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ZDM - Mathematics Education
https://doi.org/10.1007/s11858-025-01686-z
ORIGINAL PAPER
One Model to Fit Them All? A Mixture Distribution Path Analysis of
Preservice Early Childhood Teachers’ Beliefs, Emotions, and Knowledge
Simone Dunekacke1·Lars Meyer-Jenßen2·Michael Eid3
Accepted: 10 April 2025
© The Author(s) 2025
Abstract
Previous studies have applied a variable-centered approach to conduct extensive investigations of preservice early childhood
teachers’ (PECTs’) epistemic beliefs in the domain of mathematics (application-related beliefs, process-related beliefs, static
orientation), enjoyment of mathematics, mathematics anxiety, mathematical content knowledge, and mathematics pedagog-
ical content knowledge. However, person-centered approaches, which have been fruitfully applied to other constructs and
domains concerning pre- and inservice teachers, have not yet been applied to the aforementioned constructs. We addressed
this research gap by investigating relationships between mathematics-related beliefs, emotions, and knowledge in terms of
the well-established control-value theory in combination with a mixture distribution path analysis. About 1,851 PECTs took
part in the study. Participants worked on tests and questionnaires during regular class time in teacher education. The results
yielded two latent classes with structural differences in the coefficients of the path model, which we termed the application
and static learning classes. In Class 1, higher levels of application-related beliefs were in line with lower levels of anxiety
and higher levels of knowledge. In Class 2, higher levels of static orientation were in line with lower levels of enjoyment
and higher levels of anxiety and knowledge. These novel results indicate two pathways for learning, with implications for
research and practice. For research, the results are interesting with regard to static orientation and show the need for further
research. For practice, they indicate the need to respect individual differences even during teacher education.
Keywords Control-value theory ·Early childhood teachers ·Mixture distribution path analysis ·Knowledge ·Emotions ·
Beliefs
1 Introduction
The importance of early mathematics education has been
emphasized repeatedly over the last two decades (Early et
al., 2007; OECD, 2018). This view has also been associated
with increased research activity, including research on the
professional competence required for early childhood (EC)
teachers (e.g., Blömeke et al., 2017b; Gasteiger et al., 2020).
Research indicates that EC teachers’ affective-motivational
dispositions and knowledge (mathematical content knowl-
edge [MCK] and mathematics pedagogical content knowl-
edge [MPCK]) are important with respect to perceiving
mathematics-related learning situations (Dunekacke et al.,
2016), mathematics-related activities in preschool (Tor-
beyns et al., 2022,2024), and children’s development in
the domain of mathematics (Pohle et al., 2022). According
to Hannula (2019), when studying affective dispositions in
mathematics, it is necessary to distinguish between beliefs
and emotions. In contrast to emotions (which refer to feel-
Lars Meyer-Jenßen and Simone Dunekacke share first authorship.
We extend our gratitude to Prof. Dr. Sigrid Blömeke for her
exemplary leadership of the KomMa project, under which this study
was conducted. Her invaluable guidance and constructive suggestions,
particularly regarding the data analysis, were instrumental in shaping
this research. Prof. Blömeke’s expertise significantly enhanced the
quality and rigor of our study.
S. Dunekacke
simone.dunekacke@rptu.de
1Rheinland-Pfälzische Technische Universität Kaiserslautern
Landau, Department of Education, Landau, Germany
2Health and Medical University Erfurt, Department Psychology,
Erfurt, Germany
3Freie Universität Berlin, Department of Education and
Psychology, Berlin, Germany
S. Dunekacke et al.
ings), the term beliefs refers to the cognitive dimension of
affect.
The relationship between emotions and achievement
is well established in the context of control-value theory
(Pekrun, 2006), and the number of studies investigating
the complex relationships between beliefs, emotions, and
achievement in greater detail is growing (Rosman & Mayer,
2018; Trevors et al., 2017). Obtaining knowledge about indi-
vidual differences in learning processes is meaningful for in-
struction in different contexts, including postsecondary ed-
ucation, where the present study is located. We aimed to use
control-value theory to examine the relationships between
preservice early childhood teachers’ (PECTs’) beliefs, emo-
tions, and knowledge. To this end, we drew on a mixture
distribution approach to extend research in the field of EC
education, which has primarily applied approaches that are
based on the assumption that the parameters of a model are
the same for all members of the population. This assumption
means that the same relationships (e.g., between beliefs and
emotions or between emotions and knowledge) are expected
to apply to all individuals in the population. This assump-
tion is quite rigorous. Reasons for questioning this strict
assumption in the context of postsecondary education are
differential exploitation of learning situations (e.g., due to
motivation) or different ways of dealing with one’s emotions
(Dunekacke et al., 2021; Jenßen et al., 2021). Mixture distri-
bution approaches can identify subpopulations that have dif-
ferent model parameters (Rost & Eid, 2009; Steinley, 2023)
and could represent a promising approach for investigating
individual differences in the relationships between beliefs,
emotions, and knowledge.
In their study, Chen et al. (2025) asserted the added value
of mixture distribution approaches in the context of math-
ematics education studies in secondary school. They made
a strong case for these approaches, highlighting their abil-
ity to effectively model the mixture of emotions and be-
liefs that occur in learning and performance situations and
their impact on mathematics performance. The authors fur-
ther posited that such environments can foster adaptabil-
ity in learning responses to emotions and beliefs, thereby
facilitating the development of professional knowledge.
