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Cole Persichitte | Fuzzy Logic | Logic

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Fuzzy Cognitive Mapping: Applications in Education Jason R. Cole,1, * Kay A. Persichitte 2 ,† 1 Nashoba Regional School District, 50 Mechanic St., Bolton, Massachusetts 01740 2 Ed Tech / McKee 518, University of Northern Colorado, Greeley, Colorado 80639 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
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  Fuzzy Cognitive Mapping: Applications in Education Jason R. Cole, 1, * Kay A. Persichitte 2 ,† 1 Nashoba Regional School District, 50 Mechanic St.,Bolton, Massachusetts 01740 2 Ed Tech/McKee 518, University of Northern Colorado,Greeley, Colorado 80639 Fostering conceptual and cognitive change in learners can be difficult. Students oftencome to a learning situation with robust, implicit understandings of the material understudy. One explanation for the implicit nature of these understandings is a lack of metaknowledge about the knowledge to be acquired. Helping learners create metaknowl-edge may free paths to conceptual change. This paper proposes the use of fuzzy cognitive Ž . 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 indetail and recommendations are included for further investigations within educationalcontexts.  2000 John Wiley & Sons, Inc. ‘‘So far as the laws of mathematics refer to reality, they are not certain. And so far asthey are certain, they do not refer to reality.’’ A. Einstein I. INTRODUCTION Ž . Fuzzy cognitive mapping FCM is a tool for formalizing understandings of conceptual and causal relationships. 1 By combining conceptual mapping tools with fuzzy logic and other techniques srcinally developed for neural networks,FCMs allow for the representation and formalization of soft knowledge domains Ž . e.g., politics, education . This paper explores FCM procedures and proposestwo methodologies for developing FCMs in educational organization settings.Other potential applications in education are explored and directions for futureresearch are included.To apply FCMs in education requires a basic understanding of the theoreti-cal foundation of cognitive mapping. This paper presents a brief review of thattheoretical foundation, as well as some related research literature. Advantages *e-mail: jrcole@nrsd.net.†Author to whom correspondence should be addressed; e-mail: persi@edtech.UNCo.edu. Ž . INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 15, 1  25 2000  2000 John Wiley & Sons, Inc. CCC 0884-8173 r 00 r 010001-25  COLE AND PERSICHITTE 2of conceptual and semantic mapping are explored. Inherent weaknesses incurrent approaches are described. Most current mapping systems use a crisp Ž . approach truth values of 1 or 0 to conceptual understanding and causalmapping. Crisp logic cannot accurately represent human understanding, espe-cially in soft knowledge domains.Fuzzy logic allows us to represent truth values on a continuous scale from 0to 1, providing mathematical methods for representing concepts and causalities Ž . that are true to some degree neither wholly true nor false . Consequently, thelaw of the excluded middle does not apply in fuzzy logic. Most human reasoningis fuzzy, with crisp distinctions as the special case of fuzzy logic. For example, when we say the water is very dirty, there is no hard line between dirty and ¨   ery dirty . At some point the water is both dirty and very dirty. Fuzzy logic allows usto represent this idea mathematically and, thus, it becomes machine encodeable.FCMs combine the strengths of cognitive maps with fuzzy logic. By repre-senting human knowledge in a form more representative of natural humanlanguage than traditional concept mapping techniques, FCMs ease knowledgeengineering and increase knowledge-source concurrence. FCMs can also bemodeled on computers, thus allowing for dynamic modeling of cognitive systems. 2 Two methods for facilitating the creation of FCMs are presented in thispaper. Both are in the early stages of testing. The educational field testingincludes trials of a method for group knowledge acquisition and for individualknowledge acquisition. In addition, a computer modeling system is presented forthe development and analysis of FCMs. II. CONCEPTUAL CHANGE Learners’ naive conceptions have been well studied. 3,4 It is widely accepted within the domain of cognitive psychology that students come to school withsome 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. 5 Naive theories are based on interaction with the everyday world. A child whorepeatedly drops items on the floor is building an implicit theory of gravity. Achild who tries to manipulate her parents into taking her for ice cream isbuilding an implicit theory of human behavior.Naive understandings display many of the characteristics of implicit knowl- Ž . Ž . edge. Implicit knowledge: a is characterized by specificity of transfer, b is Ž . associated with incidental learning conditions, c gives rise to a phenomenal Ž . sense of intuition, and d remains robust in the face of time, psychologicaldisorder and secondary tasks. 6 Naive understandings meet many of the samecriteria. They are learned incidentally, they give rise to a sense of ‘‘knowing,’’and remain robust in the face of time and schooling. 3 These naive understand-ings can be very difficult to diagnose and change. 