construction and the method of analysis will be based on an equal footing of the ... different type of order, originating from the âgenealogyâ of conceptual systems;.
Science & Education, in print
Coherent knowledge structures of physics represented as concept networks in teacher education Ismo T. Koponen and Maija Pehkonen Department of Physics, University of Helsinki, FINLAND
ABSTRACT. A characteristic feature of scientific knowledge is the high degree of coherence and connectedness of its conceptual structure. This notion is also behind the widely accepted instructional method of representing the concepts as networks. We suggest here that notions of explanatory coherence and deductive coherence naturally connect the structure of knowledge to the processes which are important in constructing the concept networks. Of these processes, experimental method and modelling are shown to be closely connected with explanatory and deductive coherence, respectively. From this viewpoint, we compare here how experts and novices represent their physics knowledge in the form of concept networks, and show that significant differences between experts’ and novices’ quality of knowledge become directly reflected in the structure of the networks. The results also show how concept networks make visible both the structure of knowledge and the methodological procedures, which support its formation.
1
1. Introduction In science education one of the recurrent themes is the importance of fostering formation of organized and coherent knowledge structures during learning. The goal is often set as being the expert’s knowledge, which is thought to be well organized, coherent and consisting of a rich body of knowledge about the subject matter (Chi et al. 1981; Bransford et al. 2000; Snyder 2000; Mestre 2001). These demands are particularly clear in learning and teaching physics knowledge structures, which are often assumed to be well-organized and hierarchical structures (Chi et al. 1981; Moreira 1985; Snyder 2000; van Zele et al. 2004). Any attempt which sets as its goal to give an idea of the structure of knowledge must first concentrate on the question, what are the structural aspects which can be recognized and which are of importance. Many studies within philosophy of science have emphasised the hierarchical structure of physics knowledge, most notably the logical empiricist views (Nagel 1961; Suppe 1977) but also inherited in semantic views on theories (Suppe 1977; Giere 1988, 2006). It is more or less evident that in the strongly reconstructed form of finalized theories, the physics knowledge structures must be hierarchical. However, physics knowledge is not only hierarchically arranged but it is quite often described as webs or networks, where concepts are linked to other concepts and principles (Anderson 2001; Wilson 1999; Thagard 1988, 1992, 2000). The concepts of such networked conceptual systems are far from being independent; instead, they are fundamentally entangled and can be used only with an understanding of their interdependency. This overall coordination between concepts is the core idea in the notion of coherence (Lehrer 1990; Thagard 1992, 2000; Kosso 1992, 2009). Understanding the interconnected and networked nature of knowledge and the coherence it brings with it are in focus not only to understand the structure of science but also to understand the way in which scientific knowledge is produced and accepted (Kosso 2009). The notions concerning the organization of scientific knowledge and expert’s knowledge have had a direct impact on goals for teaching and education, where much effort has been paid onto developing instructional strategies that may help students to create organized knowledge structures. In educational settings, the use of graphical knowledge representation tools like concept maps has been one promising line of approach which is known to support the formation of such organized knowledge (Novak and Gowin 1984; Ruiz-Primo and Shavelson 1996; Trowbridge and Wandersee 1989; diSessa and Sherin 1998; Ingeç 2008). The importance of knowledge organization has thus been recognized. Consequently concept maps have been used at all levels of science teaching as well as in textbooks as a representation of knowledge. However, the interconnectedness and coherence of knowledge are only seldom, if ever, reflected in textbook images of science or in science teaching, although there should be all reasons to take these important issues into account. The research concentrating on the structure of concept maps has brought forward the notion that interconnectedness and patterns, which are more complex than hierarchical patterns, are important for learning and cognition. It has been suggested that good understanding and high quality of students’ knowledge is reflected as connected network structures (Vanides et al. 2005; Ingeç 2008; van Zele et al. 2004; Kinchin 2000, 2005) visually complex and branching structures (Slotte and Lonka 1999), logically connected chains (Machin and Loxley 2004), or as hierarchical structures (Moreira 1985). Although there are different ideas about the advantageous structure, the commonly accepted view is that the structure and content are connected.
2
In these studies, the important structural features have thus been recognized, but little has been said about the origins of these more complex structures. Only recently the issue has been raised that the cyclicity and clustering in concept maps can be related to the processes of knowledge formation. When new concepts are added onto the network, there is certain co-variation between concepts, i.e. when one concept is changed or affected, other concepts become affected too (Safayeni et al. 2005; Derbentseva et al. 2007). This “co-variation” is of course the basic feature contained also in the notion of coherence. In this study, we develop further the background theory for construction of the concept networks – as well as the method to analyse them. The method of construction and the method of analysis will be based on an equal footing of the explicated background theory of coherent knowledge. In organization one of the most central issues is the question about the coherence of the conceptual networks (Kosso 2009). The concept network with a high quality of content should represent concepts and principles, which are coherent enough to allow coherent explanations, coherent deductions, and coherent analogies within the system (Thagard 1992, 2000). If the conceptual system possesses these coherent features, it has the possibility of producing predictions, explanations and, therefore, of bringing understanding. In this picture, an empiricist method becomes a method globally supporting the acquisition of coherence (Koponen and Mäntylä 2006; Mäntylä and Koponen 2007; Koponen 2007; Kosso 2009). Within the above-outlined coherentist framework, conceptual structures can be represented as concept networks, where connections between concepts are formed through the explanatory and deductive coherence. However, it is not yet enough to provide overall notions of coherence and network structure of knowledge, but one must show how the coherence can be recognized and evaluated. In this study, we examine the uses of concept map -like network representation of physics knowledge and their structural features in representing organized, coherent and interconnected knowledge structures. In order to recognize important structural and topological features signifying coherence, we have examined the structural features of concept networks drawn by experts in physics instruction and by novices (university students). The structural features are analysed by using graph theoretical methods. It is shown that experts’ networks are characterized by coherence and hierarchies, which are built into the network-structures. In novices’ concept networks similar features are found only in the best cases. Many novices produce concept networks with only partial coherence and a fractured organizing hierarchy, and in some cases there is no structure at all. Finally, we discuss the implications of these notions for physics teacher education.
2. Coherence of knowledge and coherent conceptual structures The question concerning the structure of knowledge is a long standing problem in philosophy of science. The most well-known and detailed attempts to answer this question are probably given within logical empiricism, which displays knowledge as a hierarchically organized system of concepts, laws and propositions (Nagel 1961; Suppe 1977). The hierarchical view is inherited also in the semantic view on theories, which sees a hierarchically ordered set of models at the core of knowledge systems (Suppe 1977; Giere 1988). However, within knowledge systems there is also a
3
different type of order, originating from the “genealogy” of conceptual systems; concepts, laws and principles are based on precedants, which affect how the conceptualisation proceeds and how new knowledge becomes justified and annexed into an existing body of knowledge. In this case the knowledge system is naturally seen as an interconnected network or web of concepts and laws, which is more complex than a hierarchically arranged system (Thagard 1988, 1992, 2000; Kosso 1992, 2009; Wilson 1999; Anderson 2001). The networked system of knowledge thus has structural properties related to the interconnectedness of the networks’ elements making it the ”seamless” or ”tightly interwoven” web of knowledge (Anderson 2001). These kinds of structural properties related to the structure of knowledge are not captured by the notion of hierarchy alone; instead, we need to turn to properties related to coherence and consistency. The coherence of knowledge is thoroughly discussed in contemporary philosophy, at least from the point of view of epistemic justification of knowledge, principles of reasoning and also from the point of view of truth (see e.g. BonJour 1985; Harman 1986; Lehrer 1990). Still the notion of coherence itself has remained somewhat ambiguous (for a detailed discussion, see Thagard 2000). What coherentist views agree on, however, is that the justification for accepting a certain hypothesis comes from its capacity to give, together with other hypotheses, explanations in a systematic, consistent and non-contradictory way (BonJour 1985; Haack 1993). The advantage of the notion of coherence is that it is related to the system of knowledge as a whole, not only in a piecemeal way to a local set of propositions or models. The coherentist view of knowledge sees the knowledge structures as globally connected systems. In addition, the inclusion of new knowledge takes place so that large parts of the structure are involved and the structure itself is affected (Kosso 1992, 2009). The coherence of hypotheses are an important feature in their acceptance and justification as a part of the system (Kosso 1992). Not all hypotheses, however, can be characterized as having an equal importance in the process of acceptance and justification. Those hypotheses which are central in giving explanations for observations or for outcomes of experiments need to have a special role, because only through them does the system of knowledge become connected to reality (Haack 1993; Thagard 1992; Kosso 1992). The idea of a special role of coherence in the context of giving explanations has been elaborated further by Thagard (1988, 1992, 2000, 2007), who on this basis introduces the idea of explanatory coherence, which is concerned with giving coherent explanations of real phenomena. In addition to explanatory coherence, Thagard distinguishes five other types of coherence, of which the deductive coherence is concerned with logical consistency of the knowledge system. In what follows, only the explanatory and deductive coherence are discussed in more detail. The explanatory coherence is quite naturally connected to the methodology of the experiments. The deductive coherence, on the other hand, is closely related to the deductive use of models and model-type symbolic relations. The motivation behind this restriction is the notion that, in teaching and instruction, experiments and modelling are central procedures connected to the construction and use of knowledge. Experiments and modelling are often discussed as separate activities but very seldom from the point of view of how they combine to create an organized web of concepts and laws.
