Sunday, November 23, 2014

Rules Based Fuzzy Cognitive Mapping: Applications in Education

Naveen H Gouda
Department of electronics and communication, SCEM Mangalore, IEEE



Abstract Fostering conceptual and cognitive change in learners can be difficult. Students often come to a learning situation with robust, implicit understandings of the material under study. One explanation for the implicit nature of these understandings is a lack of metaknowledge about the knowledge to be acquired. Helping learners create metaknowledge may free paths to conceptual change. This paper proposes the use of fuzzy cognitive maps [FCMs] as a tool for creating metaknowledge and exploring hidden implications of a learner’s understanding. Two specific educational applications of FCMs are explored in detail and recommendations are included for further investigations within educational contexts.

I.      Introduction
Fuzzy cognitive mapping [FCM] is a tool for formalizing understandings of conceptual and causal relationships. By combining conceptual mapping tools with fuzzy logic and other techniques originally developed for neural networks, FCMs allow for the representation and formalization of soft knowledge domains [e.g., politics, education]. This paper explores FCM procedures and proposes two methodologies for developing FCMs in educational organization settings. Other potential applications in education are explored and directions for future research are included.
To apply FCMs in education requires a basic understanding of the theoretical foundation of cognitive mapping. This paper presents a brief review of that theoretical foundation, as well as some related research literature. Fuzzy logic allows us to represent truth values on a continuous scale from 0 to 1, providing mathematical methods for representing concepts and causalities that are true to some degree [neither wholly true nor false]. Consequently, the law of the excluded middle does not apply in fuzzy logic. FCMs combine the strengths of cognitive maps with fuzzy logic. By representing human knowledge in a form more representative of natural human language than traditional concept mapping techniques, FCMs ease knowledge engineering and increase knowledge-source concurrence. FCMs can also be modelled on computers, thus allowing for dynamic modelling of cognitive systems.
     Two methods for facilitating the creation of FCMs are presented in this paper. Both are in the early stages of testing. The educational field testing includes trials of a method for group knowledge acquisition and for individual knowledge acquisition. In addition, a computer modelling system is presented for the development and analysis of FCMs.
II.    CONCEPTUAL CHANGE
It is the processes whereby concepts and relationships between them change over the course of an individual person’s lifetime or over the course of history. Learners’ naive conceptions have been well studied. It is widely accepted within the domain of cognitive psychology that students come to school with some form of conceptualization of the natural world and their place in it. Frequently, however, these conceptions are not scientifically accurate. Instead, these conceptions represent a theory that is useful in everyday experience. Naive theories are based on interaction with the everyday world. A child who repeatedly drops items on the floor is building an implicit theory of gravity. A child who tries to manipulate her parents into taking her for ice cream is building an implicit theory of human behaviour.
Naive understandings display many of the characteristics of implicit knowledge. Implicit knowledge: (a) is characterized by specificity of transfer, (b) is associated with incidental learning conditions, (c) gives rise to a phenomenal sense of intuition, and (d) remains robust in the face of time, psychological disorder and secondary tasks. Naive understandings meet many of the same criteria. They are learned incidentally, they give rise to a sense of ‘‘knowing,’’ and remain robust in the face of time and schooling. These naive understandings can be very difficult to diagnose and change.
Much of the research on naive understandings has been in science education. The difficulty of changing conceptions of the natural world that have been formed over many years is well documented. Situations in which students are unable to process that which they are not expecting create opportunities for implicit learning. For example, when presented with science instruction, students often are looking to confirm what they ‘‘know’’ already, with the result both extremely selective attention and distortion of information provided during instruction. Implicit knowledge concentrates on the process of acquiring the knowledge. Furthermore, it involves learning without awareness. In order to make implicit knowledge available to the learner, some structured task must be available to elicit the knowledge from the learner. Concept, or cognitive, mapping represents a possible tool for developing such a structured environment.
III.    Concept Mapping
A concept map is a type of graphic organizer used to help students organize and represent knowledge of a subject. Concept maps begin with a main idea (or concept) and then branch out to show how that main idea can be broken down into specific topics. The current use of concept mapping within education has its roots in research conducted at Cornell that focused on conceptual changes in students over a 12 year
period. This research required a method to compare conceptions over time and between learners. The Cornell researchers developed a system of representing conceptual knowledge graphically: circles for concepts and arrows for the links between them.
  A review of the literature provides several definitions of concept maps, also known as cognitive maps. Two factors are common in these definitions: all of the authors reviewed define a cognitive map as a graphical representation. Most include some aspect of subjectivity. A graphical representation is fundamental to the idea of concept mapping. In one of the earliest references, Axelrod developed a system for representing causal relationships in social science domains. The system represented concepts in sociology and political science as nodes in a directed graph. The nodes were connected by arrows that were assigned to represent positive or negative causal relationships. The other common factor in the definition of cognitive mapping is the subjectivity of the map. Irvine describes concept mapping as the individual’s diagrammatic interpretation of ideas. The definition from Park and Kim concisely encapsulates many of these definitions. The cognitive map graphically represents interrelationships among a variety of factors. It is a representation of the perceptions and beliefs of a decision maker or expert about his/her own subjective world, rather than objective reality.  Since the development of the graphical system at Cornell, there have been several studies conducted on the efficacy of using concept maps as teaching and learning tools. For example, Jegede, Alaiyemola, and Okebukola report that students in Nigeria who used concept maps documented significantly higher mean scores on an achievement test for the subject matter studied. Other studies have concentrated on the use of concept mapping in teacher education. Novak notes that most science teachers understand science to be a large body of information to be mastered, as opposed to a method for constructing new knowledge. Novak reports that concept mapping plays an important role in facilitating the change of science teachers’ perception of science and the purpose of science education. There are several factors that contribute to the power of cognitive mapping in learning. Johnson, Goldsmith, and Teague describe two values categories of cognitive maps: the stimulus value and the structural advantage. The stimulus value is inherent in the graphical representation. Learners can easily see the global organization of the represented concepts. Graphical representations also allow for the organization of complex domains for learners and designers alike. The network structure of a concept map allows the simultaneous display of all the important relationships. The structural advantage is relative to the assessment of pair-wise ratings. Johnson, Goldsmith, and Teague report that assessing network representations of student understanding resulted in more valid measures than assessing pair-wise comparisons of concepts Thagard proposes a system of conceptual change based on his historical research of scientific revolutions. The system delineates five levels of conceptual change, ranging from the simple to the complex:
1.        Addition of concepts.
2.        Deletion of concepts.
3.        Simple reorganization of concepts in the kind-hierarchy or part-hierarchy which results in new kind-relations and part-relations.
4.        Revisionary reorganization of concepts in the hierarchies, in which old kind-relations or part-relations are replaced by different ones.
5.        Hierarchy reinterpretation, in which the nature of the kind-relation or part-relation that constitutes a hierarchy changes.

