The Role of Intelligence and Self-Concept for Teachers’ Competence

Research on intelligence and competence has developed widely independent of each other. The present paper aims at relating these traditions and at integrating the dominant models to fill gaps in the respective theories. We test the structural models derived from this integration in a series of confirmatory factor analyses and a latent moderated structural equations approach using teachers as an example. The data reveal that both fluid intelligence (gf) and domain-specific knowledge affect teachers’ ability to solve the domain-specific items. Teachers’ academic self-concept related to mathematics explains individual differences beyond gf. An interaction effect between gf and self-concept exists for teachers’ pedagogical content and general pedagogical knowledge, but not for their mathematics knowledge. This finding indicates that a positive self-concept cannot compensate for a lack of gf, but it supports the acquisition of domain-specific knowledge in case of high gf, probably because it facilitates overcoming challenges.


Introduction
The term intelligence is used in cognitive psychology to describe a variety of cognitive abilities at different hierarchical levels. More general abilities are assumed to underly more specific ones (Jensen 1998). In educational research in contrast, there are hardly any studies using the term intelligence. Instead, a competing terminology has become prominent that uses the term competence as the disposition underlying human behavior (see, e.g., Blömeke et al. 2015).
The aims of the present paper are two-fold: Firstly, we intend to relate the competence terminology prominent in education to the intelligence terminology prominent in cognitive psychology and to integrate the dominant models from these two traditions. Both academic disciplines could benefit from such an integration because it has the potential to fill gaps in the respective theories. Secondly, we intend to test the models we can derive from an integrated theory using teachers as an example.

The Cattell-Horn-Carroll Theory of Cognitive Abilities
There are still some unresolved incongruencies between different traditions of modeling intelligence. Nevertheless, it is possible to summarize that the Cattell-Horn-Carroll (CHC) theory of cognitive abilities has evolved as a theory widely accepted in cognitive psychology and supported by validity evidence (Flanagan and Harrison 2012;Schneider and McGrew 2012).
The CHC theory is based on Carroll's (1993) three stratum model and Cattell's (1963) and Horn's (1968) extended gf -gc model. It describes the hierarchical organization of cognitive abilities where broader mental processes are placed on higher levels and more task-specific abilities on lower levels in the system of human cognitions. Cognitive abilities on higher levels are hypothesized to affect the acquisition of abilities on lower levels. Consequently, individuals with high achievement in one domain often show high achievement in other domains (Gustafson and Undheim 1996).
There are a variety of cognitive abilities included in the CHC model. From an educational perspective, the most important distinction is between those that are difficult to change because they are, to a large extent, caused by biological and neurological factors and those that are mostly developed through opportunities to learn. Gf (fluid intelligence) is the ability to process information and to analytically distinguish between objects. Inductive and deductive reasoning are crucial indicators and regarded as particularly relevant when a situation is new, and automatic processes no longer work (Schneider and McGrew 2012). Gf is a dimension commonly regarded as relatively difficult to change, and if so, only over long time (Ritchie and Tucker-Drob 2018).
In contrast, gc (crystallized intelligence) includes a broad class of cognitive abilities mostly acquired through opportunities to learn. A recent development has been to split up this construct into cognitive abilities related to the broader culture and society on the one hand (called gc) and domain-specific knowledge (called gkn) on the other hand (Schneider and McGrew 2018). The latter dimension is an umbrella term for "specialized knowledge (knowledge not all members of a society are expected to have)" (Schneider and McGrew 2012, p. 123). It is acquired through "intensive systematic practice and training (over an extended period of time)" and is maintained through "regular practice and motivated effort (a.k.a., expertise)" (McGrew 2009, p. 6). Gkn includes a range of distinct cognitive abilities (Flanagan and Dixon 2014).
Another CHC dimension that is developed through opportunities to learn, already included in Cattell's (1963) and Horn's (1968) extended gf -gc model, is gq (quantitative knowledge). Gq describes the "knowledge related to mathematics" (Schneider and McGrew 2012, p. 127).
From an educational perspective, it is crucial to identify cognitive abilities that are mostly developed through opportunities to learn and to distinguish them from those that are more difficult to change. Only then can we design education in line with the effort needed to support human development. Moreover, only then we can evaluate the effects of education properly because the proportion of variance explained by general cognitive ability can be controlled for (McClelland 1973).
Disentangling teacher cognitions in such a way is urgently needed. The sample used in this study was drawn from the population of early childhood teachers because, to our knowledge, there has not yet been a study that has attempted this, even though the value of education has been repeatedly questioned for this group of teachers (Strauss 2005). Without clarifying the role of general cognitive abilities, it is almost impossible to evaluate to what extent such criticism is justified or how to design teacher education so that it supports the development of teacher competence.

