MSc Brain and Cognitive Sciences
Second Research Project
A network analysis of childrens’ concepts
of floating and sinking
Katharina M¨
uller
10867562
supervised by
Prof. Dr. Maartje Raijmakers
Contents
1 Abstract 2 2 Introduction 3 3 Methods 8 3.1 Participants . . . 8 3.2 Procedure . . . 8 3.3 Materials . . . 9 3.4 Analyses . . . 9 3.4.1 Transcription . . . 9 3.4.2 Features . . . 10 3.4.3 The networks . . . 11 3.4.4 Network analyses . . . 11 4 Results 12 4.1 Predictions . . . 12 4.1.1 Object difficulty . . . 124.1.2 Features connected to floating vs. sinking . . . 12
4.1.3 Accuracy of participants . . . 13
4.2 Explanations . . . 13
4.2.1 General language analyses . . . 13
4.2.2 Features per set . . . 14
4.3 Weighted network . . . 14
4.4 Binary networks . . . 16
4.5 Individual di↵erences in network structure . . . 19
5 Discussion 21 5.1 Predictions . . . 22
5.2 Explanations . . . 22
5.3 Knowledge domains . . . 23
5.4 Theory theory vs. knowledge in pieces . . . 24
5.5 Conclusion . . . 26 6 Acknowledgments 26 7 Appendix 31 7.1 Exclusion of participants . . . 31 7.2 Materials . . . 31 7.3 Feature categories . . . 33 7.4 Object difficulty . . . 35
1
Abstract
Before children learn the formal rules governing physical processes, they already possess naive conceptions of these processes. According to theory theory, these misconceptions are in themselves coherent and can be equated to scientific theories, while advocates of knowledge in pieces argue that children’s naive concepts about scientific topics represent pieces of fragmented knowledge. In this present study, children between 8 and 12 years predicted and explained buoyancy of 32 objects. Networks were built from the overlap between features given to explain buoyancy; nodes were the respective objects.
The weighted network showed local clusters, each containing objects of specific material or object type. Nodes with the most connections represented the most canonical objects. Networks had small-world structure, indicating a high amount of feature overlap in local clusters. Most importantly, individuals di↵ered largely in their network structure; some networks could be translated to a coherent theory of buoyancy in the tradition of theory theory, i.e. networks with equally strong connected nodes, while others showed fragmented knowledge domains in the tradition of knowledge in pieces, i.e. networks containing many subgraphs, and yet others networks could be categorized as in between those two structures. We formalized these network di↵erences as being dependent on the amount of clusters per network. The amount of clusters and therefore also the network structure as predicted by theory theory or knowledge in pieces did not dependent on accuracy. We conclude that theory theory and knowledge in pieces are two endpoints of the same scale. Semantic networks are a powerful tool to investigate children’s naive concepts about scientific processes.
2
Introduction
In formal science education, children are taught about the law of Archimedes, stating that an ob-ject immersed in water is acted upon by an upward buoyant force that is equal to the weight of water displaced by the object. In simplified terms, this means that an object whose density (weight
divided by volume) is lower than the density of the fluid (e.g. water = 1.0 g/cm3), floats while an
object whose density is higher than the density of the fluid sinks. Every material has its unique density, determining thus the buoyancy of all objects of this material - in principle. Boats are made of metals much denser than water; to understand why boats float, one has to refer back to the law of Archimedes to understand that the amount of displaced water depends not only on the density of an object, but also on its shape.
Even before children formally learn about the physics of buoyancy, they possess some knowledge about floating and sinking. This conceptual knowledge can stem from various sources, i.e. obser-vations, personal experiences, science exhibitions, or from abstract information as told in books or by adults. How conceptual knowledge develops from these heterogeneous sources has been of great interest for developmental psychologists, cognitive scientists and science educationists. To answer this question, it is studied how children’s naive concept of scientific theories is organized, that is their knowledge before they acquire formal scientific education. There has been a debate about the structure of the naive knowledge between two camps: theory-theorists versus those advocating the view of knowledge in pieces.
Theory theory vs. knowledge in pieces
Theory-theorists claim that even naive knowledge of physical concepts is an abstract, coherent system consisting of causal rules, and in that it resembles scientific theories (Vosniadou and Ioan-nides (1998); Gopnik (2003); Gopnik and Wellman (1994); Carey (1999)). Moreover, there are similarities between cognitive development and the formation of scientific theories; children use the same cognitive devices as scientists: Hypotheses are tested, evidence is gathered, and if the theory is falsified, alternative hypotheses are searched. As Gopnik (2003) puts it nicely, in science, we typically tackle just the problems we have not solved in childhood. Following this stance, sci-ence learning is interpreted as a process in which children’s theories based on everyday experisci-ences are continuously enriched and restructured until the correct scientific perspective is accomplished (Vosniadou & Ioannides, 1998). In the view of theory-theorists, children make use of a theory of naive physics, which does not necessarily share all content with the current scientific view, but is in itself a consistent theory (Vosniadou & Brewer, 1992). For instance, a child might believe buoyancy to be completely dependent on an object’s weight and thus consistently judge the heavier object to be sinking and lighter objects to be floating, independent of the object to be judged. Later during development, this theory gets di↵erentiated, by including an objects’ volume, and, as a last step, they should understand the di↵erence between weight and density (Piaget, 1930).
On the other side of the debate is the view of knowledge in pieces, which argues that before a formal science education, children’s concepts are theoryless and consist of fragmented, loosely connected pieces of knowledge acquired from everyday experience (DiSessa (1988); DiSessa and Sherin (1998)). DiSessa (1988) argues that intuitive physics is a fragmented collection of ideas, loosely connected and reinforcing, having nothing of the commitment or systematicity that one
but are not able to integrate them into a coherent model. For instance, children might be aware of a coin sinking in water and wood usually floating on water, but do not understand the underlying principle of density. Which rule children apply, depends on the context or objects to be judged. Di↵erent studies have shown children’s knowledge of physical concepts to be fragmented and incon-sistent, and these fragments seem to develop independently from each other and at di↵erent rates (Straatemeier, van der Maas, and Jansen (2008); Nobes et al. (2003); DiSessa (1988)). It is claimed that the transition from naive concepts to scientific knowledge involves more than a mere update of the content, instead, a major structural shift from knowledge in pieces towards a systematic theory is taking place (DiSessa, 1988).
Which criteria to use to di↵erentiate between the two perspectives is not a trivial question. According to diSessa (1988), this can be tested by asking children questions which share the same underlying physical property, but di↵er in context or phrasing. Knowledge in pieces would predict that children give di↵erent answers to these similar questions, while theory theory would predict children to widely stick to their pre-defined theory and thus to give strongly related answers. Di↵erent studies have produced evidence supporting either the view of theory theory (Frappart, Raijmakers, and Fr`ede (2014); Vosniadou and Brewer (1992); Ioannides and Vosniadou (2002); Smith, Carey, and Wiser (1985)) or that of Knowledge in pieces (DiSessa, Gillespie, and Esterly (2004); Nobes et al. (2003); Straatemeier et al. (2008); Franse, van Schijndel, Visser, and Raijmakers (n.d.)); and in some cases also for both, dependent on the knowledge domain addressed (Raijmakers, van Es, van Schijndel, & Franse, n.d.), the time in the learning process (Schneider & Hardy, 2013), or external influences (Vosniadou, Skopeliti, & Ikospentaki, 2005);.
