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

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 written on the outside.


Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 33

23

25

24

26

32

34

36

35

Sensorimotor system

Single representation (prediction)

Single representation

Single representation

Single representation /representational mapping

**Person Content:** verbal *(gestures, manipulations, gaze directions)* **Complexity Nr**

Researcher "Even with a longer tube?" 27 Boy "Yes" *(Looks at his syringe)* 28 Researcher *(Pushes her piston in, it works)* "So now it works as well" 29 Boy *(Pauses, pushes the piston of his syringe in, then pulls it out)* 30 Researcher "So it has to do with the tube or something like that..." 31

Researcher "But then, what is in it? And how is it possible that we can move one by pushing the other?"

Boy "Because this is attached *(Touches the end of the tube)* and then it can move out" *(Pulls the piston out)*

Researcher "You know what; we can also attach a longer tube! *(Gets* 

Boy *(Gets the end of the tube and attaches it to his syringe)* "I

Boy "Yes, because the tube is attached to this one *(Looks at* 

Boy *(Keeps on following the tube with his finger, can't reach for* 

Researcher *(Follows the last bit of the tube with her finger)* "Yes, but what is going through the tube?"

*(Follows the tube until halfway)*

*5.1.1. Person-context dynamics – social construction* 

Boy "That... *(Pauses and looks at the tip of his syringe)* "The sigh

*syringe while he pushes the piston back in)*, and it is

attached to here *(Points at the point where syringe and tube are connected)*, and then goes *(Makes a gesture for pushing the piston in)* this *(Points at the tip of the syringe)*, it goes like this" *(Follows the tube from the tip until he is halfway)*

*the last bit, so follows it in the air)* "The tube, it goes like

is going through the tube *(Gestures for pushing the piston in)* "And then it goes, like this, and this, and this"

Researcher "The sigh is going through the tube and flows to mine?" 37 Boy "Yes" *(Pulls the piston of his syringe out)* 38 **Table 1.** Excerpt of a session from our longitudinal project in which a boy (4 years, 6 months) explores

An important part of these context dynamics is the social part of the context, meaning the people around the child. Thus, the development of the child's understanding occurs in

Researcher "I see…what do you mean when you say 'this'?" 33

think still the same."

this"

the syringes task together with a researcher.

*a longer tube)* "What do you think will happen then?"

up?"

work?"

goes backwards!"

**Person Content:** verbal *(gestures, manipulations, gaze directions)* **Complexity Nr**

Boy *(Looks at his own syringe)* "I don't know" No level 2 Researcher "But what do you think?" 3 Boy *(Looks from the researcher to his syringe)* "Uhm..." No level 4

1

5

7

11

17

19

18

20

zero

zero

6

8

10

Single representation (prediction)

Single representation

Sensorimotor system

*one syringe to the boy)* "I attached the tube to these. What

piston out *(Touches the piston)*, and pushed the other piston in *(Points down to the other piston)*. Then I attached the tube. What do you think will happen if I push this one in?" *(Gestures as if she is pushing down)*

*one hand, while his other hand pushes the end of the piston* 

Researcher "Really? Why does that happen?" 9

Boy *(Shakes his head)* "No". 12 Researcher *(Pushes her piston in, pauses)* "Were you right?" 13 Boy *(Watches his own syringe as the piston pulls out)* "Yes" 14 Researcher "Can you do it as well?" *(Holds her syringe up)* 15 Boy *(Looks at both syringes, pushes the piston of his syringe in)* 16

*(Points at the piston of the boy's syringe)* and then this one

Boy *(Pushes piston in and pulls it out)* "I don't know" No level,

What do you think is inside the syringes and tube?"

