Finding the Key Concepts of Students’ Knowledge

A Network Analysis of Coherence and Contingency of Knowledge Structures

Ismo T. Koponen

and Maija Nousiainen

Department of Physics, University of Helsinki, Helsinki, Finland

Keywords: Conceptual Structures, Coherence, Networks.

Abstract: The desired outcome of learning science is students’ expert-like subject knowledge, which is expected to be

at the same time well-organized, coherent and contingent. However, it has proved difficult to find ways to

represent these features and to identify the key conceptual elements or concepts that are responsible for them.

In this study concept networks constructed by physics students’ representing their views of the relatedness of

physics concepts are analyzed in order to clarify how coherence and contingency can be captured and

measured. The data consist of concept networks (N=12) constructed by physics students, representing

relationships between physics concepts of electricity and magnetism. The networks are first analyzed

qualitatively for their epistemic acceptability. The structure of the concept networks is then analyzed

quantitatively using a network graph theoretical approach. The analysis picks out a handful of key concepts

which all play a central role in all of the concept networks examined. From the physics point of view these

key concepts are relevant ones (most of them having to do with fields), which indicates the relevance and

power of the method in describing knowledge structures.

1 INTRODUCTION

Expert-like knowledge can be characterized as well

organized and utilizable; such knowledge is coherent

and contingent at the same time (Derbentseva et al.,

2007; Koponen and Nousiainen, 2013). Coherence is

obviously connected to structural organization of

knowledge (BonJour, 1985; Thagard, 2000), but

exactly what such coherence might mean and how to

recognize it, is rarely discussed. Contingency of

knowledge refers to different and alternative ways of

introducing concepts by using the support of already

known concepts (Scheibe, 1989). In order to study the

coherence and contingency of students’ knowledge

some suitable representational vehicles are needed to

illustrate the relationships obtaining between

concepts. One tool for representing pre-service

teachers’ conceptual understanding is provided by

concept networks (Koponen and Nousiainen, 2013;

Nousiainen, 2013; Börner et al., 2009).

The question of the structure of knowledge as it is

represented in a concept network is related to

principles of map design. A concept map which is

meant to be an expression or representation of

epistemically justified knowledge, as well as

communicable to others through argumentation,

needs specific rules and certain norms (Nousiainen,

2013 and references therein). Such concept networks

provide a lot of information on how students conceive

the structure and content of subject matter

knowledge, and are in fact related to question how

learner construct ontologies in learning. By visual

inspection only, however, it has proved very difficult

to quantify the essential differences (Koponen and

Nousiainen, 2013).

We present here an analysis of concept networks,

which takes into account the epistemic justification of

knowledge as it is represented in the networks, and

which also pays attention to the structure of the

networks. As a consequence, coherence and

contingency as epistemic and structural notions

become defined and yield to operationalization. The

method that combines qualitative analysis with

detailed quantitative methodology is advantageous in

the study of large, connected sets of conceptual

elements.

We use here the context of electricity and

magnetism as a specific example to show how

coherence and contingency emerge as a special type

of connectedness or relatedness of concepts, and how

the key concepts provide this coherence and

contingency. The research questions posed and

Koponen, I. and Nousiainen, M..

Finding the Key Concepts of Students’ Knowledge - A Network Analysis of Coherence and Contingency of Knowledge Structures.

In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2015) - Volume 2: KEOD, pages 239-244

ISBN: 978-989-758-158-8

239

answered in the present study the following:

 How can the key concepts providing coherence

and contingency be identified in students’

representations of their knowledge?

 What are the key concepts providing coherence

and contingency in the case of electricity and

magnetism?

These questions are answered using a sample of

concept networks and written supplementary reports

produced by physics students’, third year university

level. The analysis is based on a combination of

qualitative and quantitative methods. Qualitative

analysis is used in assessing the epistemic

acceptability of knowledge expressed in the networks

and written reports. Quantitative analysis based on

network theory is then used to identify the key

concepts providing coherence and contingency.