Children’s knowledge of buoyancy
The concept of floating and sinking has been studied at length from perspectives ranging from cognitive development (Piaget, 1930), conceptual change (Schneider & Hardy, 2013), to science education (Kallery (2015); Hsin and Wu (2011); Tenenbaum, Rappolt-Schlichtmann, and Zanger (2004)). Children as young as three years old have been shown to successfully predict which ob-jects will float on water and which obob-jects will sink (Kohn, 1993). However, three year-olds were significantly less consistent in their judgments on the same object over multiple trials than four-and five year-olds, four-and even children until the age of five sometimes make judgments below chance level, and on rare occasions, also adults (Kohn (1993); Kloos, Fisher, and Van Orden (2010)).
Even though three year olds have been shown to have at least some implicit understanding of density, weight systematically biased judgments that were based on density (Kohn, 1993). Heavier objects were more often predicted as sinking than lighter objects with the same density, and lighter objects were more often predicted as floating than heavier objects with the same density.
Other researchers have additionally shown that children attend more to an object’s weight than its density, and that this may lead to inaccurate judgments, and children, even in grade eight, have been unable to di↵erentiate between weight and density (Smith, Maclin, Grosslight, and Davis (1997); Smith et al. (1985)). It has been suggested however this is due to framing e↵ects: when objects are presented in pairs, of which one will float and the other will sink, children appear to attend more to weight than to density; when objects are presented in isolation, then children seem to compare volume and weight of that object and thus making more accurate judgments according
to density (Kloos et al., 2010).
Instead of assessing strategies from the floating and sinking predictions, one could also ask par-ticipants to explain why they think these objects float or sink. The overarching, scientific answer would be to refer to Archimedes’ law, i.e. an object’s density or the amount of water displaced. Next to the few cases where children mention density, they also refer to mass, volume, or the mate-rial of the object (Tenenbaum et al. (2004); Hsin and Wu (2011); Franse et al. (n.d.)). In addition, children also refer to irrelevant facts such as color, texture, or anthropomorphic explanations (Butts, Hofman, and Anderson (1993); Meindertsma, Van Dijk, Steenbeek, and van Geert (2014)). Weight is very often used as a sole explanation of buoyancy judgments (Franse et al. (n.d.); Meindertsma et al. (2014); Smith et al. (1985); Hsin and Wu (2011)). The level of explanation or knowledge has been hypothesized and shown to depend on the age of the child (Piaget (1930); Tenenbaum et al. (2004)). Interestingly, explanations and predictions give di↵erent indications of the used strategies: When predicting if an object of unknown material will float or sink, children integrate mass and volume implicitly, but when verbally explaining, they point mostly to an object’s weight (Franse et al., n.d.). It thus seems as if implicit knowledge is partly driving the prediction; children seem to be either not aware of the interplay between weight and volume (i.e. density), or they merely cannot articulate it. A big problem in comparing developmental research on floating and sinking, and density in general, stems from the vast variety of methodology used, especially in the design of the objects to be judged. Some studies used objects with varied density, volume and weight, some in a more (Kohn (1993); Kloos et al. (2010)) or less systematic fashion (Hsin and Wu (2011); Schneider and Hardy (2013)). Other studies employed di↵erent sets of objects, consisting of e.g. hollow objects and solid objects (Kallery (2015)), or daily objects (Meindertsma et al., 2014). Tenenbaum (2004) and Janke (1995) chose a compensation approach and asked their participants how many sandbags are needed until a milk carton would sink in water (Tenenbaum et al., 2004), or how much cargo a ship can carry before it sinks (Janke, 1995).
Franse et al. (n.d.) systematically tested the e↵ect of di↵erent objects on children’s predictions and justifications by using three di↵erent sets of objects: consisting of either known objects, cubes of known materials or cubes of unknown materials. It was found that participants predicted the buoyancy behaviour more accurately for known materials and known objects than for abstract objects, and seemingly used di↵erent explanations for di↵erent sets. It is however not clear how the buoyancy knowledge of children is structured. Do children possess di↵erent knowledge domains for di↵erent objects? According to the inherence heuristics, when being prompted to give an explanation, people tend to give answers about the inherent features of the object or topic at hand which are relevant for the problem at hand (Cimpian, 2015). Thus, children’s explanations about why objects sink or float should also be mostly about features that ”describe its basic structure and constitution” and are relevant for buoyancy (Cimpian, 2015). How are the shared features of objects that children consider relevant for buoyancy structured for a great variety of objects?
Network analysis
ei-(Kallery (2015); Meindertsma et al. (2014); Hsin and Wu (2011); Tenenbaum et al. (2004)), or they are fitted to a latent class model (LCA; Schneider and Hardy (2013); Straatemeier et al. (2008); Franse et al. (n.d.); Raijmakers et al. (n.d.); van Schijndel, Visser, van Bers, and Raijmakers (2015)). With LCA, it is statistically tested how many classes are needed to best fit the data, and thus the categories are not constrained to a pre-defined set; instead, alternative and unexpected categories can be detected. Both LCA and the inclusion of explanations into pre-defined categories assume a latent variable underlying the data. In this case, the predictions participants make are assumed to underlie latent strategies, such as ”scientific strategy”, ”mass” or ”mass x volume” (Raijmakers et al. (n.d.); van Schijndel et al. (2015)). Observed variables (here, predictions or explanations) are assumed to be conditionally independent of each other. That is, any correlation between two explanations stems from them underlying the same strategy. With this, possible relationships be-tween variables get lost (Schmittmann et al., 2013). In the context of buoyancy explanations, it is conceivable that claiming that a wooden branch floats because it is made of wood might cause the child to claim that a wooden cube also floats because it is made of wood. In this scenario, both the explanation ”wood floats” and the object at hand (the wooden stump) might have an influence on the buoyancy explanation of another object which it shares features with.
These issues can be resolved assuming a network of directly related causal entities, i.e. objects. According to Cimpian (2015), when prompted to explain a given situation, people first retrieve information about the main constituents of the explanandum. In the case of buoyancy judgments, that would be the object placed in the water and possibly water itself. Thus, the object will gen-erate an explanation, and thereby it receives a causal role.
The network approach is not new to brain and cognitive sciences (e.g. Steyvers and Tenenbaum (2005); Rumelhart, Hinton, and Williams (1986); for reviews: Bullmore and Bassett (2011); Sporns, Chialvo, Kaiser, and Hilgetag (2004); van den Heuvel and Sporns (2013)). A network is a graph consisting of nodes that are connected to one another via edges. In semantic networks, nodes could represent entities, while edges could represent their relation, e.g. features shared by two nodes (e.g. Beckage, Smith, and Hills (2011); Hills, Maouene, Maouene, Sheya, and Smith (2009)). Two nodes sharing an edge are called neighbours; a node with all its neighbours is a neighbourhood. Semantic networks are networks representing relationships between concepts, and can therefore used to study knowledge ( Steyvers and Tenenbaum (2005); Quillian (1967); Rumelhart and Norman (1973)). Nodes represent concepts (e.g. bird, shoe, sock) and are connected via edges representing attributes or uses of this concept (e.g. has wings, used for walking, is clothing). The structure of a semantic network can provide information about the (hierarchical) organization of concepts into categories (e.g. Hills et al. (2009); Beckage and Colunga (n.d.); Steyvers and Tenenbaum (2005)).