Relation of the Competence Model to the CHC Model
In contrast to intelligence models that intend to identify and classify cognitive abilities in a top-down manner, competence models intend to identify and classify the dispositions underlying observable behavior. Spencer and Spencer (1993, p. 9) defined competence in this sense as "an underlying characteristic of an individual that is causally related to criterion-referenced effective and/or superior performance in a job or situation. Underlying characteristic means that the competency is a fairly deep and enduring part of a person's personality." One could describe this modeling approach as a bottom-up approach, since it first identifies the criterion and then tries to identify the traits involved.
The most important goal with modeling competencies in this way is to identify and describe those competence dimensions that are learnable and thus can be influenced by education. Koeppen et al. (2008) defined competence in this sense as "domain-specific cognitive dispositions that are required to successfully cope with certain situations or tasks, and that are acquired by learning processes" (p. 68). Blömeke et al. (2015) extended this understanding of competence as purely cognitive by suggesting considering dispositions as a multi-dimensional set of not only cognitive, but also affective-motivational-volitional characteristics. In their argumentation, they refer back to Snow's (1994) concept of two pathways that contribute to achievement, namely a cognitive and a commitment pathway, and the broad range of studies that support such a concept. Students' cognitive abilities are typically the strongest predictors of student achievement. However, including motivational and similar characteristics increases the predictive validity, although to a smaller extent, particularly with respect to teacher-set grades, but less so with respect to standardized test scores (Lavrijsen et al. 2021;Steinmayr et al. 2019). Academic self-concept proved to be the most influential construct in this context (ibid.).
The Blömeke et al. (2015) competence model further clarified that these cognitive and affective-motivational dispositions are domain-specific, but in a general way, beyond single tasks and situations within this domain. The dispositions are stable continuous traits that, in turn, underlie task-and situation-specific cognitive skills not organized in an academic-disciplinary way, as is knowledge, but along the specific demands of narrowly defined situations or tasks. Blömeke et al. (2015) propose regarding the transformation of dispositions into observable behavior as fully or partially mediated by task-and situationspecific cognitive skills.
From this description, it should be clear that there is strong overlap between the CHC and the Blömeke, Gustafsson, and Shavelson competence models. The cognitive dimension of competence is conceptualized similarly to gkn as domain-specific and learnable. It is highly specialized knowledge that has been acquired during a long process of education and requires regular practice. We are not the first ones to point out such an overlap of competence and intelligence models. Previously, Wilhelm and Nickolaus (2013) and Trapp et al. (2019) had done this, although they did not specify the overlap in detail.
The affective-motivational-volitional dimension of the Blömeke, Gustafsson, and Shavelson competence model overlaps with the investment traits as conceptualized by  and Ziegler et al. (2012). The investment theory has provided evidence for the existence of traits that support the development of certain cognitive abilities because these support "the tendency to seek out, engage in, enjoy and continuously pursue opportunities for effortful cognitive activities" (von Stumm et al. 2011, p. 225).  examined a set of personality and motivational traits and provided evidence for their effects on gc beyond the effects of gf. Ziegler et al. (2012) extended this research with the openness-fluid-crystallized-intelligence model by providing evidence for an interaction effect of gf and personality traits, in particular, openness. However, very few studies exist that test the interaction hypothesis (e.g., (Zhang and Ziegler 2015) with respect to the interaction of openness and gf ; (Lechner et al. 2019) with respect to the interaction of openness, domain-specific interest, and gf ).
The core differences between the intelligence and competence traditions are, first, the absence of general cognitive abilities that are difficult to change, in particular of gf, in the Blömeke, Gustafsson, and Shavelson model. Although this is a decision that was made intentionally, since the objective was to identify traits that can be influenced by education, this absence could play out in an unintended negative way. An evaluation of the effects of education could be hampered by hidden third-variable (e.g., gf ) effects. Conceptually, it can be argued that gf is less relevant for experts because their specialized knowledge can only be gained through specific opportunities to learn during many years of education. However, specialized knowledge also represents cognitive traits where investments should be crucial, since we know from previous research that gf shows effects in all cognitive domains. Many studies and meta-analyses demonstrate, for example, that gf is able to explain substantial amounts of variance in domain-specific school achievement (e.g., Brunner 2008;Gustafsson 1994). Second another core difference between the two traditions is the role of the criterion. Both the CHC and the competence models are structural theories that examine the dimensions and facets of the respective constructs. While no criterion per se is considered in the intelligence tradition, modeling competence first identifies the criterion in terms of human behavior and then examines the underlying dispositions.
Third, the investment theory has so far focused on selected personality and motivational traits. Self-related cognitions have not been included, even though they would be considered a dimension of competence models given that studies on predictors of student achievement point to the relevance of self-related cognitions when it comes to explaining individual differences (e.g., Marsh and Martin 2011). Most recently, Demetriou et al. (2019) examined relationships between school achievement, gf, and self-concept using a sample of 10-to 17-years-olds. They found a decreasing effect of gf on school achievement but an increasing effect of self-concept with age. Since we intend to examine the cognitive structure of teachers, this result is very relevant.