4
2.1. EXPLANATORY COHERENCE AND EXPERIMENTS Explanatory coherence deservedly plays a special role, because through it the conceptual system becomes connected to real phenomena. Explanatory coherence is obtained when hypotheses and propositions are used consistently in regard to their targets of explanation, which can be e.g. observations or experimental results. When explanatory coherence is achieved, all those hypotheses that together explain a certain specific situation must retain their relations with each other when they are used in related but different situations (Thagard 1992, 2000). This simply means that in a coherent system the co-variation of concepts is constrained and that this rigidity is a hallmark of coherence. Of course, such coherence and rigidity is necessarily also a requirement of validity of the scientific knowledge (Kosso 1992; 2009). The explanatory coherence is readily extended to cover laboratory experiments and the explanations given to the data produced in such experiments. In this case it is evident that the explanatory coherence becomes connected to the methodology of designing experiments, of acquiring knowledge based on interpretation of experiments and of justifying such knowledge. The close connection between the coherence of knowledge and the experimental methods has recently been noted also by Kosso (2009), who speaks in favour of a “global empiricist” method. The empirical method globally provides the coherence of the structures, rather than functioning locally in justifying or falsifying isolated hypothesis. The experiments are always designed within the framework of existing theoretical systems, their results are always interpreted by embedding the results in the existing frameworks and in this embedding – in the most favourable cases – the theoretical framework becomes altered and acquires better coherence (Kosso 1992, 2009; Koponen and Mäntylä 2006; Koponen 2007). Also from the point of view of physics teaching and instruction the experiments of most interest are the so-called quantifying experiments, where the concept is operationalized and made measurable through the pre-existing concepts. Although there are many different types of experiments that are relevant for teaching and learning, the quantitative experiments have a very special role for the construction of conceptual and theoretical systems. This special role comes from the close connection of quantitative experiments and theory. In teaching and learning, quantitative experiments can be used to support the construction of the meaning of concepts and form new relations between quantities in a conceptual system. Of course, there are many different types of experiments but, as we have recently discussed (Koponen and Mäntylä 2006; Mäntylä and Koponen 2007), the quantitative, operationalizing experiments have a special role owing to their methodological controllability and these kinds of experiments are also valuable ones in advanced physics instruction. The quantitative experiments that operationalize concepts and laws thus have an important role in physics education, because they contribute to constructing the meaning of concepts by producing an interwoven web of concepts and laws. In operationalizing experiments a new concept or law is always constructed sequentially, starting from the already existing ones, which provide the basis of an experiment’s design and interpretation. This methodological process not only creates the web of concepts and laws but it also does it in a controllable way. In its most idealized (but still sufficiently truthful) form the new concept and law C is formed on the basis of two pre-existing concepts A and B so that the operationalization creates C on the basis of the relations AC and BC, which also requires that A and B can be related as 5
AB. There is then a triangular and inductive-like mutual dependence ABCA, which we have previously described as an inductive-like cycle or a cycle of generative justification (Koponen & Mäntylä 2006). The use of concepts A and B is such that triangular co-variation between concepts A, B and C is also now the basic form of coherence, which ensures that also C has explanatory coherence in regard to the experiment in question. This kind of triangular interdependency has been argued as being important for both the cognitive stability of knowledge patterns (Leve 2006) and an essential feature of dynamic, functional knowledge (Safayeni et al. 2005; Derbentseva et al. 2007). Therefore, such a triangular inductive-like pattern, although highly idealized, is arguably a basic central pattern indicating coherence and, furthermore, related to explanatory coherence. The concept network, which gradually becomes constructed through repeated use of operationalizing experiments, naturally has explanatory coherence with respect to all those experiments used to construct it. In what follows, we concentrate on these kinds of experiments in the construction of coherent concept networks for purposes of learning and teaching. It will be shown that the coherence produced by operationalizing experiments can also be recognised in the structure of concept networks designed for purposes of representing physics knowledge. 2.2. DEDUCTIVE COHERENCE AND MODELING Models are the core components of knowledge structures, as well as in the use of knowledge itself. There are of course many different views on how models operate and what they actually are, but common to all views is that there is a certain degree of interdependence between models and the rules with which to produce and use models (Hestenes 1992; Giere 1988, 2006; Morrison and Morgan 1999; Koponen 2007). Mathematical and logical consistency in the construction and use of models is a selfevident requirement of consistent and coherent use of models and modelling. In this work we concentrate on the use of models in giving explanations or description in connection to experiments. Typically, a model may be an idealized and symbolic representation of dependencies found in an experiment (e.g. an idealized interaction law for point charges or masses) or it may be a description of dependencies taking place in experiment (e.g. the formation of a homogeneous electric field in a capacitor system). In this mode of usage models are always produced from theories following a set of modelling rules (Hestenes 1992; Giere 1988, 2006; Sensevy et al. 2008). Deductive coherence is closely related to the use of models and modelling. The most common way to use models in physics teaching is namely a deductive way of providing explanations and predictions (Hestenes 1992; Sensevy et al. 2008). In this case, model construction is carried out with comprehensible rules subordinated to higher level theory, and the constructed models are then validated by matching them with experimental results (Hestenes 1992; see also Koponen 2007). In such a use of models it is fairly evident how modelling will provide the deductive coherence for the knowledge structures. There are other modes of model usage, where models are more inherently the basis of experiments’ design and interpretation, as we have discussed in more detail elsewhere (Koponen 2007). In this latter mode, deductive coherence naturally becomes connected to the gradual fitting of experiments and models so that a closure of understanding of the experimental phenomena within the range of existing models becomes possible, a process we have called matchmaking (Koponen 2007). However, this kind of more complex modelling is beyond the scope of the present study where only the more realistic and theory sub-ordinated modelling is taken into account. 6
The use of models subordinated to theory is simple enough to yield a more formal description. In the simplest case, the modelling procedure sequentially uses existing concepts to produce a better understanding of the use of the concept or to define it better. In this case the structure concept or principle A is used to model a certain situation (e.g. through idealization) so that a new concept, law or principles B and C become hypothesized, and which then become connected to A through the modelling procedure. This cycle CABC is structurally identical to a triadic cycle found in the case of operationalizing experiments. In the case of modelling, however, it often happens that concept A is generalized or a new concept is hypothesized so that B can be used as a basis to form several new concepts, laws or principles This kind of deductive pattern has a directed hierarchical backbone, which starts with a dyadic directed connection AB and continues then to a set of concepts {C1,…CN} as a directed spoke B{C1,..CN}. Then the pattern AB{C1,..CN} constitutes a directed tree with N branches. The modelling process based on the deductive use of a concept is naturally connected to the idea of deductive coherence, which has an important role in giving shape to the internal consistency of a conceptual system. In deductive coherence as outlined by Thagard (2000) the cohering elements are mathematical propositions, axioms and theorems. The basic instance of deductive coherence is when e.g. a theorem is deduced from an axiom, in which case the axiom and theorem cohere symmetrically with each owing to the nature of the construction process. Again the coherence is achieved through a constrained co-variation between the concepts concerned in the deduction. The coherence when achieved gives mutual support to both theorems and axioms (Thagard 2000). This kind of deductive coherence as a basis for justifying mathematics knowledge structures has been defended also by Russell (1973) and Kitcher (1983). In the case of modelling, deductive coherence is never this simple, but a similar type of mutual justification through the consistency of the construction process can be recognised. We have here discussed experiments and models as separate processes but this has been done mostly for the sake of simplification. It can be argued that in the authentic practices of physics, models and experiments are much more a complex and essentially bi-directional process, where models are used to make a match between experiments and theory. This is a process of mutual matching, where models representing the experimental data, and on the other hand, the deductively produced theoretical models, are sequentially adjusted and transformed so that they eventually become structured into a mutually compatible form (Koponen 2007; Morrison and Morgan 1999). This process is a global one, where the interdependence of concepts and laws is essential in order to make the matching process possible; no results can be judged in isolation or annexed to the system independently of the system of knowledge already existing. The operationalizing experiments self-evidently contain this component of bi-directionality because the experiments are designed on the basis of theory and the results are also interpreted in terms of models. However, this subprocess is not described here explicitly, but is rather understood as a part of experimental procedures. Therefore, the mutual interdependence of experiment and its interpretation in terms of models also create a strong deductive coherence on the whole system of knowledge. 