We also adopted these arguments in the following for the
mathematics-related training of EC teachers, where un-
til now, the relationships between beliefs, emotions, and
achievement have only been investigated separately (cf.
Dunekacke et al., 2016; Jenßen et al., 2021), and only
two studies have applied a mixture distribution approach
by modeling EC teachers’ competencies with latent profile
analysis (Im & Choi, 2020; Zhu et al., 2023).
However, from a practical point of view, it seems plausi-
ble to assume that there are subpopulations with structural
differences in model parameters in this context. For exam-
ple, it is feasible that some PECTs are more capable of re-
flecting on their anxiety and regulating it, whereas others
might not succeed in doing so, for example, due to nega-
tive experiences in previous educational settings (Boyd et
al., 2014). Disregarding such differences could lead to the
use of inappropriate learning contexts during teacher edu-
cation and might unfavorably influence preservice teachers’
later work with children.
2 Theoretical Background
Next, we present theoretical and empirical findings on
affective-motivational dispositions (beliefs and emotions)
and on EC teachers’ knowledge. We also discuss the rela-
tionships between the variables with reference to control-
value theory.
2.1 PECTs’ Beliefs About Mathematics
The study of beliefs has been the focus of research for many
years, with a particular emphasis on beliefs about the na-
ture of mathematics and on teaching and learning mathemat-
ics (Li et al., 2024; Vesga-Bravo et al., 2022). In line with
Richardson (1996), we define beliefs “as psychologically-
held understandings, premises or propositions about the
world that are felt to be true” (p. 103). Beliefs control one’s
awareness of content during learning situations and thus
also the process of knowledge development. We focused
on beliefs about the nature of mathematics, with the un-
derstanding that such beliefs have evolved considerably in
recent years, as evidenced by the plethora of research ac-
tivities in this field (see Törner, 1996; Vesga-Bravo et al.,
2022). Vesga-Bravo et al. (2022) focused on the epistemo-
logical foundations of mathematical knowledge (e.g., abso-
lutism, constructivism) and the question of learning math-
ematics in the school context (constructivism, traditional-
ism). By contrast, Törner (1996), focused more on func-
tional and practical or application-related aspects. On the ba-
sis of these priorities, we refer to Törner’s work here. These
aspects have been employed on numerous occasions for EC
teachers in Germany (Benz, 2012; Dunekacke et al., 2016;
Thiel, 2010), and their emphasis on application orientation
makes them particularly well-suited for daily mathematics
instruction. Törner (1996) identified two overarching orien-
tations, namely, the dynamic orientation and the static orien-
tation. The dynamic orientation includes process-related and
application-related beliefs and addresses a more evaluative
and inquiry-based perspective on the nature of mathemat-
ics. The static orientation includes formalistic and scheme-
related beliefs and thus a more algorithm-based perspective
on mathematics.
To the best of our knowledge, only Zhu et al. (2023)in-
vestigated EC teachers’ beliefs in the domain of mathemat-
ics, specifically beliefs about teaching and learning. They
One Model to Fit Them All? A Mixture Distribution Path Analysis of Preservice Early Childhood Teachers’ Beliefs.. .
found three profiles: a constructivist profile, a profile with
mixed high constructivist and traditional beliefs, and one
with mixed low constructivist and traditional beliefs. Partic-
ipants’ profile membership was related to their professional
experience. Those with more professional experience were
more likely to be found in profiles with high constructivist
beliefs.
2.2 PECTs’ Emotions Toward Mathematics
Achievement emotions refer to emotions in learning and
achievement-related situations (Pekrun & Perry, 2014). In
general, emotions can be experienced as pleasant or un-
pleasant and as activating or deactivating (Barrett & Russell,
1998). In this context, enjoyment is defined as a pleasant and
activating emotion, whereas anxiety is unpleasant and acti-
vating. Because of their differences in polarity (pleasant vs.
unpleasant), the two emotions are negatively correlated with
one another (for PECTs: r=−.67; Jenßen et al., 2021).
Although both emotions are part of PECTs’ emotional
experiences in teacher education (Jenßen et al., 2021),
the majority of EC teachers stated that they experience
pleasant feelings (e.g., enjoyment) when facing mathemat-
ical requirements (Sumpter, 2020). When understood as
an achievement emotion, enjoyment encourages individu-
als to approach specific learning and achievement situations
(Ainley & Hidi, 2014). It also promotes the maintenance
of learning processes (Schukajlow & Rakoczy, 2016) and
leads to the use of deeper learning strategies (Muis et al.,
2015). Conversely, anxiety reduces cognitive capacity and
goes along with shallow learning strategies (Muis et al.,
2015). It limits PECTs’ self-confidence and self-efficacy in
mathematics (Bates et al., 2013; Gresham & Burleigh, 2019;
Thiel & Jenßen, 2018). Research has shown that mathemat-
ics anxiety does not seem to prevent PECTs from working
in early childhood education and care (ECEC) institutions,
even though mathematics is a fundamental learning field in
ECEC (Jenßen, 2022). PECTs attribute their anxiety about
mathematics to their prior experiences with mathematics in
school (Boyd et al., 2014).
