory
and research because of the lack of within-person research that has
applied experience sampling rather than the between-person approach
used in most research (see Marsh & Craven, 2006). In addition to our
two focal research questions, we aimed to examine interindividual
(between-person) differences in the observed intraindividual relations
between ASC and perceived achievement. Specifically, we explored
whether interindividual differences in the observed within-person
associations between MSC and perceived achievement could be
explained by students’gender, reasoning ability, or mathematics
grades. This resulted in our third research question, which focused on
the between-person, interindividual level:
RQ 3: Are everyday skill-development and self-enhancement
dynamics generalizable across or moderated by students’gen-
der, reasoning ability, and mathematics grades?
Gender, reasoning ability, and mathematics grades have all been
shown to play predominant roles in the formation of MSC: Students
with higher reasoning ability and those obtaining better report-card
grades in mathematics typically report higher levels of MSC (Möller et
al., 2020), whereas gender disparities in MSC to the disadvantage of
girls and young women have repeatedly been found in previous
research regardless of students’actual mathematics performance (e.g.,
Frenzel et al., 2007;Niepel et al., 2019). Overall, previous between-per-
son research on longitudinal relations between ASC and achievement
(e.g., Valentine et al., 2004;Wu et al., 2021) led us to expect to find
state-based skill-development (RQ 1) and self-enhancement effects
(RQ 2) between MSC and perceived achievement in students’everyday
life. Further, previous between-person findings on the generalizability
of skill-development and self-enhancement effects across gender and
ability levels (e.g., Marsh et al., 2005;Seaton et al., 2015;Valentine et
al., 2004) led us to expect that both effects would be largely generaliz-
able across gender, reasoning ability, and obtained mathematics grades
(RQ 3). We applied multilevel confirmatory factor analyses (MCFA)
and dynamic structural equation modeling (DSEM; Asparouhov et al.,
2018; see the Method section below) to address our research questions.
Method
Procedure and Participants
In the present study, we drew a sample of N= 372 students
(34.1% young men, based on n= 301) whose data were collected
as part of the larger “Dynamics of Academic Self-Concept in
Everyday Life”(DynASCEL) project,
1
which focused on the
everyday dynamics of ASC. Students attended one of 18 different
classes at six different academic-track schools (Gymnasium) in
Grade 9 (n= 308) or Grade 10 (n= 64) in four different federal
states of Germany (Baden-Württemberg, Mecklenburg-Vorpom-
mern, Nordrhein-Westfalen, Rheinland-Pfalz). The average self-
reported age was 15.3 years (SD = .68; range: 13.3 to 17.4; n=
298). Students’participation was voluntary, and written parental
consent was obtained for all participating students. Students could
skip prompts or single questions. All procedures were approved by
the ethics review panel of the University of Luxembourg and by
all involved education authorities.
The data collection took part over a 5-week period at the respec-
tive schools. In Week 1, students completed a background inven-
tory (paper-pencil format). In Weeks 2 to 4, students were given a
smartphone as a hub for experience sampling over 3 weeks (e-di-
ary approach). In Week 5, students completed a shorter postques-
tionnaire (paper-pencil format). In the present study, we focused
on the e-diary data on state MSC and perceived achievement in ev-
ery single mathematics lesson that we collected across the 3-week
period from Weeks 2 to 4. In addition, we drew on data collected
in Week 1 (i.e., background inventory) to obtain information about
students’gender, reasoning ability, and mathematics grades (see
the Measures section below).
To obtain data on state MSC and perceived achievement in every
single mathematics lesson, students received prompts through an au-
ditory signal that asked them to complete a brief electronic question-
naire on the smartphone at the end of each mathematics lesson.
Prompts were programmed using the movisensXS software (Movis-
ens GmbH, 2017) following the timetable of each respective class
with the number of mathematics lessons (i.e., measurement points)
thus varying from class to class (M= 10.11 mathematics lessons;
SD = 3.39; range: 3 to 16). As can be expected from intensive longi-
tudinal designs, there are many reasons for missing values. These
included, for example, students’or mathematics teachers’sick leave,
exams, excursions, or other obligations, as well as technical issues
(e.g., dead battery, students left smartphone at home). Per design, stu-
dents were instructed not to respond to prompts when they had not
had classroom instruction or were absent from school. We obtained
responses to 2,702 prompts (students * mathematics lessons), repre-
senting 71.4% of the previously programmed prompts. Of these
2,702 accepted prompts (i.e., with at least one item answered), 97.5%
provided complete data (i.e., six items per prompt; see Measures sec-
tion below).
Measures
E-Diary
State Mathematics Self-Concept. We assessed students’
state MSC after every single mathematics lesson across a time-pe-
riod of 3 weeks. State MSC was assessed with three items based
on the Self-Description Questionnaire (SDQ; Marsh et al., 1983),
which is considered to be one of the best self-concept instruments
1
Data from the larger research project have been and will be used in
other articles addressing different research questions (Dörendahl et al.,
2021). The intensive longitudinal data examined in this study have not
previously been reported in other articles.
STATE MATH SELF-CONCEPT AND PERCEIVED ACHIEVEMENT 1383
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available (e.g., Byrne, 2002). Three-item short-form (trait) MSC
instruments based on the SDQ have been shown to be psychomet-
rically sound for educational research purposes (Gogol et al.,
2014) and are commonly used in longitudinal MSC research (e.g.,
Marsh et al., 2015,2018;Möller et al., 2011;Niepel et al., 2014b).
Specifically, we adapted the MSC items to the specific demands of
experience sampling by beginning every item with the passage
“Currently, I think that ....”Students responded on a 6-point Lik-
ert scale ranging from 0 (false)to5(true). The item wordings
were “Currently, I think that I am good at mathematics,”“[...]
work in mathematics is easy for me,”and “[...] I learn quickly in
mathematics,”such that higher item scores indicated higher state
MSC. The original German-language item wordings are listed in
Table S1 in the online supplemental material.
Lesson-Specific Perceived Achievement in Mathematics. As
we did for state MSC (see the previous section), we assessed stu-
dents’perceived achievement in terms of their lesson-specificcom-
prehension and learning progress after every single mathematics
lesson across a time-period of 3 weeks. Three items were used to
assess perceived achievement. Students responded on a 6-point Likert
scale, ranging from 0 (false)to5(true). The item wordings were “I
was able to follow the last lesson well,”“I understood a lot in the last
lesson,”and “I learned a lot in the last lesson,”such that higher item
scores indicated better lesson-specific perceived achievement in
mathematics. Similar items have been commonly used in e-diary
studies (e.g., Peterson & Miller, 2004;Shernof et al., 2017;Shernoff,
Ruzek, & Sinha, 2017;Shernoff, Sannella, et al., 2017)aswellasin
previous between-person research (e.g., Yoon et al., 2020) to measure
perceived (learning) achievement (see Richmond et al., 1987). The
original German-language item wordings are listed in Table S1 in the
online supplemental material.
Background Inventory
Reasoning Ability. We applied the Intelligenz-Struktur-Test-
Screening (IST-Screening; Liepmann et al., 2012) to assess stu-
dents’reasoning ability. The IST-Screening is an economic (less
than 30 min) reasoning ability measure that includes three groups
of tasks consisting of verbal analogies, number sequences, and fig-
ural matrices (each consisting of 20 items). It is based on the Intel-
ligenz-Struktur-Test (IST; Amthauer, 1970;Liepmann et al.,
2007), an intelligence test that is widely used in Germany
(Schmidt-Atzert & Amelang, 2012). The IST-Screening exists in
two parallel versions, A and B; in the present study, we used Ver-
sion A. It was presented in a paper-pencil format in the week
before the e-diary assessment began. Liepmann et al. (2012)
reported an internal consistency of a= .87 for the full-scale rea-
soning ability composite score. In our subsequent analyses, we
used the full-scale reasoning ability composite raw score; the
observed reliability in the present study was x= .77 (a= .76).
Mathematics Grades. Students reported their grades in math-
ematics as obtained from their most recent report card. Research
on the validity of self-reported grades in Germany suggests that
self-reported school grades are not subject to systematic reporting
biases (Dickhäuser & Plenter, 2005;Sparfeldt et al., 2008). In Ger-
many, a 6-point grading system is used; we reverse-scored the
grades in the present study such that higher values indicated better
school grades in mathematics (i.e., ranging from 1 = unsatisfactory
to 6 = very good).
Data Analysis
Multilevel Confirmatory Factor Analysis (MCFA)
Prior to our main analyses, we conducted a MCFA of the intra-
individual e-diary data using the statistical software Mplus 8.3
(Muthén & Muthén, 1998–2019) to inspect the psychometric prop-
erties of the e-diary measures (Kim et al., 2016). Utilizing MCFA,
we accounted for timepoints nested within students by explicitly
modeling the factor structure on the within-person level (Level 1;
i.e., state-like factors) as well as on the between-person level
(Level 2; trait-like factors). To control for class-level effects, we
included 17 dummy variables (based on 18 classrooms) at Level
2.
