**5.3. Understanding is micro-genetically variable**

In our example (see table 1), micro-genetical variability is seen in the child's understanding of how the material works. First, in fragment 10 the boy names a single cause for what happens: "Because we attached the tube". This is an answer on a sensorimotor system level; he gives a single, observable causal explanation for the phenomenon, not taking the volume of the syringes or the air into account (see also the third column of table 1). Over the course of the interaction, he briefly regresses to "I don't know" (fragments 18 and 20; no skill level), and restores his previously gained skill level again in fragment 24: "Because this [the tube] is attached". From there, he further constructs his understanding, and eventually reaches a higher level in fragment 36: "The sigh is going through the tube", for which he needs a representation of the role of air in the system.

In Figure 2 a time-serial illustration of the fluctuations in the boy's answer levels during the air pressure task is depicted. The graph shows how the understanding of the boy fluctuates over time. While Skill Theory's level 4 (single representation) is mostly observed during the interaction, the boy also regularly shows understandings at level 3 (sensorimotor system). Even though his understanding seems to increase in complexity over time (on average the boy reaches level 4 more often in the second half of the interaction), his understanding often regresses to level 3 and to incorrect/irrelevant understandings. Hence, understanding is not a fixed entity, but varies over time, even within a single task.

36 Current Topics in Children's Learning and Cognition

the cyclical or reciprocal character of causality occurs.

previous sessions influence subsequent sessions.

representation of the role of air in the system.

**5.3. Understanding is micro-genetically variable** 

influences between the child and environment are bidirectional, meaning that not only the action of the researcher influences the next (re)action of the student, but also that the previous interaction influences the next interaction. Iterativeness is thus the form in which

In our example (table 1), the iterative nature of the process is not only illustrated by how the researcher and child react to what has been said previously throughout the whole transcript, but also by how the child's understanding develops during the interaction. With regard to the prediction he makes in the first half of the interaction, the child goes from "I don't know" (fragments 2 and 4; no skill level) to "This one goes up like this" (fragment 6; single representation). This change in understanding is constructed in reaction to what the researcher said right before in fragment 5. With regard to the explanation of the boy why this happens, his understanding goes from "Because this [the tube] is attached" (fragment 24; sensorimotor system), to "Something goes like this [through the tube]" (fragment 32; sensorimotor system/single representation), to "The sigh is going through the tube" (fragment 36; single representation/representational mapping)." The statement that the tube is attached, which the researcher repeats and emphasizes in fragments 19 and 31, leads to the conclusion that there must be something flowing inside the tube. Since there is no water in the tube fragments 21

and 22), or anything else visible for that matter, it must be "sigh" (fragment 36).

This step-wise refining of the boy's understanding, in which each previous step is the beginning of the next step, illustrates the iterative nature of the process nicely. Not only does iterativeness occur on the conversation level (what the child says depends on what the researcher said previously and vice versa), it also occurs on the complexity level of understanding (each understanding of the child depends on the previous understanding). Finally, the iterative nature of the process can also be seen over sessions, meaning that

In our example (see table 1), micro-genetical variability is seen in the child's understanding of how the material works. First, in fragment 10 the boy names a single cause for what happens: "Because we attached the tube". This is an answer on a sensorimotor system level; he gives a single, observable causal explanation for the phenomenon, not taking the volume of the syringes or the air into account (see also the third column of table 1). Over the course of the interaction, he briefly regresses to "I don't know" (fragments 18 and 20; no skill level), and restores his previously gained skill level again in fragment 24: "Because this [the tube] is attached". From there, he further constructs his understanding, and eventually reaches a higher level in fragment 36: "The sigh is going through the tube", for which he needs a

In Figure 2 a time-serial illustration of the fluctuations in the boy's answer levels during the air pressure task is depicted. The graph shows how the understanding of the boy fluctuates over time. While Skill Theory's level 4 (single representation) is mostly observed during the interaction, the boy also regularly shows understandings at level 3 (sensorimotor system). The short-term intra-individual variability influences the variations in development we can see on the long term (Fischer & Bidell, 2006; Van Geert & Fischer, 2009). If micro-genetical variability is associated with reaching higher-level skills (Howe & Lewis, 2005; Thelen, 1989), long-term trajectories of understanding may differ between children showing more periods of variability versus children showing little periods of variability within short-term interactions. This also makes sense in combination with the property *Iterativeness*, as a shortterm interaction showing a broad range of skill levels makes it more likely that skill levels subsequently move toward a higher level (cf., a phase transition), compared to a previous interaction showing a narrow range of skill levels. After all, the interaction with a broad range of skill levels yields more possibilities for the next interaction than an interaction with a narrow range. In conclusion, as Howe and Lewis (2005) mention, understanding gets form over various instances and in turn, drives long-term developmental change. This connection between the short- and long-term scale of development brings us to the next property, that of interconnected timescales.