Latent profiles of EC teachers’ mathematics-related en-
joyment and anxiety have not yet been directly investi-
gated. Pino-Pasternak and Volet (2018) identified four clus-
ters of preservice primary school teachers’ attitudes toward
learning science. Interestingly, participants in the so-called
promising profile showed high levels of perceived difficulty
and anxiety but also high levels of self-efficacy, interest,
and enjoyment. Compared with the other clusters, affiliation
with the promising cluster was also more stable over time.
2.3 PECTs’ Mathematical Knowledge
Professional knowledge can be understood as the outcome
of learning processes during teacher education. With respect
to professional knowledge, Shulman (1986) introduced the
distinction between MCK and MPCK for schoolteachers.
This distinction is also well-established in existing research
on pre- and inservice EC teachers (Bruns et al., 2021;
Dunekacke & Barenthien, 2021; Linder & Simpson, 2018).
EC teachers’ MCK is understood as general knowledge
about mathematical facts, concepts, and rules in different
mathematical domains such as numbers or geometry (Bruns
et al., 2021; Jenßen et al., 2019). MPCK reflects aspects
of learning and teaching mathematics in EC (McCray &
Chen, 2012). Hence, MPCK includes knowledge about chil-
dren’s development in the domain of mathematics as well
as knowledge about teaching mathematics to young chil-
dren, such as typical instructional strategies or how to de-
sign learning environments in play-based situations to foster
children’s learning in mathematics (Clements et al., 2011;
Jenßen et al., 2019).
A few studies have used a person-centered approach to
investigate pedagogical content knowledge in several do-
mains. Yang et al. (2023) identified three profiles of Chi-
nese preservice preschool teachers’ technological pedagog-
ical content knowledge (TPACK). Profile membership was
predicted by the participants’ level of education, gender, and
professional interests. Holzberger et al. (2021) investigated
preservice teachers’ knowledge, beliefs, self-efficacy, and
self-regulation. They found three different profiles (highly
knowledgeable and engaged, low mindset, less knowledge-
able).
A study by Blömeke et al. (2020) identified four profiles
of German secondary school teachers’ knowledge, skills,
and beliefs. There were also qualitative differences between
two profiles: They were similar in MCK, mathematics-
related skills, general pedagogical knowledge, and dynamic
beliefs but differed significantly in static orientation and be-
liefs about mathematics as an innate and fixed ability. Im
and Choi (2020) conducted a profile analysis of EC teachers’
mathematics-related knowledge and skills. The four profiles
indicated that the participants, Korean preschool teachers,
had similar (low) levels of knowledge but showed qualita-
tive differences in skills by demonstrating high sensitivity to
mathematics in children’s play but asking only questions of
low complexity, or vice versa.
2.4 Relationships Between PECTs’ Achievement,
Emotions, and Beliefs
PECTs’ knowledge, emotions, and beliefs are inherent parts
of their professional competence and their development
(Dunekacke et al., 2022). If preservice teachers are viewed
as learners during their teacher education, the relationships
between these dispositions can be conceptualized in terms
of control-value theory (Jenßen et al., 2023; Pekrun, 2006;
Pekrun & Perry, 2014).
S. Dunekacke et al.
This well-established theory describes the relationship
between achievement and emotions as a function of learn-
ers’ control and value appraisals regarding learning and
achievement situations. In line with this theory, emotions
are not the direct result of a situation. Instead, they repre-
sent the result of evaluations of (a) the controllability of the
situation (control appraisal) and (b) the significance of the
situation or the domain (e.g., mathematics) that the situation
is related to (value appraisal). These appraisals can be seen
as self-related cognitive evaluations that mediate between
aspects of the situation or features of the specific domain
and emotions (Pekrun, 2006). For instance, when solving a
problem in mathematics, individuals experience a variety of
emotions depending on their appraisals of the task or math-
ematics in general (Goldin, 2014).
The individual might appraise their control over the out-
come (success or failure) as relatively low if they believe
they must follow specific rules to solve the task (static be-
lief). Conversely, they might appraise higher controllability
if they are convinced that multiple solution paths exist. In
this case, the person holds a more flexible belief about the
nature of mathematics. Similarly, the individual will assess
the value of mathematics, for example, by evaluating math-
ematics as a significant domain for everyday life due to its
high applicability.
Depending on how the person appraises the controllabil-
ity (control appraisal) and the importance (domain value)
of mathematics while working on a mathematical task,
they will experience different emotions about the domain
of mathematics (Pekrun, 2006). With high perceived con-
trol and high value, they will typically experience enjoy-
ment, whereas with low perceived control and high value,
they might experience anxiety (Pekrun, 2006). Pleasant
emotions (e.g., enjoyment) are positively associated with
achievement, for example, because they enhance elaborative
learning, whereas unpleasant emotions (e.g., anxiety) lower
achievement, for example, due to reduced self-regulation
(Pekrun et al., 2011).
Thus, in the context of PECTs’ learning during their
teacher education, beliefs can be seen as generalized ap-
praisals of a specific domain such as mathematics. Addition-
ally, preservice teachers’ achievement in a specific domain
captures their acquisition of specific knowledge in this do-
main over the course of teacher education.