2 COHERENCE, CONTINGENCY

AND KEY CONCEPTS

The question of the organization of knowledge is

closely related to the ways conceptual knowledge is

acquired and justified using existing conceptual

knowledge and existing concepts. The ways that

concepts are used in that process tie them together,

provide meaning and eventually lead to an

interwoven web of concepts wherein they are related.

Coherence of such a web of knowledge arises from

the mutual support of relations and from the epistemic

justification of such relations. In addition, for

coherence to be useful and interesting, the knowledge

system must also be contingent (Scheibe, 1989;

BonJour 1985).

The coherence we are interested in here is the

coherent relatedness of concepts and other possible

elements of conceptual knowledge, for example

models (cf. BonJour, 1985; Thagard, 2000). These

coherent relations are based on specific types of

situations: using concepts either in the context of

describing or explaining the outcomes of

experiments, or using concepts as parts of models

which describe or generalize experimental results

(see Nousiainen 2013 and references therein).

Coherence with regard to experiments and

experimental observations ensures that a conceptual

system can be used in giving explanations and

making predictions of observed features of real

systems. The use of concepts is systematic and

symmetric in the sense that concepts retain their

mutual dependencies and relations in different

situations (BonJour, 1985; Thagard, 2000).

In educational settings, coherence is established

through instruction and argumentation, rather than

through genuine discovery. In physics education,

instructional settings providing means of introduction

of new concepts are most often different kinds of

laboratory experiments or modelling activities. The

experiments discussed here cover laboratory

experiments and the explanations which are given to

data produced in such experiments. The models and

modelling of relevance here is the most common way

to use models in physics teaching, namely providing

explanations and predictions (see Nousiainen, 2013,

and references therein).

Coherence which is produced through the above-

mentioned use of experiments and models connects

concepts to each other symmetrically so that if

concept A is connected to B, and B to C, a connection

between A and C also becomes established. In

practice, these types of connections give rise to

cyclical basic patterns, of which a 3-cycle of three

concepts is the most common (Koponen and

Nousiainen, 2013; Nousiainen, 2013). Therefore, in

what follows, coherence will be operationalized

through special counting of such cyclical, mutually

supporting connections.

The contingency of knowledge refers to the

different possible conceptual paths with which

concepts are related to each other successively, thus

providing different and alternative ways of

introducing concepts using the support of already

known concepts. Coherence and contingency are both

important aspects of scientific knowledge and are

expected to increase when the body of knowledge

expands (Scheibe, 1989; Chen et al., 2009). In

teaching and learning, contingency answers to the

questions of how, and in how many ways, new

concepts are introduced and justified on the basis of

concepts which have already been learned. This kind

of knowledge is an important part of the learner’s,

conceptual knowledge (Koponen and Nousiainen,

2013; Nousiainen, 2013). Contingency, as a

qualitative notion, is therefore related to the

multiplicity of ways a given concept participates in

connecting other concepts (BonJour, 1985; Scheibe,

1985). Different concepts in the web have thus

different epistemic and structural roles in providing

coherence and contingency.

The key concepts of the network are those ones

which provide the coherence and contingency of the

whole conceptual system. Coherence and

contingency, however, are notions that refer mainly

to structure. Reference to the epistemic content of

knowledge is also needed in order to represent

reliable knowledge. The key concepts should also

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

240

have strong epistemic status in the network,

recognized from their role in providing epistemically

well-justified connections.