It is widely claimed that categorical knowledge - knowledge about di↵erent properties of an object categorizing it together with other similar objects - can be inferred from the structure of semantic networks, e.g. from the way features connect objects or whole concepts. In a study by Hills et al. on early-learned nouns (2009), highly connected nouns such as zebra, mouse, squirrel and dog are members of the class mammals, while a gira↵e does not share many features with any other nouns, and thus it is not included in the group of mammals. As children learn the shared properties of objects (Hills et al. (2009); Siegler, DeLoache, and Eisenberg (2011)), neighborhoods become increasingly coherent and distinct from other neighbourhoods. This process have been equated with the development of a knowledge domain. By comparing networks of children of
dif-ferent ages or with di↵erent levels of knowledge, we can study the development of these knowledge domains.
Semantic networks often show a small-world structure (Steyvers and Tenenbaum (2005); Hills et al. (2009)), even in children as young as 15 months (Beckage et al., 2011). Small-world networks are sparsely connected networks with strong local clustering and short average path lengths be-tween nodes (Travers and Milgram (1969); (Steyvers & Tenenbaum, 2005)). These properties of small worlds are achieved via nodes connecting two local clusters. Other examples of small-world networks are social networks (Travers & Milgram, 1969), links between webpages (Bar-Yam, 1997) or scientific collaboration networks (Newman & Newman, 2001).
This graph-theoretic approach has been applied to study category learning and semantic knowl-edge in children (Hills et al. (2009); Beckage et al. (2011)), the transfer of knowlknowl-edge between di↵erent people (Allen, James, & Gamlen, 2007), and even to simulate learning on a neuronal level (Rumelhart et al., 1986). Networks have furthermore been used increasingly to study psychological constructs (Schmittmann et al. (2013); Borsboom and Cramer (2013); Costantini et al. (2015)). However, no study has been conducted on the learning of scientific concepts.
In this present study, we will construct a semantic network of children’s concepts of floating and sinking, following the approach of Hills et al. (2009). The nodes of the network will represent the objects children judge on their buoyancy, and edges will be the shared features that children mention in order to explain why they think the objects sink or float. Objects come from four di↵er-ent sets, similar to those of Franse et al. (n.d.), and systematically share features with each other. Children seem to apply di↵erent strategies to solve buoyancy problems for di↵erent objects (Franse et al., n.d.), indicating di↵erent knowledge domains for di↵erent objects (Siegler et al., 2011).
Network analysis on buoyancy knowledge
The mixed results from studies on buoyancy as presented above, and the current tie between The-ory theThe-ory and Knowledge in pieces seem to call for a di↵erent approach to study naive scientific concepts. In fact, this long-existing tie between the two theories might indicate that there are at least a few knowledge structures that lie conceptually in between fragmented knowledge and a theory-like knowledge. Knowledge in pieces and Theory theory might be the two endpoints of the same continuous scale. Where a person lies on this scale, might be dependent on the level of knowledge this person has about the topic at hand.
The two perspectives of Theory theory and Knowledge in pieces make di↵erent predictions on the network structure. The first hypothesizes a network where all nodes are equally strong con-nected (fig. 1A). The second on the other hand hypothesizes a network with many subclusters (fig. 1B). The amount of clustering should give the degree in which a child’s knowledge is fragmented; in the extreme case all objects are very loosely connected, indicating that the explanation of buoyancy is di↵erent for each object. Since our objects share properties with each other, some nodes might act as connecting nodes between two clusters, and thus a network with high local clustering might additionally show short average path lengths, and thus show small-world structure. A network with
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 A" B"
Figure 1: Simulated graph. A: Network with equally strong connected notes, representing theory theory-like knowledge structure. B: Network showing many subclusters, representing knowledge in pieces-like knowledge structure.
This study is foremost a proof of concept. We will show that and how we can use network anal-ysis to make a precise statement on how children’s concept of floating and sinking is constructed. This entails answering the question whether the features children mention in order to explain their buoyancy predictions structure objects into larger, meaningful categories. We will examine the emergence of neighbourhoods consisting of di↵erent objects, indicating the existence of knowledge domains. Our analyses additionally encompass the assessment of network properties essential in small-world structures (clustering coefficient and shortest path lengths). Secondly, we will discuss whether this graph-theoretic approach can give us evidence for a theory-like concept of buoyancy or for a fragmented knowledge of buoyancy, or for both. Thirdly, we will test if there exist di↵erences in network structure between age groups, gender, and proficiency groups.
3
Methods
3.1
Participants
Participants were recruited at the Science Center NEMO in Amsterdam. 58 children were tested, of which 12 were later excluded mainly due to technical difficulties (see Appendix). The remaining
46 participants (18 male, 28 female), were between 8 and 12 years old (9.5 ± 1.3, twelve 8-year
olds, fourteen 9-year olds, six 10-year olds, eleven 11-year olds, three 12-year olds) old and attended grades 3 to 8 in school (grade 2: 1, grade 3: 13, grade 4 : 10, grade 5: 9, grade 6: 11, grade 7: 2).
3.2
Procedure
The experiment took place in the Research & Development room at the Science Center NEMO. The room was isolated from the rest of the museum, to ensure little distraction and most privacy. After parents gave informed consent, the video was started and the experiment began. Parents were allowed to sit in the room, but the child faced away from parents, and to the experimenter. The female experimenter explained the experiment to the child, and emphasized that there was no right or wrong, but that she was instead very interested in how the child thought about floating
and sinking. It was emphasized that the child could use the same explanations for multiple objects. The child was presented with each object at a time that she could inspect as long as wished. The child was asked to tell the experimenter if she predicted the present object to float or sink, and was then asked to explained why she thought so. If the child paused for very long, the experimenter would ask what she was thinking. The experimenter would occasionally ask if there was anything else the child wants to add, and then go on with the next object. Previous and following objects were not visible to the child. There was no pre-defined order of objects. During the whole experiment, there was a bucket of water on the table.
After completing the experiment, the child was allowed to test the objects in the bucket of water, and was given further explanation about buoyancy.
3.3
Materials
Following the approach of Franse et al. (n.d.), we constructed four sets consisting of 32 objects in total. The sets of objects were designed to trigger strategies most commonly found in the literature. A few objects were the same as in Franse et al. For pictures, measurements and average accuracy scores of the objects, please refer to fig. A1 to A4.
Set 1: Abstract objects. Eight cubes made out of clay, present in four di↵erent sizes. Each of these four di↵erent cubes exists in two di↵erent densities; either with styrofoam (version A: floating) or metal (version B: sinking) hidden inside. This set was designed such that density did not depend on material, and thus to trigger the mentioning of physical features such as volume, mass, density, or shape.