Teachers' Competencies
Teachers acquire their competencies through a long process of schooling, teacher education, and professional development. This also applies to early childhood (EC) teachers used as a sample in this study (Dunekacke et al. 2021). Based on the seminal work by Shulman (1987), teachers' professional knowledge can be modelled as a three-dimensional construct that includes content knowledge, pedagogical content knowledge, and general pedagogical knowledge. With respect to content, the present study focuses on the domain of mathematics and EC teachers' task to support children's mathematical learning in early childhood education. Their professional knowledge then includes mathematics content knowledge (MCK), mathematics pedagogical content knowledge (MPCK), and general pedagogical knowledge (GPK) (Dunekacke et al. 2021).
MCK is the knowledge about numbers, sets, and operations; shape, space, and change; quantity, measurement, and relations; data, combinatorics, and chance (Bruns et al. 2021;Dunekacke et al. 2015a). In the case of EC teachers, this dimension has strong conceptual overlap with gq, since the level of mathematics does not exceed the level of mathematics learned during schooling. There are rarely opportunities to learn mathematics in EC teacher education that go beyond this level of mathematics. However, note that in case of other teacher groups, for example those trained for teaching mathematics in uppersecondary schools, MCK acquired during teacher education would include elements of university mathematics. This type of MCK would have to be classified as gkn because it is highly specialized mathematical knowledge which we cannot expect that all or even many members of a society have.
The MPCK of EC teachers is the knowledge of how to diagnose children's developmental state in mathematics and how to design an informal learning environment that supports the mathematical learning of children between age 3 and 6 Lee 2010;Torbeyns et al. 2019). GPK includes general foundations from educational theory, psychology, and instructional research related to early childhood and learning processes of 3-to 6-year-olds Malva et al. 2019).
This framework, the alignment of the framework and its measures, as well as the inferences to be drawn from these measures have been validated in a range of studies Dunekacke et al. 2015aDunekacke et al. , 2015bJenßen et al. 2015aJenßen et al. , 2015bBlömeke et al. 2022). The data for these studies were collected in Germany. However, the framework also reflects the professional tasks and opportunities to learn of EC teachers in other countries (e.g., Clements et al. 2004;NAEYC 2009).
As both the Blömeke, Gustafsson, and Shavelson model and the investment theory indicate, a purely knowledge-based approach to explaining individual differences in EC teachers' gq (MCK) and gkn (MPCK and GPK) may be limited given the long educational process underlying the development of these knowledge dimensions. Academic self-concept can be defined as an individual's belief about their abilities. Such self-related cognitions have been found to be relevant for a broad range of developmental and educational outcomes ). According to Marsh (1990) and Bong and Skaalvik (2003), academic self-concept is a domain-specific construct and needs to be examined with scales specific for the cognitive construct under investigation. Modeling the dispositions underlying human behavior in such a way may represent a more holistic perspective, thus enabling the explanation of more variance.