Research has already applied the basic assumptions of
control-value theory to university students by integrating be-
liefs about the nature of knowledge, emotions, and knowl-
edge in a specific domain (see Trevors et al., 2017, for an ap-
plication to university students’ learning in the field of text
comprehension). In this context, emotions can be seen as
mediators between epistemic beliefs and knowledge (Ros-
man & Mayer, 2018).
Previous studies have not investigated the complex re-
lationships between the beliefs, emotions, and knowledge
of PECTs in the context of mathematics learning. Instead,
they have focused only on individual parts of the relation-
ship structure. For example, relationships between PECTs’
epistemic beliefs and knowledge have been investigated but
without considering emotions as mediators in the sense of
control value theory (Pekrun, 2006). These studies indicate
that a static orientation toward mathematics has a negative
relationship with MCK, whereas application-related beliefs
and process-related beliefs are positively related to MCK
(Jenßen et al., 2022). Furthermore, research indicates a pos-
itive relationship between MPCK and application-related
as well as process-related beliefs (Dunekacke et al., 2016;
Jenßen et al., 2022). No relationship between a static orien-
tation and MPCK has been found.
Only a few studies have investigated relationships be-
tween PECTs’ beliefs and emotions in mathematics (for
anxiety: Aslan, 2013;Boydetal.,2014). However, these
studies did not investigate epistemic beliefs. By contrast,
significantly more studies have looked at the relationship
between PECTs’ emotions and knowledge in mathematics.
The majority of these studies have focused on mathemat-
ics anxiety (Jenßen, 2022). As is the case for mathematics
anxiety and mathematical knowledge in general, a medium
negative correlation between the two constructs can also
be assumed for PECTs (Jenßen et al., 2021). In line with
control-value theory (Pekrun, 2006; Pekrun & Perry, 2014),
it can be assumed that anxiety negatively affects achieve-
ment in mathematics, although low achievement can also
be assumed to foster mathematics anxiety (Carey et al.,
2016). Studies on PECTs’ mathematics anxiety have inves-
tigated its relationship with MCK. Thiel and Jenßen (2018)
investigated the effects of PECTs’ mathematics anxiety on
their grades on an exam that tested both MCK and MPCK
at the end of teacher education and also found a negative
effect. Jenßen et al. (2021) found that mathematics anxi-
ety was negatively related to PECTs’ MCK and MPCK,
whereas enjoyment was related only to MCK (positively, in
line with theoretical assumptions). However, another study
with PECTs found a small positive relationship between en-
joyment and MPCK (Blömeke et al., 2017a). A study on
the development of Norwegian PECTs’ enjoyment during an
exam revealed that enjoyment was positively related to their
exam grade in mathematics (including MCK and MPCK;
Blömeke et al., 2019).
Only one study so far has investigated PECTs’ epistemic
beliefs (and only dynamic beliefs), enjoyment, and knowl-
edge in mathematics. This study with German PECTs re-
vealed a positive relationship between enjoyment and dy-
namic beliefs about mathematics, and both factors positively
predicted MCK and MPCK (Blömeke et al., 2017a). To our
knowledge, no studies have investigated effects of PECTs’
epistemic beliefs on their anxiety in mathematics.
One Model to Fit Them All? A Mixture Distribution Path Analysis of Preservice Early Childhood Teachers’ Beliefs.. .
Fig. 1 Path model of mathematics-related beliefs, emotions, and
knowledge. Note:AO=Application-related beliefs; PO =Process-
related beliefs; SO =Static orientation; Enjoyment =Mathematics-
related enjoyment; Anxiety =Mathematics anxiety; MCK =Mathe-
matical content knowledge; MPCK =Mathematics pedagogical con-
tent knowledge
In summary, the literature supports the theoretical as-
sumptions of control-value theory (Pekrun, 2006) in the con-
text of mathematics for PECTs. However, the entire inter-
play between epistemic beliefs (static, application-related,
process-related), enjoyment, and anxiety as well as MCK
and MPCK has yet to be tested in a unifying model.
3 The Present Study
We reanalyzed data from the KomMa project (Blömeke et
al., 2017b) to develop a measure of preservice teachers’
mathematics-related professional knowledge, namely, their
MCK and MPCK, and gather knowledge about its structure
and level. We did not formulate hypotheses on PECTs’ pro-
files with respect to their mathematics-related beliefs, emo-
tions, or knowledge.
As outlined, recent research has combined beliefs and
emotions as predictors of academic achievement (Rosman &
Mayer, 2018). Figure 1presents the current state of research
on preservice teachers’ mathematics-related beliefs, emo-
tions, and knowledge based on studies using the variable-
centered approach. The expected directions of relationships
are shown if they have been found in at least one prior study.
We investigated whether this structure applied in the same
way to all participants.
However, all the expected relationships were based on
studies that assumed the same relationships for all partici-
pants in the sample, which is quite a rigorous empirical as-
sumption. Nascent research using mixture distribution ap-
proaches has indicated that this assumption might not be ap-
propriate in the context of pre- and inservice EC and school
teachers (Im & Choi, 2020; Zhu et al., 2023). The findings
showed that solutions with multiple profiles fit the data bet-
ter and reflected the expected quantitative and qualitative
differences. Moreover, membership in certain profiles was
explained by or related to other variables. For example, a
favorable motivational profile for choosing a teaching ca-
reer was associated with more enjoyment in teaching (Lo-
hbeck & Frenzel, 2022). There is also evidence that, for
instance, the role of static orientation can differ between
profiles (Blömeke et al., 2020). Within a study, there is an
assumption that there are no associations between the vari-
ables within the different profile classes. However, this inde-
pendence assumption might be too strong when considering
the associations between different constructs such as in the
model in Fig. 1. In our study, we assumed that there might be
different subpopulations (latent classes) for whom different
path models explain the covariance between the variables in
Fig. 1when a mixture distribution path model is applied.
