**2.1. Intertwining person-context dynamics**

Vygotsky (1934/1986) already pointed out that children develop understanding in close cooperation with their teachers and the material. His concept of the zone of proximal development is a dynamically changing concept, in which teacher and child co-construct the child's development. This means that the child's skills and understanding are constructed by a series of actions guided by the educator, instructions and tool-use, which are then internalized and personalized (cf., Van Geert, 1998; Van Geert & Steenbeek, 2005).

From a dynamic systems perspective, understanding is seen as a process of intertwining person-context dynamics (Thelen & Smith, 1994), meaning that the social (e.g., the science teacher) and material environment (e.g., materials used in science class) play an active part in the process and cannot be viewed separately, or merely as an outside-based influence. In fact, these elements are intertwined across time, in a continuous person-environment loop: at any moment in time, one component (e.g., the child) affects the other (e.g., the teacher) and the other affects the first, thus creating the conditions under which both components will operate during the next moment in time (Steenbeek, 2006). For example, interactions between a child, a researcher, and the syringes-task will organize toward certain distributed patterns of understanding at that moment (in real time), which eventually evolve toward stable attractors on a longer time scale (Thelen, 1989; Halley & Winkler, 2008). Hence, understanding is an active process of what the child constructs in interaction with (not just within) a specific environment, in which each individual contribution is virtually meaningless if not viewed in light of the interaction (Van Geert & Fischer, 2009). Merged together, person and context become what Fogel and Garvey (2007) call a "cooperative unit", in which both components not only contribute to the process of development, but are highly intertwined and form an unique process together.

Representationalists, such as Fodor (1981) hold the idea that understanding takes the form of internal structures (representations) within the child's mind. A child's scientific understanding thus consists of a collection of these internal structures which represent scientific facts and concepts, which are activated and used to coordinate our behavior toward the current environment (Haselager, de Groot, & van Rappard, 2003). In this case, a

<sup>1</sup> Actually, the dynamic systems approach has many more properties or "tools" (Howe & Lewis, 2005) to study development. However, we highlighted these four specific properties to illustrate how this approach sheds new light on the study of understanding scientific concepts.

concept or representing model of the air pressure task would be represented in the child's mind, and this representation would guide the child's behavior as he or she is working on the actual air pressure task.

Terms such as "concept" or "representation" are actually more or less undefined, and derive their meaning from a particular theoretical framework. From a representationalist (or information-processing) view, these words refer to internal entities responsible for our thinking or actions toward the environment. From a dynamic view, however, these words refer to processes, perception and action structures, that emerge within a specific environment (Van Geert & Fischer, 2009). Perceiving, acting and thinking are conscious processes that take a particular shape in the stream of consciousness of the participants, such as a child and the researcher (van Gelder, 1995; 1998). This shape is governed by the participants' actions on the objects, such as the syringes, or on physical representations of the syringes, such as prints or drawings, within their current context, and should not be identified with a retrieval of internally stored representations (Van Geert, 2011). We can construct much of this stream of consciousness by carefully watching the ongoing interaction between child and environment in terms of the intertwining of various forms of verbal and non-verbal behavior, such as eye and head movements, gestures, pointing, verbal descriptions, manipulations of the materials, etcetera. The child's current understanding of the concept at issue (for instance, the flow of air through two syringes connected by a tube), is the child's continuously changing state of mind, or stream of consciousness, as he picks up and reacts to whatever goes on in the current dynamic interaction. Thus, despite the fact that the process of constructing an understanding is a distributed process, involving the intertwining of person and context, understanding can still be specified as an individual and "internal" process corresponding with the individual child's ongoing state of mind, but only as a changing state that unfolds in this active process (Van Geert, 2011). Hence, representations are structures that emerge during a specific interaction in a specific environment, and are not internal symbolic structures which guide behavior.

#### **2.2. Iterativeness**

24 Current Topics in Children's Learning and Cognition

complexity level of understanding (Fischer & Rose, 1999).

highly intertwined and form an unique process together.

approach sheds new light on the study of understanding scientific concepts.

**2.1. Intertwining person-context dynamics** 

genetical variability)<sup>1</sup>

Applying a dynamic approach to the study of understanding scientific concepts means that several properties of this approach have to be taken into account. Below, four properties (intertwining person-context dynamics, iterativeness, interconnected time scales, and micro-

e.g., scientific concepts) will be discussed. In section 5, the properties will be illustrated in light of an empirical example, in combination with Skill Theory's framework to measure the

Vygotsky (1934/1986) already pointed out that children develop understanding in close cooperation with their teachers and the material. His concept of the zone of proximal development is a dynamically changing concept, in which teacher and child co-construct the child's development. This means that the child's skills and understanding are constructed by a series of actions guided by the educator, instructions and tool-use, which are then

From a dynamic systems perspective, understanding is seen as a process of intertwining person-context dynamics (Thelen & Smith, 1994), meaning that the social (e.g., the science teacher) and material environment (e.g., materials used in science class) play an active part in the process and cannot be viewed separately, or merely as an outside-based influence. In fact, these elements are intertwined across time, in a continuous person-environment loop: at any moment in time, one component (e.g., the child) affects the other (e.g., the teacher) and the other affects the first, thus creating the conditions under which both components will operate during the next moment in time (Steenbeek, 2006). For example, interactions between a child, a researcher, and the syringes-task will organize toward certain distributed patterns of understanding at that moment (in real time), which eventually evolve toward stable attractors on a longer time scale (Thelen, 1989; Halley & Winkler, 2008). Hence, understanding is an active process of what the child constructs in interaction with (not just within) a specific environment, in which each individual contribution is virtually meaningless if not viewed in light of the interaction (Van Geert & Fischer, 2009). Merged together, person and context become what Fogel and Garvey (2007) call a "cooperative unit", in which both components not only contribute to the process of development, but are