3 EMPIRICAL SAMPLE

The context of the research reported here is an

advanced-level course for pre-service physics

teachers (third or fourth year students) in which

electricity and magnetism were discussed. Students

were asked to concentrate on their discussions and

reflections on the central concepts, laws, models and

experiments they thought important in forming a

well-organized picture of the content and structure of

electricity and magnetism. The design of the concept

networks discussed here is based on special kinds of

nodes representing the knowledge. The nodes in these

networks represent: 1) quantities, 2) laws, 3) models,

and 4) experiments. The linking words are describe

possible procedures and actions how nodes are

connected. Students were required to provide

epistemic justification for every node-link-node chain

they draw and describe it in a supplementary written

report. Other details of the design rules to construct

the concept networks are reported elsewhere

(Nousiainen, 2013). The empirical sample analyzed

in this study consists of 12 such representations. Here,

only the final versions are considered because the

final stage of the students’ understanding of the

relatedness of concepts is of interest in finding the key

concepts.

Figure 1: An example of s student’s network which contain

55 different concepts. Altogether 121 concepts were found

in students’ networks. Only schematic, network-like

overview is shown, the text in boxes is not meant to be read.

4 ANALYSIS METHOD

The analysis is a combination of qualitative and

quantitative methods. The qualitative analysis is

carried for epistemic justification of network nodes

and links, while the quantitative analysis is used to

operationalize coherence and contingency and to

identify the key concepts.

The epistemic validity of knowledge concerns the

epistemic acceptability of explanations students

provide in their written reports. Attention is paid only

to following four epistemic dimensions: 1) ontology,

2) facts, 3) methodology and 4) valid justification.

These four criteria form a suitable basis for the

analysis of epistemic acceptability of knowledge

represented in concept networks (Nousiainen, 2013).

These criteria form cumulative, hierarchical ladders

and values 1-4 are used indicative of the order in

which the above epistemic “norms” are fulfilled. In

addition to the epistemic analysis of the nodes, also

the links were evaluated using similar kind of

taxonomy applied to links. The epistemic analysis of

written reports and linking words is the only

interpretative part of the analysis. It produces the data

for the quantitative analysis in the form of epistemic

weights of nodes and links in the concept networks.

Coherence and contingency depend on the degree

of the epistemic justification and on the overall

connections of the system of knowledge, where the

key concepts have a special role. In order to find out

which concepts are the key concepts, we must

operationalize the properties of coherence and

contingency. For this, the information contained in

the concept network itself must be suitably

formalized so that the network yields to quantitative

analysis.

Formalization of Networks. After the interpretative

analysis, the important information contained in the

networks is carried by node and link strengths. All the

values were normalized to range from 0 to 1 so that

epistemically strong nodes and links have strengths

0.75–1, while weak nodes and links have small values

0.0–0.25. The strength of node i is denoted by s

while w

is the strength of the link from node i to node

j. In addition to strength, the node carries a tag τ

which specifies the type of the node, either

conceptual, experiment or model. The information on

epistemic strengths is simplified in further analysis by

rescaling link strengths so that the epistemic strength

of an initiating node and a link emerging from it are

aggregated to form a new weight w

→ s

motivated by the notion that directed links w

pass

information from node i to node j. The directional

weighting is important, because the ordering of nodes

depends on the order of argumentation represented in

networks. However, network is symmetrized w

= w

For final analysis and direction is taken into account

Finding the Key Concepts of Students’ Knowledge - A Network Analysis of Coherence and Contingency of Knowledge Structures

241

in weights.

Coherence and Contingency Operationalized. The

networks are now described fully adjacency matrix W

with [W]

. Coherence is related to closed cycles,

and the more there are such cycles a given node

(concept) participates in, the larger is the coherence

such a node provides for the network. Without further

mathematical details we note that coherence can be

operationalized as subgraph centrality SC

(Estrada et

al., 2012; Benzi and Klymko, 2013),













where k is the given node. Contingency is

operationalized similarly by counting open walks

between nodes p and q such that a given node k is

involved in the walk. The more there are such walks,

the more there are contingent paths from p to q

supported by k (i.e. more alternatives to connect p and

q via k). Contingency is then operationalized as

communicability betweenness centrality BC

(Estrada

et al., 2012; Benzi and Klymko, 2013),

WWW

qpq



  



  









where C is a normalization factor C=(N-1)(N-2). The

subgraph centrality and betweenness centrality are

centrality measures taking into account the whole

structure and its connectivity, based on information

flow, and thus better suited for purposes of

characterizing coherence and contingency than other

measures of betweenness and centrality, not directly

related to information flow (see Estrada et al., 2012).