2
We used the Mplus MLR estimator, which is robust against
mild violations of normality and allowed us to deal with missing
data (Kaplan, 2009).
In a first step, we calculated intraclass correlations (ICCs) to esti-
mate the amount of variation in MSC and perceived achievement
across Levels 1 and 2. Second, we tested a series of different models
to analyze the measures’factor structure across both levels. To evalu-
ate the model fit, we considered the comparative fit index (CFI), the
Tucker-Lewis index (TLI), the root mean square error of approxima-
tion (RMSEA), and the standardized root-mean-square residual
(SRMR). We used the recommended cut-off values (CFI $.95;
TLI $.95; RMSEA #.06; SRMR #.08; Kline, 2005).
To examine the model fit for each level separately, we employed
partially saturated models (Janis et al., 2016). In doing so, we speci-
fied (a) the hypothesized two-factor structure as well as (b) a more
parsimonious one-factor structure at Level 1, while specifying a satu-
rated model (i.e., item variances and covariances only) at Level 2 and
vice versa (c and d). In a third step, to ensure a meaningful interpreta-
tion of the constructs across levels (Stapleton et al., 2016), we tested
for cross-level invariance by restricting the factor loadings of the cor-
responding items to be equal across Levels 1 and 2, and we freely
estimated the factor variances at Level 2 (Jak & Jorgensen, 2017).
Competing models were compared based on decreases in model fit
and differences in the Akaike information criterion (AIC) and Bayes
information criterion (BIC) with a preference given to the model with
the lower value. Finally, we calculated level-specific reliabilities in
terms of the Level 1 and Level 2 omega coefficients in freely estimat-
ing all factor loadings and fixing the factor variances to 1 at both lev-
els (Geldhof et al., 2014).
Dynamic Structural Equation Modeling
To address our research questions, we conducted DSEM
(Asparouhov et al., 2018) in Mplus 8.3 (Muthén & Muthén,
1998–2019). Before we began, we ensured that there were no
mean trends in state MSC and perceived achievement (i.e., neither
variable consistently increased or decreased over the 3-week pe-
riod) that needed to be incorporated into our model (McNeish &
Hamaker, 2020). To this end, we calculated a linear regression
with time as the only predictor at Level 1 (McNeish & Hamaker,
2020) in Mplus. The effects were close to zero and not statistically
significant (for state MSC: b= .03, p= .222; for perceived
2
The observed intraclass correlations (ICCs) for Level 3 (i.e., different
classrooms) were rather small, ranging from ICC = .033 to ICC = .076 for
perceived achievement and from ICC = .021 to ICC = .024 for MSC.
1384 NIEPEL, MARSH, GUO, PEKRUN, AND MÖLLER
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achievement: b= -.04, p= .091), suggesting no linear trend over
time.
We specified a multilevel first-order vector autoregressive
(VAR(1)) model (Hamaker et al., 2018), a multilevel extension of
a time series model. The VAR(1) model can also be thought of as
a multilevel extension of a cross-lagged panel model that allows
for interindividual differences in means and lagged effects
(Hamaker et al., 2018, p. 826). Data were decomposed into
within-person (Level 1) and between-person (Level 2) compo-
nents. At Level 1, we specified the cross-lagged relations between
state MSC and lesson-specific perceived achievement (i.e., by
using manifest mean scores of the three indicators at each time
point). We regressed state MSC (MSC
t
) on lesson-specific per-
ceived achievement at the previous time point (Achievement
t-1
)to
test the skill-development effects on a lesson-to-lesson basis
(RQ1). We regressed lesson-specific perceived achievement
(Achievement
t
) on state MSC at the previous time point (MSC
t-1
)
to test for self-enhancement effects on a lesson-to-lesson basis
(RQ2). For the autoregressive paths from MSC
t-1
(Achievement
t-1
)
to MSC
t
(Achievement
t
) at Level 1, it is important to note that these
paths indicate the amount of carryover (or inertia) from one lesson
to the next for each student (Hamaker et al., 2018). Therewith,
these two autoregressive paths indicate how quickly students
return to their habitual, trait-like MSC (or habitual level of per-
ceived achievement) after experiencing situation-specific ups and
downs in their state MSCs (perceived achievements). Put differ-
ently, the larger the carryover across mathematics lessons, the
more the current state depends on the previous lesson’s state, and
the longer it takes to return to the trait level. Students’mean levels
for MSC and perceived achievement, which can be interpreted as
their trait scores, are modeled at Level 2 (see next paragraph).
At Level 2, we estimated the interindividual variances, fixed
effects, and intercorrelations of six variables. More specifically,
we estimated the two mean values for MSC and perceived
achievement (i.e., trait scores for MSC and perceived achieve-
ment), the two autoregressive parameters for MSC and perceived
achievement (indicating carryover), the skill-development effect
from perceived achievement to MSC, and the self-enhancement
effect from MSC to perceived achievement. DSEM is based on
Bayesian estimation, and missing data are sampled from their con-
ditional posterior in this kind of analysis. For our analyses, we
used Mplus’default priors. As the time intervals between consecu-
tive measurement points (i.e., mathematics lessons) varied in ac-
cordance with each class’s timetable, we controlled for the time
intervals by using the TINTERVAL option implemented in Mplus.
To this end, we specified a time interval variable, which indicated
the time difference for every measurement point in hours from
the very first prompt per class. The Mplus code for our specified
VAR(1) model can be found in the online supplemental material.
In the last step, we explored whether interindividual differences
in the observed within-person associations between MSC and per-
ceived achievement were generalizable across persons in relation
to gender, reasoning ability, and grades (RQ3). To this end, we
saved the factor scores of the Level 2 (skill-development and self-
enhancement) effects by using a multiple imputation approach
(Graham et al., 2003) to impute 50 data sets in Mplus containing
the imputed values for the factor scores. The imputed data sets
were then used to calculated bivariate correlations between the
factor scores and gender, mathematics grades, and reasoning
ability using maximum likelihood estimation. We chose this
approach to decrease model complexity and to ensure model con-
vergence. To handle missing values in the student background var-
iables, most of which occurred due to technical problems
(percentages of missing values: 19.1% for gender; 17.7% for rea-
soning; 20.7% for mathematics grades), we used full-information
maximum likelihood (FIML) implemented in Mplus.
Results
Preliminary Analyses
Before we computed the main analyses, we examined the ICCs,
the latent factor structure, the cross-level invariance, and the reli-
abilities of the intraindividual e-diary measures (state MSC and
lesson-specific perceived achievement) by means of an MCFA.
The ICCs were ICC
ACH1
= .384, ICC
ACH2
= .373, and ICC
ACH3
=
.370 for the perceived achievement items, and ICC
MSC1
= .744,
ICC
MSC2
= .737, and ICC
MSC3
= .739 for the MSC items. The
ICCs indicated that students’perceived achievement showed
stronger intraindividual variation across the observed 3-week pe-
riod than MSC did. Most of the variance in perceived achievement
originated from Level 1 (variation within students). In contrast,
most of the variance in MSC originated from Level 2 (variation
between students).
Table 1 presents the results of testing alternative factor structures
across the two levels. Inspections of the fits of the partially saturated
models suggested a very good approximation to the data for the two-
factor solutions on both Levels 1 and 2 (see Table 1, Models a and
c). By contrast, the more parsimonious one-factor solutions exhibited
poor fits at both levels (Models b and d). In addition to the partially
saturated models, we ran further MCFAs to inspect the latent factor
structure. As such, we specified Models e and f, which assumed two
latent factors on one level but only one general latent factor at the
other level (see Table 1). Finally, we specified Model g, which
assumed a two-factor structure at both levels. The fit indices, as well
as differences in the AIC and BIC, indicated that model g provided a
better approximation to the data than Models e or f (see Table 1), in
line with the results we got from the partially saturated model test
approach (Models a to d). Together, our tests pointed to a two-factor
structure at both levels, indicating that MSC and perceived achieve-
ment were empirically distinguishable constructs at the within- and
between-person levels.
After establishing the two-factor structure at both levels, we
tested for cross-level invariance in a next step. To this end, we
built upon Model g (see Table 1), which consisted of two intercor-
related latent factors (MSC and perceived achievement) with three
indicators per construct. The six items were constrained to load
only on their respective factor, and the latent constructs were not
allowed to correlate across levels. In specifying Model h, we re-
stricted the factor loadings of the corresponding items to be equal
across Levels 1 and 2. The overall fit suggested an excellent over-
all approximation to the data (see Table 1). Compared with Model
g, no meaningful decreases in the CFI, TLI, RMSEA, or SRMR
could be detected; we observed higher AIC but lower BIC values.