The Structure of EC Teachers' Competencies
Based on the theory described above, gf, gq/MCK, and gkn (MPCK and GPK), as well as academic self-concept, are core dispositions involved in EC teachers' work of supporting children's mathematical development. However, there is ambiguity with respect to their structure and hierarchy. Carroll (1993) had placed more specialized cognitive abilities on the third, or lowest, level of his taxonomy. In contrast, the most recent CHC theory places gf, gq, and gkn on the second level and narrower abilities on the level below (Flanagan and Dixon 2014).
Evidence for the relation of gf to the other constructs has been provided by comparing first-order models with hierarchical models. The latter typically fit better to the data (Demetriou et al. 2019). In one-dimensional first order models, g explains variance in all indicators without distinguishing between the different constructs. In correlated firstorder models, specific constructs are distinguished but still placed on the same level. In hierarchical models, gf also influences the specific constructs and thus, indirectly, their indicators (Gignac 2008). In this case, the effects of gf are fully mediated by the specific factors (Yung et al. 1999). Alternatively, gf directly influences the indicators of the different specific constructs as part of a hierarchical bifactor (Holzinger and Swineford 1937) or nested-factor model (Gustafsson and Balke 1993). The exact role of academic self-concept is also an open question. In an early study, Gose et al. (1980) provided evidence for the additional potential of self-concept to explain variance in reading, language, and mathematics student achievement beyond gf. Later on, a study by Schicke and Fagan (1994) supported this result with respect to several student cohorts at different grade levels. Further studies revealed similarly positive relations of self-concept to domain-specific achievement beyond the effects of gf (Lavrijsen et al. 2021;Steinmayr et al. 2019). In addition, Guo et al. (2016) pointed to a potential interaction of self-concept and gf.
Not surprisingly, given the absence of gf in competence models and corresponding research, the organization of EC teachers' domain-specific cognitions and their relation to broader general cognitions can be regarded as a desideratum. An additional reason for this desideratum might be that standardized and validated assessments covering EC teachers' professional knowledge are rare (European Commission et al. 2019). Therefore, as yet we have only limited research with respect to the structure of EC teachers' competencies, restricted to a few knowledge dimensions and to preservice EC teachers still in teacher education. The same restrictions apply to other teacher groups (Kleickmann et al. 2013 examined preservice primary and secondary school teachers; Roloff et al. 2020 studied practicing secondary school teachers).
With respect to preservice EC teachers (n = 353), a study by Jenßen et al. Nevertheless, a nested model revealed that MCK and MPCK had a significant additional impact on the ability to solve the domain-specific items. The initially strong latent correlation between MCK and MPCK, if gf was not controlled for (r = .67), could largely be explained by gf modelled as a higher-order factor. The size of the correlation was reduced substantially in the nested model (r = .30). Moreover, as typical for more complex models, the nested-factor model fit better to the data than a pure g-factor model where all indicators loaded on one general factor g only. The nested model also explained more variance. Blömeke and Jenßen (2016) used data from the same sample to examine the impact of gf on the relationship between preservice EC teachers' MCK or MPCK, respectively, and their skill for perceiving specific mathematics-related situations. A latent regression of this perception skill on the two knowledge constructs without including gf revealed strong effects (MPCK: β = .64, MCK: β = .47). The decrease in the effects of MCK or MPCK, respectively, on the skill to perceive mathematics-related situations (MPCK: β = .42, MCK: β = −.08) in a nested model including gf revealed the relevance of the latter for the interplay between the constructs, particularly with respect to the relationship between MCK and the skill to perceive math-related situations. The nested models were additionally tested against second-order models. Differences in the results were negligible.
We have not found any study that examined the cognitive structure of practicing teachers or that included GPK. Moreover, potential investment traits, such as academic self-concept, have not yet been examined. Our study adds to the body of research in both respects.

Study Aims and Hypotheses
The first aim of the empirical portion of this paper is to clarify the role of gf for practicing EC teachers' domain-specific knowledge. We test the common models against each other with the following hypotheses: H1. There are strong positive latent correlations between MCK, MPCK, and GPK if gf is not controlled for.

H3.
A pure g model, where all indicators load on one factor (see Figure 2), reveals an inacceptable model fit because it does not take into account the differential effects of gf, MCK, MPCK, and GPK on EC teachers' ability to solve the corresponding items.
of gf on the relationship between preservice EC teachers' MCK or MPCK, respectively, and their skill for perceiving specific mathematics-related situations. A latent regression of this perception skill on the two knowledge constructs without including gf revealed strong effects (MPCK: β = .64, MCK: β = .47). The decrease in the effects of MCK or MPCK, respectively, on the skill to perceive mathematics-related situations (MPCK: β = .42, MCK: β = −.08) in a nested model including gf revealed the relevance of the latter for the interplay between the constructs, particularly with respect to the relationship between MCK and the skill to perceive math-related situations. The nested models were additionally tested against second-order models. Differences in the results were negligible.
We have not found any study that examined the cognitive structure of practicing teachers or that included GPK. Moreover, potential investment traits, such as academic self-concept, have not yet been examined. Our study adds to the body of research in both respects.

Study Aims and Hypotheses
The first aim of the empirical portion of this paper is to clarify the role of gf for practicing EC teachers' domain-specific knowledge. We test the common models against each other with the following hypotheses: H1. There are strong positive latent correlations between MCK, MPCK, and GPK if gf is not controlled for.
H2. Gf is positively correlated with MCK, MPCK, and GPK (see Figure 1). Figure 2), reveals an inacceptable model fit because it does not take into account the differential effects of gf, MCK, MPCK, and GPK on EC teachers' ability to solve the corresponding items.