The Key Concepts as Importance Ranking. The

nodes (concepts) that gain high values of SC

and BC

are the key concepts providing the overall coherence

and contingency of a concept network. The key can

be recognized using a normalized geometric mean of

and BC

. by introducing importance ranking IR

(Chen et al., 2009)



max{ }max{ }

SC BC



Importance ranking ranges from 0 to 1 and does not

depend on network size, which makes it possible to

make comparisons between very different networks.

5 RESULTS

The sample of N=12 networks had each on average

59 nodes and 97 links. Altogether 121 different

concepts were identified, of which 22 were

experiments, 37 were models and 72 conceptual

nodes. For each node k in each networks we

calculated the subgraph centrality SC

(coherence)

and betweenness centrality BC

(contingency). For

comparisons, the strength D

(number of links) of a

node as the sum of weights of links connected to this

node is also calculated. The results when all nodes

(experiments, models and conceptual) are taken into

account are shown in Figure 2.

Figure 2: The subgraph centrality SCk (top) the

communication betweenness centrality BCk (bottom) for

one concept network.

From figure 2 it is seen that a small subset of

concepts (labelled 27, 66, 71, 72, 75 and 109) has

large values of D

, SC

and BC

; thus they can be

identified as concepts that provide coherence and

contingency. The measures SC

and BC

seem to have

power to discern the structurally important concepts

but the changes are substantial.

In order to identify the key concepts so that the

identification does not depend on the size of the

network we calculate the importance rankings IR

. for

each of the student networks separately. Because we

are interested to compare individual students’

representations, we need to display the data so that

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

242

each node in each network can be compared. The

importance rankings of all 121 nodes, of which about

50-60 appear in a given concept network, can be

compared by representing them as a kind of

“spectrogram”, where most important concepts are

shown as black stripes and the least important ones by

white stripes. This spectrogram is shown in Figure 3

for all concept networks, with all nodes taken into

account, and for nodes with experimental (exp) and

model-based (mod) epistemic support separately. The

spectrogram makes it possible to compare different

networks at one glance. As figure 3 shows (top row),

the darkest stripes are concepts 2, 15, 27, 28, 38, 47,

51, 63, 66, 71, 100 and 109, which means that these

concepts are important in all concept networks.

Almost as important concepts stand out to be 8, 33,

57, 69, 83, 91 and 113. The middle row illustrates that

concepts 8, 51 and 57 are mainly supported by

experiments, and the lowest row that the concepts 28,

47, 69 and 83 are backed up by models. Note that

experiment and model support need to be compared

with each other to find out which one dominates.

However, most of the important concepts are

supported equally by experiments and models and the

differences in the importance rankings are not

substantial. A summary of the key concepts common

for all concept networks is as follows: charge (2),

Coulomb’s law (8), experiment: electric field lines

(15), superposition of fields (27), electric field defined

through force (28), work (33), electric potential (38),

electric flux (47), Gauss law (51), experiment:

magnetic interaction (57), experiment: magnetic flux

density (63), magnetic flux density (66), magnetic flux

(69), magnetic field defined through torque (71),

magnetic force of a moving charged particle F=qvB

(83), magnetic field as independent entity (91),

Faraday-Henry law (100), rotational electric field

(109), Ampere-Maxwell law (113).

Figure 3: Key concepts as identified on basis of Importance Rankings IR

of nodes (concepts) for all 12 networks and 121

nodes. Dark stripes denote the highest importance rankings; the lighter the stripe the lower the ranking. Note that certain

nodes (denoted in the figure with their numbers) have high rankings in many of the networks. The up-most row is for all

concepts, the middle row for experimentally supported and the lowest row for model supported concepts.