This model allows covariances within latent classes, and the
path model’s parameters are allowed to differ between sub-
populations. Consequently, we investigated the following re-
search questions (RQs):
RQ1. Are there different subpopulations of PECTs with
respect to the relationships between mathematics-related
beliefs (application-related beliefs, process-related beliefs,
static orientation), emotions (enjoyment of mathematics,
mathematics anxiety), and knowledge (MCK, MPCK) when
these variables are included in a path model? Fig. 1presents
the anticipated path model. On the basis of existing re-
search using mixture distribution approaches in the context
of teacher education, we hypothesized that a model with
more than one latent class would fit the data better than a
one-class model.
RQ2. If there are more than two latent classes, can they be
reasonably interpreted in terms of structural differences that
justify the added value of a mixture distribution path model?
This approach is explanatory; however, on the basis of exist-
ing empirical findings on qualitative differences in profiles
in terms of mean differences (e.g., Blömeke et al., 2020;Im
& Choi, 2020; Lohbeck & Frenzel, 2022; Pino-Pasternak &
S. Dunekacke et al.
Volet, 2018), we assumed such structural differences with
respect to the relationships between variables as well. Such
differences would have a substantive and practical relevance
(Hanin & Gay, 2023). On a substantive level, they would
better reflect the heterogeneity of the population, meaning
that, in practice, learning settings could be designed differ-
entially with respect to the profiles.
4 Materials and Methods
4.1 Sample and Procedure
The presented study is based on data from n=1851 PECTs
from 44 EC teacher training institutions (out of 485 institu-
tions in 2013/14). More information about the project and
the sample was published as part of the project (Blömeke et
al. 2017a,2017b). The majority of PECTs were undergoing
training in the vocational track1(vocational track: 73.9%;
university track: 26.1%). In line with the population, the par-
ticipants were mostly female (female: 85.6%; male: 14.4%),
and their average age was M=23 years (SD =5).
Data collection took place in winter 2013/14 in regular
classes in the teacher education institution. All tests and
questionnaires were administered as paper-pencil assess-
ments. Achievement tests covering professional knowledge
were administered in a multimatrix design due to the limited
time available for data collection.
4.2 Instruments
4.2.1 Beliefs
PECTs’ epistemological beliefs about the nature of mathe-
matics were assessed with well-established scales for mea-
suring schoolteachers’ beliefs (Törner, 1996) that have also
been applied to EC teachers (Benz, 2012; Blömeke et al.,
2017a; Dunekacke et al., 2016). The scales originally de-
veloped by Törner (1996) capture EC teachers’ application-
related beliefs with six items (e.g., “Mathematics is helpful
for solving everyday problems and tasks”), process-related
beliefs with four items (e.g., “Mathematics is an activity in-
volving thinking about problems and gaining insight”), and
static orientation with four items (e.g., “Mathematics de-
mands mainly formal accuracy”). All items were answered
on a 6-point scale ranging from 1 (strongly disagree)to
1In Germany, one can become an EC teacher via a vocational track (at
a vocational school) or a university track (typically a bachelor’s de-
gree program; Dunekacke et al., 2021; Oberhuemer et al., 2010). Both
tracks aim to foster PECTs’ MCK and MPCK alongside other skills
(KMK, 2017; Robert Bosch Stiftung, 2008). However, they differ with
respect to aspects such as their orientation toward academic learning
or the number and length of internships integrated into the curriculum
(Oberhuemer et al., 2010).
6(strongly agree) and achieved a good reliability (Cron-
bach’s αapplication-oriented beliefs =.85; αprocess-oriented beliefs =
.80; αstatic beliefs =.81).
4.2.2 Emotions
PECTs’ enjoyment of mathematics was captured with a
scale developed in the KomMa project consisting of four
items (e.g., “Mathematics is enjoyable”; Jenßen et al.,
2021). The items were answered on the same 6-point scale
as the items capturing beliefs. The scale achieved good reli-
ability in the current application (αenjoyment =.89).
Anxiety was assessed with four items (e.g., “I get very
nervous doing mathematics problems”). The items were de-
veloped for learners in mathematics (Lee, 2009) and have
been applied to PECTs several times in the past (Jenßen et
al., 2021,2015). The items achieved good reliability in the
current application (αmath anxiety =.89).
4.2.3 Knowledge
PECTs’ knowledge was assessed with standardized tests de-
veloped in the KomMa project (Blömeke et al., 2017b) and
previously used in several studies to assess pre- and inser-
vice EC teachers’ professional knowledge (Gasteiger et al.,
2020; Jenßen et al., 2021). The test scores can be used to
draw valid conclusions about the content of the test, in terms
of the construct, and regarding EC teacher education in Ger-
many. All items were coded dichotomously as correct or in-
correct.