Kind-relations define members of a concept (i.e., a whale is a kind of mammal), while part-relations define the characteristics of a member (i.e. whales have flippers.) Concept maps make visible the potential for conceptual change within a learner. When created correctly and thoroughly, concept mapping is a powerful way for students to reach high levels of cognitive performance. A concept map is also not just a learning tool, but an ideal evaluation tool for educators measuring the growth of and assessing student learning. As students create concept maps, they reiterate ideas using their own words and help identify incorrect ideas and concepts; educators are able to see what students do not understand, providing an accurate, objective way to evaluate areas in which students do not yet grasp concepts fully. Pressley and McCormick’s  review of the literature on multiple representations in science revealed a common process for developing concept maps. This process is briefly outlined:

1.        Key words and phrases are identified from the reading.
2.        Key concepts are ordered from the most general to the most specific.
3.        The concepts are then clustered using two criteria. Concepts that interrelate are grouped; concepts are classified with respect to their level of abstraction (i.e. general concepts to specific ones). All of the concepts are then arranged loosely in a two-dimensional array with abstractness defining one dimension and main ideas defining the second dimension.
4.        Related concepts are then linked with lines, which are labelled to specify the relationship between concepts.

Thus, a map of the domain is created, with key ideas linked by nodes that describe the type of relationship between them (see Fig.1). The map, however, is static. It represents the crystallized, declarative knowledge of a domain. The map can be used to understand some forms of the conceptual relationships. Cognitive maps represent formal, bivalent (true or false) logical relationships. This is acceptable in ‘‘hard’’

knowledge domains like physics or mathematics where the nature of the knowledge in the domain is usually binary. In most domains, however, the knowledge base is uncertain, or fuzzy. Social studies (e.g., politics, international relations), management science, and the study of art or literature are all ‘‘soft’’ knowledge domains, where uncertainty and degrees of truth are more common. Thus, the true nature of these domains cannot be understood from bivalent, or true-false, concept maps.
Cognitive maps also represent a form of distributed intelligence. It serves the construction and accumulation of spatial knowledge, allowing the "mind's eye" to visualize images in order to reduce cognitive load, enhance recall and learning of information. This type of spatial thinking can also be used as a metaphor for non-spatial tasks, where people performing non-spatial tasks involving memory and imaging use spatial knowledge to aid in processing the task and avoid error. FCM’s are especially applicable to soft knowledge domains. several example FCM’s are given below.
Figure1. Fuzzy cognitive map.