Set 2: Known materials. Three wooden cubes and three metal cubes, in similar sizes as set 1. Densities are the same as in set 1. Set 2 was designed to trigger material features, in this case wood and metal. Objects are same as in Franse et al. (n.d.)
Set 3: Known objects. Five di↵erent known objects (wooden branches, stones, coins, balls, boats) with three di↵erent types per each object, di↵ering in material or size. Set 3 was designed to trigger references to specific objects. The big branch and the green plastic ball were the same as in Franse et al. (n.d.)
Set 4: Objects with hole. One wooden cube with a hole inside, a small ball with multiple holes inside, and a Japanese coin with a hole inside. Set 4 was designed to trigger features such as the existence of holes or the containment of air.
3.4
Analyses
3.4.1 Transcription
Explanations and predictions were transcribed from the video file into written text with the help of the program CLAN (MacWhinney & Snow, 1990), which was designed to transcribe and analyze child language data. Using CLAN, we computed the total number of words (tokens) and the total number of unique words (types) per participant.
3.4.2 Features
Explanations were scored into categories which were based on Franse’s (n.d.) scoring scheme, but were largely expanded. In total, there were 36 di↵erent categories (for a detailed overview, see appendix). It was further noted whether the scored category referred to sinking or floating. An independent person scored 20 % of the data (9 participants). In order to calculate the amount of agreement between first and second scorer, for each participant, a vector with binary values repre-senting the presence of absence of a feature was computed. These single vectors were concatenated into one large vector, for all 9 participants. Comparing the two vectors of the first and second rater resulted in Cohen’s of 0.718, or of 94.864 %. These values di↵er because of the large abundance of zeroes. According to Landis and Koch (1977), a value between 0.61 and 0.80 refers to substantial agreement.
The data were analyzed in R (2014). First, it was counted how many times each participant mentioned the same feature for each pair of objects, creating a 32 objects x 38 features-matrix for each participant. For this, antonyms such as ”small” and ”big”, and ”light” and ”heavy” were merged into one feature category, in this case ”volume” and ”weight”. The category ”rest” could not function as an edge. The number of resulting categories was 23. For a detailed overview on feature categories, refer to the Appendix.
Each feature was mentioned in total between 10 and 976 times (mean = 133.087, SD = 195.891). See fig. 2 for the detailed frequencies of all features. Each participant mentioned between 34 and 115 features for all objects (mean = 65.152, SD = 17.840). Weight is by far the most frequently mentioned feature (976 times).
In order to normalize for di↵erent numbers of features mentioned per object, we divided the overlap count by the total number of features mentioned for both objects. In the next step, the number of feature overlap per object was summed over all participants, creating a weights matrix. The weighted number of shared features per object pair ranged between 4.752 and 34.249 (mean = 16.191, SD = 5.385). Every feature contributed to the network.
w eight vol density metal w ood plastic cla y rub ber stone coin br
anch boat ball hole air
shape texture water press la
yer inside ref rest 0 200 400 600 800 1000
3.4.3 The networks
Nodes represent objects and edges represent the proportion of features shared between objects. All networks were undirected. We first created a weighted network, visualizing di↵erences in edge strengths. To investigate how the amount of shared features between objects influenced the net-work structure, we created binary netnet-works by defining thresholds (w ), defining the strengths of connections two objects must have in order to be connected (Hills et al., 2009). The networks were created using the qgraph package (Epskamp, Cramer, Waldorp, Schmittmann, & Borsboom, 2012).
3.4.4 Network analyses
Descriptive analyses of the network graph characteristics were conducted using the R packages
qgraph (Epskamp et al., 2012) and igraph (Cs´ardi & Nepusz, 2006). The following network
char-acteristics were analyzed:
• Clustering coefficient was calculated as introduced by Watts and Strogatz (Watts & Stro-gatz, 1998) (unweighted networks) or by using Zhang’s algorithm (Zhang & Horvath, 2005) (weighted networks).
• Shortest path length was calculated using Dijkstra’s algorithm (Dijkstra, 1959).
• Smallworldness index was computed by dividing the graph’s global clustering coefficient (normalized by the global clustering coefficient mean of 1000 random graphs (algorithm im-plemented according to the Erdos-Renyi model (Erdos & R´enyi, 1959)) of the same node and edge count) by the graph’s average shortest path length (normalized by the mean shortest path length of 1000 random graphs of the same size).
• Node degree is equivalent to the number of connections a node has.
• Graph density is the ratio of the number of edges and the number of possible edges. Our statistical analyses were split up in two parts: Firstly, in order to answer our research question if and how children di↵er in their reasoning about di↵erent object types, we analyzed the properties of networks resulting from lumping together the data of all participants. We used bootstrapping of 1000 random networks with the same numbers of nodes and edges to test the significance of values of network statistics. Secondly, in order to answer the question how participants di↵er in their reasoning, we analyzed graph characteristics of individual networks.
Lastly, community structure was detected implementing the fast greedy modularity optimization algorithm (Clauset, Newman, & Moore, 2004), which tries to find the densest subgraph while simultaneously optimizing the modularity score. The results of this algorithm were validated using the max statistic (Miller, 1965): The community detection results were resampled using N-1 samples (jackknifing). For each of the samples all possible mappings of cluster numbers were considered. The mapping with the maximal Cohen’s value was considered for further computation. The agreement between the N-1 community detection values were compared using Cohen’s .
4
Results
4.1
Predictions
4.1.1 Object difficulty
An object’s difficulty score is equivalent to the proportion of correct answers over all participants. Fig. 3 shows the mean accuracy of each set. The results of a one-way ANOVA show that there
are no di↵erences between the accuracies of the four di↵erent sets (F(3, 28) = 0.808, adjusted R2
= -0.019, p = 0.5), also not when considering branches, stones, coins, balls and boats separately
(F(7,24)= 1.177, adjusted R2 = 0.038, p = 0.352). Abstr act W ood Metal Br
anches Stones Coins Boats Balls Holes
Accur acy 0.0 0.2 0.4 0.6 0.8 1.0
Abstract Known materials Known objects Holes
Accur acy 0.0 0.2 0.4 0.6 0.8
Figure 3: Accuracies per set. Left: No di↵erences in difficulty between the four sets were found. Right: No di↵erences in difficulty were found if considering objects of set 3 separately.
Buoyancy of sinking objects was less difficult to predict than of floating objects (fig. 1), shown by the results of a paired t-test (t(30)= -4.370, p < 0.0001). For the clay floaters, the wooden cubes and the branches (see fig.A1, fig.A2 and fig.A3), accuracy seems higher for smaller and lighter than bigger and heavier objects. This e↵ect is also apparent for the clay sinkers, the metal cubes and the coins, but to a weaker degree.