Overall, the results suggested that the assumption of cross-level
invariance held for both constructs. The factor loadings ranged
from k= .742 to k= .934 for perceived achievement and from k=
STATE MATH SELF-CONCEPT AND PERCEIVED ACHIEVEMENT 1385
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.795 to k= .845 for MSC at Level 1. They ranged from k= .708
to k= .917 for perceived achievement and from k= .959 to k=
.972 for MSC at Level 2 (all ps,.001). The latent correlations
between MSC and perceived achievement were q= .646 (p,
.001) at Level 1 and q= .863 (p,.001) at Level 2 (the coeffi-
cients came from Model h).
Reliability was tested by calculating McDonald’sxon Levels 1
and 2, indicating good reliabilities of x= .863 for MSC and x=
.894 for perceived achievement at Level 1 and of x= .996 for
MSC and x= .950 for perceived achievement at Level 2.
Main Analyses
We examined our three research questions using a multilevel VAR
(1) model for MSC and perceived achievement within the DSEM
framework. We used 100,000 Markov chain Monte Carlo (MCMC)
iterations with two Markov chains. The results here were based on
10,000 iterations because the first half of each chain was discarded as
burn-in, and a thinning of 10 iterations was used (i.e., only one in 10
iterations was saved; Gelman et al., 2014;seeHamaker et al., 2018).
Model convergence was evaluated by applying the potential scale
reduction criterion (PSR; Asparouhov & Muthén, 2010). PSR is the
ratio of the total variance across chains and the pooled variance
within a chain. We used PSR ,1.05 as an appropriate convergence
criterion (Gelman & Rubin, 1992).TheDSEMresultedingoodcon-
vergence. PSR values were below 1.05 for each parameter. More-
over, the trace plots for each parameter did not indicate any signs of
nonconvergence.
Figure 1 represents the path diagram for the within-person part of
the model and depicts the observed results (i.e., standardized parame-
ters and their 95% credible intervals). To address our two focal
research questions, RQ 1 and RQ 2, we looked at the cross-lagged
relations. We found intraindividual effects from perceived achieve-
ment to MSC at the next mathematics lesson (ACH
t-1
to MSC
t
)and
from MSC to perceived achievement at the next mathematics lesson
(MSC
t-1
to ACH
t
). None of the parameters’CIs contained zero.
These results point to the existence and significance of skill develop-
ment effects (RQ 1) and self-enhancement effects (RQ 2) in students’
everyday life at school. Specifically, we found average individually
standardized effects at the within-person level of .186 (95% CI [.117,
.257]) for skill-development effects and of .052 (95% CI [.003, .095])
for self-enhancement effects.
Concerning the autoregressive paths, the average individually
standardized autoregressive effects were stronger for perceived
achievement than for MSC (see Figure 1). This suggests that students
have more carryover in their lesson-specific perceived achievement
Figure 1
Graphical Representation of and Results for the Within-Person Part of the
Multilevel VAR(1) Model
Note. Paths indicating lesson-to-lesson self-enhancement effects (MSC
t-1
to ACH
t
), skill-
development effects (ACH
t-1
to MSC
t
), as well as students’carryover from lesson to lesson
(MSC
t-1
to MSC
t
; ACH
t-1
to ACH
t
). Standardized model parameters are shown; credible
intervals are depicted in brackets. MSC = mathematics self-concept; ACH = perceived
achievement.
Table 1
Testing Alternative Factor Structures and Cross-Level Invariance
Model Within Between MLR v2(df) CFI TLI RMSEA SRMR
W
SRMR
B
AIC BIC
Alternative factor structures
(a) Saturated 2 factors 12.499 (8) 1.000 .993 .014 .002 .003 31,590.651 32,428.699
(b) Saturated 1 factor 158.181 (9) .985 .786 .078 .044 .040 31,734.823 32,566.969
(c) 2 factors Saturated 11.243 (8) 1.000 .995 .012 .009 .001 31,598.737 32,436.785
(d) 1 factor Saturated 848.415 (9) .918 .204 .186 .122 .004 33,438.489 34,270.635
(e) 1 factor 2 factors 1,135.797 (17) .891 .150 .156 .122 .004 33,435.386 34,220.318
(f) 2 factors 1 factor 362.261 (17) .966 .738 .087 .044 .040 31,745.567 32,530.499
(g) 2 factors 2 factors 26.695 (16) .999 .991 .016 .009 .003 31,600.308 32,391.142
Cross-level invariance
(h) 2 factors 2 factors 41.082 (20) .998 .986 .020 .015 .012 31,614.708 32,381.935
Note. MLR = Maximum likelihood estimation with robust standard errors; CFI = comparative fit index; TLI = Tucker-Lewis index; RMSEA = root
mean square error of approximation; SRMR
w
= standardized root mean square residual value for within; SRMR
b
= standardized root mean square residual
value for between; AIC = Akaike’s information criterion; BIC = Bayesian information criterion.
1386 NIEPEL, MARSH, GUO, PEKRUN, AND MÖLLER
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from one lesson to the next than is the case for their state MSC (i.e.,
carryover effects). Furthermore, the average correlation between the
within-person residuals of perceived achievement and MSC was
.484 (95% CI [.438, .534]; see Figure 1). The averaged proportion of
explained variance on the within-person level in our model was .222
(95% CI [.198, .250]) for lesson-specific perceived achievement and
.326 (95% CI [.288, .359]) for state MSC.
Table 2 depicts interindividual, Level 2 means, variances, and cor-
relations. Looking at the variance estimators, we found that students
differed not only in their (trait-like) MSC and (trait-like) perceived
achievement values but also in the strengths of their (trait-like) indi-
vidual skill-development and self-enhancement effects (i.e., between-
person variances of the within-person cross-lagged relations across
mathematics lessons). Seven out of the 15 correlations (see Table 2)
were statistically significant (their 95% credible intervals did not con-
tain zero). The two mean values were positively intercorrelated, indi-
cating that students higher in MSC also tended to report higher levels
of perceived achievement. Further, we found that the skill-develop-
ment and self-enhancement effects were negatively related. This indi-
cates that students who experienced stronger self-enhancement effects
tended to experience weaker skill-development effects. The remaining
five correlations involved the autoregressive (carryover) effects for
perceived achievement (ACH
t-1
!ACH
t
)orMSC(MSC
t-1
!
MSC
t
). Specifically, mean MSC and mean perceived achievement
were both negatively correlated with the autoregressive effects for
perceived achievement. This indicates that students with a higher trait
level for MSC (or for perceived achievement) also tended to have less
carryover in their lesson-specific perceived achievements from one
lesson to the next. Further, skill-development effects (ACH
t-1
!
MSC
t
) were positively correlated with autoregressive effects for per-
ceived achievement (ACH
t-1
!ACH
t
) and negatively correlated with
autoregressive effects for MSC (MSC
t-1
!MSC
t
). This indicates that
students who had stronger skill-development effects also tended to
have more carryover in their lesson-specific perceived achievement
from lesson to lesson. At the same time, students who had stronger
skill-development effects tended to have less carryover in their state
MSCs. Finally, we found that the self-enhancement effects
(MSC
t-1
!ACH
t
) were positively related to the autoregressive
effects for MSC (MSC
t-1
!MSC
t
). This indicates that students
who had stronger self-enhancement effects also tended to have
more carryover in their state MSCs from one lesson to the next.
Students who were generally high versus low in trait MSC (and
perceived achievement) did not seem to differ in how often they
experienced skill-development or self-enhancement effects: We
observed no relation between students’mean values and the
cross-lagged parameters.
To address RQ 3, in the last step we calculated correlations
between the factor scores of the six Level 2 variables (obtained
from the DSEM we conducted as described in the previous para-
graph) and the student background variables (gender, mathemat-
ics grades, and reasoning ability). Gender was not significantly
related to reasoning ability (r=–.109, p= .057; 0 = young men;
1 = young women) or to grades (r=–.084, p= .152); reasoning
ability and grades showed a significant positive relation (r=
.389, p,.001). Table 3 depicts the observed correlations
between factor scores and background variables. Importantly,
we did not find any significant correlations between either the
self-enhancement or the skill-development effects with the inter-
individual student characteristics. This suggests that both effects
Table 2
Level 2 Means, Variances (Unstandardized Estimates), and Correlations (Standardized Estimates) With Their 95% Credible Intervals (in Brackets)
Variable Fixed effects
(means) Random effects
(variances)
Correlations
12345
1. MSC 3.247* [3.128, 3.368] 1.300* [1.117, 1.522] —
2. ACH 3.477* [3.390, 3.562] 0.558* [0.461, 0.675] .860* [.814, .896] —
3. ACH
t-1
!MSC
t
0.125* [0.069, 0.181] 0.086* [0.057, 0.120] .130 [.290, .027] .070 [.263, .110] —
4. MSC
t-1
!ACH
t
0.091* [0.005, 0.169] 0.046* [0.019, 0.115] .030 [.460, .398] .146 [.581, .277] .560* [.806, .199] —
5. MSC
t-1
!MSC
t
0.171* [0.106, 0.242] 0.119* [0.083, 0.168] .142 [.330, .047] .206 [.405, .019] .529* [.693, .308] .823* [.576, .954] —
6. ACH
t-1
!ACH
t
0.320* [0.251, 0.386] 0.092* [0.062, 0.133] .413* [.588, .225] .453* [.693, .229] .457* [.210, .631] .143 [.541, .381] .057 [.300, .232]
Note. MSC = mathematics self-concept (indicating students’trait score); ACH = perceived achievement (indicating students’trait score); ACH
t-1
!MSC
t
= cross-lagged relation indicating students’
skill-development effect; MSC
t-1
!ACH
t
= cross-lagged relation indicating students’self-enhancement effect; MSC
t-1
!MSC
t
= autoregressive effect for mathematics self-concept indicating stu-
dents’carryover from lesson to lesson; ACH
t-1
!ACH
t
= autoregressive effect for perceived achievement indicating students’carryover from lesson to lesson.
