**3. Skill theory and understanding**

Skill Theory focuses on the complexity and variability of children's skills, which consist of actions and thinking abilities, and the way these are constructed (Fischer, 1980; Fischer & Bidell, 2006). Since skills are thinking structures mastered in a specific context, such as a science class, they hold both person-related as well as context-related characteristics (Parziale & Fischer, 1998). An example of a skill is a child's ability to understand how air pressure works while manipulating the syringes-task. This understanding is reformulated when the student works on a similar task in another environment (e.g. with different materials or without the help of the researcher). Skills are thus highly influenced by the possibilities and constraints of the situation in which the skill is used.

Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 29

, each one more complex than the

before. The third tier consists of abstractions, which are general nonconcrete rules that also apply in other situations (Schwartz & Fischer, 2005). This would be an explanation about the

previous one. The first one can be characterized as a single set, meaning a single action (or a single representation, or a single abstraction). The second level is a relation between two of these sets, which is referred to as a mapping. The third level is a system of sets, which is a relation between two mappings, in which each mapping consists of a relation between single sets. After this level, a new tier starts, which is divided in single sets, mappings and systems as well (Fischer & Bidell, 2006). For the emergence of each level, evidence of discontinuities and differences between levels has been demonstrated using analysis

Fischer and colleagues (Fischer, 1980; Fischer & Bidell, 2006; Yan & Fischer, 2002; Schwartz & Fischer, 2005; Granott & Parziale, 2002) showed that Skill Theory can not only describe and explain the development of skills on the long term, but also describe the micro-genesis of problem solving. When facing a new task or problem within a domain, even high-skilled adults go through the same cycles of development. That is, at the beginning they show skill levels that are mostly sensorimotor, which build up to more elaborate levels during the course of the task. During a task (and also during the long-term development of skills), people do not go through the skill cycles in a linear fashion. Instead, they repeatedly build up skill levels and show collapse before they obtain their highest possible level, something Yan and Fischer (2002) call "scalloping". During a task, people vary constantly within a bandwidth between their highest and lowest possible complexity levels, which is also known as the developmental range. The highest levels within the bandwidth are only reachable when the environment provides sufficient support (Fischer & Bidell, 2006; see also

Skill theory also accounts for inter-individual differences in understanding and is therefore especially suitable for describing individual developmental pathways (Fischer, Rose & Rose, 2007). Yan and Fischer (2002) showed that adults' performance on a computer task can move through a variety of pathways, each one showing nonlinear fluctuations. Of all participants, novices showed the most frequent and rapid fluctuations in performance. Experts however fluctuated less frequent in their performance, meaning that variations

In sum, a model of understanding needs some kind of ruler to determine the complexity of understanding levels children show. Skill Theory (Fischer, 1980; Fischer & Bidell, 2006) provides a content-independent ruler for understanding, which can be applied to different

2 After the 3 levels of the abstraction tier, a higher complexity level emerges, also known as 'single principles', which is the 10th level of the scale. Additionally, people function on the few highest levels usually in early adulthood, but only for their domains of expertise. For most other domains, people

relationship between pressure and volume inside a syringe.

methods based on Rasch scaling (Schwartz & Fischer, 2005).

Yan & Fischer, 2002).

followed on each other in a slower fashion.

function on a lower complexity level.

Within each tier, three levels can be distinguished<sup>2</sup>

Skill Theory explains both long- and short-term development of skills by measuring these on the same hierarchical complexity scale. This complexity scale consists of 10 levels, grouped into 3 tiers, which are sensorimotor, representational or abstract by nature. The scale can be applied to different cognitive (Fischer & Granott, 1995; Schwartz & Fischer, 2005), social (Fischer & Bidell, 2006) and language domains (Fischer & Corrigan, 1981), as it focuses on hierarchical complexity rather than content. This makes Skill Theory especially suitable to describe differences between children, as well as differences between skills in different domains for the same child (Parziale & Fischer, 1998).