Table&1A& &
Accuracy'of'sinking'objects''
Clay&
sinkers& Metal&cubes& Stones& Coins& Coin&hole& n& Total&accuracy& 0.897& 0.964& 0.978& 0.819& 0.804& 14& 0.892& Table&1B&
&
Accuracy'of'floa4ng'objects''
Clay&
floaters&Wooden&cubes& Branches& Boats& Balls& Wood&hole& Ball&hole& n& Total&accuracy& 0.636& 0.594& 0.572& 0.739& 0.826& 0.457& 0.457& 18& 0.612!
Table 1: Accuracy of sinking and floating objects.
4.1.2 Features connected to floating vs. sinking
Features participants mentioned in order to explain an object’s buoyancy did not necessarily only refer to the just-made prediction, i.e. floating or sinking. When a child was unsure about the pre-diction, he or she might give features as to explain why the just-made prediction might be wrong.
Because of that, we annotated if each feature referred to either floating or sinking. In total, par-ticipants mentioned 1241 features in reference to floating and 2285 in reference to sinking. These di↵erences were not significant (t(36) = -1.256, p = 0.214). On average, 95.484 % of the features referred to the made prediction.
4.1.3 Accuracy of participants
A child’s accuracy score refers to the percentage of correct answers given. Accuracy scores range between 0.343 and 0.813; with a mean of 0.589 (SD = 0.113; fig. 4). This was significantly higher than chance level of 0.5 (t(45) = 5.335, p < 0.001). Accuracy scores were normally distributed (Shapiro Wilk W = 0.970, p = 0.275). A linear model with gender, age and total number of words as predictors showed no significant main e↵ect of gender ( = 0.005, p = 0.890), age ( = -0.004,
p = 0.796) or number of words ( = 0.00008, p = 0.134). The overall model fit was adjusted R2 =
-0.010, p = 0.472). Accuracy Frequency 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 12
Figure 4: Frequency distribution of participant’s accuracy scores.
4.2
Explanations
4.2.1 General language analyses
Per person, the total number of unique words (types) ranged between 39 and 271 (mean = 135.283, SD = 49.568), and the total number of words (tokens) ranged between 266 and 1789 (mean = 830.652, SD = 365.861). A multiple linear regression was calculated to predict the total number of words based on age, gender, accuracy and the total number of features over all objects. A significant
regression equation was found (F(4, 41) = 9.559), with an adjusted R2of 0.432 (p < 0.0001). Age
was a significant predictor for number of words ( = 93.808, p < 0.01), total number of features was a significant predictor ( = 11.075, p < 0.0001), gender was not a significant predictor ( = 3.578, p = 0.966), and accuracy was also not a significant predictor ( = 214.674, p = 0.573).
4.2.2 Features per set
Fig. 5 shows the total number each feature was mentioned per set. Weight is mentioned most frequently for all four sets, except for set 4.
For set 1, eeight (310 times) and volume (81) were most frequently mentioned as an explanation. The scientific explanations of density and upwards pressure were given 44 times. It was referred 44 times to the material ”clay”, and it was mentioned 21 times that ”there is something inside” the cubes.
For set 2, next to weight (203) and volume (53), metal (95) and wood (56) were the most frequent. Scientific explanations were given 38 times.
In the context of set 3, next to weight (399), reference to object types is fairly often mentioned (in total: 263), as is material (in total: 161), texture (57), shape (72), containment of air (78), and to the possibility of water coming in the object (66). Scientific explanations were mentioned 64 times. The most frequently mentioned feature of set 4 is hole (90), followed by weight (64), water coming in (49) and the containment of air (13). Scientific explanations were given 2 times.
w eight vol density iron w ood plastic cla y rub ber stone coin br
anch boat ball hole air shape texture water press la
yer inside ref rest 0 100 200 300 400 w eight vol density metal w ood plastic cla y rub ber
stone coin branch boat ball hole
air
shape texture water press la
yer inside ref rest 0 50 100 150 200 250 300 w eight vol density metal w ood plastic cla y rub ber
stone coin branch boat ball hole
air
shape texture water press la
yer inside ref rest 0 50 100 150 200 w eight vol density metal w ood plastic cla y rub ber
stone coin branch boat ball hole
air
shape texture water press la
yer inside ref rest 0 100 200 300 400 w eight vol density metal w ood plastic cla y rub ber stone coin br
anch boat ball hole air shape texture water press la
yer inside ref rest 0 20 40 60 80 C" D" B" A"
Figure 5: Frequency distribution of features per set. A: abstract objects, B: known materials, C: known objects, D: hole objects.
4.3
Weighted network
Fig. 6 shows the network of object-feature connections lumped over all participants. The edges are weighted with respect to the number of features shared by two objects, normalized for the total number of features mentioned for both objects. Coloring objects according to object types (fig. 6a)
and materials (fig. 6b) makes subgraphs apparent. Visual inspection revealed that objects tend to be connected close to members of their sets, or objects with the same material. The community detection algorithm (Clauset et al., 2004) supported the results of our visual inspection (fig. 7A). Three subgraphs were detected: A including all metal objects, B including all clay and all stone objects, C including a larger cluster of all wooden objects, all boats, balls and two hole objects. The stones and clay objects in cluster B seem to belong to two subgraphs. A problem of algorithms that try to maximize modularity, as the fast greedy modularity optimization algorithm, is that they tend to merge smaller communities together in order to form larger communities (resolution limit problem) (Chen, Kuzmin, & Szymanski, 2014). Therefore, the cluster of stones is likely a separate subgraph. Similarly, the cluster D, consisting of wooden cubes and branches, is also likely a separate subgraph from the boats, balls and the two hole objects.
Using the max statistic method, we calculated a mean agreement between the N-1 detected
com-munities of Cohen’s = 1.0 (± 0.0), which refers to perfect agreement and a stable clustering
solution. a" b" c" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ball boat branch clay coin holes metal stone wood ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 clay metal plastic rubber stone wood clay metal plastic rubber stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 floater sinker floater sinker Clustering" Average"length" Density" Isolates"
0.498& 0.069& 1& 0&
A" B" C"
Figure 6: Weighted network. Grouped according to A: object types, B: materials, C: sinking and floating objects. Global clustering coefficient = 0.498, average path length = 0.069, density = 1, number of isolates = 0.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ball boat branch clay coin holes metal stone wood ball boat branch clay coin holes metal stone wood
A:!Metal!cube!2cm!3cm!6cm!Coin!japanese!20ct!2€!hole!
B:$Clay!floa:ng!2cm!3cm!6cm!8cm!Clay!sinking!2cm!3cm!6cm!8cm!Stone!small!medium!big! C:!Wooden!cube!3cm!6cm!8cm!Boat!wood!:n!plas:c!Ball!rubber!soD!football!Hole!wood!ball!
D:$Stone!small!medium!big!Ball!rubber!soD!Clay!floa:ng!3cm!!
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ball boat branch clay coin holes metal stone wood ball boat branch clay coin holes metal stone wood
A:!Metal!cube!2cm!3cm!6cm!Coin!japanese!20ct!2€!hole!Hole!wood!ball!
B:$Clay!floa:ng!2cm!6cm!8cm!Clay!sinking!2cm!3cm!6cm!8cm!Ball!football!Boat!wood!! C:!Wooden!cube!3cm!6cm!8cm!Branch!small!medium!big!