*Parameter’s credible interval does not contain zero.
STATE MATH SELF-CONCEPT AND PERCEIVED ACHIEVEMENT 1387
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operated independently from gender, reasoning ability, and obtained
mathematics grade (RQ 3). However, we did find significant relations
for students’mean MSC and mean perceived achievement with gen-
der, reasoning, and grades. These results indicate that students who
had higher reasoning test scores and obtained better mathematics
grades tended to have higher trait levels on both MSC and perceived
achievement. Male students tended to have higher state MSCs and
lesson-specific perceived achievements than female students. How-
ever, we did not find any significant association between students’
carryover effect for MSC and any of the interindividual characteris-
tics we examined. Students’carryover effects for perceived achieve-
ment were significantly negatively correlated with their mathematics
grade but not with gender or their reasoning test scores. This indi-
cates that students who obtained better mathematics grades tended to
have less carryover in their lesson-specific perceived achievement
from one lesson to the next.
Discussion
Previous research on longitudinal reciprocal relations between
ASC and achievement has emphasized interindividual differences
between students averaged across relatively long time-periods.
Our study is apparently the first to evaluate this issue by exploring
intraindividual relations between ASC and perceived achievement
over short time-periods with an experience-sampling approach.
Our overarching aim was to examine the self-enhancement and
skill-development effects on students’concrete learning situations
in school. Further, we examined whether these effects were mod-
erated by or generalized across students’gender, reasoning ability,
or mathematics grades.
Dynamics Between Mathematics Self-Concept and
Perceived Achievement From Lesson to Lesson
We found that both self-enhancement and skill-development
effects operated to form students’self-concept and perceived achieve-
ment on a lesson-to-lesson basis. Our results are thereby in line with
predictions derived from existing interindividual research on the
reciprocal effects model (e.g., Marsh & Martin, 2011;Wu et al.,
2021). However, our study differs considerably from previous
research. As highlighted earlier, self-development and self-enhance-
ment effects are formation processes at the intraindividual level, but
they have been studied almost exclusively at the between-person
level. The present study is apparently the first to show the within-per-
son effects of self-concept on perceived achievement (and vice versa)
in real time in actual learning situations. As such, we provide evi-
dence that a higher ASC leads a student to better follow and under-
stand the learning material in concrete learning situations in real time.
Also, a student uses their concrete learning experience in real time to
shape their self-concept.
The sizes of the skill-development and self-enhancement effects
observed in the present study were comparable to the effect sizes
reported in the meta-analysis on the reciprocal effects model by
Wu et al. (2021) for trait-like self-concept and achievement, with
descriptively stronger skill-development effects than self-enhance-
ment effects. According to typical guidelines for the evaluation of
effect sizes (Cohen, 1988;Kline, 2005), the observed effects could
be interpreted as small and possibly even negligible (but see also
Gignac & Szodorai, 2016). However, we would like to emphasize
that both effects are incremental (i.e., in controlling for lesson-spe-
cific carryover effects and interindividual differences in MSC and
perceived achievement), mutually reinforcing across time, and
refer to relatively short time intervals from one school lesson to
another.
Another observation from our applied within-person perspective
was that the residuals of the two within-person measures (i.e., state
MSC and lesson-specific perceived achievement) showed a sub-
stantial contemporaneous correlation. This correlation might be
due to both potential (unobserved) third variables in the within-
part of our model (e.g., lesson-specific demands, social interac-
tions in the classroom, students’mood) and potential reciprocal
effects between the two variables, which may have occurred
within school lessons (Hamaker et al., 2018).
Interindividual Differences in the Dynamics Between
Mathematics Self-Concept and Perceived Achievement
The present study examined not only intraindividual relations
between state MSC and perceived achievement but also whether
there were interindividual differences in these intraindividual rela-
tions. Our results suggest that students differ in the extent to which
they experience self-enhancement effects (current situation-spe-
cific perceived achievement is influenced by state MSC from the
previous lesson) and skill-development effects (current state MSC
is influenced by the student’s level of perceived achievement in
the previous lesson) in their daily school life. As such, we pro-
vided evidence that students differ in the extents to which (a) their
ASC is influenced by their daily learning performance (interindi-
vidual differences in skill-development effects) and (b) their ASC
influences their daily learning performance (interindividual differ-
ences in self-enhancement effects).
Further, the self-enhancement and skill-development effects were
negatively correlated with each other. Thus, students who experience
more of one tend to experience less of the other. Importantly, this is a
substantive new finding with significant implications that cannot be
readily examined with the traditional (between-person) approaches
that are typically used to study reciprocal relationships between ASC
Table 3
Correlations Between the Factor Scores and Students’Gender,
Reasoning Ability Scores, and Mathematics Grades
Variable Gender Reasoning
ability Mathematics
grade
1. MSC .272** .303** .509**
2. ACH .217** .244** .414**
3. ACH
t-1
!MSC
t
.102 .030 .095
4. MSC
t-1
!ACH
t
.034 .043 .013
5. MSC
t-1
!MSC
t
.001 .072 .065
6. ACH
t-1
!ACH
t
.138 .152 .203*
Note. Gender: 0 = young men; 1 = young women; MSC = mathematics
self-concept (within-person mean indicating students’trait score); ACH =
perceived achievement (within-person mean indicating students’trait
score); ACH
t-1
!MSC
t
= cross-lagged relation indicating students’skill-
development effect; MSC
t-1
!ACH
t
= cross-lagged relation indicating
students’self-enhancement effect; MSC
t-1
!MSC
t
= autoregressive
effect for mathematics self-concept indicating students’carryover from
lesson to lesson; ACH
t-1
!ACH
t
= autoregressive effect for perceived
achievement indicating students’carryover from lesson to lesson.
*p,.05. ** p,.001.
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and achievement. Thus, implicit in the traditional approach is the
assumption that the relative strengths of the cross-lagged paths are
the same across different students (this holds true not only for cross-
lagged panel models but also for more recent approaches using ran-
dom intercept cross-lagged panel models; see, e.g., Ehm et al., 2019).
However, our within-person evaluation suggests that this assumption
is wrong. Not only were there differences in the relative strengths of
the self-enhancement and skill-development effects, but within each
student, the two effects tended to counterbalance each other. These
findings have important implications for the design of studies that are
intended to enhance ASC, achievement, or both constructs. Our study
suggests that such interventions must account for interindividual dif-
ferences in students’individual propensity for self-enhancement or
skill-development effects to be more effective.
The interindividual student characteristics that we examined—gen-
der, reasoning ability, and mathematics grades—seem to be unrelated
to differences in skill-development and self-enhancement effects: We
found these differences to be independent from gender, reasoning
ability, and grades. These findings were in line with our expectations,
which were based on previous between-person research. Specifically,
previous research has shown that gender does not moderate relations
between achievement and ASC (Valentine et al., 2004). The present
study offers some first tentative support for this finding with intensive
longitudinal data. In a similar vein, previous between-person research
has provided evidence for the generalizability of skill-development
and self-enhancement effects across ability levels (e.g., Gorges et al.,
2018;Seaton et al., 2015). The present study supported these findings
in providing initial evidence that lesson-to-lesson skill-development
and self-enhancement effects generalize across different levels of rea-
soning ability and mathematics grades. Notably, in their recent meta-
analysis on the longitudinal relations between trait-like ASC and
achievement, Wu et al. (2021) found that skill-development effects
generalized across different achievement levels. In at-risk and poor-
performing samples, however, they found evidence that self-enhance-
ment effect sizes were weaker than in unselected samples. Future
research that is aimed at replicating our results in lower performing
samplesisthereforewarranted.