2.3. CONCEPT NETWORKS Experimental procedures and modelling procedures which fulfil the requirements of coherence and which are repeatedly used to construct the concept network will bind 7
the concepts together into an interwoven web. This web of concepts then emerges out of the procedural connections, i.e. the way in which the experiments and models are used. If the procedures are used in a methodologically correct manner and in the simplified form explained in previous sections, one should recognise the triangular and hierarchical motifs in the structure of the networks. Here we discuss only the context, where knowledge is taught and learned during instruction, when the validity of knowledge is found through argument and example rather than through research. It is therefore essential to understand that the picture outlined here is not about real science and its practices, but one of scientific knowledge as it becomes displayed in education. Nevertheless, the coherentist account of knowledge, where explanatory and deductive coherence now only are taken into account, naturally leads to the idea that the connections between concepts and the principles linking the concepts are of central importance in establishing the meaning of the conceptual system. The coherence of the conceptual system is of utmost importance also for the validity of knowledge, because the coherence is an indispensable and necessary requirement of valid knowledge (Kosso 1992). Internal coherence is of course not yet a sufficient condition for valid and truthful knowledge. However, the more the knowledge structure shows explanatory coherence with respect to real phenomena and experiments, the more its validity and truthfulness is enforced. Therefore, the better the coherence the better is the quality of knowledge. The viewpoint where concepts are seen as elements of complex network-like structures and where a special role is given to interrelations between concepts, puts weight on the notion that concepts cannot be defined semantically or in isolation of other concepts; ”rules connected to concepts are parts of them as well concepts are part of the rules” (Thagard 1992, p. 30). This view approaches cognitively oriented views on concepts suggested e.g. by Minsky (1975) and Rumelhart et al. (1986). According to Minsky (1975), concepts can be seen as frames, which are understood as symbolic structures storing those attributes, which connect a concept to other concepts. A related view has been suggested by Rumelhart et al. (1986), who see concepts as schemas activated in the mind. These schemas consist of sets of connected nodes, which become activated as a whole when a concept is used. In the network view concepts are not isolated, defined entities; instead, the meaning of concepts is connected to the whole network of concepts or conceptual systems. In Thagard’s network view, conceptual systems can be ”analyzed as a network of nodes, with each node corresponding to a concept, and each line in the network corresponding to a link between concepts” (Thagard 1992, p. 30). This notion will be utilized here in constructing and analysing the concept networks. The network view of concepts has gained attention not only in cognitively oriented philosophy of science but also in scientists’ own reflections on the nature of knowledge. For example, physics knowledge has been described as a seamless conceptual web (Anderson 2001) or a web providing unity of knowledge (Wilson 1999). There are several reasons why the network view has attracted attention, and one is apparently the notion that the network is a natural way to take into account the manner in which nearby concepts are functionally related. An entangled web of concepts and rules makes possible the application of concepts in different situations by providing explanations and making predictions. The instance of providing an explanation or prediction enables the functional relation of concepts. It is this functional relation which gives the symbols meaning, and it concerns both the symbol’s relation to other symbols as well as its connection to the external world (Thagard 1988; Harman 1987). Therefore, as Thagard (1988) notes, the meaning of a
8
concept emerges not only by learning the definitions and rules, but also in the ways in which the concepts and rules in various problems are applied. In order to accomplish their function, concepts need to relate to other concepts in ”various inductive, hierarchical, non-definitional ways with other concepts. That is how meaning emerges” (Thagard 1988, p. 70). These inductive and hierarchical interrelations of concepts are based on local connections but are essentially global and holistic features of the network. The question centres on the ”architecture of knowledge” (Kosso 1992), captured now in the level of network representation. Therefore, understanding the global structure is necessarily a part of understanding the concepts themselves. The global structure of the conceptual network not only carries information about the methodological procedures in making the connections, but serves as an essential part of the nature of the conceptual knowledge as well.
3. Construction of concept networks: Rules produce structures and coherence The structure of knowledge and the coherence of knowledge, even when recognized, is a difficult issue to discuss in a more definite and detailed way. If we may assume that structures of representations of knowledge carry enough messages of the structures of knowledge, we can focus on certain types of network representations and structures. The question about the structures of representations is central also in regard to learning and instruction. Working with and building networks can, in some sense, be seen as a continuous conceptual change where new conceptual systems are obtained by addition and deletion of nodes, agglomeration of branches of the network, or restructurisation of the network as a whole (Thagard 1992; diSessa and Sherin 1998). Therefore, it is of importance for learning and instruction to develop tools for making conceptual structures visible, so that it becomes possible to evaluate how concepts are linked together. One possibility is to use concept maps (Novak and Gowin 1984), but then, if one is interested in coherence, attention must be paid to the rules which are used in constructing and evaluating the maps. Moreover, the rules must be closely connected to the underlying conception of knowledge, and not merely to ideas connected to the learner’s cognitive processes. This aspect of the theory of knowledge is usually lacking from most suggested applications of concept maps, and therefore we propose here an alternative to traditional concept mapping. In order to produce epistemologically well-defined content and structure in a concept network representing the connections between concepts or laws, the connections can not be free associations. It is not necessary to restrict the connections to simple propositions, as suggested by some researchers advocating the use of concept maps (Novak and Gowin 1984; Ruiz-Primo and Shavelson 1996; van Zele et al. 2004; Slotte and Lonka 1999). The idea of explanatory and deductive coherence as discussed here and their centrality in structuring the propositional system as a whole suggests that connections themselves need rather to be certain procedures, e.g. experiments or deductions, which establish the connections. When the connections are made on the basis of procedures, the interrelations of concepts to each other reflect the constraints for which the real experiments and the logically and mathematically possible procedures allow. The connections are then no longer free associations or subjective beliefs, but are based instead on scientifically acceptable procedures. Of course, not all possible procedures are tractable in representations meant for purposes of teaching and instruction, and therefore we have retained only the quantitative and
9
operationalizing experiments and deductive modelling procedures. Moreover, regarding deductive modelling procedures, simple logical deduction and definitions are also taken into account. The concept networks studied here are constructed in such a manner that the nodes and edges of the network constitute the conceptual elements and rules which are used to construct the networks. Nodes are the basic elements of the concept networks and they can be: 1. 2. 3.