A child's understanding within a domain, as an emergent process in real-time, can be viewed along two dimensions: the first being the dimension of content (the subject), the second of complexity (the complicatedness). In order to evaluate children's understanding (of, for example, air pressure), we need a fair ruler to determine how elaborate their understanding is, and to evaluate whether they need extra help in some areas. One of the most powerful characteristics of Skill Theory (Fischer, 1980) is that it extracts complexity from content, resulting in a content-independent ruler of understanding. Because of the content-independent nature of the way Skill Theory approaches understanding (or other skills), it enables researchers to compare understanding across multiple time points, contexts, persons, and for different age ranges.

According to Fischer (1980) and Fischer and Bidell (2006), development in a particular domain goes through 10 levels of skills hierarchically grouped into three tiers that develop between 3 months and adulthood. The first tier consists of sensorimotor skills: simple connections of perceptions to actions or utterances. An example is a statement that two syringes are attached to a tube. Sensorimotor skills form the basis of the skills in the two subsequent tiers, i.e. they are the building blocks of the higher levels. The second tier constitutes of representational skills, these are understandings that go beyond current simple perception-action couplings, but are still based on them. Hence, the term representation refers to the coordination of several sensorimotor skills at the same time, not to an internal symbolic structure (Fischer, 1980). Within the context of the air pressure task for example, the child can predict what will happen if the piston is pushed in without literally touching or manipulating the syringe. Nonetheless, what he or she predicts depends on the material context, and on the sensorimotor skills that he or she mastered before. The third tier consists of abstractions, which are general nonconcrete rules that also apply in other situations (Schwartz & Fischer, 2005). This would be an explanation about the relationship between pressure and volume inside a syringe.

28 Current Topics in Children's Learning and Cognition

**3. Skill theory and understanding** 

Skill Theory focuses on the complexity and variability of children's skills, which consist of actions and thinking abilities, and the way these are constructed (Fischer, 1980; Fischer & Bidell, 2006). Since skills are thinking structures mastered in a specific context, such as a science class, they hold both person-related as well as context-related characteristics (Parziale & Fischer, 1998). An example of a skill is a child's ability to understand how air pressure works while manipulating the syringes-task. This understanding is reformulated when the student works on a similar task in another environment (e.g. with different materials or without the help of the researcher). Skills are thus highly influenced by the

Skill Theory explains both long- and short-term development of skills by measuring these on the same hierarchical complexity scale. This complexity scale consists of 10 levels, grouped into 3 tiers, which are sensorimotor, representational or abstract by nature. The scale can be applied to different cognitive (Fischer & Granott, 1995; Schwartz & Fischer, 2005), social (Fischer & Bidell, 2006) and language domains (Fischer & Corrigan, 1981), as it focuses on hierarchical complexity rather than content. This makes Skill Theory especially suitable to describe differences between children, as well as differences between skills in

A child's understanding within a domain, as an emergent process in real-time, can be viewed along two dimensions: the first being the dimension of content (the subject), the second of complexity (the complicatedness). In order to evaluate children's understanding (of, for example, air pressure), we need a fair ruler to determine how elaborate their understanding is, and to evaluate whether they need extra help in some areas. One of the most powerful characteristics of Skill Theory (Fischer, 1980) is that it extracts complexity from content, resulting in a content-independent ruler of understanding. Because of the content-independent nature of the way Skill Theory approaches understanding (or other skills), it enables researchers to compare understanding across multiple time points,