Finding the Key Concepts of Students’ Knowledge - A Network Analysis of Coherence and Contingency of Knowledge Structures

243

The similarity of different networks can now be

examined on the basis of the importance rankings.

This examination shows that in nearly all concept

networks the field concepts (28, 71, 91 and 109) are

the most central ones, and for them, experimental

support and model support are equally important. If

we focus on this core set of key concepts, the

networks are similar. In addition to this core set, there

is a handful of almost as important concepts (8, 33,

57, 69, 83, 91 and 113) which appear in many of the

networks. Although there is much variation between

students, there are, however, also many shared key

concepts.

6 CONCLUSIONS

Good organization of content knowledge is here

approached from the assumption that coherence and

contingency are two important qualitative features of

well-organized knowledge. These kinds of relations

are noted to be central for the functionality of

conceptual knowledge (Derbentseva et al. 2007;

Koponen and Nousiainen, 2013; Nousiainen, 2013).

The method presented here allows us to analyze key

concepts which provide the coherence and

contingency. Coherence is operationalized through

cyclical connections between concepts. Contingency

is operationalized as connected, not cyclical, paths

between given concepts. The key concepts were

found by forming an importance ranking on the basis

of these operationalized measures. The importance

rankings brought forward a small set of key concepts

which have a more important role than other concepts

in providing the coherence and contingency for the

whole set of concepts. These concepts turn out to be

meaningful from the point of view content, too, which

is of course a satisfying finding and not trivially

expected in this kind of learning context. In all cases

epistemic support from experiments and models was

found of equal importance, although slightly

differently for different key concepts.

Importance rankings also allow us to compare

networks: if the same nodes have high importance

rankings in two different concept networks, it means

that the networks are similar to some extent. The

analysis carried out here showed that all the 12

networks inspected here had much similarity in the

way they all emphasized the centrality of field

concepts.

In summary, our results suggest that concept

networks, if properly analyzed, contain valuable

information of how students organize conceptual

structure in physics. In particular, with network based

methods it becomes possible to identify the key

concepts that provide coherence and contingency of

such concept networks. This kind of knowledge is

important to understanding how human learners

construct ontologies in learning, how these ontologies

may differ, and how learning environments can

support the ontology construction by suitable

visualizations.

REFERENCES

Benzi, M., Klymko, C., 2013. Total communicability as

centrality measure, Journal of Complex Networks, 1,

124-149.

BonJour, L., 1985. The Structure of Empirical Knowledge.

Cambridge, Massachusetts: Harvard University Press.

Börner, K., Scharnhorst, A. 2009. Visual

conceptualizations and models of science. Journal of

Informetrics, 3, 191-209.

Chen, C., Chen, Y., Horowitz, M., Hou, H., Liu, Z.,

Pellegrino, D., 2009. Towards an explanatory and

computational theory of scientific discovery. Journal of

Informetrics, 3, 191-209.

Derbentseva, N., Safayeni, F., Cañas, A., 2007. Concept

Maps: Experiments on Dynamic Thinking. Journal of

Research in Science Teaching, 44, 448–465.

Estrada, E., Hatano, N., Benzi, M., 2012. The Physics of

Communicability in Complex Networks. Physics

Reports, 514, 89–119.

Koponen, I. T., Nousiainen, M., 2013. Pre-service Physics

Teachers’ Understanding of the Relational Structure of

Physics Concepts. International Journal of Science and

Mathematics Education, 11, 325–357.

Nousiainen, M., 2013. Coherence of Pre-service Physics

Teachers’ Views of the Relatedness of Physics

Concepts. Science & Education, 22, 505–525.

Scheibe, E., 1989. Coherence and Contingency: Two.

Neglected Aspects of Theory Succession. Noûs, 23, 1–16.

Thagard, P., (2000). Coherence in Thought and Action.

Cambridge, Massachusetts: The MIT Press.

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

244