Concepts or quantities. Laws, i.e. particular laws or law-like relations. Fundamental principles.
Of these elements, laws could be taken as particular experimental laws (inferred from measurements) or law-like predictions in specific situations (derived from theory). Fundamental principles feature as the highest-level principles or axioms of theory. All laws are thus relations between concepts, produced by following well-defined procedures or are otherwise formed in a definitional way (principles). Edges, which connect the nodes, are procedural rules, which can be: 4. 5.
Experimental procedures, which are operational definitions of a concept or laws or their demonstrations. Modelling procedures, which can also be simple definitions or logical deductions.
The procedures therefore play a central role in conferring coherence on the structure. In regard to coherence, experimental procedures need to possess explanatory coherence. The procedures need to be such ones, that their theoretical interpretations provide explanations for the regularities found in the measurement data. The logical procedures need to possess deductive coherence, whereby these are mostly modelling procedures, in which the definition of a concept or law is given as a law-like relation; or then, a model is produced to describe a certain situation or the making of a prediction in deductive fashion. The rules define what kind of elements we are allowed to use in the construction of concept networks and also how we can combine these elements. Connected to rule 4 is the basic triangular motifs ABCA of experimental procedures (see section 2.1) and connected with rule 5 are the triangular motif CABC and the hierarchical motif AB{C,…,E} of modelling procedures (see section 2.2). These motifs are shown graphically in Fig. 1.
10
Fig. 1. The basic triangular motif corresponding to inductive-like experimental procedures (left), and triangular (middle) and hierarchical (right) motifs corresponding to deductive-like modelling. . Rules 1–5 for the construction of the concept networks will provide explanatory and deductive coherence, but they also restrict the possibilities of what kinds of knowledge can be represented in these networks. Although, this is a restriction, it is not too severe and does not prevent displaying essential and useful information in the form of a concept network. The most severe restriction of the rules adopted here is that our conceptual structures will be limited to laboratory phenomena and experiments in the form discussed in more detail by Koponen and Mäntylä (2006) and Koponen (2007). Similarly, the deductive coherence contained in the structures is limited to logical procedures and definitions. In representing typical target knowledge in physics instruction and learning this seems to be adequate and serves as a starting point in order to demonstrate the advantages of a viewpoint emphasizing coherence and the ways to make it explicit in instruction. 4. Recognizing the coherence of the novice’s and expert’s knowledge The basic questions one needs to face in trying to foster the formation of coherent knowledge structures is, first, whether or not the assumed coherence and structure can be somehow recognized within the concept network. Second, one should discover whether or not the structure is related to the content of the knowledge the network is meant to represent. The basic argument behind the construction and analysis of the concept networks runs then as follows. 1. The coherence of knowledge, which is restricted in explanatory and deductive coherence, is a requirement reflected in the form of experimental and modelling procedures, as explained in section 2.1 and 2.2. 11
2. The coherence requirement enforces on the interrelationships certain recognizable relationships, which on the idealized level can be described as either triangular or hierarchical basic motifs (see Fig. 1). 3. Different types of coherence are connected to different motifs, which can then be made measurable. We can thus operationalize the notion of coherence through the motifs. The attempt to recognize coherence in knowledge representations is, however, a bit more demanding a task than just e.g. counting the number of motifs. The problem of course is that networks are made of interconnected motifs and there is no unique way to decompose the network on the basis of motifs. Instead, we need to formulate global measures which reliably measure the important features contained in the motifs. This final step that completes the operationalizing of the notions of coherence are discussed in the next section. Before introducing this final step, however, we place the general ideas presented here in a specific context of representing knowledge in the domain of electrostatics. This helps to embody the general notions of coherence, procedures and motifs. In order to clarify the point regarding the usefulness of these general notions, we have examined the concept networks which were constructed by experts and novices. 4.1. EXPERTS’ CONCEPT NETWORK Three experts, who were physics instructors, were given the task to represent connections between concepts of electrostatics in a concept network. In order to have representations, which could be evaluated and analysed as unambiguously as possible, the experts were introduced to the use of rules for connections (links or edges) and rules to make a distinction between different entities in the concept network. The experts were familiar with the idea of concept mapping and had used it in their teaching to some degree before. Following the rules outlined in the previous section, experts produced a preliminary network which, after some discussion, they modified over several stages. The final version which they agreed on and which they thought contained the essential elements of electrostatics is shown in Figure 2. In this representation nodes are concepts (boxes), laws and principles (boxes with thick borders). The links are either procedures of the operationalizing experiments (E) or the modelling procedures classified as logical deductions (L) or symbolic definitions (D). In addition, conceptual (geometrical) models are shown with ellipses, and entity-like objects with rounded boxes. It should be noted that the numbering in links denotes the sequence of steps when construction of the networks proceeds.
12
The experiments shown in the concept network are standard student laboratory experiments done in the context of electrostatics but are now formulated so that they support the concept formation as explained in sect. 2.2. and discussed in detail elsewhere (Koponen and Mäntylä 2006; Mäntylä and Koponen 2007). For example, the triangular motif leading to Coulomb’s law (10) initially requires qualitative notions of a charge (1) and repulsive/attractive electrical forces (2) caused by a charge (or charging). Then a particular idealized experiment, the Coulomb’s experiment (E2) with spherical capacitors can be designed. The outcome of this experiment, when strength of charging (amount of charge) in capacitors is changed as well as their distance is varied, allows hypothesizing the symbolic form of Coulomb’s law. This finally enables measurability of the charges, in turn, defining the quantity of the charge through Coulomb’s law. This completes the final part of the quantitative and operationalizing experiment E2. Experiment E17 leading to capacitor law and quantification of the capacitance is very similar in structure. Some experiments, like E8 and E20 are considerably simpler, because in them only one aspect in the situation
Fig. 2 The experts’ concept network for electrostatics. In the network is shown: concepts (boxes), laws and principles (boxes with thick borders), definitions (D) and logical deductions (L), and quantitative experiments (E). A conceptual (geometrical) model is shown as an ellipse, and entity-like objects as rounded boxes. is changed: In E8 the test charge (small charge vs. large one) is introduced and its position is changes, thereby enabling to define the electric field strength. and in E20 the test charge is displaced in the field between the plates of a planar capacitor system, enabling to generalize the mechanical work for displacements if electric field. Most of the logical deductions are the deductive modelling examples, designed to explain and clarify certain experimental situations. But often these are already included as a part of the experimental procedures; for example, in E2 the model of radial force-lines and point charges is involved, and in E17 the homogeneous field
13