The MCK test consisted of 24 items covering various
mathematical content areas (numbers, geometry, quantity
and relations, data). MPCK was measured with 28 items ad-
dressing mathematical learning in formal and informal set-
tings and how to diagnose children’s mathematical compe-
tence. The test scores were scaled using two-parameter lo-
gistic item response theory models (Blömeke et al., 2017b).
All tests achieved good reliability (αMPCK =.87, αMCK =
.88).
4.3 Data Analysis
To address RQ1, we used Mplus 8.2 (Muthén & Muthén,
1998–2017) to conduct mixture distribution path analysis
following the guidelines given by Ferguson et al. (2020). In
accordance with the path model we proposed, we considered
a number of variables, including those pertaining to beliefs
(application-related beliefs [AO], process-related beliefs
[PO], and static orientation [SO]), emotions (mathematics-
related enjoyment [enjoyment] and mathematics anxiety
[anxiety]), and knowledge (MCK and MPCK), in addition
to the relationships that were shown above.
In order to determine the number of classes, we esti-
mated a series of mixture models with an increasing number
One Model to Fit Them All? A Mixture Distribution Path Analysis of Preservice Early Childhood Teachers’ Beliefs.. .
Table 1 Descriptive results for all variables
Min; Max MSD
Application-related beliefs (AO) 1.00; 6.00 4.24 0.90
Process-related beliefs (PO) 1.00; 6.00 3.83 0.93
Static orientation (SO) 1.00; 6.00 4.20 0.75
Mathematics-related enjoyment (Enjoyment) 1.00; 6.00 3.59 1.29
Mathematics anxiety (Anxiety) 1.00; 4.00 2.58 0.82
Mathematical content knowledge (MCK) 20.05; 78.68 50.00 10.00
Mathematics pedagogical content knowledge (MPCK) 14.45; 74.14 50.00 10.00
Table 2 Fit indices for different numbers of latent classes
#Class AIC BIC BICadj En-tro-py Class prob. (in %) Class size (in %) LMR LRT (p)
1 48780.472 48973.794 48862.599
2 48509.372 48884.969 48668.935 .45 78.4/84.9 41.9/58.1 .062
348395.69 48953.569 48632.695 .48 72.8/80.5/77.8 52.2/21.5/26.3 .522
Note. AIC =Akaike Information Criterion, BIC =Bayesian Information Criterion, BICadj =sample-size adjusted Bayesian Information Crite-
rion, LMR LRT =Lo-Mendell-Rubin likelihood ratio test.
of latent classes and no further restrictions within classes.
To determine model fit and interpretability, we evaluated
the solutions using the Akaike Information Criterion (AIC),
Bayesian Information Criterion (BIC), and sample-size ad-
justed Bayesian Information Criterion (BICadj), preferring
the BIC (Ferguson et al., 2020), and the Lo-Mendell-Rubin
likelihood ratio test (LMR LRT; Nylund et al., 2007).
We addressed RQ2 through a detailed examination and
interpretation of the identified classes when they were
present.
5Results
5.1 Descriptive Results
Table 1presents descriptive statistics for the sample. Knowl-
edge test scores (MPCK, MCK) were standardized to M=
50 (SD =10; Blömeke et al., 2017b). With respect to be-
liefs, enjoyment, and anxiety, participants showed on aver-
age high levels of application- and process-oriented beliefs
and enjoyment but also high levels of static-oriented beliefs
and anxiety.
5.2 Mixture Distribution Path Analysis
Table 2present fit indices for the estimated mixture distri-
bution solutions. The smallest values of the AIC and BICadj
arose for the three-class solution, whereas the BIC favored
the two-class solution. According to the LMR LRT, the two-
class solutions did not fit the data better than the one-class
solution even though the pvalue was close to .05. Thus, the
different fit indices favored different solutions (between one
and three classes), which is often the case. According to the
simulation studies by Fonseca and Cardoso (2007) and Ny-
lund et al. (2007), the BIC seems to be the best information
criteria for latent profile and related models. Thus, we based
our selection of the number of classes on the BIC coefficient
and decided on a two-class model. Moreover, the two-class
model led to a solution in which the two classes were rela-
tively large and similar in size, showing that there were sub-
stantive subgroups. Moreover, the results for the two-class
solution were very interesting from a theoretical point of
view and offered new theoretical insights. Whereas entropy
was somewhat advantageous for the three-class solution, it
was not much different from the two-class solution. The en-
tropy values are rather low. However, entropy is a measure
indicating whether individuals can be reliably assigned to la-
tent classes. It is, however, not a good measure for detecting
the correct number of latent classes (e.g., Tein et al., 2013).
We do not aim to assign individuals to classes but to separate
the classes with respect to the parameters of the path model.
Because we were not interested in assigning individuals to
latent classes for further analysis, we think that the level of
entropy was appropriate for the current purpose (Muthén,
2008). For RQ1, the results showed that a two-class solution
was superior to a one-class solution.
A table with all the coefficients that were estimated for
the two-class solution is available in the electronic supple-
ment. We highlight and discuss significant paths in Sect. 5.3
when describing the classes in detail.