Carryover Effects From Lesson to Lesson
We found evidence for within-person carryover effects for state
MSC and perceived achievement from one lesson to the next.
These findings suggest that situation-specific ups and downs in
students’state MSC continue to affect the next lesson before stu-
dents return to their habitual levels of MSC. Similarly, it suggests
that situation-specific ups and downs in students’lesson-specific
perceived achievement continue to affect the next lesson before
students return to their typical levels of perceived achievement.
Further, our results suggest that students differ in the extent to
which they experience carryover effects. Again, this is a finding
that could not be tested with typical between-person approaches.
As such, we provided evidence that students differ in how strongly
their current state self-concept (lesson-specific perceived achieve-
ment) depends on their previous lesson’s state and how long it
takes to return to their habitual trait level.
Here, we found some relations with specific student characteristics.
Concerning carryover effects for perceived achievement, students
with higher trait MSC levels, higher mean perceived achievement
levels, and better mathematics grades tended to have less carryover in
their perceived achievements from one lesson to another. Such stu-
dents thus seemed to return to their typical levels of perceived
achievement more quickly than students with less confidence in their
own abilities and lower achievement levels. Conversely, students
with stronger skill-development effects seemingly tended to have
more carryover in their perceived achievement levels from lesson to
lesson. Thus, students who do not return to their typical levels of per-
ceived achievement as quickly as their peers after experiencing les-
son-specific ups and downs tend to simultaneously experience
stronger daily influences of their perceived achievements on their
self-concepts.
Similarly, with regard to carryover effects for state MSC, we
found that students with stronger self-enhancement effects seem-
ingly tended to have a stronger carryover for state MSC from les-
son to lesson. Thus, students who do not return to their typical
levels of self-concept as quickly after experiencing lesson-specific
ups and downs tend to simultaneously experience stronger daily
influences of their self-concepts on their perceived achievements.
Conversely, students with weaker skill-development effects seem-
ingly tend to have a stronger carryover for state MSC from lesson
to lesson. Thus, students who do not return to their typical levels
of self-concept as quickly after experiencing lesson-specific ups
and downs tend to simultaneously experience weaker daily influ-
ences of their perceived achievements on their self-concepts.
Overall, our study points to the need for further research on carry-
over effects in ASC research in general, their role in ASC forma-
tion, and their relations to students’characteristics.
State Academic Self-Concept
ASC is commonly considered a construct that is characterized
by high stability, even over long periods of time (Jansen et al.,
2020; see also Orth et al., 2021). However, our results provide
strong evidence for substantial situation-specificfluctuations in
ASC as experienced by students in their everyday life at school.
These results are consistent with hallmark conceptual work on
self-concept that already described situation-specific aspects of
self-concept (James, 1890, p. 307; Shavelson et al., 1976) as well
as with existing empirical work related to our study (e.g., Malm-
berg & Martin, 2019;Tsai et al., 2008). The positive correlations
we found between students’averaged state MSC and their reason-
ing ability and mathematics grades, as well as the observed gender
differences in averaged state MSC in favor of male students, corre-
sponded to previous findings on the nomological network of (trait)
MSC, thus providing further evidence for the construct validity of
state MSC.
Shavelson et al.’s (1976) multidimensional, hierarchical model
of self-concept is nowadays often considered to be the starting
point of modern empirical research on ASC (Marsh, 2006;Marsh
et al., 2019). In the original Shavelson et al. (1976) model, there
was an implicit hierarchy of stability, from general school ASC
(e.g., “I am good at most school subjects”) at the apex, to domain-
specific ASCs (e.g., “I am good at mathematics”) at the next level,
with even more task-specific ASCs at lower levels of the hierar-
chy. At the base of this hierarchy, however, the authors claimed a
situation-specific level, where “self-concept varies greatly with
variation in situations”(Shavelson et al., 1976, p. 414). Appa-
rently, this level was thought to reflect state ASCs.
STATE MATH SELF-CONCEPT AND PERCEIVED ACHIEVEMENT 1389
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We propose a need to conceptually disentangle the dimension
of domain level or task specificity in ASC from the situation-spec-
ificity dimension in ASC. Whereas the first dimension can be
thought of as falling on a continuum that ranges from more general
school to more task-specific (e.g., general school ASC vs. MSC),
the latter can be thought of as falling on a trait-to-state continuum
(e.g., trait ASC vs. state ASC). Notably, many person variables
typically regarded as between-person phenomena have been
shown to exhibit substantial within-person variability (Podsakoff
et al., 2019)—including, for instance, supposedly stable traits such
as the Big Five (Fleeson, 2001). The notion that, in principle, any
person variable can be thought of on a trait-to-state continuum and
could potentially even be operationalized as both a trait and a state
has been advocated for and has garnered ample empirical support
(Rauthmann, 2021). With this in mind, it would make sense to
assume that even the general school ASC can in principle also be
operationalized as both a trait and a state. In future ASC theory and
research, we thus suggest that the two dimensions—domain-level
(or task) specificity and situation (or temporal) specificity—should
be thought of as conceptually distinct from each other. Further
research is needed to explicitly integrate the dimension of situation
specificity into a future, expanded taxonomy of ASC.
Limitations
As is characteristic of intensive longitudinal approaches, our ex-
perience-sampling data can be characterized as highly ecologically
valid. Our data were temporally ordered by design, thus allowing
for a fine-grained assessment of students’real-time experiences in
their everyday life. Nevertheless, the omitted variable problem
persisted for the within-person part, according to which a time-
varying variable not included in the analyses could have caused
the observed relations (Hamaker et al., 2018). As such, support for
implicit causal interpretations must be made with appropriate cau-
tion (Grosz et al., 2020).
Furthermore, we asked the students about their perceptions of their
everyday learning progress for each individual mathematics lesson to
operationalize students’lesson-specific perceived achievements. We
acknowledge that this particular operationalization represents only an
approximation of students’actual achievement in that lesson. In ASC
research, report card grades or standardized achievement tests are of-
ten used to represent the achievement component. In an experience-
sampling design, both measures are arguably not or only partially ap-
plicable as daily assessments while maintaining ecological validity.
Our empirical examination of the factorial validity demonstrated that
situation-specific, state ASC can clearly be empirically distinguished
from students’perceptions of their lesson-specific achievement. The
latter is subject to much stronger lesson-to-lesson fluctuations. This
suggests that the students did indeed distinguish accurately between
their own abilities (self-concept) and the learning progress they
achieved (achievement) in that particular lesson. Nevertheless, it
would be useful to replicate the study by using alternative criteria to
measure students’achievement. For example, teachers’ratings might
be a viable alternative. However, student gains from lesson to lesson
for each student might not be readily visible to teachers, and teacher
ratings are subject to other judgment biases (Loibletal.,2020).
Hence, we leave this concern as a direction for further research.
Our study does not provide a direct replication of the classic
between-person reciprocal effects model (Marsh, 1990;Marsh &
Craven, 2006;Wu et al., 2021) at the within-person level. A large
body of research has established the reciprocal effects model using
between-person, long-term approaches and more objective achieve-
ment indicators such as grades or test scores. The present study
diverges from this research tradition not only in the level of analysis
(within-person and short-term vs. between-person and long-term), but
also in the nature of the achievement indicator (perceived achievement
vs. more objective achievement indicators). Further research is indi-
cated here, linking the findings of the present study to the ongoing
debate on the juxtaposition of within-person and between-person per-
spectives on the reciprocal effects model (see, e.g., Ehm et al., 2019).
Finally, in the current study, we focused on German secondary
school students attending the academic track (i.e., the Gymnasium,
which 44% of German students attend after elementary education;
Autorengruppe Bildungsberichterstattung, 2018). Future research
drawing on data from other educational systems, age groups, or
ability tracks, and other cultural contexts is needed to test the
results’generalizability.
Conclusion
ASC plays a prominent role in students’everyday life for their
personal academic development. In the comparatively long tradi-
tion of research on ASC in educational psychology, there is clearly
a lack of experience-sampling studies that have examined the psy-
chosocial intraindividual processes postulated by ASC theory over
time in an ecologically valid setting in real time. We are off to a
good start with this study. Our results indicate that a student’s
ASC does indeed influence their achievement formation in every-
day school life and that this, in turn, is influenced by previous
achievement. Our results should thereby also be seen as a call and
starting point for further research, not only to use within-person,
experience-sampling approaches to replicate previous results in
ASC research, but also to better understand situation-specificfluc-
tuations in ASC and student motivation more broadly in concrete
learning situations, which, as our results suggest, students actually
experience. We strongly believe that this will significantly help to
advance ASC theory and research toward providing a better under-
standing of students’learning experiences.
References
Amthauer, R. (1970). Intelligenz-Struktur-Test [Intelligence structure test].
Hogrefe.