Boy *(Pauses for a long time, looks around)* "I don't know" No level,

Researcher "I think there's no water in it" *(Shakes her syringe)* 21 Boy "No" *(Starts shaking the syringe)* 22

Researcher *(Attaches the two syringes by a small transparent tube, gives* 

Researcher *(Pauses)* "You said they are the same. I pulled this

Boy "Then this one will go up like this." *(Holds his syringe in* 

*on the table, then he moves his hands up)*

Researcher *(Points to this syringe the boy holds)* "Is that one going

Boy "Yes, and then that one is going down" *(Points at the piston of the syringe the researcher is holding)*

*finger to the tip of his syringe)*

Boy "Because we attached the tube." *(Follows the tube with his* 

Researcher "I see… If we would take away the tube, it wouldn't

Researcher "How is this possible? You're pushing it over there

Researcher "OK, but it has something to do with the tube, you said.

do you think will happen if I push mine in?"


**Table 1.** Excerpt of a session from our longitudinal project in which a boy (4 years, 6 months) explores the syringes task together with a researcher.

## *5.1.1. Person-context dynamics – social construction*

An important part of these context dynamics is the social part of the context, meaning the people around the child. Thus, the development of the child's understanding occurs in

interaction with the social environment (e.g. the teacher), and it is this interaction that drives the process of understanding, enabling the student to receive adaptive assistance and make progress step by step (Hirsch-Pasek, Golinkoff, Berk, & Singer, 2009; Van Geert & Steenbeek, 2005). In our example (see table 1), the child constructs his answers together with the researcher. The researcher's questions are guided by, and on their turn guide, the child's answers. An illustration of this can be seen in fragments 2 to 6 of table 1. After the boy answers he does not know what happens with the syringe he is holding if the researcher pushes the piston of the other one in, the researcher asks him "What do you think?" In this way, she is trying to get the boy to make predictions, encouraging him to hypothesize. In response, the boy looks around and does not answer the question. The researcher, in turn, helps him getting started by summarizing what he said before and by a verbal repetition of her actions with the task material. After having heard the adult's repetition of her actions, the boy starts to construct an answer on a higher complexity level than before. In terms of Skill Theory, this answer can be classified as a single representation, as he makes a prediction that goes beyond simpler perception-action couplings (skill levels, when applicable, are indicated the right column of table 1).

Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 35

approach. Scaffolding is an interactive process in which the student makes progress using the help of a teacher, which scaffold-level should be adapted to the student's level in order to have the right effect. One of the most interesting properties of this dynamic model is that it accounts for transactions between teacher and student, and that it portrays a dynamic, real-time combination of both the student's performance level and the scaffold-level of the teacher. One of the parameters in the model is the *optimal scaffolding distance*, a bandwidth which differs among individuals and contexts, within which help stimulates learning. Within that bandwidth, the optimal scaffolding distance is the distance between the pupil's level and the level of help or scaffolding for which the learning effect is maximal. Just like in our model of understanding, the actions of student and teacher form a unique process built

In addition to the social context, the material context (such as the syringes) also plays an important role in the process of understanding. The syringes should not be conceived of as fixed or monolithic things, but are instead part of the emerging dynamics. Even an unmovable material object is dynamic in terms of its effect on the child, in the sense that the child continuously changes his angle of vision towards the object and thus sees different parts of the object. The dynamic and intertwining nature of the material context is even more strongly illustrated by the syringes task, in which the child or the adult manipulate the

In the example (table 1), the syringes and tube are frequently touched by the boy and the researcher to emphasize or guide their verbal expressions (see fragments 5, 6, and 10). The best illustration of this, however, can be found in fragments 32 to 36. In this fragment, the boy uses the material extensively, after which a higher level of complexity emerges: he transitions from a sensorimotor systems level to a single representation/representational mappings level. Note how the boy substitutes words for gestures and pointing in fragments 32 and 34, following the process of what happens with his hands. Parallels can be drawn with fragment 5, in which the researcher is talking the boy through what happened before. In fragments 32 and 34, however, the boy uses the material instead of the researcher's words to construct his understanding. Before fragment 32, he predicted that one piston comes out when you push the piston of the other syringe in. However, so far, he was not able to explain why. Now, using his hands to examine the syringe, he is able to represent the process, and concludes that "it" is going through the tube. Eventually, guided by the researcher's question "But what is going through the tube?" which seems to suggest that he is on the right track, he is able to replace the word "this" in his explanation for "sigh".