H3. A pure g model, where all indicators load on one factor (see
H4. Gf explains variance in MCK, MPCK, and GPK, but there are positive (partial) correlations between MCK, MPCK, and GPK after controlling for gf in a bifactor (S-1) model (see Figure 3).
The second aim of the empirical portion of this paper is to clarify the role of academic self-concept for EC teachers' ability to solve the domain-specific items:

H5. We hypothesize that EC teachers' self-concept related to mathematics explains individual differences in teachers' MCK, MPCK, and GPK beyond gf.
H6. Academic self-concept moderates the influence of gf on MCK, MPCK, and GPK (see Figure  4).  The second aim of the empirical portion of this paper is to clarify the role of academic self-concept for EC teachers' ability to solve the domain-specific items: H5. We hypothesize that EC teachers' self-concept related to mathematics explains individual differences in teachers' MCK, MPCK, and GPK beyond gf.

Sample
Practicing EC teachers were the target population of the present study. 210 EC teachers took the test battery. They were recruited by contacting EC institutions from all municipalities in two German federal states (Berlin and Brandenburg) via email and asking for voluntary participation. Therefore, they do not constitute representative samples. EC teachers were surveyed online without any time limitation. If they desired, they could postpone the second part of the survey until the next day. The teachers were, on average, M = 28 years old (SD = 7.4; min-max = 21-57). The majority (84%) indicated female as gender, while 16% indicated male as gender. Almost all EC teachers in the sample (93%) spoke German as their first language. About one-third of the sample (34%) had been trained at EC colleges, while the other two-thirds had received their training at post-secondary schools. Their median grade in their last class of school mathematics (a variety of levels, depending on the school year and type of class) was a 3 on a scale from 1 (best) to 6 (worst), with 4 as the minimum passing grade. These characteristics satisfactorily reflect the heterogeneity of EC teachers in Germany.

Measures of EC Teachers' Competencies
Similar to Zhang and Ziegler (2015), we used figural reasoning as an indicator of gf. This was assessed with the figural facet of the intelligence structure test on the basis of the Berlin Model of Intelligence Structure (BIS) (Jäger 1982). Validation studies revealed that the figural subtest of the Intelligence-Structure Test 2000R ) provides test scores corresponding to visual-based thinking Ziegler et al. 2012). The figural facet of the screening version (IST-Screening; Liepmann et al. 2012) consists of 20 items. A CFA applying the WLSMV estimator for categorical data revealed a good fit to the data and supported our one-dimensional assumption (χ 2 = 191.0, df = 170, p = .13; RMSEA = .03, 90%CI[.00; .05]; CFI = .94). Scale reliability as indicated by McDonald's omega for congeneric scales (McDonald 1999) was good (ω = .90, SE = .02). We could therefore build item parcels using a factorial approach by sequentially assigning items to three parcels based on factor loadings, starting with the highest loading. This procedure led to a perfectly identified congeneric measurement model with standardized factor loadings between .64 and .78, applying the MLR estimator.
With respect to each dimension of EC teachers' professional knowledge, large item pools were developed in collaboration with academic and practical experts. Item selection was completed on the basis of both conceptual considerations (American Educational Research Association et al. 2014) and a series of cognitive labs, unstandardized pre-pilot, standardized pilot, and validation studies Dunekacke et al. 2015aDunekacke et al. , 2015b. The resulting tests consisted of multiple-choice, bundled, and open-response items. In all cases, gender-neutral language was used to reduce the risk of stereotype threats (Cadinu et al. 2005). For item examples, see Appendix A.
EC teachers' academic self-concept was assessed with three domain-specific items related to mathematics. The statements were formulated negatively to maximize variance. We recoded the negatively worded statements to positive ones to facilitate the interpretation. McDonald's omega was good (ω = .90, SE = .02). A one-factor congeneric measurement model revealed standardized factor loadings between .82 and .93. The three items are (our translation): "Nobody is capable of everything. I just do not have any talent for math;" "Mathematics does not particularly suit me;" "When it comes to things in mathematics that I do not understand, I know from the start: I will never understand this." See Table 1 for correlations of all parcels.

