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Good conceptual understanding of physics is based on understanding what the key concepts are and how they are related. This kind of understanding is especially important for physics teachers in planning how and in what order to introduce concepts in teaching; connections which tie concepts to each other give direction of progress—there is “flux of information” so that what was learned before provides the basis for learning new ideas. In this study, we discuss how such ordering of concepts can be made visible by using concept maps and how they can be used in analysing the students’ views and ideas about the inherent logic of the teaching plans. The approach discussed here is informed by the recent cognitively oriented ideas of knowledge organisation concentrating on simple knowledge organisation patterns and how they form the basis of more complex concept networks. The analysis of such concept networks is then very naturally based on the use of network theory on analysing the concept maps. The results show that even in well-connected maps, there can be abrupt changes in the information flux in the way knowledge is passed from the initial levels to the final levels. This suggests that handling the information content is very demanding and perhaps a very difficult skill for a pre-service teacher to master.

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Correspondence to Ismo Koponen.



Theoretical Quantities Used in Analysis

In a map (network) of n nodes, the variable a ij indicates the connections between nodes i and j so that if nodes are connected, then a ij  = 1 and if there is no connection, then a ij  = 0. These elements form a nxn-dimensional matrix, the so-called adjacency matrix, a. The justification of the knowledge represented by the links can be taken into account by giving a weight from 0 to 1 so that if L levels of the epistemic justification are fulfilled, the weight is given by w ij  = (L/4)a ij , where L = 0,1, …4 is the number of epistemic levels achieved. The weights for connections between nodes i and j are then set from 0.25, 0.50, 0.75 or 1.0 on the basis of their epistemic classification. The matrix w then gives the weights which describe the degree of the validity of the link, i.e. the plausibility of the knowledge represented in the maps. All quantities D k , C k , Φ k , and Ψ k of interest can now be calculated from matrices a and w. These quantities are central for characterising the connectedness of networks and their capability to pass on information. Within the literature of network theory, in particular, they discussed clustering extensively (Kolaczyk, 2009; Boccaletti et al., 2006), but they also discussed the information flux (Karrer & Newman, 2009). Their mathematical definitions in terms of the connectivity matrix a are given in Table 4. Here, however, we do not discuss the derivation or properties of these quantities further. The definitions can be used for weighted links simply by replacing a ij by weights w ij . The only exception is the clustering, where only the node, which is responsible for transitivity, i.e. the link between neighbouring nodes, is weighted and the term a ij a jk a ik is replaced by a ij w jk a ik so that only the link jk, which is responsible for the clustering, is taken into account. When the data are analysed and represented in terms of the quantities defined in Table 4, their phenomenological interpretation will be clear enough without the technical details or proofs.

Table 4 Mathematical definitions of variables for measuring the topology of the networks

Empirical Data for All Concept Maps, G1 – G6

In the main text, empirical data are given in detail only for representative cases G1, G2 and G3.

Here, all data are given. In Table 5, a complete listing of the concepts and laws is given in the order in which they appear in the maps (compare with Table 1).

Table 5 Complete listing of concepts and laws and their order of appearance in the maps

The concept maps in the form of embedded graphs are given in Figure 6. Numbering refers to the concepts in Table 4. These graphs contain all relevant empirical information of the pre-service students’ concept maps discussed in the main text.

Figure 6
figure 6

All six concept maps in spring-embedded forms, G1G6

The results of the analysis of epistemic weights using grades from 1 to 4 are given as relative weight factors w = 0.25, 050, 075 and 1.0 which are given in list form for concepts 1 – 34 in Tables 6 and 7. In a few cases, there are connections which refer backwards, but this represents deviation from the principle to introduce new concepts on the basis of old ones. These links are set to zero weights.

Table 6 Weights of the links in graphs G1 – G6

The average values of the clustering and fluxes for graphs G1 – G3, based on the unweighted (all links either 1 or 0) links and weighted links with weights given in Tables 6 and 7, are shown in Figure 7 for maps G4, G5 and G6 (maps G1 – G3 are shown in the main text). The distributions of the values are shown in Figure 8. For all maps G1 – G6, the average values X = {D,C,Φ, Ψ} and corresponding dispersions δ X  = σ X /X, where σ X is the standard deviation, are given in the main text in Table 3.

Figure 7
figure 7

Node-by-node (nodes 1 – 34) values of degree D, clustering C, and fluxes Φ and Ψ for maps G4 – G6. The weighted values are shown as star symbols; unweighted values are given as bullets

Figure 8
figure 8

Frequency distributions of degree D, clustering C, and fluxes Φ and Ψ for maps G1 – G3. The weighted values are shown as greyscale bars; unweighted values are given as white bars

Table 7 Weights of the links in graphs G1 – G6

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  • concept maps
  • knowledge organisation
  • teacher education
  • teaching plans