According to Fischer (1980) and Fischer and Bidell (2006), development in a particular domain goes through 10 levels of skills hierarchically grouped into three tiers that develop between 3 months and adulthood. The first tier consists of sensorimotor skills: simple connections of perceptions to actions or utterances. An example is a statement that two syringes are attached to a tube. Sensorimotor skills form the basis of the skills in the two subsequent tiers, i.e. they are the building blocks of the higher levels. The second tier constitutes of representational skills, these are understandings that go beyond current simple perception-action couplings, but are still based on them. Hence, the term representation refers to the coordination of several sensorimotor skills at the same time, not to an internal symbolic structure (Fischer, 1980). Within the context of the air pressure task for example, the child can predict what will happen if the piston is pushed in without literally touching or manipulating the syringe. Nonetheless, what he or she predicts depends on the material context, and on the sensorimotor skills that he or she mastered

possibilities and constraints of the situation in which the skill is used.

different domains for the same child (Parziale & Fischer, 1998).

contexts, persons, and for different age ranges.

Within each tier, three levels can be distinguished<sup>2</sup> , each one more complex than the previous one. The first one can be characterized as a single set, meaning a single action (or a single representation, or a single abstraction). The second level is a relation between two of these sets, which is referred to as a mapping. The third level is a system of sets, which is a relation between two mappings, in which each mapping consists of a relation between single sets. After this level, a new tier starts, which is divided in single sets, mappings and systems as well (Fischer & Bidell, 2006). For the emergence of each level, evidence of discontinuities and differences between levels has been demonstrated using analysis methods based on Rasch scaling (Schwartz & Fischer, 2005).

Fischer and colleagues (Fischer, 1980; Fischer & Bidell, 2006; Yan & Fischer, 2002; Schwartz & Fischer, 2005; Granott & Parziale, 2002) showed that Skill Theory can not only describe and explain the development of skills on the long term, but also describe the micro-genesis of problem solving. When facing a new task or problem within a domain, even high-skilled adults go through the same cycles of development. That is, at the beginning they show skill levels that are mostly sensorimotor, which build up to more elaborate levels during the course of the task. During a task (and also during the long-term development of skills), people do not go through the skill cycles in a linear fashion. Instead, they repeatedly build up skill levels and show collapse before they obtain their highest possible level, something Yan and Fischer (2002) call "scalloping". During a task, people vary constantly within a bandwidth between their highest and lowest possible complexity levels, which is also known as the developmental range. The highest levels within the bandwidth are only reachable when the environment provides sufficient support (Fischer & Bidell, 2006; see also Yan & Fischer, 2002).

Skill theory also accounts for inter-individual differences in understanding and is therefore especially suitable for describing individual developmental pathways (Fischer, Rose & Rose, 2007). Yan and Fischer (2002) showed that adults' performance on a computer task can move through a variety of pathways, each one showing nonlinear fluctuations. Of all participants, novices showed the most frequent and rapid fluctuations in performance. Experts however fluctuated less frequent in their performance, meaning that variations followed on each other in a slower fashion.

In sum, a model of understanding needs some kind of ruler to determine the complexity of understanding levels children show. Skill Theory (Fischer, 1980; Fischer & Bidell, 2006) provides a content-independent ruler for understanding, which can be applied to different

<sup>2</sup> After the 3 levels of the abstraction tier, a higher complexity level emerges, also known as 'single principles', which is the 10th level of the scale. Additionally, people function on the few highest levels usually in early adulthood, but only for their domains of expertise. For most other domains, people function on a lower complexity level.

time scales of development, and takes both the role of context, as well as inter- and intraindividual variability into account.

Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 31

**Figure 1.** A conceptual (action) model of understanding based on principles derived from dynamic

In the next sections, we illustrate the model and the four properties by using an example (see table 1) derived from our empirical study focusing on the long-term development of understanding air pressure (and other scientific concepts, such as gravity) in three to seven year old children. Table 1 is an excerpt of a transcribed session in which a boy (4 years, 6 months) and a researcher explore the syringes task mentioned in the introduction. The transcript starts right after the point in which the researcher and the boy explored the exterior of the syringes. That is, they compared them in size and examined the numbers

**5. An empirical example and illustration of the model** 

systems theory and Skill Theory

written on the outside.
