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A Solution for Effective Teaching using Fuzzy

Cognitive Maps (COBFCMS)

Mr. A. Ra jku mar, Mrs. Jose Parvee na, Dr. C. Jaya lat ha, Dr.A.praveen pra kash

Abs tractThe necessity to use various tools instructional media and technology f or effective teaching using combined overlap block f uzzy cognitive maps (COBFCMS) def ined by W.B. Vasantha Kandasw amy is analyzed in this paper. The comb ined overlap block FCM’s def ined in this method become eff ective w hen the number of concepts can be grouped and are large in numbrs. In this paper w e have analyzed the various tools needed f or eff ective know ledge transf er. In this paper, w e analyzed the eff ective teaching method s and pedagogical practices and to develop new insights to serve their needs by f uzzy cognitive maps. Th is paper has f ive sections: First section gives the inf ormation about development of f uzzy cognitive maps, second section gives preliminaries of f uzzy cognitiv e maps, and combined overlap block f uzzy cognitive maps, in section three w e describe the proble m, in section f our w e explain the method of determining their hidden pattern and the f inal section gives the conclusion based on our studies.

Inde x TermsFCMS, CBFCMS, Tea chi ng Methods .


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olitical scientist R.Axelr od[1] intr oduced cognitive maps for r epr esenting social scientific knowledge and descr ibing the methods that ar e used for decision making in social and political systems. Then B. Kosko[2,3,4] enhanced the pow er of cognitive maps consider ing fuzzy values for the concepts of the cognitive map and fuzzy degr ees of interr elationships between concepts. FCMs can successfully r epr esent knowledge and human exper ience, intr oduce concept to r epr esent the essential elements and cause the effect r elationships among the concepts to model the behavior of any system. It is a very convenient, simple and power ful tool, which is used in numer ous fields such as social economic and medical etc. the purpose of study is to identify r isk gr oups, In
this case w e ar e discussing about the
teaching methods for the teacher s how it is useful for the pur pose of teaching.w e ar e identify the best teaching methods
and w e ar e conculced the effects of it.
Hence fuzzy tools alone has the capacity to analyze these concepts. Hence it is chosen her e.

———— ——— ——— ——— ———

A.Rajkumar is currently working as asst prof. in Hindustan university, Chennai

N.Jose parveena is currently working as asst prof. in K.C.G. College of

Technology, Chennai

Dr. C. Jayalatha is currently working as Prof. in Hindustan university,


Dr. A. Praveen Prakash is currently working as Prof. in Hindustan

university, Chennai


Fuzzy cognitive maps (FCMs) ar e mor e applicable when th e data in the first place is an unsupervised one. The FCMs w or k on the opinion of experts. FCMs model the w orlds as a collection of classes and causal r elation between classes.

Definition 2.1: An FCM is a dir ected graph with concepts like policies, events etc. As nodes and causalities as edges. It r epr esents causal r elationship betw een concepts.

Definition 2.2: W hen the nodes of the FCM ar e fuzzy sets then they ar e called as fuzzy nodes.

Definition 2.3: FCMs w ith edge w eights or causalities fr om the set {-1,0,1} ar e simple

Definition 2.4: The edges eij take values in the fuzzy causal interval [-1,1]. eij =0 indicates no causality eij >0 indicates causal increase Cj increases as Ci increases (Or Cj Decreases as Ci Decreases). E <0 indicates causal decrease or negative causality. C Decreases as C increases (And or C j Increases as Ci Decreases). Simple FCMs have edge values in {-1,0,1}. Then if causality occurs, It occurs to a maximal positive or negative degree. Simple FCMs provide a quick first approximation to an expert stand or printed causal knowledge. If i ncrease (Or decrease) in one concept leads to increase(or decrease) in another, Then we give the value 1.If there exists to relation between the two concepts, The value 0 is given. If increase (or decrease) in one concept decreases(or increases) another, then we give the value -1. Thus FCMs are described in this way. Consider the or concepts C1, ,…, Cn Of the FCM. Suppose the directed graph is drawn using edge weight eij ε {0,1,-1}. The matrix E be defined by E= (eij ), Where the eij is the weight

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of the directed edge Ci,cj. E is called the adjacency matrix of the FCM, also known as the connection matrix of the FCM. It is important to note that all matrices associated with an FCM are always square matrices with diagonal entries as zero.
Definiton 2.5: Let C1 , C2 ,…. Cn be the nodes of an FCM. Let A=(a1 , a2 ,…., an ). Wher e aiε {0,1}. A is called the instantaneous state vector and it denoted the on off position of the node at an instant
ai =0 if ai is off=1
ai =1 if ai is on, wher e i=1,2,….,n.
Definition 2.6: Let C1 ,C2 , ……, Cn be the nodes of an FCM. LetC1 C2 ,C2 C3 ,…, CiCj, be the edges of the FCM (i j ). Then, the edges form a dir ected cyclic. An FCMs said to be cyclic if it
possesses a dir ected cyclic. An FCM is said to be a cyclic if it does not possess any dir ected cyclic.

Definition 2.7: An FCM w ith cycles is said to have a feedback.

Definition 2.8: Wher e ther e is a feedback in an FCM, i.e., When the causal r elations flow thr ough a cycle in a r evolutionary way, The FCM is called a dynamical system.