Arens, A. K., Becker, M., & Möller, J. (2017). Social and dimensional
comparisons in math and verbal test anxiety: Within- and cross-domain
relations with achievement and the mediating role of academic self-con-
cept. Contemporary Educational Psychology,51, 240–252. https://doi
.org/10.1016/j.cedpsych.2017.08.005
Arens, A. K., Marsh, H. W., Pekrun, R., Lichtenfeld, S., Murayama, K., &
vom Hofe, R. (2017). Math self-concept, grades, and achievement test
scores: Long-term reciprocal effects across five waves and three
achievement tracks. Journal of Educational Psychology,109(5),
621–634. https://doi.org/10.1037/edu0000163
Asparouhov, T., & Muthén, B. (2010). Bayesian analysis using Mplus:
Technical implementation. Mplus Technical Report. http://statmodel
.com/download/Bayes3.pdf
Asparouhov, T., Hamaker, E. L., & Muthén, B. (2018). Dynamic structural
equation models. Structural Equation Modeling,25(3), 359–388. https://
doi.org/10.1080/10705511.2017.1406803
1390 NIEPEL, MARSH, GUO, PEKRUN, AND MÖLLER
This document is copyrighted by the American Psychological Association or one of its allied publishers.
Content may be shared at no cost, but any requests to reuse this content in part or whole must go through the American Psychological Association.

Autorengruppe Bildungsberichterstattung. (2018). Bildung in Deutschland
2018: Ein indikatorengestützter Bericht mit einer Analyse zu Wirkungen
und Erträgen von Bildung [Education in Germany 2018: An indicator-
based report with an analysis of the effects and returns of education]. https://
www.bildungsbericht.de/de/bildungsberichte-seit-2006/bildungsbericht-
2018/pdf-bildungsbericht-2018/bildungsbericht-2018.pdf
Bolger, N., & Laurenceau, J.-P. (2013). Intensive longitudinal methods: An
introduction to diary and experience sampling research. Guilford Press.
Brown, G. T. L., Andrade, H. L., & Chen, F. (2015). Accuracy in student
self-assessment: Directions and cautions for research. Assessment in
Education: Principles, Policy & Practice,22(4), 444–457. https://doi
.org/10.1080/0969594X.2014.996523
Brunner, M., Keller, U., Dierendonck, C., Reichert, M., Ugen, S.,
Fischbach, A., & Martin, R. (2010). The structure of academic self-con-
cepts revisited: The nested Marsh/Shavelson model. Journal of Educa-
tional Psychology,102(4), 964–981. https://doi.org/10.1037/a0019644
Byrne, B. M. (2002). Validating the measurement and structure of self-con-
cept: Snapshots of past, present, and future research. American Psycholo-
gist,57(11), 897–909. https://doi.org/10.1037/0003-066X.57.11.897
Calsyn, R. J., & Kenny, D. A. (1977). Self-concept of ability and perceived
evaluation of others: Cause or effect of academic achievement? Journal
of Educational Psychology,69(2), 136–145. https://doi.org/10.1037/
0022-0663.69.2.136
Chesebro, J. L., & McCroskey, J. C. (2000). The relationship between stu-
dents’reports of learning and their actual recall of lecture material: A
validity test. Communication Education,49(3), 297–301. https://doi.org/
10.1080/03634520009379217
Cohen, J. (1988). Statistical power analysis for the behavioral sciences.
Academic Press.
Dickhäuser, O., & Plenter, I. (2005). “Letztes Halbjahr stand ich zwei”.
Zur Akkuratheit selbst berichteter Noten [On the accuracy of self-
reported school marks]. Zeitschrift Für Pädagogische Psychologie,
19(4), 219–224. https://doi.org/10.1024/1010-0652.19.4.219
Dörendahl, J., Scherer, R., Greiff, S., Martin, R., & Niepel, C. (2021).
Dimensional comparisons in the formation of domain-specific achieve-
ment goals. Motivation Science,7(3), 306–318. https://doi.org/10.1037/
mot0000203
Dormann, C., & Griffin, M. A. (2015). Optimal time lags in panel studies. Psy-
chological Methods,20(4), 489–505. https://doi.org/10.1037/met0000041
Eccles,J.S.(2009).WhoamIandwhatamIgoingtodowithmylife?Perso-
nal and collective identities as motivators of action. Educational Psycholo-
gist,44(2), 78–89. https://doi.org/10.1080/00461520902832368
Ehm, J. H., Hasselhorn, M., & Schmiedek, F. (2019). Analyzing the devel-
opmental relation of academic self-concept and achievement in elemen-
tary school children: Alternative models point to different results.
Developmental Psychology,55(11), 2336–2351. https://doi.org/10.1037/
dev0000796
Fleeson, W. (2001). Toward a structure- and process-integrated view of
personality: Traits as density distribution of states. Journal of Personal-
ity and Social Psychology,80(6), 1011–1027. https://doi.org/10.1037/
0022-3514.80.6.1011
Frenzel, A. C., Pekrun, R., & Goetz, T. (2007). Girls and mathematics—A
“hopeless”issue? A control-value approach to gender differences in
emotions towards mathematics. European Journal of Psychology of
Education,22(4), 497–514. https://doi.org/10.1007/BF03173468
Geldhof, G. J., Preacher, K. J., & Zyphur, M. J. (2014). Reliability estima-
tion in a multilevel confirmatory factor analysis framework. Psychologi-
cal Methods,19(1), 72–91. https://doi.org/10.1037/a0032138
Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation
using multiple sequences. Statistical Science,7(4), 457–472. https://doi
.org/10.1214/ss/1177011136
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., &
Rubin, D. (2014). Bayesian data analysis (3ed.). Taylor & Francis.
Giannakos,M.N.,Sharma,K.,Papavlasopoulou,S.,Pappas,I.O.,&
Kostakos, V. (2020). Fitbit for learning: Towards capturing the learning ex-
perience using wearable sensing. International Journal of Human Com-
puter Studies,136, 102384. https://doi.org/10.1016/j.ijhcs.2019.102384
Gignac, G. E., & Szodorai, E. T. (2016). Effect size guidelines for individ-
ual differences researchers. Personality and Individual Differences,102,
74–78. https://doi.org/10.1016/j.paid.2016.06.069
Goetz, T., Bieg, M., Lüdtke, O., Pekrun, R., & Hall, N. C. (2013). Do girls
really experience more anxiety in mathematics? Psychological Science,
24(10), 2079–2087. https://doi.org/10.1177/0956797613486989
Gogol, K., Brunner, M., Goetz, T., Martin, R., Ugen, S., Keller, U.,
Fischbach, A., & Preckel, F. (2014). “My questionnaire is too long!”
The assessments of motivational-affective constructs with three-item
and single-item measures. Contemporary Educational Psychology,
39(3), 188–205. https://doi.org/10.1016/j.cedpsych.2014.04.002
Gorges, J., Neumann, P., Wild, E., Stranghöner, D., & Lütje-Klose, B.
(2018). Reciprocal effects between self-concept of ability and perform-
ance: A longitudinal study of children with learning disabilities in inclu-
sive versus exclusive elementary education. Learning and Individual
Differences,61,11–20. https://doi.org/10.1016/j.lindif.2017.11.005
Graham, J. W., Cumsille, P. E., & Elek-Fisk, E. (2003). Methods for han-
dling missing data. In A. S. Velicer & W. F. Velicer (Eds.), Research
methods in psychology (pp. 87–114). Wiley.
Grosz, M. P., Rohrer, J. M., & Thoemmes, F. (2020). The taboo against
explicit causal inference in nonexperimental psychology. Perspectives
on Psychological Science,15(5), 1243–1255. https://doi.org/10.1177/
1745691620921521
Guay, F., Marsh, H. W., & Boivin, M. (2003). Academic self-concept and
academic achievement: Developmental perspectives on their causal
ordering. Journal of Educational Psychology,95(1), 124–136. https://
doi.org/10.1037/0022-0663.95.1.124
Guo, J., Marsh, H. W., Parker, P. D., Morin, A. J. S., & Dicke, T. (2017).
Extending expectancy-value theory predictions of achievement and aspi-
rations in science: Dimensional comparison processes and expectancy-
by-value interactions. Learning and Instruction,49,81–91. https://doi
.org/10.1016/j.learninstruc.2016.12.007
Hamaker, E. L., & Wichers, M. (2017). No time like the present: Discover-
ing the hidden dynamics in intensive longitudinal data. Current Direc-
tions in Psychological Science,26(1), 10–15. https://doi.org/10.1177/
0963721416666518
Hamaker, E. L., Asparouhov, T., Brose, A., Schmiedek, F., & Muthén, B.