S. Dunekacke et al.
Fig. 2 Model for Latent Class 1. Note:AO=Application-related be-
liefs; PO =Process-related beliefs; SO =Static orientation; Enjoy-
ment =Mathematics-related enjoyment; Anxiety =Mathematics anx-
iety; MCK =Mathematical content knowledge; MPCK =Mathemat-
ics pedagogical content knowledge
Fig. 3 Model for Latent Class 2. Note:AO=Application-related be-
liefs; PO =Process-related beliefs; SO =Static orientation; Enjoy-
ment =Mathematics-related enjoyment; Anxiety =Mathematics anx-
iety; MCK =Mathematical content knowledge; MPCK =Mathemat-
ics pedagogical content knowledge
5.3 Description of Latent Classes
Figure 2presents the model for Latent Class 1, showing that
41.9% of the participants belonged to this particular class.
To avoid cluttering the figure, only significant effects are
shown. All effects can be found in the ESM. For RQ2, we
describe the differences between the two classes in detail.
The associations and effects are conditional on the other
variables in the model. For reasons of simplicity, we de-
scribe the associations without always mentioning that they
are conditional associations.
As the figure shows, application- and process-related be-
liefs were associated with enjoyment and anxiety. The static
orientation was not related to mathematics-related emotions
but was negatively related to MCK as expected and neg-
atively related to MPCK, an unexpected finding. MPCK
was positively related to application-related beliefs, as ex-
pected. Against our expectations, there was no significant re-
lationship between MCK and application-related or process-
related beliefs. Also against our expectations, there was a
negative relationship between MPCK and process-related
beliefs in this latent class. Enjoyment was not related to
MCK nor to MPCK, another finding that went against our
expectations. However, anxiety was negatively related to
MCK, but there was no significant relationship with MPCK.
Overall, in Latent Class 1, the dynamic beliefs demonstrated
a mixed pattern. Application-related beliefs were associ-
ated with higher enjoyment and MPCK and lower anxiety.
Process-related beliefs were associated with both higher en-
joyment and higher anxiety but lower MPCK. By contrast, a
static orientation was associated with lower knowledge. We
propose that this latent class be called the application learn-
ing class.
Figure 3presents the path model of mathematics-related
beliefs, emotions, and knowledge for Latent Class 2, consti-
tuting 58.1% of the total number of participants.
As expected, application-related and process-related be-
liefs, which both reflect a dynamic orientation toward math-
ematics, were associated with higher levels of enjoyment.
Application-related beliefs also exhibited significant rela-
tionships with MPCK, MCK, and mathematics anxiety.
Process-related beliefs were linked to enjoyment and anx-
One Model to Fit Them All? A Mixture Distribution Path Analysis of Preservice Early Childhood Teachers’ Beliefs.. .
iety but to none of the outcomes. Interestingly, and contrary
to expectations, in this latent class, there was a positive rela-
tionship between a static orientation and the two outcomes
of MCK and MPCK. Furthermore, static orientation had a
positive relationship with anxiety and a negative relationship
with enjoyment. For the relationships between emotions and
knowledge, anxiety’s expected negative relationships with
MCK and MPCK arose, comparable to Class 1. In addition,
enjoyment was not correlated with MCK or MPCK, consis-
tent with the findings for Class 1.
Class 2 was thus characterized by mixed significance for
static orientation, which was surprisingly positively corre-
lated with achieved knowledge and anxiety. However, this
pattern is concurrent with an unfavorable impact on the ex-
pression of enjoyment. Class 2 can be characterized by its
unexpected associations with static orientation. Thus, we
propose that Latent Class 2 should be called the static learn-
ing class.
6 Discussion
Research based on control-value theory (Pekrun, 2006)is
well-established in the context of education (Rosman &
Mayer, 2018). However, research has not yet applied a
mixture distribution path analytic approach to investigate
mathematics-related beliefs, emotions, and achievement in
terms of control-value theory, even though the benefit of
mixture distribution approaches for research on pre- and in-
service (EC) teachers has been shown (Blömeke et al., 2020;
Im&Choi,2020; Zhu et al., 2023). Our aim was to apply
a mixture distribution path analysis to investigate whether a
path model including mathematics-related epistemological
beliefs, emotions, and knowledge differs between different
subgroups of PECTs. We found two latent classes that ex-
hibited structural differences in the path model parameters.
Latent Class 1 was called the application learning class.
The acquisition of MPCK for PECTs in this class is re-
lated to higher levels of application-oriented beliefs, which
also go with lower levels of anxiety. This finding is in line
with existing research (Rosman & Mayer, 2018) because
application-oriented beliefs are part of the dynamic orien-
tation of epistemological beliefs about the nature of math-
ematics and can thus be seen as a more evaluative and
inquiry-based perspective on mathematics. However, the ac-
quisition of MCK is also strongly guided by the unpleas-
ant emotion of anxiety. This relationship might be explained
by the fact that (preservice) EC teaching can be understood
as a mathematics-avoiding profession (Jenßen, 2022). When
it comes to acquiring knowledge, participants might benefit
from learning opportunities that address their application-
oriented beliefs. Such activities might give PECTs personal
experience with EC mathematics activities, supported by
guidance and reflection (Bruns et al., 2017).