4 Much of the research on naive understandings has been in science educa-tion. The difficulty of changing conceptions of the natural world that have beenformed over many years is well documented. Situations in which students are   APPS AND FCMs 3unable to process that which they are not expecting create opportunities forimplicit learning. For example, when presented with science instruction, stu-dents often are looking to confirm what they ‘‘know’’ already, with the resultboth extremely selective attention and distortion of information provided duringinstruction.Berry and Dienes 6 argue that implicit learning is in part characterized by alack of metaknowledge. That is, we are unaware that we know what we know.The knowledge is unavailable in free recall tasks, but is generally available inperformance, on forced choice tests and constrained answer tasks. 6 In order tomake implicit knowledge available to the learner, some structured task must beavailable to elicit the knowledge from the learner.Concept, or cognitive, mapping represents a possible tool for developingsuch a structured environment. The next section explores some of the methodscurrently used to create graphical formalisms of concepts. III. CONCEPT MAPPING To promote conceptual change, some researchers have proposed usinggraphical representations of a specified conceptual domain. The current use of concept mapping within education has its roots in research conducted at Cornellthat focused on conceptual changes in students over a 12 year period. 7 Thisresearch required a method to compare conceptions over time and betweenlearners. The Cornell researchers developed a system of representing concep-tual knowledge graphically: circles for concepts and arrows for the links betweenthem. A review of the literature 8  10 provides several definitions of concept maps,also known as cognitive maps. Two factors are common in these definitions: Ž . a all of the authors reviewed define a cognitive map as a graphical representa- Ž . tion, and b most include some aspect of subjectivity. A graphical representa-tion is fundamental to the idea of concept mapping. In one of the earliestreferences, Axelrod 8 developed a system for representing causal relationships insocial science domains. The system represented concepts in sociology andpolitical science as nodes in a directed graph. The nodes were connected byarrows that were assigned to represent positive or negative causal relationships.The other common factor in the definition of cognitive mapping is the subjectiv-ity of the map. Irvine 9 describes concept mapping as the individual’s diagram-matic interpretation of ideas. The definition from Park and Kim 10 conciselyencapsulates many of these definitions. The cognitive map graphically represents interrelationships among a variety of factors. Itis a representation of the perceptions and beliefs of a decision maker or expert abouthis r her own subjective world, rather than objective reality. 11 Since the development of the graphical system at Cornell, there have beenseveral studies conducted on the efficacy of using concept maps as teaching andlearning tools. For example, Jegede, Alaiyemola, and Okebukola 12 report that  COLE AND PERSICHITTE 4students in Nigeria who used concept maps documented significantly highermean scores on an achievement test for the subject matter studied. Otherstudies have concentrated on the use of concept mapping in teacher education.Novak 7 notes that most science teachers understand science to be a large bodyof information to be mastered, as opposed to a method for constructing newknowledge. Novak reports that concept mapping plays an important role infacilitating the change of science teachers’ perception of science and thepurpose of science education.There are several factors that contribute to the power of cognitive mappingin learning. Johnson, Goldsmith, and Teague 13 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 theglobal organization of the represented concepts. Graphical representations alsoallow for the organization of complex domains for learners and designers alike.The network structure of a concept map allows the simultaneous display of allthe important relationships. 13 The structural advantage is relative to the assess-ment of pair-wise ratings. Johnson, Goldsmith, and Teague 13 report that assess-ing network representations of student understanding resulted in more validmeasures than assessing pair-wise comparisons of concepts.Thagard 5 proposes a system of conceptual change based on his historicalresearch of scientific revolutions. The system delineates five levels of conceptualchange, ranging from the simple to the complex: Ž . 1 Addition of concepts. Ž . 2 Deletion of concepts. Ž . 3 Simple reorganization of concepts in the kind-hierarchy or part-hierarchy whichresults in new kind-relations and part-relations. Ž . 4 Re ¨  isionary reorganization of concepts in the hierarchies, in which old kind-rela-tions or part-relations are replaced by different ones. Ž . 5 Hierarchy reinterpretation , in which the nature of the kind-relation or part-rela-tion that constitutes a hierarchy changes. Ž Kind-relations define members of a concept i.e., a whale is a kind of  . Ž mammal , while part-relations define the characteristics of a member i.e., .  whales have flippers . Concept maps make visible the potential for conceptualchange within a learner. Thagard, 5 himself, makes good use of concept maps todemonstrate the conceptual change undergone by the scientists he studied.Pressley and McCormick’s 14 review of the literature on multiple representa-tions in science revealed a common process for developing concept maps. Thisprocess is briefly outlined: Ž . 1 Key words and phrases are identified from the reading . Ž . 2 Key concepts are ordered from the most general to the most specific . Ž . 3 The concepts are then clustered using two criteria . Concepts that interrelate are Ž 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 looselyin a two-dimensional array with abstractness defining one dimension and mainideas defining the second dimension.
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