D:$Stone!small!medium!big!Ball!rubber!soD!Clay!floa:ng!3cm!!
A B
Figure 7: Networks with detected subgraphs. A: Weighted network, B: Binary network (w = 20). Within the dashed lines lie possible further subclusters not detected by the community detection algorithm. Members of detected subgraphs are given in the two boxes underneath each graph.
Grouping objects according to whether they sink or float (fig. 6c) visually splits the graph into one subgraph consisting of floaters and another subgraph consisting of sinkers. Similar to that, fig. A5 shows the least difficult objects (> median object accuracy of 0.578) are also mostly grouped together in the graph, apart from the more difficult objects (< 0.578). This is not surprising, since sinkers are also the least difficult objects, while floaters are the most difficult ones (except for the wooden boat and the big branch).
The network is fully connected (density = 1). Within the 75th percentile of node degrees are all abstract clay objects (besides the floating cube with 8 cm of edge length), and the metal cube of 6 cm edge length. Within the 25th percentile of node degrees are the tin and plastic boat, the 20 cent and 2 coin, the green ball, and all hole objects. Further networks characteristics can be found in fig. 6.
4.4
Binary networks
The weighted graph is fully connected (density = 1). In order to make structure and subgraphs apparent, we transformed the weighted graph into binary graphs, with increasing thresholds that feature weights must have in order to be an edge (Hills et al., 2009). Unweighted graphs are furthermore necessary for bootstrap statistics on the clustering coefficient, and therefore also the
small-world index.1
1To compute the clustering coefficient for a weighted graph by Zhang and Horvath (Zhang & Horvath, 2005), the graph needs to be of the object type qgraph (Epskamp et al., 2012); a simulated random graph according to the Erdos-Renyi model (Erdos & R´enyi, 1959) however is of the type igraph (Cs´ardi & Nepusz, 2006). Therefore, the clustering coefficient of the weighted network cannot be compared to a random graph.
Fig. 8 shows a series of object-feature networks with increasing thresholds. At w = 10, the network is densely connected and there is hardly any apparent structure. At w = 15, nodes with smaller degrees become separated, such as the metal and hole objects and two of the boats. The clay and wooden objects remain rather closely connected together. At w = 20, the network is sparsely connected, contains two isolates, and structure is clearly apparent. Objects belonging to the same set or consisting of the same material are connected closely together, with clay and wooden objects appearing more clustered than metal objects.
The clear separation into sinking and floating objects (and, by implication, also of difficult and less difficult objects) as apparent in the weighted networks diminishes in the binary networks. This information probably got lost when transforming the weighted graphs into binary graphs.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 clay metal plastic rubber stone wood clay metal plastic rubber stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ball boat branch clay coin holes metal stone wood ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 floater sinker floater sinker 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 clay metal plastic rubber stone wood clay metal plastic rubber stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ball boat branch clay coin holes metal stone wood ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 floater sinker floater sinker B" C" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 clay metal plastic rubber stone wood clay metal plastic rubber stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ball boat branch clay coin holes metal stone wood ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 floater sinker floater sinker A"
into di↵erent subsets. For the following network statistics, we consider only the binary networks. For the network with w = 20, only the connected network was considered and thus the two isolates were excluded from the analysis.
With increasing threshold w, the global clustering coefficient decreases, the average path length increases and the graph density decreases (table 2). Since the graph with w = 20 already contains two isolates, it is not recommended to increase the threshold any further. Table 2 shows that the binary networks have a small-world index that is significantly higher from that of 1000 random networks with the same number of nodes and edges.
Table&2&
!
Network!sta+s+cs!for!binary!networks!
w! Clustering& Random&
clustering& Average&length& Random&length& Density& Small<&worldness& Random&Smallworldness& Isolates&
10! 0.918*& 0.889&
(p&<&0.001)& 1.111& 1.111&&(p&=&1)&& 0.889& 1.033*& 1.000&(p&<&0.001)& 0&
15& 0.728*& 0.521&
(p&<&0.001)& 1.520*& 1.480&(p&<&0.001)& 0.520& 1.363*& 0.999&(p&<&0.001)& 0&
20& 0.688*& 0.206&
(p&<&0.001)& 2.644*& 2.002&(p&<&0.001)& 0.237& 2.513*& 1.015&(p&<&0.001)& 2&
Table 2: Network statistics for the binary networks. * Indicates a significant di↵erence (p < 0.001) for the clustering coefficient, average path length and small-world index from the random population, using a one-sample t-test.
A small-world structure entails higher local clustering than a random network of the same size would exhibit, while the path lengths between nodes are similar to those of a random network. In the networks with w = 15 and w = 20, the average path lengths are significantly higher than those of random graphs; nonetheless, the small-world index is significantly higher than of a random graph. The properties of small-world networks - high clustering, while path lengths are short - are achieved through the existence of nodes connecting di↵erent clusters. This means that two clusters are only indirectly connected via a node that is directly connected with the two clusters. In the network with w = 20, the metal cube of 6cm (node 14) connects the cluster of metal and hole objects with that of clay objects, the wooden cubes of 3 and 6cm (9 and 10) act as connecting nodes between wooden and clay objects, and the small stone (23) connects the stone cluster with the clay cluster.
Nodes with relatively high degrees (here defined as 10 edges) are the eight clay objects, two
wooden cubes (2cm and 3 cm), one metal cube (6 cm) and the big wooden branch. In order to account for unequal set sizes, we divided the total number of nodes per set (with distinguishing between di↵erent known objects) by the number of members per set, and calculated the node de-grees of all object-nodes for each set. Fig. 9 shows that, even when controlling for unequal set sizes, abstract objects still have the highest node degrees, followed by the known materials made out of wood and metal, and the wooden branches. Abstract objects and known materials are also the objects with the most canonical features: cubes in four di↵erent sizes, and two di↵erent
densi-ties. The more specific objects of set 3 and 4 have smaller node degrees (with the exception of the branches). Abstr act W ood Metal Br
anches Stones Coins Boats Balls Holes
Node degrees per set
0 2 4 6 8 10 12
Figure 9: Node degrees per set, controlled for uneven set sizes.
The community detection algorithm detected four subgraphs in the binary graph with w = 20 (fig. 7 B). They are similar to the three detected subgraphs of the weighted network, in that all metal objects (expect the metal boat) are connected in one cluster; all wooden objects (besides the wooden boat) form one cluster, and all the clay objets (besides the floating clay cube of 3cm) form a cluster. It might be that the just mentioned objects are more closely clustered to one another and thus robust in their division into subgraphs to increasing thresholds than those objects that have a di↵erent cluster membership between weighted and binary network. We tested for the robustness of the community detection algorithm by using the max statistic method, we calculated a mean agreement between the N-1 detected communities of Cohen’s = 0.734 (SD = 0.050), which, according to Landis and Koch (1977), can be translated to a substantial amount of agreement (0.61 - 0.80). The community detection algorithm was significantly more robust on the weighted graph than on the binary graph (t(88) = 25.626, p < 0.001).