emerges with analogy to a gravitational field. Another type of modelling is used e.g. in the case where the electrical field (16) is first introduced in L6 as a concept redundant to an electrical force (the force is divided by smaller charge) but then idealized and generalized so that it is understood and defined (L7) as an entity created by a charge and felt by another charge. This definition can be tested based on Coulomb’s law, which then in turn enforces the position of Coulomb’s law as a basic law of electrostatics. On the concept of field (16 and 6) one can base the notion potential energy of the electrical interaction, by using an analogical model with a gravitational field (L10). This completes a triangular cycle consisting of nodes 16, 6 and 8 and links L6, L7 and L10. The analogical model with a gravitational field is also used in triangle consisting of nodes 6, 8 and 10, where the principle of energy conservation is used as an already understood principle in mechanics, but now applied in the case of electrical field (L10). When these structures are viewed a bit differently, we can recognize the hierarchical deductive chains, e.g. one starting from 16 and going to 8 and then from 8 to 7 and 9, through links L6, L15 and L11, where e.g. L15 generalizes the idea of mechanical work for the electrical field by using a line integral representation. Other similar deductive chains are easily recognized in Fig. 2. Similar types of networks as shown in Fig. 2 were produced also by physics teacher students, who produced collaboratively in small groups (up to 3 persons in each) 20 concept networks of the same topic, with similar concepts. All the students were already familiar with concept mapping and the use of concept maps. The students who produced the networks were in a teacher preparation course (third year studies) and had studied the standard first year university courses on electricity and magnetism and electromagnetism, but they had not done any advanced studies. Therefore, the concept networks constructed by students are expected to be more like novices’ concept networks. The students were given a list of concepts and laws that contained the same concepts and laws as the experts’ network. On the basis of the concept list, the students were asked to represent the connections and also to define the nature of the connections, i.e. whether they were experimental procedures or logical procedures. In addition, a description of the procedure was required as a sequel. 4.2. STRUCTURE OF EXPERTS’ CONCEPT NETWORK The interesting feature of the experts’ concept network is the high connectivity of concepts and the tendency of some concepts to attract edges. However, more detailed analysis of the concept networks is difficult without methods to analyse the topology of the networks. It needs to be noted that the layout of the network, i.e. the specific way it is drawn, should not be emphasised too much. Instead, the important connections are inherent dependencies coded in the topology of the structure and they are not immediately visible from a given layout. The topology of the concept network can be examined more closely by using the methods of formal graph theory. The steps followed to analyze the structure are: 1. Coding the network to a connectivity (adjacency) matrix. 2. Using different representation modes i.e. different embeddings. 3. Calculating characteristic measures of topology for the network. The coding of the network’s nodes and edges is done by using only the binary values 0 and 1, so that no other qualities but the existence of connection and its direction are coded. The content of the network will then of course be reduced, because only 14
elementary structural aspects will remain. However, this kind of reduction and separation of structure and content is necessary if we want to give a more precise meaning to the notions of network-like structure and coherence at the structural level. The redrawing and transformation of networks was done using Combinatorica –program (Pemmaraju and Skiena 2006), which allows using different rules to redraw the networks and to compare their topological features. We used two different graphembedding methods. The first one is the spring-embedding. Networks are then redrawn as undirected structures but in such a manner that the “energy” of edges (edges taken as springs with tension) is minimized. This reveals how tightly certain concepts are tied together. The second one is the tree (or root) embedding. Networks are redrawn as an ordered hierarchical tree, selecting a certain node as a root. Then nodes and edges are rearranged so that nodes that are equidistant from the root are on the same hierarchical level (Pemmaraju and Skiena 2006). The experts’ concept network shown in Fig. 2 is first redrawn as a springembedded form and next as tree-embedded forms, as shown in Fig. 3. From these embeddings we can see that the experts’ network is revealed as a rather connected web-like structure, where certain concepts hold the structure together. It also contains “hidden” hierarchies, which become visible by choosing a certain concept as a root of the hierarchical trees as shown in Fig. 3 This, of course, is as expected, because the network is drawn following well-defined rules, where concepts and laws follow either from experimental procedures or from logical procedures. However, the hierarchy alone is not yet a sufficient condition for coherence. What is needed in addition to hierarchy is sufficient clustering of nodes (seen as triangles around nodes in embedded structures).
Fig. 3 The experts’ concept network drawn as a spring-embedded structure and two examples of tree-embedded structures, which show three examples of the inbuilt hierarchies. Numbers refer to concepts in Fig.1.
15
The clustering property is closely related to the coherence, because the triangular substructures shown in Fig. 1 are now the basic structural elements connected to interdependence and co-variation of the concepts A, B and C. Structurally these kind of triangular structures are the basic motifs, which bring with them the connectedness and coherence. Another important feature is the hierarchical backbones, reflecting the hierarchical ordering and deductive-like directionality created by the hierarchical motif shown in Fig. 1. Having recognised hierarchy and clustering as two important features of the concept networks, we need to find ways to operationalize these properties. In order to measure how certain nodes (concept) cluster other nodes (concepts) around it, we have calculated several well-defined measures characterizing the networks (for detailed definitions, see da Costa et al. 2007). The measures of most importance are (subscript indicates that the measure is for a node k locally): 1. Degree of node Dk which measures the number of connections (links) on a given node. 2. Clustering coefficient Ck, which measures the number of triangles and their connectedness around a given concept. 3. Subgraph centrality SCk, which measures the number of subgraphs that constitute closed paths traversing through a given node. 4. Transit efficiency Tk, which measures the relative ease of passing through a given node and is a kind of modified inverse geodesic distance or the so-called efficiency in a network. 5. Hierarchy Hk, which measures degree of hierarchy and is calculated as a hierarchy level weighted sum of connections within a given level. The hierarchy measure is thus very similar to that introduced by McClure et al. (1999) but now the scoring rewards of the number of levels (through weighted sum). One advantage of this hierarchy measure is that perfect tree-like hierarchy without intralevel connections will have Hk = 0 while fully connected structure with only one hierarchy level will have Hk = 1. Sophisticated hierarchies, which are tree-like with a number of intralevel connections (typical to structures with cycles), will always have Hk > 1. The mathematical definitions of the local measures of the topology are given in Table I. Table I. Mathematical definitions of measures for local and global topology of the networks. In the definitions aij is the element of the adjacency matrix a and dij is the element of matrix d of the shortest paths and N is the number of nodes. Observable
Definition
Degree of node
Dk
Clustering
Ck
Subgraph centrality
SCk
Transit efficiency
Tk
ki ,in + ki ,out , k = number incoming/outgoing edges
∑ (∑
k> j k
(∑
aij aik a jk / ∑
) (∑
(a k )ii / k ! /
j ,k
(d ji + dik ) / d jk
Hierarchy
Hk
∑
Importance
Ik
Ck × SCk × Tk
j
k> j
aij aik , a = adjacency matrix j
)
Di / N −1
)
, d=matrix of shortest paths
j n j /( N − 1) , j=hierarchy level with nj cross links
16
The observables in Table I give information of the different but closely related structural aspects of the concept network and they are therefore first calculated for each node in the network. It is found that coefficients Ck, SCk and Tk more or less correlate, because they are measures of different aspects of the centrality of the node (concept) in the whole structure. This suggests that we can reduce the information by requiring that the node, which is structurally important and clusters other nodes around it, has a high value of all observables Ck, SCk, and Tk and thus define the importance Ik of the node in regard to its clustering capability as a product I k = C k × SCk × Tk (1) Table II gives the correlations between different observables defined in Table I. From the results it is seen that clustering measures have high positive correlations with each other but significant negative correlation with hierarchy. The correlations in Table II show that the observables Ik and Hk now measure structurally different aspects of the node. Moreover, the results justify the notion that Ik is a reasonable measure that describes the overall importance of the node’s structural clustering capacity. Table II. Correlations between values of the observables as defined in Table I and calculated for the experts’ concept network shown in Figure 2. The mean value of hierarchy is H=1.01.