Figure 2. FCM
IV.  FUZZY COGNITIVE MAPS
    Fuzzy cognitive maps (FCMs) and concept maps have similar applications in education. There are two important differences between traditional concept maps and FCMs: fuzzy logic and feedback. Uncertain or soft knowledge domains can be represented with fuzzy logic. Kosko originally developed FCMs to represent concepts in military science, but his since gone on to demonstrate their usefulness in representing arguments in sociology and political science. Figure 1 illustrates the structure of the FCM. The map consists

of nodes connected by "edges" or links. Nodes represent factors such as "temperature" or "on-ramp congestion." Unlike

Bart Kosko's original FCM design, the software and assignments I present assume the nodes have three potential states (positive, negative, and neutral) instead of two states, since three produces more interesting behaviour. The links between nodes represent promoting/inhibiting relationships. For example, a link between two nodes might represent a relationship like, "as soot in the atmosphere increases then temperature often decreases." Since several nodes may interact, the link weights are "stereotypical" or "fuzzy" in terms of their effects on other nodes. Thus, words like "often" and "typically" and "inhibits" are more accurate than "cause" or "affect". Fuzzy logic is a system for representing uncertainty, or possibility. The formal extension of the original possibility theory created in the 1920s by Lukasiewicz was developed by Zadeh. A generalization of traditional, bivalent, Aristotelian logic, fuzzy logic creates a system for mathematically representing systems with natural linguistic variables (e.g., tall, little). Traditional binary logic requires that a statement must be either true or false. An animal can be a cat or it is not a cat. The world of traditional logic is black or white. This is known as the law of the excluded middle: there is no option between 100% true and 100% false see Fig. 3. The problem with a binary system is that it does not allow for accurate representations of the way humans perceive and represent their world. Formal representations of human knowledge, like those described in the previous section, typically rely on binary logic.
Figure 3. Binary logic representation.
Fuzzy logic ignores the law of the excluded middle and allows for representations that are both true and false.1 Terms like age, height, and intelligence are fuzzy variables; they have no exact definition and allow for degrees of membership. Fuzzy variables, or linguistic variables, have fuzzy values (e.g., young, short, bright).
    A fuzzy set is a generalization of an ordinary set by allowing a degree of membership for each element. A membership degree is a real number on [0, 1]. In extreme cases if the degree of membership is 0 the element does not belong, and if 1 the element belongs 100% to the set.
    Thus, a person might be considered tall to a 0.9 degree if they were 6’5” tall. Membership value is determined by a function that represents changing membership as a value changes (see Fig. 4)
V.    CAUSALITY AND FCMS
Like traditional causal concept maps, FCMs have nodes that represent variable concepts. The links between the concepts are signed police vigilance or theft to represent the nature of the relationship between nodes. Fuzzy logic allows the representation of fuzzy concepts and degree of causality. Feedback allows the user to explore the hidden properties of the map. By creating a formal representation of causality, FCMs can be used to create and explore models of dynamic events and search for causal explanation.
The most important difference between the concept maps which Pressley and McCormick describe and FCMs is the temporal and causal nature of the FCM. FCMs express causality over time and allow for causality effects to fluctuate as input values change. Nonlinear feedback can only be modelled in a time-based system. FCMs are intended to model causality, not merely semantic relationships between concepts. The latter relationships are more appropriately represented with semantic networking tools like Sem Net. By modelling causality over time, FCMs facilitate the exploration of the implications of complex conceptual models, as well as representing them with greater flexibility.
There is evidence to suggest that temporal mapping may improve learning. Lambiotte et report that providing students with maps of procedures or processes, rather than semantic conceptual maps, improved learning; especially for lower ability students. FCMs are a tool for representing a dynamic process and modelling the process in real-time.
Figure 4. Fuzzy and crisps sets of  “Tall” people
    