4.5
Individual di↵erences in network structure
In order to test our hypothesis that individuals might di↵er in their network structure, in that some show a network structure that is in line with theory theory, while others’ networks are consistent with knowledge in pieces, and yet other children’s networks might lie in between both concepts, we visually inspected all individual’s networks. For a network to fall in the category of theory theory, it had to be equally strongly connected on all nodes, with no apparent subgraphs. A network falling in the category of knowledge in pieces had to show apparent subgraphs. Networks falling in between both categories were equally strongly connected on all nodes of a significant part of the graph, while other nodes started to form subgraphs. Fig. 10 shows examples of those three structures.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood A" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2324 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood C" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood B" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood ● ● ● ● ● ● ● ● ● ball boat branch clay coin holes metal stone wood
Figure 10: Individual networks categorized as A: theory theory, B: between theory theory and knowledge in pieces, C: knowledge in pieces.
We formalized where on the scale between theory theory and knowledge in pieces a network lies by considering the amount of clusters detected by the fast greedy modularity optimization al-gorithm. Isolated nodes were counted as a separate cluster. We calculated a linear regression to predict the number of clusters based on age and accuracy scores. The regression equation (F(2,43)
= 0.200) was not significant, with an adjusted R2 of -0.037 (p = 0.819). Accuracy was not a
sig-nificant predictor ( = -1.611, p =0.553), and age was not a sigsig-nificant predictor ( = -0.040, p = 0.866.
Fig. 11 visualizes that the number of subclusters does not di↵er between accuracy scores; this is most prominently the case for networks with 5 subclusters, with the according accuracy scores covering the whole range. Participants with 6 cluster-networks (0.481) showed the lowest average accuracy scores; the highest accuracy scores were found with the two participants with 8 cluster-networks (0.656). The overall accuracy mean was 0.589.
0.4 0.5 0.6 0.7 0.8 2 N=4 3 N=6 4 N=14 5 N=10 6 N=5 7 N=2 8 N=2 9 N=1 10 N=1 11 N=1 Clusters Accu ra cy
Figure 11: Boxplot showing the relation between number of clusters and accuracy scores. N refers to the amount of individual networks with the respective amount of clusters.
In order to further assess if network characteristics di↵er between the genders, age groups or di↵erent accuracy scores, we calculated three multiple linear regressions to predict global clustering coefficient, average shortest path length and average node degrees based on age, gender and accu-racy. We found no significant regression equation to predict global clustering coefficient (adjusted
R2= -0.023, p = 0.579). The three predictors could furthermore not explain the variance in average
shortest path length (adjusted R2 = -0.032, p = 0.664), or in average node degree (adjusted R2 =
-0.038, p = 0.718). For the model statistics, please refer to Table 3 in the Appendix.
5
Discussion
Our results show that network analysis can be successfully applied to study children’s concepts of buoyancy. The main findings of the present results are the following:
1. Features mentioned in order to justify buoyancy predictions categorize objects into subgraphs of di↵erent object types and materials.
2. The binary graphs exhibit small-world structure that is robust to threshold increase. 3. Nodes with the highest number of edges seem to represent the most canonical objects. 4. Individual’s networks were categorized as supporting knowledge in pieces, theory theory or
falling in between the two frameworks.
In the following, we will discuss these findings and others in more detail and with respect to prior findings.
5.1
Predictions
Children were better at predicting the buoyancy of objects that float vs. those that sink (table 1). There is no implication in the literature that children have a bias towards sinking, or difficulty pre-dicting floating objects per se. An explanation for this finding might be that more of our floaters
had densities close to the density of water at 1.0 g/cm3, than our sinkers (see Fig. A1 to A4).
Objects much more or much less dense than water have been shown to be more accurately judged than objects with densities closer to the density of water (Kohn, 1993). With decreasing volume and weight but constant density, accuracy appears higher for the floaters. Similarly but at a smaller e↵ect, with increasing volume and weight but constant density, accuracy appears to increase for the clay sinkers, the metal cubes and the coins. This is in line with the findings of Kohn (1993) and Franse et al. (n.d.) who showed that when predicting if an object with density close to water sinks or floats, children perform better with heavier or lighter objects, respectively. This e↵ect becomes
smaller the further away from 1.0 g/cm3the object’s density is; this explains why accuracy of the
sinkers at 2.8 g/cm3 only slightly increases with increasing weight.
This means by implication that the buoyancy of wooden objects is more difficult to judge than that of metal objects. When teaching especially young children about buoyancy, metal should be used with care, since children might underestimate the densities an object can have in order to float on water. Once the compensation of weight and volume is understood, it might be valuable to employ small and relatively light metal objects to teach about the denstiy of di↵erent materials.
Accuracy did not seem to di↵er between age groups. This seems at first surprising, as age e↵ects have been found in previous studies on buoyancy and density (Franse et al. (n.d.); Kohn (1993); Kloos et al. (2010); Siegler and Chen (2008)). However, in those studies, participants were younger (Kohn (1993); Kloos et al. (2010)), or covered a broader range (Franse et al. (n.d.); Kloos et al. (2010)) than in our study. Similar age groups were used by Siegler and Chen (2008), however they found an age e↵ect by splitting up participants in two age groups (one with mean age of 7.7 years, SD = .62 and the other with mean age of 9.6 years, SD = .67); this is not comparable to our continuous age variable. It is thus conceivable that we would have found age e↵ects, if we had included participants at a broader age range. Lastly, the low mean accuracy score indicated that the objects might be relatively difficult for children between 8 and 12 years; for further studies it would be worth including older children, or even adults.
5.2
Explanations
Consistent with the literature (Franse et al. (n.d.); Meindertsma et al. (2014); Smith et al. (1985); Hsin and Wu (2011)), weight is the most frequently given explanation to predict if an object sinks or float (fig. 2). Similarly to Franse et al. (n.d.), material was frequently mentioned for set 2, reference to object types was frequently mentioned for set 3 and reference to holes was frequently mentioned for set 4.
a few children mentioned that something was ”inside” the cubes, and that they were made of clay. Furthermore, the possibility of water coming in seems to have been of concern to the participants for objects with hole, besides the containment of air.
5.3
Knowledge domains
Features mentioned to explain the buoyancy of di↵erent objects categorized these objects into di↵erent subgraphs. Subgraphs - coherent node neighborhoods that are distinct from other neigh-bourhoods - have been equated with the development of knowledge domains, (Siegler et al. (2011); Hills et al. (2009)). At least three communities of highly clustered subgraphs were detected that
included nodes representing the same object type and/or the same material.2
This means that within each of these subgraphs, children used the same features to judge if objects will sink or float. By implication, this also means that children used largely distinct explanations for distinct type of objects and materials.
Furthermore, the nodes with the most connections (node degrees) are also the most canonical objects: cubes made out of either clay or metal. Connectedness might be related to the generaliz-ability of these objects’ features to other objects. The generaliz-ability to generalize from facts about a few objects to other objects sharing similar features, is a crucial part of inductive reasoning. Objets that share a number of features might likely also be similar in other features. There is evidence that children between 3 and 5 years rely solely on shared perceptual features between objects when generalizing to other objects (Sloutsky, Kloos, and Fisher (2007); Godwin and Fisher (2015)). Ac-cording to the similarity-based account, children first label objects acAc-cording to their perceptual similarities, and only later learn to generalize based on linguistic categories (Sloutsky and Fisher (2004); Fisher and Sloutsky (2005)).