(2018). At the frontiers of modeling intensive longitudinal data:
Dynamic structural equation models for the affective measurements
from the COGITO study. Multivariate Behavioral Research,53(6),
820–841. https://doi.org/10.1080/00273171.2018.1446819
Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. P. P. (2015). A critique
of the cross-lagged panel model. Psychological Methods,20(1),
102–116. https://doi.org/10.1037/a0038889
Huang, C. (2011). Self-concept and academic achievement: A meta-analy-
sis of longitudinal relations. Journal of School Psychology,49(5),
505–528. https://doi.org/10.1016/j.jsp.2011.07.001
Jak, S., & Jorgensen, T. D. (2017). Relating measurement invariance,
cross-level invariance, and multilevel reliability. Frontiers in Psychol-
ogy,8, 1640. https://doi.org/10.3389/fpsyg.2017.01640
James, W. (1890). The principles of psychology (Vol. I). Henry Holt and
Co. https://doi.org/10.1037/10538-000
Janis, R. A., Burlingame, G. M., & Olsen, J. A. (2016). Evaluating factor
structures of measures in group research: Looking between and within.
Group Dynamics,20(3), 165–180. https://doi.org/10.1037/gdn0000043
Jansen, M., Lüdtke, O., & Robitzsch, A. (2020). Disentangling different sour-
ces of stability and change in students’academic self-concepts: An integra-
tive data analysis using the STARTS model. Journal of Educational
Psychology,112(8), 1614–1631. https://doi.org/10.1037/edu0000448
STATE MATH SELF-CONCEPT AND PERCEIVED ACHIEVEMENT 1391
This document is copyrighted by the American Psychological Association or one of its allied publishers.
Content may be shared at no cost, but any requests to reuse this content in part or whole must go through the American Psychological Association.

Kaplan, D. (2009). Structural equation modeling: Foundations and exten-
sions. Sage.
Kim, E. S., Dedrick, R. F., Cao, C., & Ferron, J. M. (2016). Multilevel fac-
tor analysis: Reporting guidelines and a review of reporting practices.
Multivariate Behavioral Research,51(6), 881–898. https://doi.org/10
.1080/00273171.2016.1228042
Kline, R. B. (2005). Principles and practice of structural equation model-
ing. Guilford Press.
Kurucay, M., & Inan, F. A. (2017). Examining the effects of learner-
learner interactions on satisfaction and learning in an online undergradu-
ate course. Computers & Education,115,20–37. https://doi.org/10
.1016/j.compedu.2017.06.010
Liepmann, D., Beauducel, A., Brocke, B., & Amthauer, R. (2007). Intelli-
genz-Struktur-Test 2000R [Intelligence structure test 2000R]. Hogrefe.
Liepmann, D., Beauducel, A., Brocke, B., & Nettelnstroth, W. (2012).
Intelligenz-Struktur-Test- Screening [Intelligence structure test-screen-
ing]. Hogrefe.
Loibl, K., Leuders, T., & Dörfler, T. (2020). A framework for explaining
teachers’diagnostic judgements by cognitive modeling (DiaCoM).
Teaching and Teacher Education,91, 103059. https://doi.org/10.1016/j
.tate.2020.103059
Malmberg, L. E., & Martin, A. J. (2019)., April Processes of students’
effort exertion, competence beliefs and motivation: Cyclic and dynamic
effects of learning experiences within school days and school subjects.
Contemporary Educational Psychology,58, 299–309. https://doi.org/10
.1016/j.cedpsych.2019.03.013
Marsh, H. W. (1990). Causal ordering of academic self-concept and aca-
demic achievement: A multiwave, longitudinal panel analysis. Journal
of Educational Psychology,82(4), 646–656. https://doi.org/10.1037/
0022-0663.82.4.646
Marsh, H. W. (2006). Self-concept theory, measurement and research into
practice: The role of self-concept in educational psychology. British
Psychological Society.
Marsh, H. W., & Craven, R. G. (2006). Reciprocal effects of self-concept and
performance from a multidimensional perspective: Beyond seductive pleasure
and unidimensional perspectives. Perspectives on Psychological Science,1(2),
133–163. https://doi.org/10.1111/j.1745-6916.2006.00010.x
Marsh, H. W., & Martin, A. J. (2011). Academic self-concept and aca-
demic achievement: Relations and causal ordering. The British Journal
of Educational Psychology,81(Pt. 1), 59–77. https://doi.org/10.1348/
000709910X503501
Marsh, H. W., & O’Mara, A. (2008). Reciprocal effects between academic
self-concept, self-esteem, achievement, and attainment over seven ado-
lescent years: Unidimensional and multidimensional perspectives of
self-concept. Personality and Social Psychology Bulletin,34(4),
542–552. https://doi.org/10.1177/0146167207312313
Marsh, H. W., & Shavelson, R. (1985). Self-concept: Its multifaceted, hier-
archical structure. Educational Psychologist,20(3), 107–123. https://doi
.org/10.1207/s15326985ep2003_1
Marsh, H. W., Lüdtke, O., Nagengast, B., Trautwein, U., Abduljabbar,
A. S., Abdelfattah, F., & Jansen, M. (2015). Dimensional comparison
theory: Paradoxical relations between self-beliefs and achievements in
multiple domains. Learning and Instruction,35,16–32. https://doi.org/
10.1016/j.learninstruc.2014.08.005
Marsh, H. W., Pekrun, R., Murayama, K., Arens, A. K., Parker, P. D.,
Guo, J., & Dicke, T. (2018). An integrated model of academic self-con-
cept development: Academic self-concept, grades, test scores, and track-
ing over 6 years. Developmental Psychology,54(2), 263–280. https://doi
.org/10.1037/dev0000393
Marsh, H. W., Pekrun, R., Parker, P. D., Murayama, K., Guo, J., Dicke, T., &
Arens, A. K. (2019). The murky distinction between self-concept and self-effi-
cacy: Beware of lurking jingle-jangle fallacies. Journal of Educational Psy-
chology,111(2), 331–353. https://doi.org/10.1037/edu0000281
Marsh, H. W., Relich, J. D., & Smith, I. D. (1983). Self-concept: The con-
struct validity of interpretations based upon the SDQ. Journal of Person-
ality and Social Psychology,45(1), 173–187. https://doi.org/10.1037/
0022-3514.45.1.173
Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S.
(2019). The centrality of academic self-concept to motivation and learn-
ing. In K. A. Renninger & S. E. Hidi (Eds.), The Cambridge handbook
of motivation and learning (pp. 36–62). Cambridge University Press.
https://doi.org/10.1017/9781316823279.004
Marsh, H. W., Trautwein, U., Lüdtke, O., Köller, O., & Baumert, J. (2005).
Academic self-concept, interest, grades, and standardized test scores: Recip-
rocal effects models of causal ordering. Child Development,76(2),
397–416. https://doi.org/10.1111/j.1467-8624.2005.00853.x
McNeish, D., & Hamaker, E. L. (2020). A primer on two-level dynamic struc-
tural equation models for intensive longitudinal data in Mplus. Psychologi-
cal Methods,25(5), 610–635. https://doi.org/10.1037/met0000250
Molenaar, P. C. M. (2004). A manifesto on psychology as idiographic sci-
ence: Bringing the person back into scientific psychology, this time for-
ever. Measurement: Interdisciplinary Research and Perspectives,2(4),
201–218. https://doi.org/10.1207/s15366359mea0204_1
Möller, J., Retelsdorf, J., Köller, O., & Marsh, H. W. (2011). The recipro-
cal internal/external frame of reference model: An integration of models
of relations between academic achievement and self-concept. American
Educational Research Journal,48(6), 1315–1346. https://doi.org/10
.3102/0002831211419649
Möller, J., Zitzmann, S., Helm, F., Machts, N., & Wolff, F. (2020). A meta-anal-
ysis of relations between achievement and self-concept. Review of Educational
Research,90(3), 376–419. https://doi.org/10.3102/0034654320919354
Movisens GmbH. (2017). movisensXS (Version 1.3.0-1.3.4) [Mobile App].
https://www.movisens.com
Murayama, K., Goetz, T., Malmberg, L.-E., Pekrun, R., Tanaka, A., &
Martin, A. J. (2017). Within-person analysis in educational psychology:
Importance and illustrations. British Journal of Educational Psychology
Monograph Series II,12,71–87.
Muthén, L. K., & Muthén, B. O. (1998–2019). Mplus (Version 8.3) [Com-
puter software]. https://www.statmodel.com
Niepel, C., Brunner, M., & Preckel, F. (2014a). Achievement goals, aca-
demic self-concept, and school grades in mathematics: Longitudinal re-
ciprocal relations in above average ability secondary school students.
