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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