Understanding the scientific theory of buoyancy might then also depend on children’s ability to generalize from their ideas about the buoyancy of one object to other objects. Object-nodes with a lot of connections to other object-nodes might be the most important objects for changing childrens knowledge, because they seem to have the most impact on other objects. Indeed, there is evidence that it is not enough to let children test if di↵erent objects float or sink, in order to change their concept of buoyancy (Tenenbaum et al. (2004); Butts et al. (1993)). Instead, actively searching for what floating objects and sinking objects have in common has been shown to be a significant factor (Butts et al. (1993); Hsin and Wu (2011)). Starting with canonical objects might be a good approach.
Loosely connected nodes represented the boats, the balls, the ball with holes and the wooden cube with holes. These objects seem to share the least features with other objects. Apart from the wooden cubes, they are all known objects of less canonical shape. Moreover, in contrast to the other object types of set 3, each of the balls and the boats are made of a di↵erent material. This emphasizes again the importance of material for buoyancy decisions. Lastly, real-life boats cannot be explained by referring to weight or density, but they float because of their shape. From real-life experience, most children are likely aware of this, but it might be challenging for them to 2 Since the community detection algorithm was more robust with the weighted graph, we will only consider the
explain it. Thus, these objects seem to have been treated by the children as specific cases, instead of being argued for with similar features as other object types. Objects that do not share many perceptual features with other objects might be unsuitable candidates to start teaching children about buoyancy, but they might be valuable challenges for children with a more advanced concept about buoyancy.
The small-world structure was robust to increases in threshold. The property of small-worldness is consistent with other semantic networks on early-learned nouns (Hills et al. (2009); Beckage et al. (2011)) and multiple associative networks (Steyvers & Tenenbaum, 2005), and multiple real-world networks (Travers and Milgram (1969); Bar-Yam (1997); Newman and Newman (2001)). Objects were designed to share features with objects of other sets, which was likely responsible for the short path lengths between clusters.
5.4
Theory theory vs. knowledge in pieces
According to theory theory, naive knowledge of physical concepts is a coherent, albeit not necessarily scientifically correct, system. Children might for instance have the misconception that all objects that feel heavy sink, while those that feel light float. If this misconception is applied constantly -meaning in di↵erent contexts -, then this is referred to as a coherent theory. The according net-work of this type of naive concept would be equally connected on all nodes (fig. 1A). Advocates of the rivaling view of knowledge in pieces argue that before children learn the scientifically correct concept, their notion of buoyancy consists of fragmented, loosely connected pieces of knowledge. These pieces of knowledge could for instance refer to objects of di↵erent materials, sizes, or shapes. Thus, dependent on the context and the objects to be judged, children apply di↵erent rules. The according network would consist of multiple subgraphs that are distinct from other neighboring subgraphs (fig. 1B).
We found individual networks of both types (fig. 10). Moreover, some networks could be cat-egorized as falling in between the two theories; networks that were on most parts of the graph evenly connected, but showed a few small subgraphs. This supports our hypothesis, that knowledge in pieces and theory theory could in fact be seen as two endpoints of the same scale. This would explain why both theories have been seemingly equally supported by empirical studies (Vosniadou and Brewer (1992); Ioannides and Vosniadou (2002); Smith et al. (1985); DiSessa et al. (2004); Nobes et al. (2003); Franse et al. (n.d.); Raijmakers et al. (n.d.)), notably of which none used a network approach to find evidence for the respective theoretical framework.
Consequently, we formalized where on the scale between theory theory and knowledge in pieces a network lies as the number of clusters, including isolated nodes. Networks with 6 clusters belonged on average to children with accuracy scores below average, which is in line with the view of knowl-edge in pieces predicting that children with poor understanding of a scientific topic have fragmented knowledge of the topic. However, networks with 5 clusters for instance covered the whole range of accuracies. Additionally, the number of clusters was not predicted by accuracy scores and age, which is not in line with knowledge in pieces, which predicts that once children fully understand a scientific concept, they possess mental models of the concept.
Theory theory on the other hand predicts that even children with a very inaccurate concept of a scientific process possess a mental model of the scientific concept. Evidence for this comes from the
finding that children with networks containing as few as 2 clusters showed both lower and higher accuracy scores. However, already the mere existence of highly clustered networks is not predicted by theory theory.
Concluding, children’s network structure did not predict their accuracy scores in the prediction task. These findings do not fully support either of the two theoretical frameworks.
According to Cimpian (2015), people tend to explain problems by referring to the inherent properties of the object at hand; with ’inherent’ referring to features that, if altered, would pro-duce a real change in the object. What these inherent features relevant to buoyancy are, di↵ers between object types. Individuals with mental model-like network structures applied the same rule for almost all objects, which might indicate that they ignored inherent features of many objects and instead focused on what they conceive as relevant for buoyancy. For instance, a child might have always explained buoyancy by weight and ignored more salient features such as material, shape or object type that he or she found irrelevant for buoyancy. However, since the performance on the prediction task did not di↵er between degrees of clustering, it is not clear if this ability to ignore salient, but possibly irrelevant features is important for children’s understanding of buoyancy. Individuals with network structures lying in between fragmented knowledge and mental models seem to apply one rule for many objects, but treat a few objects as special cases by referring to di↵erent features. For future studies, it would be worthwhile to analyze if object-nodes belonging to small clusters are predicted with lower accuracy than object-nodes belonging to bigger clusters. This means that the network structure of a child depends partly on the objects types included in the study. With less variant objects, more children might refer less to the objects’ inherent properties and instead apply the same rule for more objects.
Theory theorists claim that mental models are resistant to fluctuations (Wellman & Gelman, 1998), that is, unless children underwent a conceptual change, their concept of buoyancy will be the same if tested again some time after the experiment. For future research, it would therefore be interesting to assess if children who showed a mental model-like network structure exhibit the same structure when tested again on the same objects.
Furthermore, in order to test if, as proposed by knowledge in pieces (DiSessa & Sherin, 1998) children indeed develop from a concept consisting of fragmented pieces of knowledge to a coherent theory, the topic of conceptual change should be investigated from a network perspective. When a child who currently shows fragmented knowledge of buoyancy gets a better understanding of the concept, her network should develop closer to a coherent-like structure. For this, the e↵ects of a teaching intervention could be studied, in which the child is for instance encouraged to seek which features are important for floating and sinking (e.g. Butts et al. (1993); Hsin and Wu (2011)). Lastly, in order to verify the validity of network analysis to study children’s naive concepts of scientific processes, this graph-theoretic approach should be applied for other topics where there is dispute between mental models and fragmented knowledge, e.g. force (Ioannides and Vosniadou (2002); DiSessa et al. (2004)) or the Earth (Vosniadou and Brewer (1992); Straatemeier et al. (2008)). An important step in this developing research is to validate the formalization method introduced in this report (i.e. amount of clusters per network) to decide an individual’s position on