In figure 1, the iterative character of the understanding dynamics between student and researcher is shown in that each previous action of the student has an influence on the subsequent (re-)action of the researcher, and vice versa. Over time, each session has an influence on the subsequent session of this student-researcher pair, which implies that the

syringe, and are thus changing the nature of the object in line with their activities.

**5.2. Second property: Understanding is an iterative process** 

of bi-directional relationships (Fogel & Garvey, 2007).

*5.1.2. Person-context dynamics – the material context* 

Two things are important here. First, the researcher is responding to the boy in this way, because he did not know the answer. Had the boy given the answer, she may had pushed the piston in, or asked him to elaborate on his answer. Because the boy does not know the answer, she needs an approach to determine whether he really has no idea, and if so, how she can help him to make a prediction based on what he knows about the syringes. In order to do this, she tries out two different approaches. First, she asks him what he thinks, which can be a starting point for further elaboration on his side. When the boy does not reply, she decides to help him to get started by giving some information about what they have done and seen before. The boy now hypothesizes what happens if the piston of one of the syringes is pushed in. The answer to the question "What do you think will happen?" (see fragment 1 of table 1) is therefore the product of the interaction between the boy and researcher. In her reactions to the boy's "I don't know" the researcher is trying to guide his understanding. In turn, after hearing the researcher's summary, the boy constructs his understanding. What happens with regard to the boy's understanding during the interaction with the researcher is not mere retrieval of earlier gathered knowledge, or a reaction to a trigger (whether it be the syringe itself or the questions), but a (re)construction of knowledge through a constellation of interactions with researcher and material. If we look at understanding while it occurs in real time, we can only study the person-context aggregation that results from this interactive process and cannot distinguish the unique contribution of the individual components (Van Geert & Fischer, 2009). Even though one can describe what the child does in answer to a specific action or expression of the adult; it is not possible to distinguish the adult's or child's contribution to the (variance in) understanding during the task.

Parallels can be drawn with other teacher-student interactions, such as in scaffolding during instructions in arithmetic lessons. In their model of scaffolding, Van Geert and Steenbeek (2005) model the process of scaffolding during an arithmetic class taking a dynamic systems approach. Scaffolding is an interactive process in which the student makes progress using the help of a teacher, which scaffold-level should be adapted to the student's level in order to have the right effect. One of the most interesting properties of this dynamic model is that it accounts for transactions between teacher and student, and that it portrays a dynamic, real-time combination of both the student's performance level and the scaffold-level of the teacher. One of the parameters in the model is the *optimal scaffolding distance*, a bandwidth which differs among individuals and contexts, within which help stimulates learning. Within that bandwidth, the optimal scaffolding distance is the distance between the pupil's level and the level of help or scaffolding for which the learning effect is maximal. Just like in our model of understanding, the actions of student and teacher form a unique process built of bi-directional relationships (Fogel & Garvey, 2007).

## *5.1.2. Person-context dynamics – the material context*

34 Current Topics in Children's Learning and Cognition

applicable, are indicated the right column of table 1).

understanding during the task.

interaction with the social environment (e.g. the teacher), and it is this interaction that drives the process of understanding, enabling the student to receive adaptive assistance and make progress step by step (Hirsch-Pasek, Golinkoff, Berk, & Singer, 2009; Van Geert & Steenbeek, 2005). In our example (see table 1), the child constructs his answers together with the researcher. The researcher's questions are guided by, and on their turn guide, the child's answers. An illustration of this can be seen in fragments 2 to 6 of table 1. After the boy answers he does not know what happens with the syringe he is holding if the researcher pushes the piston of the other one in, the researcher asks him "What do you think?" In this way, she is trying to get the boy to make predictions, encouraging him to hypothesize. In response, the boy looks around and does not answer the question. The researcher, in turn, helps him getting started by summarizing what he said before and by a verbal repetition of her actions with the task material. After having heard the adult's repetition of her actions, the boy starts to construct an answer on a higher complexity level than before. In terms of Skill Theory, this answer can be classified as a single representation, as he makes a prediction that goes beyond simpler perception-action couplings (skill levels, when