Data Analysis
The hypotheses H1 and H2 were analyzed using a CFA with correlated first-order factors for each construct (e.g., Eid et al. 2017). Hypothesis H3 was tested assuming a one-factor-congeneric measurement model for all observed variables. Hypothesis H4 was analyzed using a bifactor (S-1) model (Eid et al. 2017;Eid et al. 2018) with the gf factor as a general reference factor and three correlated specific factors for MCK, MPCK, and GPK.
Hypothesis H5 was analyzed using a multivariate latent regression analysis with the factors of MCK, MPCK, and GPK as dependent variables and the factors of gf and self-concept as independent variables. In order to analyze hypothesis H6, this multivariate regression model was extended by including interactions between gf and self-concept in a latent moderated structural equations approach (LMS; Klein and Moosbrugger 2000). Moreover, in this case, a maximum likelihood estimator with robust standard errors was used by applying a numerical integration algorithm. As suggested in the literature, we included the main effects of all variables to avoid the confounding of the main and interac-tion effects or the possibility that changes in the scale could result in arbitrary estimates .
All analyses were carried out with the Mplus 8.4 software package (Muthén andMuthén 1998-2019) using the robust maximum likelihood estimator (MLR). Missing data existed on two constructs, gf and academic self-concept, because teachers had dropped out after the knowledge tests. The missingness was handled by using the full-information maximum likelihood (FIML) procedure implemented in Mplus. The FIML procedure provides unbiased parameter estimates and standard errors under the missing-at-random assumption (Enders and Bandalos 2001). Testing the missing-completely-at-random assumption with the background information available revealed that most variables did not display significant differences for teachers with full as compared to those with missing data. This applied to teachers' age, the level of their school degree, and their cultural capital, as well as to the education background of their fathers and mothers. However, significant differences existed  Table 2) if gf is not controlled for. In particular, MPCK and GPK were empirically almost indistinguishable (r = .95). The two domains related to mathematics were less strongly, but still substantially correlated with each other (MCK/MPCK: r = .54; MCK/ GPK: r = .48). There were also high latent correlations between gf and the three knowledge dimensions. The correlation was particularly high for MCK (r = .84), and lower, but equally high for MPCK and GPK (r = .52). Hence, our analyses support hypothesis H1 and H2. 1 Table 2. CFA model with correlated first-order factor: latent correlations and standard errors (in parentheses).

H3: One-Dimensional Structure of EC Teachers' Cognitive Abilities
In accordance with our hypothesis (H3), the data revealed an unacceptable model fit for the pure g-factor model where just one factor was hypothesized to underly all 14 observed variables, i.e., the indicators of gf, MCK, MPCK, GPK (χ 2 = 898.5, df = 77, p < .001; RMSEA = .23, 90%CI[.21; .24]; CFI = .67). The standardized factor loadings varied between .37 for two of the gf parcels and >.90 for the three MPCK parcels. For the gf parcels, the variance explained by the underlying factor was very low (<0.2) and barely significant. The modification indices suggest many correlated error variables, in particular of indicators belonging to the same construct, showing that the structure is multidimensional. Regarding EC teachers' cognitive abilities as one-dimensional is thus not supported by the data, which is in line with the well-fitting model with correlated first-order factors presented in Table 1, and our hypothesis H3.

H4: Hierarchical Structure of EC Teachers' Cognitive Abilities: Bifactor (S-1) Model
The bifactor (S-1) model fit the data very well (χ 2 = 70.3, df = 63, p = .25; RMSEA = .02, 90%CI[.00; .05] CFI = 1.00). The percentage of variance explained by the gf factor was very high for the indicators of MCK (between 54 and 64 percent), and lower, but still substantive, for the indicators of MPCK (between 20 and 24 percent) and PCK (between 13 and 21 percent). The correlations between the specific factors are partial correlations (after controlling for gf ). These correlations (see Table 3) are smaller than the zero-order correlations presented in Table 2, and not statistically significant for MCK and GPK. Therefore, as a whole, H4 has to be rejected. However, it is noteworthy that the correlation between the two pedagogical knowledge dimensions is still very high. These results show that the general gf factor can explain, to a relatively large degree, the correlations between the content knowledge (MCK) and the pedagogical knowledge (MPCK, GPK), but not the correlations between the pedagogical knowledge dimensions. Table 3. Bifactor model: Latent partial correlations between the specific factors (standard errors in parentheses).

