2.1. 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 (
Gustafsson 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.
2.2. 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 situation-specific 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
Ackerman (
1996) 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).
Ackerman (
1996) 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.
2.3. 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 upper-secondary 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 (
Dunekacke et al. 2015b;
Lee 2010;
Torbeyns et al. 2020). GPK includes general foundations from educational theory, psychology, and instructional research related to early childhood and learning processes of 3- to 6-year-olds (
Blömeke et al. 2017;
Malva et al. 2019).
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 (
Marsh et al. 2016). 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.
2.4. 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 first-order 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. (
2019) revealed a strong impact of
gf (in terms of verbal, numerical, and figural intelligence assessed with the screening version of I-S-T 2000R;
Liepmann et al. 2012) on MCK and MPCK. 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.