Contemporary Educational Psychology,39(4), 301–313. https://doi.org/
10.1016/j.cedpsych.2014.07.002
Niepel, C., Brunner, M., & Preckel, F. (2014b). The longitudinal interplay
of students’academic self-concepts and achievements within and across
domains: Replicating and extending the reciprocal internal/external
frame of reference model. Journal of Educational Psychology,106(4),
1170–1191. https://doi.org/10.1037/a0036307
Niepel, C., Stadler, M., & Greiff, S. (2019). Seeing is believing: Gender diversity
in STEM is related to mathematics self-concept. Journal of Educational Psy-
chology,111(6), 1119–1130. https://doi.org/10.1037/edu0000340
Orth, U., Dapp, L. C., Erol, R. Y., Krauss, S., & Luciano, E. C. (2021). De-
velopment of domain-specific self-evaluations: A meta-analysis of lon-
gitudinal studies. Journal of Personality and Social Psychology,120(1),
145–172. https://doi.org/10.1037/pspp0000378
Pekrun, R., Goetz, T., Titz, W., & Perry, R. P. (2002). Academic emotions
in students’self-regulated learning and achievement: A program of qual-
itative and quantitative research. Educational Psychologist,37(2),
91–105. https://doi.org/10.1207/S15326985EP3702_4
Pekrun, R., Murayama, K., Marsh, H. W., Goetz, T., & Frenzel, A. C.
(2019). Happy fish in little ponds: Testing a reference group model of
achievement and emotion. Journal of Personality and Social Psychol-
ogy,117(1), 166–185. https://doi.org/10.1037/pspp0000230
Peterson, S. E., & Miller, J. A. (2004). Quality of college students’experi-
ences during cooperative learning. Social Psychology of Education,
7(2), 161–183. https://doi.org/10.1023/B:SPOE.0000018522.39515.19
1392 NIEPEL, MARSH, GUO, PEKRUN, AND MÖLLER
This document is copyrighted by the American Psychological Association or one of its allied publishers.
Content may be shared at no cost, but any requests to reuse this content in part or whole must go through the American Psychological Association.

Podsakoff, N. P., Spoelma, T. M., Chawla, N., & Gabriel, A. S. (2019).
What predicts within-person variance in applied psychology constructs?
An empirical examination. Journal of Applied Psychology,104(6),
727–754. https://doi.org/10.1037/apl0000374
Rauthmann, J. F. (2021). Capturing interactions, correlations, fits, and transac-
tions: A person-environment relations model. In J. F. Rauthmann (Ed.), The
handbook of personality dynamics and processes (1st ed., pp. 427–522).
Elsevier. https://doi.org/10.1016/B978-0-12-813995-0.00018-2
Reis, H. T. (2014). Why researchers should think “real-world”: A concep-
tual rationale. In M. R. Mehl & T. S. Conner (Eds.), Handbook for
research methods for studying daily life (pp. 3–21). Guilford Press.
Retelsdorf, J., Köller, O., & Möller, J. (2014). Reading achievement and read-
ing self-concept—Testing the reciprocal effects model. Learning and
Instruction,29,21–30. https://doi.org/10.1016/j.learninstruc.2013.07.004
Richmond, V. P., McCroskey, J. C., Kearney, P., & Plax, T. G. (1987).
Power in the classroom VII: Linking behavior alteration techniques to
cognitive learning. Communication Education,36(1), 1–12. https://doi
.org/10.1080/03634528709378636
Ross, J. A. (2006). The reliability, validity, and utility of self-assessment.
Practical Assessment, Research & Evaluation,11, Article 10. https://doi
.org/10.7275/9wph-vv65
Rovai, A. P., Wighting, M. J., Baker, J. D., & Grooms, L. D. (2009). De-
velopment of an instrument to measure perceived cognitive, affective,
and psychomotor learning in traditional and virtual classroom higher
education settings. Internet and Higher Education,12(1), 7–13. https://
doi.org/10.1016/j.iheduc.2008.10.002
Schmidt-Atzert, L., & Amelang, M. (2012). Psychologische Diagnostik
[Psychological assessment]. Springer. https://doi.org/10.1007/978-3-642
-17001-0
Schurtz, I. M., Pfost, M., Nagengast, B., & Artelt, C. (2014). Impact of social
and dimensional comparisons on student’s mathematical and English subject-
interest at the beginning of secondary school. Learning and Instruction,34,
32–41. https://doi.org/10.1016/j.learninstruc.2014.08.001
Seaton, M., Marsh, H. W., Parker, P. D., Craven, R. G., & Yeung, A. S.
(2015). The reciprocal effects model revisited: Extending its reach to
gifted students attending academically selective schools. Gifted Child
Quarterly,59(3), 143–156. https://doi.org/10.1177/0016986215583870
Shavelson, R. J., Hubner, J. J., & Stanton, G. C. (1976). Self-concept: Vali-
dation of construct interpretations. Review of Educational Research,
46(3), 407–441. https://doi.org/10.3102/00346543046003407
Shernof, D. J., Ruzek, E. A., Sannella, A. J., Schorr, R. Y., Sanchez-Wall,
L., & Bressler, D. M. (2017). Student engagement as a general factor of
classroom experience: Associations with student practices and educa-
tional outcomes in a university gateway course. Frontiers in Psychology,
8, 994. https://doi.org/10.3389/fpsyg.2017.00994
Shernoff, D. J., Ruzek, E. A., & Sinha, S. (2017). The influence of the
high school classroom environment on learning as mediated by student
engagement. School Psychology International,38(2), 201–218. https://
doi.org/10.1177/0143034316666413
Shernoff, D. J., Sannella, A. J., Schorr, R. Y., Sanchez-Wall, L., Ruzek,
E. A., Sinha, S., & Bressler, D. M. (2017). Separate worlds: The
influence of seating location on student engagement, classroom experi-
ence, and performance in the large university lecture hall. Journal of
Environmental Psychology,49,55–64. https://doi.org/10.1016/j.jenvp
.2016.12.002
Shin, N. (2003). Transactional presence as a critical predictor of success in
distance learning. Distance Education,24(1), 69–86. https://doi.org/10
.1080/01587910303048
Sparfeldt, J. R., Buch, S. R., Rost, D. H., & Lehmann, G. (2008). Akkura-
tesse selbstberichteter Zensuren [The accuracy of self-reported grades in
school]. Psychologie in Erziehung Und Unterricht,55(1), 68–75.
Stapleton, L. M., Yang, J. S., & Hancock, G. R. (2016). Construct meaning
in multilevel settings. Journal of Educational and Behavioral Statistics,
41(5), 481–520. https://doi.org/10.3102/1076998616646200
Trautwein, U., & Möller, J. (2016). Self-concept: Determinants and conse-
quences of academic self-concept in school contexts. In A. A.
Lipnevich, F. Preckel, & R. D. Roberts (Eds.), Psychosocial skills and
school systems in the 21st century: Theory, research, and practice (pp.
187–214). Springer. https://doi.org/10.1007/978-3-319-28606-8_8
Trull, T. J., & Ebner-Priemer, U. (2014). The role of ambulatory assess-
ment in psychological science. Current Directions in Psychological Sci-
ence,23(6), 466–470. https://doi.org/10.1177/0963721414550706
Tsai, Y.-M., Kunter, M., Lüdtke, O., & Trautwein, U. (2008). Day-to-day
variation in competence beliefs. How autonomy support predicts young
adolescents’felt competence. In H. W. Marsh, R. G. Craven, & D. M.
McInerney (Eds.), Self-processes, learning, and enabling human poten-
tial (pp. 119–143). Information Age Publishing.
Valentine, J. C., & DuBois, D. L. (2005). Effects of self-beliefs on academic
achievement and vice-versa: Separating the chicken from the egg. In H. W.
Marsh, R. G. Craven, & D. M. McInerney (Eds.), International advances in
self research (Vol. 2, pp. 53–78). Information Age Publishing.
Valentine, J. C., DuBois, D. L., & Cooper, H. (2004). The relation between self-
beliefs and academic achievement: A meta-analytic review. Educational Psy-
chologist,39(2), 111–133. https://doi.org/10.1207/s15326985ep3902_3
Wu, H., Guo, Y., Yang, Y., Zhao, L., & Guo, C. (2021). A meta-analysis
of the longitudinal relationship between academic self-concept and aca-
demic achievement. Educational Psychology Review. Advance online
publication. https://doi.org/10.1007/s10648-021-09600-1
Yoon, S., Kim, S., & Kang, M. (2020). Predictive power of grit, professor sup-
port for autonomy and learning engagement on perceived achievement within
the context of a flipped classroom. Active Learning in Higher Education,
21(3), 233–247. https://doi.org/10.1177/1469787418762463
Zirkel, S., Garcia, J. A., & Murphy, M. C. (2015). Experience-sampling
research methods and their potential for education research. Educa-
tional Researcher,44(1), 7–16. https://doi.org/10.3102/0013189X1
4566879
Received December 30, 2020
Revision received August 5, 2021
Accepted August 22, 2021 n
STATE MATH SELF-CONCEPT AND PERCEIVED ACHIEVEMENT 1393
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