Latent Class 2 was called the static learning class.Its
main difference with the application learning class was the
positive relationship between a static orientation and knowl-
edge in the static learning class. High levels of static orien-
tation in this class went with higher MCK and MPCK but
also higher anxiety. A static orientation means that a person
places great weight on applying general algorithms instead
of finding solutions that fit the specific task at hand. Fur-
ther research is necessary to investigate this relationship and
its consequences in more detail. One possible explanation
might be that high levels of static orientation give preservice
teachers feelings of control or safety but are also associated
with more anxiety about failure because of the strict em-
phasis on correctness (Brunner & Star, 2024), for example,
when applying algorithms that demand correctness. To in-
terpret this finding carefully, it might indicate a dilemma for
participants who belong to this class, which goes with higher
knowledge but also with higher anxiety. Our study used a
cross-sectional design, which precludes the ability to make
any statements about causal relationships between variables.
In future research, this design limitation should be taken into
account with regard to Class 2, with the aim of facilitating
a more profound understanding of the optimal application
of opportunities to learn, whether these opportunities are re-
lated to beliefs or emotions.
Our study comes with some limitations that should be
kept mind when interpreting our results. First, our analysis
was based on a sample from German EC teacher education
and can thus be generalized only to countries with similar
EC teacher education systems (e.g., Ireland, European Com-
mission et al., 2019). Furthermore, as previously mentioned,
we reanalyzed data from 2013. Conditions of EC teacher ed-
ucation might have changed since then, and thus, PECTs’
learning classes might have changed as well. Whether any
changes affect the relationships between beliefs, emotions,
and knowledge is an open question. Recent studies have in-
dicated increased learning opportunities in mathematics in
the EC teacher training curriculum since 2013 (Dunekacke
& Barenthien, 2023). Nevertheless, our findings raise con-
cerns about the quality of these learning opportunities and
the methods through which they can be further developed.
To the best of our knowledge, there are no extant results
available for the EC teacher training context, thus rendering
our study a valuable source of preliminary insights in this
area. Moreover, the contemporary discourse on the signifi-
cance of early mathematical education (SWK, 2022) under-
scores the relevance of our findings.
7 Implications and Conclusion
As mentioned before, mixture distribution approaches can
be used to identify patterns of shared behavior between
S. Dunekacke et al.
participants and thus test the rigorous empirical assump-
tion that relationships hold for all individuals in the pop-
ulation. One can question this assumption because partici-
pants in institutional learning settings such as postsecondary
education differ in terms of their motivation, for example.
Applying a mixture distribution path analysis to PECTs’
mathematics-related epistemic beliefs (application-related
beliefs, process-related beliefs, static orientation), emotions
(enjoyment, anxiety), and knowledge (MCK, MPCK), tak-
ing into account assumptions from control-value theory
(Pekrun, 2006) and its applications (Rosman & Mayer,
2018), leads to two latent classes with qualitative differ-
ences: an application learning class and a static learning
class.
Whereas (preservice) EC teachers’ mathematics-related
beliefs, emotions, and knowledge have been investigated for
more than 10 years (Bruns et al., 2021; Dunekacke & Bar-
enthien, 2021; Jenßen, 2022; Linder & Simpson, 2018), our
results contribute to the research literature in three ways.
First, with respect to the role of the static orientation, ap-
plying the mixture distribution approach allowed us to iden-
tify an interesting and potentially practically relevant differ-
ence between the two classes in terms of significant rela-
tionships with EC teachers’ knowledge. We are the first to
find such relationships. By contrast, a strong static orienta-
tion has sometimes been critically discussed (Benz, 2012;
Dunekacke et al., 2016). Thus, it is necessary to learn more
about the relationships between knowledge, emotions, and a
static orientation found for participants in the static learning
class and its practical implications.
Second, our results indicate that there might be differ-
ent pathways for learning, which is an important insight for
teacher educators. Our results point to individual differences
between participants in teacher education not only with re-
spect to their level of knowledge but also with respect to
their learning paths. Individuals in Class 1 will likely ben-
efit greatly from repeatedly reflecting on the importance of
mathematics in general but also in the context of early edu-
cation in the classroom. For example, a discussion of chil-
dren’s mathematics-related questions in the context of prac-
tical experiences could be beneficial for reducing their anx-
iety and expanding their MPCK. For individuals in Class 2,
static beliefs seem to go hand in hand with higher knowl-
edge but also with higher anxiety. This negative emotional
side effect could be reflected in the preservice teachers by
the aim that only the positive effect on MPCK remains. Con-
sequently, we suggest that reflection on the connection be-
tween static orientation and anxiety could be helpful in this
class. However, further research is required to establish the
causality of these relationships before any further recom-
mendations can be made.
Third, the present analysis was grounded in control-value
theory. The results indicate that the anticipated relationships
were not consistent across all individuals. This inconsis-
tency concerns the relationship between beliefs and knowl-
edge, but in particular, it calls into question the central sig-
nificance of emotions in learning processes. Our results indi-
cate that emotions do not have a significant effect on knowl-
edge. However, our study is not suitable for adequately map-
ping processes with regard to beliefs, emotions, and knowl-
edge (Pekrun & Marsh, 2022). Further studies should inves-
tigate these aspects.
Supplementary Information The online version contains supplemen-
tary material available at https://doi.org/10.1007/s11858-025-01686-z.
Funding Information Open Access funding enabled and organized by
Projekt DEAL.
Declarations
Competing Interests The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, pro-
vide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included
in the article’s Creative Commons licence and your intended use is not
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