**6. Discussion**

From a theoretical point of view, we discussed a number of dynamic properties in combination with Skill Theory's ruler of cognitive development. We argued that using these properties and ruler give both educators and researchers important means to get a grip on how children's understanding of scientific concepts builds up over time. More specifically, it helps to understand how children organize their knowledge in concordance with the context, i.e. the teacher, and highlights the importance of being aware of teachers' accounts in conversations with children, for example during a science lesson.

There are many different types of knowledge generation processes, one of which is the socially situated process between adult, child and task that we are discussing here. When a child is assessed or diagnosed, a different process of knowledge generation occurs. In these instances, the child is asked to construct knowledge without the help of an adult, but usually in interaction with a particular symbolic substrate, such as a piece of paper to draw on, or the structure of language that the child is using to describe knowledge. It is however wrong to think that only the latter process (in which the child works without help) is a reflection of the child's "real" knowledge. In fact, both the co-constructed as well as the individually constructed knowledge reflect the child's "real" understanding. Variations in complexity levels within one type of knowledge generation, but also between different types of knowledge generation, illustrate the intrinsic variation of understanding as such.

The model we proposed helps in re-conceptualizing the process of understanding in individual children, and the underlying mechanisms of change in their understanding. The latter is especially important, since "Developmental psychologists are not simply interested in the stable states achieved by individuals along their lifespan, but also about the mechanisms of change that lead from one state to the next." (Howe & Lewis, 2005, p.248). The advantage of a dynamic systems approach to the study of understanding is that it makes the development of understanding more transparent and no longer limited to an invisible process inside the individual learner (Van Geert & Fischer, 2009). Instead, it enables us to closely monitor interactions between child and environment to determine how the outcome (a form of understanding at some point) is constructed in real time.

**6. Discussion** 

question and the material, but without further interference.

in conversations with children, for example during a science lesson.

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

From a theoretical point of view, we discussed a number of dynamic properties in combination with Skill Theory's ruler of cognitive development. We argued that using these properties and ruler give both educators and researchers important means to get a grip on how children's understanding of scientific concepts builds up over time. More specifically, it helps to understand how children organize their knowledge in concordance with the context, i.e. the teacher, and highlights the importance of being aware of teachers' accounts

There are many different types of knowledge generation processes, one of which is the socially situated process between adult, child and task that we are discussing here. When a child is assessed or diagnosed, a different process of knowledge generation occurs. In these instances, the child is asked to construct knowledge without the help of an adult, but usually in interaction with a particular symbolic substrate, such as a piece of paper to draw on, or the structure of language that the child is using to describe knowledge. It is however wrong to think that only the latter process (in which the child works without help) is a reflection of the child's "real" knowledge. In fact, both the co-constructed as well as the individually constructed knowledge reflect the child's "real" understanding. Variations in complexity levels within one type of knowledge generation, but also between different types

of knowledge generation, illustrate the intrinsic variation of understanding as such.

outcome (a form of understanding at some point) is constructed in real time.

The model we proposed helps in re-conceptualizing the process of understanding in individual children, and the underlying mechanisms of change in their understanding. The latter is especially important, since "Developmental psychologists are not simply interested in the stable states achieved by individuals along their lifespan, but also about the mechanisms of change that lead from one state to the next." (Howe & Lewis, 2005, p.248). The advantage of a dynamic systems approach to the study of understanding is that it makes the development of understanding more transparent and no longer limited to an invisible process inside the individual learner (Van Geert & Fischer, 2009). Instead, it enables us to closely monitor interactions between child and environment to determine how the

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).

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 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,

by making the dynamic principles that play a crucial role in the development of understanding accessible for a broader public of educators. Based on the short-term interaction patterns we see emerge, and the implications this has for the long term, we can eventually construct adaptive teaching programs, lessons and materials for science education, which are better adapted to children's current levels of understanding and how this understanding develops in interaction.

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An example of an adaptive educational and assessment (computer) program is Mathgarden (van der Maas, Klinkenberg, & Straatemeier, 2010), an educational computer game with a wide range of sums children that can play at school or at home. Children's responses (the short-term child-computer interactions) are frequently analyzed and reported to their teachers by means of error analyses, individual growth curves, and comparisons between the particular child and his classmates (or the broader population of peers). The program itself uses the child's data by varying the complexity of the sums adaptively, depending on the percentage of right answers, but also on the child's reaction time. Moreover, using the responses and reaction times of all individual children, the items of Mathgarden are arranged (and get frequently re-arranged) in terms of complexity. This program shows how multiple short-term interactions provide information about the individual's long-term development and how this information can inform educational practice. These kinds of adaptive teaching and assessment programs translate dynamic principles into concrete materials that help children to develop their understanding in an optimal way.

In conclusion, as Vygotksy (1934/1986) already noted: "To devise successful methods of instructing the schoolchild in systematic knowledge, it is necessary to understand the development of scientific concepts in the child's mind. No less important than this practical aspect of the problem is its theoretical significance for psychological science." (p. 146). We think that by studying the development of children's understanding of scientific concepts using a model based on properties derived from dynamic systems theory and Skill Theory an important contribution to both this applied and scientific goal is made.