Role of Academic Self-Concept
The second aim of the empirical portion of this paper was to examine the role of self-related cognitions for the ability to solve domain-specific items. In the first step, we extended the CFA model with correlated first-order variables presented in Table 2 by including a factor for self-concept. This model fit the data approximately well (χ 2 = 145.3, df = 109, p < .05; RMSEA = .04, 90%CI[.02; .06]; CFI = .99). Self-concept is significantly correlated with all other constructs (see Table 4). Whereas there were only comparatively small correlations with MPCK and GPK, the correlations were stronger with respect to MCK and gf. Table 4. CFA model with correlated first-order factor: latent correlations and standard errors (in parentheses). The multivariate latent regression model fit the data well (χ 2 = 145.3, df = 109, p < .05; RMSEA = .04, 90%CI[.02; .06]; CFI = .99). In line with our hypothesis (H5), EC teachers' academic self-concept explained individual differences in teachers' MCK beyond gf (see Table 5). However, in contrast to our hypothesis, the regression coefficient was not significantly different from 0 for MPCK and GPK. Both independent variables together explained a substantive amount of variance in all dependent variables, but the variance explained in the dependent variables differed by construct: while the variance in MCK could, to a large extent, be explained by gf and self-concept (R 2 = .76), the proportion was lower in the case of MPCK and GPK (R 2 = .27 in both cases). Table 5. Latent multivariate regression model: standardized regression coefficients for the effects of the independent variables gf and academic self-concept on MCK, MPCK, and GPK (standard errors in parentheses).
A latent moderated modeling approach including an interaction effect between gf and academic self-concept (see Figure 4) revealed no moderator effect with respect to MCK (see Table 6). This means that H6 must be rejected with respect to this dimension. In contrast, the interaction effect was significant for both MPCK and GPK. This result suggests that in the case of high gf, the effect of a positive self-concept is stronger than in the case of low gf. Table 6. Latent moderated structural equation model: standardized regression coefficients for the effects of the independent variables gf and academic self-concept, and their interaction on MCK, MPCK, and GPK (standard errors in parentheses).
In line with these results, the variance explained in MCK was not higher in the latent moderated approach than in the model without an interaction term, while it was substantially higher in the case of MPCK (R 2 = .36) and GPK (R 2 = .34).

Discussion
Confronting theories stemming from cognitive psychology and education may benefit both traditions. The CHC theory uses an established terminology and provides a broad picture of how to think about the structure of human cognitions (Schneider and McGrew 2012). A comparison with the Blömeke, Gustafsson, and Shavelson model of competence (2015) revealed that substantial conceptual overlap exists between gq and MCK on the one hand and gkn and MPCK and GPK on the other hand. However, from an educational point of view, the CHC theory neglects, to some extent, the final objective: explaining human behavior and identifying ways to influence it. The Blömeke, Gustafsson, and Shavelson model of competence, in contrast, emphasizes behavior as the criterion and relates the dispositions needed to show this behavior to the criterion. However, this model does not include broader general cognitive abilities, such as gf, beyond the domain-specific ones.
The latter is a challenge not many educators may have yet realized. Relations between achievement domains are at risk for appearing as excessively strong if a common underlying factor is neglected. The effects of opportunities to learn can hardly be evaluated properly if general cognitive abilities are not partialled out. In fact, our study revealed interesting insights with respect to the role of gf for the population of EC teachers. A bifactor (S-1) model fitted the data well, which means that both general cognitive ability and domainspecific knowledge influenced EC teachers' ability to solve the domain-specific items. Competence models are therefore advised to include gf -not as a cognitive ability that can be influenced by teacher education to a larger extent, but as a necessary precondition to be considered when evaluating the effects of education.
The benefits but also the challenges of teacher education or professional development are otherwise at risk of remaining hidden, as the following example may reveal: to be able to use their knowledge in practice requires that the different dimensions of EC teachers' professional knowledge be deeply integrated (Tirosh et al. 2011). In particular, the integration of MCK with the two pedagogical dimensions has been regarded as critical. At first glance, EC teacher education seems to succeed in this respect, given latent correlations around .50 both for MCK and MCPK and for MCK and GPK. However, it now turns out-as we have demonstrated in this article-that these relations were, to a large extent, caused by an underlying factor (gf ), reducing estimates to around .20 once gf was controlled for. Thus, the initial correlations did not reflect integration. It then seems to be important to develop educational practices that have the potential to overcome the disintegration of MCK, MPCK, and GPK, in particular, since we know from previous research that a similar phenomenon of seemingly strong latent correlations also applies to pre-service EC teachers (Jenßen et al. 2019). The latent correlation between MCK and MPCK of this target population was at .67, if gf was not controlled for, but only at .30 otherwise.
The investment theory could also benefit from being confronted with models of competence. Defining a criterion and trying to identify underlying traits that support its development may expand the range of characteristics examined. Our data revealed, for example, that academic self-concept affects professional knowledge and can thus be regarded as an investment trait. An interesting, and in case of GPK, even surprising finding was that the interaction effect of gf and academic self-concept was significant for both MPCK and GPK, given that self-concept was domain-specifically operationalized. This finding indicates that a positive self-concept cannot compensate for a lack of gf, but that it contributes to gaining MPCK and GPK in the case of high gf. The underlying mechanism here is probably related to the general finding that a positive self-concept makes it easier to overcome challenges. Moreover, the result may indicate that domain-specific self-concept is related to a broader, more general type of academic self-concept, as pointed out by Marsh (1990).