Mean
D 3.29
C 0.21
SC 1.35
T 1.10
C
0.49
SC
0.95
0.53
T
0.85
0.42
0.87
I
0.76
0.78
0.88
0.71
H
-0.64
-0.43
-0.69
-0.66
I 0.42
-0.64
17
Fig. 4 Hierarchy Hk plotted against importance Ik for each node of the experts’ concept network. Three clusters A (triangle), B (circle) and C (box) shown are found by using agglomerative clustering analysis (see text).Nodes with high a hierarchy index are not always the concepts which have the largest clustering capability or which are central for several sub-graphs. This feature is seen particularly clearly if for each node Hk is plotted against Ik as shown in Figure 4. The data in Figure 4 can be grouped in three distinct clusters by the agglomerative clustering analysis (Ruskeepää 2004). Three different distance criteria (Euclidean, Squared Euclidean and Manhattan) then leads to exactly similar clustering, shown in Figure 4. The nodes are thus clustered in three distinct classes A–C on the basis of their role in the topological structure of the network. We can refer to these three classes as clustering, hierarchy and coherence classes, which each have different characteristic features. It is now of interest to see in more detail what is actually the physical content of the concepts that belong to each class. A. The clustering class consists of nodes which have high importance Ik > 0.8 but only moderate hierarchy 0.5 < Hk < 0.8 values. These nodes (concepts) are highly central but do not necessarily produce overarching hierarchies. In this class are concepts: potential energy (8), electric field potential (9), and voltage (3), where numbering refers to Figure 2. It is evident, that these concepts are physically highly important ones in connecting energy and field concepts. B. The hierarchy class contains nodes, with Hk > 0.8, but with low clustering capacity indicated by low values Ik < 0.2. Nodes in this class thus produce the most extensive hierarchical tree-like structures and order many other concepts as subordinated branches. In this class are the concepts as the addition and conservation of charges (12, 13) and the electric charge (1), which by nature are ordering principles or fundamental properties. In addition the concepts as the capacitance (2) and capacitor law (11) are found in this class. These kinds of concepts with high hierarchy but low importance for clustering are kinds of terminal concepts, either a starting 18
point or the end point of links. They are found at the far end of the chain of connections if inductively produced, or at the beginning of a chain if used as a principle in deductions. C. The coherence class is so named, because the concepts in this class have notable coherence with respect to each other; they are equally well clustering as hierarchically ordering with significant importance 0.2 < Ik < 0.8 and hierarchy 0.6 < Hk < 1.4. In this class belong most of the concepts in the expert’s network, for example such central concepts as the electric field (16), the electric force (4), Coulomb’s law (10) and the principle of energy conservation (14). Structurally, these concepts play a very central role and they are responsible for producing the overall coherence of the structure. The different concepts shown in the network in Figure 2 thus fall in different classes and it can be seen that the measures of hierarchy Hk and importance Ik together reveal much information about the structure of the networks; moreover the measures also directly relate to the content relevance of the concepts. These results for the experts’ concept network justify the claim that structural features carry significant information concerning the concept’s role in the network. On the other hand, the concept’s role in bringing coherence to the conceptual system as a whole is reflected on the local topological features. On the basis of these conclusions, in what follows, we will use the average values of degree D = (∑ k Dk ) / N , importance I = (∑ k I k ) / N and hierarchy H = (∑ k H k ) / N , respectively, as the central global measures to examine the conceptual coherence of concept networks. These global measures for the experts’ concept network are D=3.3, H=1.01 and I=0.42. We can now use these observables as operational means for structural analysis, whereby good quality of content is reflected as high values of H and I.
4.3. STRUCTURAL CLASSES OF NOVICES’ CONCEPT NETWORKS The students’ concept networks (the total number used in the analysis is 40) were analysed using the same methods as used for experts’ concept network. For each network we calculated the averages of degree D, hierarchy H and importance I of nodes. In the students’ concept networks D, H and I were mostly much lower than in the experts’ concept network, indicating that the coherence is not nearly as good in the students’ as in the experts’ concept network. Closer inspection revealed that in students’ concept networks not all concepts with high H and I were significant ones in the physics point of view and that in expert’s network the same concepts could have values close to I =0 or H=0. In order to reduce the information for comparison of students’ and experts’ concept networks, we have sorted – or rather filtered – the students’ networks by taking a “projection” of the observables calculated from students’ networks on the same observables in the experts’ network. This is done representing the observables as vectors X and calculating the projected value Xc of observables as a vector dot-product X c = (1/ L) ( X expert ⋅ X novice )1/ 2 ; L = ( X expert ⋅ X expert )1/ 2 , (2) where L is the normalization factor chosen so that projected values (length of the vectors) are the same for original ones for the experts’ network. Values of Xc close to 1 now require that the value set of observables is close to each other in the students’ and experts’ networks, while the value close to 0 means that there are no structural similarities. In addition to local measures for each node, the above definition of contraction can be used for global measures as well (1-element vectors). This
19
projection is now the most elementary and objective content comparison, which can be done on a structural basis, that is, without subjective interpreters or subjective interpretations. Note that we have not chosen to make the comparison on the basis of detailed topological similarity by using e.g. the adjacency matrix, which would correspond to a comparison of that suggested by McClure et al. (1999), i.e. comparing the connections of each concept to neighbouring concepts (also occasionally called semantic connectivity). This kind of comparison emphasises close and detailed similarity and here a more flexible structural similarity measured through observables like H and I is desirable, without requiring detailed correspondence in connections (i.e. no detailed similarity of semantic fields). On the basis of the projected variables we form a single quantity, which is a measure for the quality of structural coherence and the quality of content of knowledge, therefore called the quality factor Q ≡ Ic × Hc
(3)
The quality factor Q is now constructed so that all four measures needed to establish the overall coherence need to be large in good networks, but that in addition the projection now also takes into account the content of the networks as compared with the experts’ concept network. With D and Q we can represent each network so that the quality of its content is described through Q and the richness of its content is described by D (where more connections mean more content). The results for all 58 networks are shown in Figure 5.
Fig. 5 Quality of knowledge represented in the maps as measured by a knowledge quality factor Q and its dependence on the degree D (richness of the content).
20
In terms of the quality factor Q and the richness of knowledge D the students’ concept networks form a continuum, where the best networks approach the experts’ network. Of course, this is partly due to the fact that the experts’ network is used as a basis upon which to make the projection. It should be noted that all physically acceptable connections found in the students’ networks are also contained in the experts’ network. The behaviour of Q versus D suggests that there is an interesting change in structure of networks around 2.3 < D < 2.6, indicated by a rise in Q. This notion is confirmed by calculating the correlation coefficient for data D > 2.4, which is only 0.16 indicating that in this region, Q is nearly constant and independent of D. The existence of step in D ≈ 2.5 is expected, because in this region triadic cycles begin to appear, essential for the formation of clustered structures. By contrast, in the structurally poor networks only chain-like features are present. The networks in regions D < 2.3; 2.3 < D < 2.6 and D > 2.6 are also visually different. We can characterize the classes as webs (I), necklaces (II) and chains (III), with typical examples shown in Figure 6. The averages of the topological measures of the contracted observables in each of these classes are given in Table III and a typical example of a network in each class in Figure 6. Of the networks only 11 (19 %) out of 58 were in the class of webs, while there were 22 (38 %) necklaces and 25 (43 %) were simple chains (note that some of the chains are overlapping in Fig. 5). Table III. Structural measures D, H, I and Q for networks. Classification as webs, necklaces and chains is done on the basis of agglomerative clustering for D and Q. The total number of networks was 58 with number N in each class as indicated.
Degree D Quality Q DQ Hierarchy H Importance I
Expert map Webs N=11 (19%) 3.29 2.80 ± 0.24 0.59 0.32 ± 0,08 1.94 0.88 1.01 0.80 ± 0.18 0.42 0.40 ± 0.09
Necklaces N=22 (38%) 2.38 ± 0.18 0.24 ± 0.07 0.58 0.62 ± 0.10 0.34 ± 0.08
Chains N=25 (43%) 2.06 ± 0.18 0.03 ± 0.07 0.07 0.28 ± 0.30 0.06 ± 0.11
I Webs. The students’ concept networks, which score the highest quality and richness of knowledge values, look like connected webs when they are drawn in a spring embedded form. They are in many respects similar to the experts’ concept network in Figure 2. The webs are tightly connected, which is related to good coherence of concepts, and there are very few separate branches or spokes. In the webs there is also number of relevant inbuilt hierarchies. One example is shown in Figure 6 (left). In comparing the web-like networks to the experts’ concept network (Fig. 3) we can see however that the hierarchies are not as well defined as in the case of the experts’ concept network and the hierarchies are partially broken. Moreover, in nearly all cases the clustering capacity of concepts is significantly lower than in the experts’ concept network. However, in some cases the structural quality Q has high values, even if D is significantly lower than in the expert’s network. This reveals that even the best of the students’ concept networks do not contain as much coherent connection between concepts as the experts’ network contains.