Representationalists, such as Fodor (1981) hold the idea that understanding takes the form of internal structures (representations) within the child's mind. A child's scientific understanding thus consists of a collection of these internal structures which represent scientific facts and concepts, which are activated and used to coordinate our behavior toward the current environment (Haselager, de Groot, & van Rappard, 2003). In this case, a

1 Actually, the dynamic systems approach has many more properties or "tools" (Howe & Lewis, 2005) to study development. However, we highlighted these four specific properties to illustrate how this

internalized and personalized (cf., Van Geert, 1998; Van Geert & Steenbeek, 2005).

and examples of their application to the study of understanding (of

Within the process that results from an intertwining between person and context, understanding emerges through iteration, that is, every step in understanding is based on the previous one and embedded in the current context. More precisely, iterativeness (sometimes referred to as recursiveness) involves a series of computational operations, in which the input of the next operation is the output of the previous one. For instance, if a child determines that an empty syringe contains air, he can build on this knowledge by trying out what happens if he joins two of these syringes together by using a tube. Understanding changes through repeated interactions, instead of being the retrieval of a complete representation that is already there in memory. During a teaching interaction, each previous action of the child has an influence on the subsequent (re-)action. In other words, the existing understanding is the basis for the emergence of the next understanding as it develops in the interaction.

In its simplest possible form, a dynamic systems model specifies the change in a variable (L) over time (t) as a function of the current level of the variable: L t+1 = *f* (Lt). The function *f*  refers here to the change in 'understanding', but can specify any sort of influence or mechanism of change (Steenbeek, 2006). Understanding does not consist of particular moments within the interaction (e.g. when the child answers), but is in fact the whole iterative process itself, and every interaction unit is a component of this holistic understanding process during a particular problem solving event. Even though understanding consists of the whole iterative process, the child's answers are a reflection of the child's ongoing state of mind within that process and reveal his or her understanding at that very moment in time.

Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 27

and long-term scales interact, in that repeated (iterative) processes on the short term time scale influence processes on the long-term time scale (Lewis, 2000). In addition, the emergence of large-scale patterns also influences what happens on the short-term time scale, by shaping the structure and function of the interaction on the short term (Lewis & Granic, 2000; Smith & Thelen, 2003; Van Geert & Steenbeek, 2005; Steenbeek, 2006). The underlying idea is that all levels of the developing system interact with each other in a self-organizing way, and consist of nested processes that unfold over many time scales, from milliseconds to

As a result of the iterative organization of the components and the intertwining between child and context that mark the process of children's understanding, we can observe microgenetical variability. This means that the complexity of children's understanding fluctuates within very short periods of time, e.g. during one task. While studying the processes of developmental change, it is crucial to take many observations (adopting a microgenetic research method) to detect the subtle changes that constitute understanding and its development (Siegler & Crowly, 1991; Kuhn, 1995). Researchers note that, driven by bidirectional interactions with the environment, the complexity of children's understanding can increase during a task, but also temporally decrease, for example when the task difficulty increases, when the teacher's support decreases, or when children encounter something unexpected while working on a task. Understanding can change gradually or abruptly in a stage-like pattern in a short timeframe, even during a single task (Yan &

Researchers have suggested that this variation is an important factor in development, since an increase in variability may be related to the ability to reach higher levels of skill (Howe & Lewis, 2005; Thelen, 1989), or, more generally, to a transition to another pattern of behavior (i.e., attractor) (e.g., Thelen & Smith, 1994; Van Geert, 1994). The variability on the shortterm (e.g. during the syringes-task or during a science lesson) can therefore yield important information about how the developmental pathways of understanding will be shaped on

In order to capture the complexity of understanding and variations in complexity over a short and longer time periods, we can use Skill Theory's framework of cognitive development (Fischer, 1980; Fischer & Bidell, 2006). This framework can be used on both the long- and short-term time scale and is compatible with a dynamic systems approach. Even more so, Skill Theory could be considered as a specific dynamic system's theory applied to human skill development, since it assumes skills are built in an iterative and hierarchical way, i.e. each skill level builds on the previously obtained skill level. Moreover, skills are highly context-dependent and fluctuate over time, that is, they depend on the constraints

and affordances of the context in which they are mastered (Fischer & Bidell, 2006).

years (Thelen & Smith, 1994; Lewis, 2000).

**2.4. Micro-genetical variability** 

Fischer, 2007; Siegler & Crowly, 1991).

the long term.

As Howe and Lewis (2005) point out, the iterative nature of the process of understanding can also explain some of the differences between children. When children's understanding depends on interactions, and each interaction is based on the previous one, small differences between children's initial states of understanding can grow bigger over several interactions. This is particularly so if the process takes the form of a positive feedback loop amplifying idiosyncratic properties of the answers, i.e. properties that are typical of a particular child. For example, if the child focuses on only one syringe and the researcher's follow-up questions center on that syringe as well, the difference between this child and another child who focuses on both syringes grows bigger. However, if the process takes the form of a negative feedback loop reducing the idiosyncrasies, small differences in initial states will most likely remain small over the course of the problem-solving process. This would be the case if the researcher switches the focus of her follow-up questions to the other syringe, thereby scaffolding the child towards a more complete picture of the task. The difference between this child and the child who initially focused on two syringes then becomes smaller.