VI.  IMPLEMENTATION
Representing and manipulating FCMs mathematically is not difficult. A given FCM with Cn. concepts can be represented in an N*N matrix. Causality is represented by some nonlinear usually sigmoidal. edge function Ci ,Cj ., which describes the degree to which Ci causes Cj. The edge function occurs over the bipolar interval wy1, 1x, as edges can be inhibitory or excitory. Using the notion of disconcepts; C., the unit interval [0, 1] can be retained. Thus, what results is a matrix with causality between concepts represented by some real number between 1 and 0. A row, i, represents the causality between concept i and all other concepts in the map. No concept is assumed to cause itself, thus the diagonal is zeroed. See Table I for an example of a simple FCM matrix. Traditional cognitive mapping relies on a crisp valued edge function of [1, 0] or [1,-1]. Thus, when an expert creates a concept map, they sign the edges as either positive or negative. FCM representation create a weighted edge function over [1, 0] or [1,-1]. Fuzziness, therefore, allows the developer to capture more fine grain information about the representation. It is possible to ask the expert to assign a real number weight to the edge, but this is difficult and usually unnecessary. Instead, an expert can use linguistic modifiers, which are then converted into fuzzy functions.
 Fuzzy inputs can be processed systematically via fuzzy causal algebra. Expert systems development typically involves only one expert due to difficulties with maintaining a tree structure and search limitations. Kosko has developed a mathematical method for combining the FCMs of multiple experts to represent a ‘‘field’’ view. Imagine combining FCMs from educational experts all over the world; creating a giant FCM for generating curriculum and research. FCMs can be combined by representing them as matrices. Each expert’s map is represented as a matrix the size of the total number of unique concepts presented by all of the experts. The matrices are then added together, with the common links naturally achieving more weight. For example, Figure 5 represents an individual’s FCM generated during a pilot test of the group process described later. The corresponding matrix is presented in Table II. The addition of a second participant see the matrix in Table III. is represented in Figure 6.
In this example, both participants used the same concepts, as those were generated in a group session.             

Figure5. FCM of the use of distance education for participant #2.








Figure 6. FCM of the use of distance education for participants #2 and #3 .
    
The resulting additive matrix is found in Table III. Notice that the links representing agreement have larger values, and the areas of disjunction or disagreement have smaller values. If there are k experts, and only one expert includes a given edge, the maximum value for that edge is 1/k. The results can then be normalized  averaged. over [0, 1] or [-1, 1].Graphics may be full colour.  All colours will be retained on the CDROM.  Graphics must not use stipple fill patterns because they may not be reproduced properly.  Please use only SOLID FILL colours which contrast well both on screen and on a black-and-white hardcopy, as shown in Fig. 1.
In this system, if two experts perfectly disagree, they cancel each other out i.e., if expert A says that edge  is 1 and expert B says that ei is y1, the resulting edge equals 0.. Large sample


sizes tend to produce stable connection strengths.22 Representations of the knowledge of multiple experts has long

been a goal in expert systems development. By creating fuzzy knowledge structures, FCMs finally allow us to achieve this goal. Indeed, if large sample sizes produce stable connection strengths, then the more experts, the better. An initial input activates the matrix. The initial state of the concepts is entered as a fuzzy vector. Inference proceeds by nonlinear spreading activation. The initial activation is allowed to reverberate through the system until it converges on a limit cycle. See Table IV. The limit cycle may be a point solution, a cyclical attractor, or a chaotic strange attractor. In other words, the output may be a steady state (A, A, A, . . . ), a cycle  (A, B,C, A, B, C, . . . ), or a chaotic attractor ( A, C, B, D, B, A, D, C, . . . .) Figure 7 presents a graph of the output from one FCM. The transition to the limit cycle is evident as each line straightens and the system converges on the limit cycle. Since FCMs are machine moveable and dynamic, they are also machine tuneable. As the FCM is run through multiple what-if scenarios, adjustments to the nodes and edges can be made to gradually force the map to fit an expected output. Kosko proposes a system of ‘‘adaptive inference through concomitant variation.’’ Instead of using simple Hebbian neural learning algorithms to tune the model.
    Hebb's principle can be described as a method of determining how to alter the weights between model neurons. The weight between two neurons increases if the two neurons activate simultaneously—and reduces if they activate separately. Nodes that tend to be either both positive or both negative at the same time have strong positive weights, while those that tend to be opposite have strong negative weights. Kosko uses a differential Hebbian learning law that measures changes in the environmental parameters. A simple Hebbian law would be something like:

where ;Xi: is the activation level of some node i, yXi: is the passive causal decay parameter, Cj: is a sigmoid function,      +j Cj ( Xj) eji: is the path-weighted internal feedback, Ii: is the external output.
A differential Hebbian learning law, on the other hand, is represented by:
             
where ejisthe edge function between concept Ci and Cj     Fig. 1.
                       