**Figure 2.** Time-serial illustration of the variability in the boy's understanding during the air pressure task, measured by using Skill Theory (Fischer, 1980). For this boy, levels on the y-axis range from 1 (single sensorimotor set) to 4 (single representation). A -1 score represents an incorrect or irrelevant answer.

## **5.4. Fourth property: Interconnected timescales**

Three months later, the researcher returns with the syringes and the tube. The researcher starts by asking "Do you remember what we had to do with this?" In response, the boy immediately grasps the material and attaches the tube to the syringes. Then he replies: "Yes, when you push this one in, the air will go over here". He doesn't need more time to think about the process in a stepwise fashion: That it works like this because the tube is attached, that there must be something going through that tube, etcetera. Based on the previous interaction, he now knows that air is going through the tube and makes the pistons move. Note, however, that this is not a mere retrieval from memory. The boy first attaches the syringes to the tube, and answers afterwards. Moreover, the question of the researcher is phrased in a way that encourages him to think about what they did before. Even though the researcher's role is not as prominent as it was in the previous interaction, the social context still plays a role in the construction of understanding. However, three months earlier, the understanding was clearly a co-construction between child and researcher. Now the child can directly introduce this understanding to the interaction, triggered by the researcher's question and the material, but without further interference.

Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 39

In an applied sense, it is of great importance for parents, (science) teachers, and other practitioners to have knowledge about how children grasp varied concepts and how their understanding develops over time. By having this knowledge, they will be able to challenge children in their current level of understanding in order to promote children's optimal developmental trajectories with regard to cognitive understanding, and by doing so, promote children's optimal development in a broader sense. Departing from the idea of understanding as a process of change in which the child and the (social and material) context intertwine, the ways and complexity levels at which educators interact with their pupils have an important influence on the development of understanding. With regard to iterativeness, it is important for educators to acknowledge that how understanding changes at one moment in time depends on the understanding at a previous time point. That is, from a dynamic systems perspective, there are no internal operations on representations of knowledge that cause intellectual growth. Understanding organizes on the spot, and gets internalized over time through multiple interactions with the environment. Regarding micro-genetical variability, it is important for educators to understand that the highest complexity level on which children operate (e.g. when they learn about scientific concepts) can change rapidly during short-term interactions, not only when the environment or the amount of support visibly changes. Finally, a better understanding of the temporal stream of understanding will help educators to become aware of their own role in the long-term learning process, and may help them to change their actions when necessary or wanted. Students who are engaged in (scientific discovery) learning need adequate support to construct their knowledge (Alfieri, Brooks, Aldrich, & Tenenbaum, 2010).We claim that teachers' awareness of their own role is an important indicator for the quality of their

support, which is a crucial factor in improving children's learning (McKinsey, 2007).

capability of the highest standards in all fields of expertise are very much needed.

An important next step in the study of the development of children's understanding of scientific concepts as a dynamic system is to try to map individual learning trajectories and build a dynamic simulation model, based on a general theory of action or agent behavior on interacting time scales, and a general theory of mechanisms of change (see van Geert, 1994; Van Geert & Steenbeek, 2008; Steenbeek, 2006). With the help of such a simulation model, the important role of the (science) educator in the emergence of understanding can be unravelled. As a result, such a simulation model will have an important educational value,

We need to work further on completing the empirical picture of possible trajectories of understanding that can emerge in individual children and investigate how these are related to processes on the short-term time scale. This will help us to differentiate components that build up to children's successful and unsuccessful learning trajectories with regard to scientific understanding. This knowledge will also help science educators to teach children to successfully master scientific concepts, as children's understanding of scientific concepts is not always accurate (Grotzer, 2004). When children have more expertise in science, feel confident about this, and enjoy science lessons, this may eventually boost the current number of young people pursuing a scientific academic career. In order to maintain economic growth, people with a scientific education who can ensure continuous technical