21
Fig. 6 Novices’ concept networks. Examples of webs (left) with D=2.9; Q=0.23, necklaces (middle) with D=2.2; Q=0.10 and chains (right) with D=1.8; Q=0.0. The tree-embeddings starting from node 6 (electric field strength) are given in lower panel. II Necklaces. In some cases students’ concept networks have a topology that consists of triangular cycles as subsets, but the subsets are quite loosely connected. Moreover, the structures have many spoke-like extensions. These kinds of webs shown in Figure 6 (middle) we have called necklaces. In principle, necklaces have many good structural features locally but some important connections are missing. This deficiency is also seen in tree-like redrawn networks, where hierarchies are now severely broken, as can be seen in the lower panel in Figure 6. III Chains. In the worst cases, where the quality and richness of knowledge factors have very low values, the structures consist of linear branches and resemble simple chains, as shown in Figure 6 (right). In these structures, there is very little connectivity. Hierarchies in these cases are trivial branching hierarchies or simple taxonomical hierarchies, with no intra-level connections. The missing connections are directly connected to poor interrelatedness and weak coherence between concepts. These structures are not an adequate representation of conceptual structure. The classification of structures in connected webs, necklaces and chains brings forward the notion that coherence in the structures is reflected on the overall topology. It is now evident that both the values D and Q measure the quality of the network. The value Q is related to the quality of the structure and can be interpreted to measure the overall coherence of the structure, related to explanatory and deductive coherence through two types of triangular motifs and to deductive coherence through the hierarchy motif. The value of D is simply related to the richness of content. The links contain essential knowledge about the methods of physics concept formation and therefore the structure is also directly related to the quality of the knowledge students have at their command. It should be noted that it is quite possible that networks with small values of D may also have high Q (one example is seen in Fig. 5 and Fig 6), but 22
in this case the structure consists of local sub-structures which are triangularly connected, as shown in Fig 6 (middle). Therefore, it is natural to require that in an expert-like network both D and Q need to be high. For the experts’ network the product is DQ = 1.94, considerably lower in the case of necklaces with an average value DQ=0.58 and for chains simply DQ is almost zero, DQ = 0.07. 5. Discussion and conclusions The two central results of this work are that coherence is a hallmark of a high quality of knowledge and that there is a connection between the contents of the concept networks and their structure. However, if we ask how a good and coherent structure can be recognized, and on what aspects we need to focus attention to, the situation is not so simple. The basic idea emphasised here is that in order to make coherence a well-defined and useful notion, for the purposes of designing educational solutions, there must be a chain of connections from coherence to the operational measures used to characterize the networks. These connections are summarized in Table IV. Table IV: Steps 1–5 needed to make the quality of knowledge a measurable feature. Quality of knowledge is understood as coherence and measured as structural coherence. Basic requirements and their realizations are summarized and the section where they are discussed is given. Requirement 1. Defining the types of the coherence. Defining quality as coherence 2. Embodying coherence types in the level of methodological practices or procedures 3. Connecting coherence types to structural properties 4. Operationalizing structural properties 5.Operationalizing quality of knowledge
Realization Explanatory coherence. (EC) Deductive coherence (DC) Experimental procedures (EP) EC Modelling procedures MP DC Explicit rules for procedures EP EC clustering. MP DC hierarchy. clustering triangular motifs and I hierarchy feed-forward motif and H Overall coherence Quality Q=IxH Richness of content D
Sect. 2.2. 2.3 2.2. 2.3 3 4.2. 4.3 4.3.
Steps 1–5 summarized in Table IV show that the connection between the quality of the contents of the concept networks and their structural features can be connected. The first three steps 1–3 connect the coherence of knowledge to the structural features of clustering (connectedness) and hierarchy and make the coherence discernible in terms of the basic motifs. Step 4 introduces several observables, which can be used to measure the structural features. Of these observables the degree D directly measures the richness of content while the quality of the structure is measured by the clustering importance I (taking into account the clustering and cyclicity) and by the hierarchy H. It is shown by a local structural analysis that in the experts’ concept network, high values of I and H are always associated with concepts (nodes) of central importance in regard to the subject content. This encourages formation of the quality factor Q=IxH, which can be taken to measure the overall coherence of the structure and thus makes contact with 23
the coherence of knowledge as it is represented in the network. In this way the background theory of coherence makes it possible to formulate the measures for clustering and hierarchy of nodes so that their connection to the underlying coherence of knowledge is subsequently taken into account. The final step 5 connects the measures D and Q with the quality of knowledge. The quality of knowledge, on the other hand, is associated with coherence that is measured by factor Q. Of course Q is not merely a measure of structure because high Q indicates that structurally central concepts are also valid and correct by their content (as they are in experts’ network). Increasing D requires good command on possible and acceptable experimental or modeling procedures, because the explicated rules require the students to represent only those aspects they can argue for by following the procedural rules. Consequently, the results can be interpreted so that high Q with high D indicates expertise over the subject matter and that networks with high D and Q can be taken as representations of knowledge with good quality. The structure and quality of the networks is also seen in the visual appearance of the networks. The experts’ concept network looks like a tightly woven web with a high of clustering of concept and well-defined internal hierarchies. In the novices’ concept networks, three qualitatively different types of networks can be distinguished: webs, necklaces and chains. Only in the class of web-like structures are the features of the networks somewhat similar to the experts’ network. In two other classes, novices’ networks consist of loosely connected web-like sub-structures. These kinds of structures also feature little coherence and content (as measured by Q and D). We have stressed here the importance of having well-defined rules and principles in order to make connections between concepts. An often used criterion for the quality of knowledge in traditional analyses of concept maps is the number of cross links and the requirement that links have some verbs or attributes attached to them (Novak and Gowin 1984; Ruiz-Primo and Shavelson 1996). In our case, these requirements are simply insufficient – on the contrary, just increasing the cross links would not necessarily improve the coherence. The design principle introduced here – which connects explanatory and deductive coherence to methodological procedures of quantitative experiments and modelling, respectively – produces concept networks which are simultaneously connected and hierarchical. If we have rules to create links (edges) and the rules which are based on procedures, we will have representations, where content and structural aspects are strongly coupled. Therefore, coherence can be associated with the internal and inbuilt regularities in the concept networks and these regularities can be made visible by suitable analysis of the networks. These assumptions are here justified through a detailed examination of the experts’ and novices concept networks. The results discussed here can be embedded into and made understandable within the framework of the coherentist view of knowledge (Thagard 1992, 2000; Kosso 1992). In this view, the concept and principles associated with knowledge obtain their credibility and reliability from the coherence of the conceptual structure. The coherence is also quite naturally an important criterion for valid knowledge, although internal coherence and consistency alone are not yet sufficient conditions for the validity of knowledge (Kosso 1992). However, the explanatory coherence connects the knowledge structures to real phenomena and laboratory experiments and is therefore of central epistemic importance. Through this connection – when explanatory coherence is achieved – the coherent knowledge structures possess the value of true knowledge in so far as any such condition of truthfulness can be assessed. Of course, in the cases discussed here, the networks contain explanatory
24
coherence in a restricted sense; that is, only with respect to experiments and experimental procedures (the standard student experiments used in teaching) contained in the network. In addition to the explanatory coherence, the structures discussed here have deductive coherence connected to logical procedures, which are important criteria for credibility of any theoretical system. The coherence of knowledge as discussed in the present study is a much reduced form of all possible forms of coherence characterizing scientific knowledge (cf. Thagard 2000). We have deliberately restricted the scope of coherence here in order to make the structures more tractable for a detailed structural analysis. Nevertheless, the analysis given here and the way the connections are built in, show that there is a direct connection between the topology and hierarchy of the concept networks and their coherence. Networks with good coherence have rich internal hierarchy and a well-connected topology. An interesting possibility contained in the network view is its potential for uses to monitor conceptual development. Thagard (1992) has applied the networkview to understanding historical conceptual change or historical conceptual development. It seems that in this case the network view leads to a deepened understanding of how explanatory coherence guides the evolution of the conceptual structures. In Thagard’s work the focus is on addition and deletion of nodes and in restructurization of links, both processes being driven by constraint satisfaction for better explanatory coherence. It is quite evident, that a similar description could also be possible in learning and in monitoring the learning, and in finding typical features of the conceptual change during learning (diSessa and Sherin 1998). This seem to open the possibility of improving traditional physics teaching and instruction, and to develop concept networks, which are truly useful representations for expressing physics knowledge, its structure and the relation of structure to methodological procedures. Such an extended view on concept networks escalates the possibilities of concept mapping from personal tools of thinking to the more inter-subjective level, where they begin to share more and more structural aspects and contents of physics knowledge itself and can be used to communicate such knowledge. In this form they can eventually also begin to function as effective tools for learning, representing and communicating the content knowledge of physics in higher education.