Figure 7. Graph of an activated matrix converging on a limit cycle.

     
 Simulations show that while the simple Hebbian learning law which correlates activations to output. Produces ‘‘spurious causal conjectures,’’25 the differential Hebbian law causes eji to converge on an exponentially weighted average of correlated change. Using such techniques, FCMs can be gradually improved over time. Nonsalient improvements that would probably have been missed by a human observer mat be added to the map as it becomes more and more accurate.

VII.ADVANTAGES
    The previous sections have explored several differences between the concept mapping process described by Pressley and McCormick and the FCM methodologies described in this paper. First, concepts in an FCM are not arrayed according to abstractness or centrality of the idea. The centrality of an idea can be naturally determined after the map has been completed. Centrality becomes a function of the number of links to and from a given node and the weight of those links. The abstractness of an idea can be interpreted as a function of its fuzziness. The more abstract an idea, the more fuzzy subsets it contains. Hierarchical conceptual relationships can be embedded within a FCM node. The node then becomes an embedded FCM within the larger framework. The resulting signal strength from the node is a function of the embedded processing. These features offer several advantages to FCMs over traditional mapping methods. FCMs have these specific advantageous characteristics:
·         FCMs capture more information in the relationship between concepts.
·         FCMs are dynamic.
·         FCMs express hidden relationships.
·         FCMs are combinable.
·         FCMs are tunable.

FCMs could also facilitate data collection for machine monitoring of student performance. Wallace and Mintzes report that traditional concept mapping techniques are very sensitive to changes in students’ conceptual frameworks. Further research will demonstrate whether this holds for FCMs as well. The FCM framework, however, gives developers the power to begin to compare student FCMs with expert FCMs. Two simple comparisons are measuring the centrality of a given node, and the effect, both direct and indirect, one node has on another. Kosko has developed several methods for categorizing the importance and centrality of nodes within an FCM. A most simple measure is to sum the weights of the edges coming into and leading out of a given node.
VIII.             FCMs IN PRACTICE
Potential applications of FCMs are very broad. For the purposes of this paper, we will investigate two categories of potential applications. First, FCMs may be used in an organisational context to promote investigation by participants of their individual, deeply held assumptions, and as a tool for facilitating the adoption of new innovations. Second, FCMs have potential application in intelligent tutoring systems. FCMs in an Organisational Context The power of FCMs to reveal hidden patterns in complex conceptual maps can be exploited to promote institutional learning. Arie De Geuss, former planner for Shell Oil, defines institutional learning as changing shared mental models. Developing shared representations of current and future mental models is a complex task. Strategic decision making environments are complex, unstructured, and not readily quantifiable. Cognitive maps have been used as a decision making tool in international relations, administrative science, management science, and operations research. FCMs can be used to make the mental maps of management teams, and others, visible. When the implicitly held assumptions of the participants are laid bare, the process of exploring and changing mental models may be facilitated. Long-term change comes from building the models and participating in the
process. Planning and decision making can also be facilitated as the models are tuned and adjusted over time. FCMs may not be the ‘‘magic bullet’’ for organizational learning, but the technique definitely shows promise as a tool for investigating organizational paradigms and promoting organizational learning

IX.  Conclusions
This paper has discussed the potential usefulness of fuzzy cognitive mapping in educational organization settings. The development of graphical tools to facilitate conceptual change is an important endeavor for educational technologists and facilitators of systemic change. By combining the capability of fuzzy logic to represent soft knowledge domains with dynamic modeling capabilities, the FCM framework has tremendous potential for contribution to the development of useful cognitive tools. FCMs are an extension of earlier concept mapping paradigms, yet they represent a significant advance over earlier, bivalent, static mapping systems. There is much research to be done on the application of FCMs to education and instructional technology. It is heartening to know that there are a growing number of traditional engineering disciplines that are using FCMs to capture and represent expert knowledge resulting in the use of that knowledge in productive and meaningful ways. This paper will hopefully stimulate educational researchers to recognize the unique applicability of fuzzy logic to our field. As our understanding of the complexity of human learning increases, we must embrace new ways of describing, facilitating, and supporting that learning. Fuzzy cognitive maps represent one step in that direction.


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