Limitations
Our study was the first one that examined the cognitive structure of practicing EC teachers. It was also the first one that included academic self-concept as a potential investment trait and examined its interaction with gf. However, given that research on the relation of intelligence and competence is scarce and we only used one sample of EC teachers in Germany, it would be important to replicate our findings with other teacher samples, in particular from other countries and other domains (e.g., oral language or science) to avoid making too far-reaching conclusions. Mathematics is only one domain among others where teachers are expected to support children's development. Relationships between competence and intelligence, as well as relationships between the different knowledge dimensions, could be different in the case of other domains or other groups of teachers, for example, those teaching in upper-secondary school. Testing whether our findings reflect domain-or age-specific particularities would provide a deeper understanding of the nature of teachers' cognitive abilities. The same applies to testing the generalizability of our models across countries. Such studies should then also examine whether and which abilities are able to predict the intended long-term outcomes, such as child development.
Another limitation of our study was the operationalization of gf as figural reasoning. Although this has been done previously (Zhang and Ziegler 2015), a risk exists that such a narrow representation underestimates relations to constructs based on verbal stimuli ) such as GPK. The bifactor (S-1) model revealed in the current operationalization a strong effect of gf on GPK. We cannot rule out the possibility that the effect would have been even stronger in case of a broader operationalization that included verbal intelligence. Since it was not possible for us, due to limited administration time, to include such a facet of intelligence, we cannot test this with our current dataset, and we encourage further studies on this relationship.

Conclusions
Beyond EC teachers, the cognitive structure of other teacher populations such as primary, lower-or upper-secondary teachers is largely an open question. Can we reasonably assume that their cognitive structure is similar to that of EC teachers? Differences in opportunities to learn are huge, starting from different entry requirements to different length and depth of domain-specific teacher education, with EC teachers having by far the lowest number of opportunities to learn, both during teacher education (Gasteiger et al. 2020) and professional development (Nasiopoulou et al. 2021). In that sense, one could even speculate that gf might be more relevant in case of EC teachers than of others-but this is an empirical question.

Institutional Review Board Statement:
The study was conducted according to the guidelines of the Declaration of Helsinki. Ethical review was carried out as part of the application for funding process. Ethical approval was waived for this observational study since no interventions were carried out.
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study that the data can be used for research purposes.

Data Availability Statement:
The data presented in this study are available on request from the authors due to privacy restrictions.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A
Item examples from the knowledge tests. J. Intell. 2022, 10, x FOR PEER REVIEW 14 of 18 include such a facet of intelligence, we cannot test this with our current dataset, and we encourage further studies on this relationship.

Conclusions
Beyond EC teachers, the cognitive structure of other teacher populations such as primary, lower-or upper-secondary teachers is largely an open question. Can we reasonably assume that their cognitive structure is similar to that of EC teachers? Differences in opportunities to learn are huge, starting from different entry requirements to different length and depth of domain-specific teacher education, with EC teachers having by far the lowest number of opportunities to learn, both during teacher education (Gasteiger et al. 2020) and professional development (Nasiopoulou et al. 2021). In that sense, one could even speculate that gf might be more relevant in case of EC teachers than of others-but this is an empirical question.

Institutional Review Board Statement:
The study was conducted according to the guidelines of the Declaration of Helsinki. Ethical review was carried out as part of the application for funding process. Ethical approval was waived for this observational study since no interventions were carried out.
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study that the data can be used for research purposes.

Data Availability Statement:
The data presented in this study are available on request from the authors due to privacy restrictions.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A
Item examples from the knowledge tests.     Since MPCK and GPK represent conceptually different constructs, it is not appropriate to collapse these, despite very strong latent correlations. Moreover, a follow-up analysis revealed that collapsing is not supported from an empirical point of view because the fit of a uni-dimensional model was worse than a model with two distinct constructs.
You are playing a dice game with three children. Please explain, in short, why their mathematical learning in the following field is fostered: Numbers and operations (e.g., calculating):  Since MPCK and GPK represent conceptually different constructs, it is not appropriate to collapse these, despite very strong latent correlations. Moreover, a follow-up analysis revealed that collapsing is not supported from an empirical point of view because the fit of a uni-dimensional model was worse than a model with two distinct constructs.