**2.4. Micro-genetical variability**

26 Current Topics in Children's Learning and Cognition

that very moment in time.

smaller.

**2.3. Time scales** 

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

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

The property of interconnected time scales entails that the dynamics of long-term development of understanding are intrinsically related to the dynamics of short-term processes of understanding (Thelen & Smith, 1994; Lewis, 2000). That is, in order to get a grip on long-term changes in understanding of children, it is worthwhile to focus on the short-term (micro-genetic) process, and examine properties of that process, such as

Iterativeness occurs on the short term as well as on the long term, meaning that on the short term (e.g. during one interaction between child and teacher in science class), each step in understanding is based on the previous step in understanding, while on the long term each interaction builds on the preceding interaction (e.g. the interaction during last week's science class). In this way, the same mechanisms are sculpting the development of understanding over a shorter and longer period. Thelen and Corbetta (2002) indicate that the general principles underlying behavioral change work at multiple time scales. The short-

variability (Granott, Fischer, & Parziale, 2002; Steenbeek, 2006).

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 & Fischer, 2007; Siegler & Crowly, 1991).

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 the long term.

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).