Two things are important here. First, the researcher is responding to the boy in this way, because he did not know the answer. Had the boy given the answer, she may had pushed the piston in, or asked him to elaborate on his answer. Because the boy does not know the answer, she needs an approach to determine whether he really has no idea, and if so, how she can help him to make a prediction based on what he knows about the syringes. In order to do this, she tries out two different approaches. First, she asks him what he thinks, which can be a starting point for further elaboration on his side. When the boy does not reply, she decides to help him to get started by giving some information about what they have done and seen before. The boy now hypothesizes what happens if the piston of one of the syringes is pushed in. The answer to the question "What do you think will happen?" (see fragment 1 of table 1) is therefore the product of the interaction between the boy and researcher. In her reactions to the boy's "I don't know" the researcher is trying to guide his understanding. In turn, after hearing the researcher's summary, the boy constructs his understanding. What happens with regard to the boy's understanding during the interaction with the researcher is not mere retrieval of earlier gathered knowledge, or a reaction to a trigger (whether it be the syringe itself or the questions), but a (re)construction of knowledge through a constellation of interactions with researcher and material. If we look at understanding while it occurs in real time, we can only study the person-context aggregation that results from this interactive process and cannot distinguish the unique contribution of the individual components (Van Geert & Fischer, 2009). Even though one can describe what the child does in answer to a specific action or expression of the adult; it is not possible to distinguish the adult's or child's contribution to the (variance in)

Parallels can be drawn with other teacher-student interactions, such as in scaffolding during instructions in arithmetic lessons. In their model of scaffolding, Van Geert and Steenbeek (2005) model the process of scaffolding during an arithmetic class taking a dynamic systems In addition to the social context, the material context (such as the syringes) also plays an important role in the process of understanding. The syringes should not be conceived of as fixed or monolithic things, but are instead part of the emerging dynamics. Even an unmovable material object is dynamic in terms of its effect on the child, in the sense that the child continuously changes his angle of vision towards the object and thus sees different parts of the object. The dynamic and intertwining nature of the material context is even more strongly illustrated by the syringes task, in which the child or the adult manipulate the syringe, and are thus changing the nature of the object in line with their activities.

In the example (table 1), the syringes and tube are frequently touched by the boy and the researcher to emphasize or guide their verbal expressions (see fragments 5, 6, and 10). The best illustration of this, however, can be found in fragments 32 to 36. In this fragment, the boy uses the material extensively, after which a higher level of complexity emerges: he transitions from a sensorimotor systems level to a single representation/representational mappings level. Note how the boy substitutes words for gestures and pointing in fragments 32 and 34, following the process of what happens with his hands. Parallels can be drawn with fragment 5, in which the researcher is talking the boy through what happened before. In fragments 32 and 34, however, the boy uses the material instead of the researcher's words to construct his understanding. Before fragment 32, he predicted that one piston comes out when you push the piston of the other syringe in. However, so far, he was not able to explain why. Now, using his hands to examine the syringe, he is able to represent the process, and concludes that "it" is going through the tube. Eventually, guided by the researcher's question "But what is going through the tube?" which seems to suggest that he is on the right track, he is able to replace the word "this" in his explanation for "sigh".

## **5.2. Second property: Understanding is an iterative process**

In figure 1, the iterative character of the understanding dynamics between student and researcher is shown in that each previous action of the student has an influence on the subsequent (re-)action of the researcher, and vice versa. Over time, each session has an influence on the subsequent session of this student-researcher pair, which implies that the

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 the cyclical or reciprocal character of causality occurs.

Using the Dynamics of a Person-Context System to Describe Children's Understanding of Air Pressure 37

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

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

**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

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,

**5.4. Fourth property: Interconnected timescales** 

a fixed entity, but varies over time, even within a single task.

of interconnected timescales.

answer.

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 previous sessions influence subsequent sessions.