25
References Anderson PW (2001) Science: A ‘Dappled World’ or a ‘Seamless Web’? Studies in History and Philosophy in Modern Physics 32:487-494 BonJour L (1985) The Structure of Empirical Knowledge. Harvard University Press, Cambridge, Massachusetts Bransford JD, Brown AL, Cocking RC (2000) How People learn: Brain, Mind, Experience, and School. National Academy Press, Washington DC da Costa LF, Rodrigues FA, Travieso G, Villas Boas PR (2007) Characterization of complex networks: A survey of measurements. Advances in Physics 56:167–242 Derbentseva N, Safayeni F, Cañas A J (2007) Concept Maps: Experiments on Dynamic Thinking. Journal of Research in Science Teaching 44: 448-465 di Sessa A, Sherin BL (1998) What Changes in Conceptual Change? International Journal of Science Education 20:1155–1191 Haack S (1993) Evidence and inquiry: towards reconstruction in epistemology. Blackwell, Oxford Harman G (1986) Change in view: principles of reasoning. MIT Press, Cambridge, Massachusetts Hestenes D (1992) Modeling games in the Newtonian World. American Journal of Physics 60:732–748 Giere RN (1988) Explaining Science: a cognitive approach. University of Chicago Press, Chicago Giere RN, Bickle J, Mauldin RF (2006) Understanding Scientific Reasoning, 5th ed. Thomson Wadsworth, Belmont, CA Ingeç SK (2008) Analysing Concept Maps as an Assessment Tool in Teaching Physics and Comparison with the Achievement Tests. International Journal of Science Education 67:1–19 Kinchin IM, Hay DB, Adams A (2000) How a qualitative approach to concept map analysis can be used to aid learning by illustrating patterns of conceptual development. Educational Research 42:43–57 Kinchin IM, De-Leij FAAM, Hay DB (2005) The evolution of a collaborative concept mapping activity for undergraduate microbiology students. Journal of Further and Higher Education 29:1–14. Kitcher P (1983) The nature of mathematical knowledge. Oxford University Press, New York Koponen IT, Mäntylä T (2006) Generative Role of Experiments in Physics and in Teaching Physics: A Suggestion for Epistemological Reconstruction. Science & Education 15:31–54 Koponen IT (2007) Models and Modelling in Physics Education: A Critical Reanalysis of Philosophical Underpinnings and Suggestions for Revisions. Science & Education 16:751–773 Kosso P (2009) The Large-scale Structure of Scientific Method. Science & Education 18:33–42 Kosso P (1992) Reading the Book of Nature. Cambridge University Press. Cambridge Lehrer K (1990) Theory of Knowledge. Routledge, London. Leve RM (2004) Informational Acquisition and Cognitive Models. Complexity 9: 31–37 Machin J, Loxley P (2004) Exploring the use of concept chains to structure teacher trainees’ understanding of science. International Journal of Science Education 26:1445–1475
26
McClure JR, Sonak B, Hoi K, Suen HK (1999) Concept Map Assessment of Classroom Learning: Reliability, Validity, and Logistical Practicality. Journal of Research in Science Teaching 36:475–492 Mestre JP (2001) Implications of research on learning for the education of prospective science and physics teachers. Physics Education 36:44–51 Minsky M (1975) A Framework for Representing Knowledge. In: Winston PH (ed) The Psychology of Vision. McGraw-Hill, New York, pp 211-277 Moreira M (1985) Concept mapping: an alternative strategy for evaluation. Assessment and Evaluation in Higher Education 10:159–168 Morrison M, Morgan M (1999) Models as mediating instruments. In Morgan MS and Morrison M (eds.) Models as Mediators, Cambridge University Press, Cambridge Mäntylä T, Koponen IT (2007) Understanding the Role of Measurements in Creating Physical Quantities: A Case Study of Learning to Quantify Temperature in Physics Teacher Education. Science & Education 16:291–311 Nagel E (1961) The Structure of Science. Routledge & Kegan, London Novak J, Gowin BD (1984 ) Learning How to Learn. Cambridge University Press, New York Pemmaraju S, Skiena S (2006) Computational Discrete Mathematics Combinatorics and Graph Theory with Mathematica. Cambridge University Press, Cambridge Ruiz-Primo MA, Shavelson RJ (1996) Problems and Issues in the Use of Concept Maps in Science Assessment. Journal of Research in Science Teaching 33:569–600 Rumelhart D, Smolensky P, Hinton G, McClelland J (1986) Schemata and sequential thought processes in PDP models. In: McClelland J and Rumelhart D (eds) Parallel distributed processing: explorations in the microstructure of cognition vol. 2. MIT Press, Cambridge, Massachusetts, pp 7-57 Ruskeepää H (2004) Mathematica Navigator: Mathematics, Statistics, and Graphs, 2nd edn. Elsevier Academic Press, New York Russell B (1973) Essays in analysis. Allen and Unwin, London Safayeni F, Derbentseva N, Cañas, AJ (2005) A Theoretical Note on Concepts and the Need for Cyclic Concept Maps. Journal of Research in Science Teaching 42:741–766 Sensevy G, Tiberghien A, Santini J, Laube S, Griggs P (2008) An Epistemological Approach to Modeling: Cases Studies and Implications for Science Teaching. Science Education 92:424–446 Slotte V, Lonka K (1999) Spontaneous Concept Maps Aiding the Understanding of Scientific Concepts. International Journal of Science Education 21:515-531 Snyder JL (2000) An investigation of the knowledge structures of experts, intermediates and novices in physics. International Journal of Science Education 22:979–992 Suppe F (1977) The Structure of Scientific Theories, 2nd edn. University of Illinois Press, Urbana Thagard P (1988) Computational Philosophy of Science. The MIT Press, Cambridge, Massachusetts Thagard P (1992) Conceptual Revolutions. Princeton University Press, Princeton NJ Thagard P (2000) Coherence in Thought and Action. MIT Press, Cambridge, Massachusetts.
27
Thagard P (2007) Coherence, Truth, and the Development of Scientific Knowledge. Philosophy of Science 74:28-47 Trowbridge JE, Wandersee JH (1989) Theory-Driven Graphical Organizers. In: Mintzes JJ, Wandersee JH and Novak JD (eds) Teaching Science for Understanding: A Human Constructivistic View. Academic Press, San Diego, pp 95-128 van Zele E, Lenaerts J, Wieme W (2004) Improving the usefulness of concept maps as a research tool for science education. International Journal of Science Education 26:1043–1064 Vanides J, Yin Y, Tomita M, Ruiz-Primo MA (2005) Using concept maps in the science classroom. Science Scope 28:27–31 Wilson EO (1999) Consilience: the unity of knowledge.Vintage Books, New York
28