Definition 2.9: Let C1 C2 , C2 C3 ,…, Ci Cj, be a cycle when Ci is sw itched on and if the causality flows thr ough the edges of a cycle and if it again causes Ci, We say that the dynamical system goes r ound and r ound. This is true for any node Ci , for i=1,2,…,n. The equilibr ium state for this dynamical system is called the hidden pattern.
Definition 2.10: If the equilibrium state of a dynamical system is a unique state vector, Then it is called a fixed point. Consider a FCM with C1 , C2 ,…, CN as nodes. For example let us start the dynamical system by sw itching on C. Let us assume that the FCM settles down w ith C1 and Cn on, i.e. the state vector r emains as (1, 0, 0,…, 0, 1). This state vector (1, 0,
0, …, 0, 1) is called the fixed point.
Definitio n 2.11: If the FCM settles down w ith a state vector r epeating in the form A1 A2 … AI A1 . Then this
equilibr ium is called limit cycle.
Definition 2.12: Finite number of FCMs can be combined together to pr oduce the j oint effect of all the FCMs. Let E1 ,E2
,…, Ep be adj acency matr ices of the FCMs w ith nodes C1 ,C2 ,…,
Cn. Then the combined FCM[5,6,7] is got by adding all the
adjacency matr ices E1 ,…, Ep . We denote the combined FCM
adjacency matr ix by E= E1 +E2 +…+Ep

Definition: 2.13 : Let P be the pr oblem under investigation. Let

{C1 ,C2 ,…, Cn} be n concepts associated w ith p(n very lar ge).
Now divide the number of concepts {C1 ,C2 ,…,Cn} into classes

S1 ,…, St wher e classes ar e such that
(1) Si Si+ 1   wher e (i=1,2,…, t-1)
(2) si=( c1,…………cn)
(3) (si) sj if i j in gener al
Now we obtain the FCM associated w ith each of the classes S1
,…, St . We deter mine the r elational matrix associated w ith
each S . Using theses matrices w e ob tain a nxn matr ix. This n x n matr ix is the matrix ass ociated with the combined overlap block FCM(COBFCM) of blacks of same sizes.
Definition 2.14: Suppose A= (a1 ,…, an ) is a vector which is passed into a dynamical system E. Then AE=)a’ 1 ,…,a’ n). After thr esholding and updating the vectors suppose w e get (b 1 ,…,
b n). We denote that by (a’1 ,a’2 ,…, a’n ) (b 1 ,b 2 ,…,b n). Thus
the symbol means that the r esultant vector has been
thr esholded and updated. FCMs have several advantages as well as some disadvantages. The main advantage of this method it is simple. It functions on experts opinion’s. when the data happens to be an unsuper vised one the FCM comes handy. This is the only known fuzzy technique that gives the hidden pattern of the situation. As w e have a very w ell known theory, which states that the str ength of the data depends on the number of exper ts opinions w e can use combined FCMs with several exper ts opinions. At the same time the disadvantage of the combined FCM is w hen the w eightages ar e 1 and -1 for the same Ci Cj. We have the sum adding to zer o thus at all times the connection matr ices E 1 ,…, Ek may not be comfortable for addition. This pr oblem w ill be easily over come if the FCM entries ar e only 0 and 1.


1. Effective delivery of conceptual knowledge : Conceptual knowledge refers to a person’s representation of the major concepts in a system. This would enable a person to tackle technical questions with relative ease. Conceptual Knowledge includes Knowledge rich in relationships and understanding and must be learnt through thoughtful reflective learning.

2. Effective knowledge transfer (EKT): EKT ia a set of activities and appr oaches that ar e undertaken to move know ledge among those who have inter ests or needs in it.

3. Innovative Pedagogical Practices: Pedagogy is the act of teaching together with its attendant discourse. It is what one needs to know, and the skills one needs to command in or der to make and j ustify the many differ ent kinds of decisions of which teaching is constit uted. As Leach and Moon (1999) expanded fur ther on w hat may define pedagogy by descr ibing a Pedagogical Setting as ‘the practice that a teacher , together with a particular gr oup of lear ners cr eates, enacts and exper iences’.

4. Knowledge base: A knowledge base is a special kind of

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dat abase for knowledge management . A know ledge base pr ovides a means for information to be collected, or ganised, shar ed, sear ched and utilised.

5. Instructional media and technology(ohp/lcd/video clips/e - learning): Use of tools r elated to Information and Communication Technology (ICT) such as computer hardwar e/software, netw or king and other technologies such as audio, video, and other multimedia tools for effective dissemination of knowledge

6. Brain storming: Brainstor ming is an infor mal way of generating topics to wr ite about, or points to j ot down on a particular topic which can be done at any time dur ing the writing pr ocess. The key to effective br ainstorming is that ther e should be no pr essur e to be Brilliant. Par t of brainstorming w ill involve a selection pr ocess.

7. Mind mapping: This is a pow er ful graphic technique which pr ovides a universal key to unlocking the potential of the brain. The Mind Map can be applied to ever y aspect of life wher e impr oved learning and clear er thinking will enhance human per formance.

8. Case study for illustration of concept: Documented study of a specific r eal-life situation or imagined scenar io, used as a training t ool in business schools and firms. Students or trainees ar e r equir ed to analy ze the pr escr ibed cases and pr esent their inter pr etations or solut ions, supported by the line of r easoning em ploy ed and assumpt ions made.

9. Modelling for practical application of concept : A r epr esentat ion of a system that allows for investigat ion of the pr opert ies of the system and, in some cases, pr ediction of fut ur e outcomes. Models ar e often used in quant itat ive analysis and t echnical analy sis, and sometimes also used in fundamental analysis.

10. Student knowledge testing methods-(GD/QA/MCQ’s): Group discussions, multiple choice questions, reas oning questions, questionnaires are various ways in which the extent of knowledge reception can be gauged. This would help

vector A1 = (1, 0, 0, …, 0), the data should pass thr ough the r elation matrix E. this is done by multiplying A1 by the matr ix E. Let A1 E= (a1 ,…., an )w ith the thr eshold oper ation that is by r eplacing ai by 1 if ai >k and ai by 0 if ai <k (k is a suitable positive integer ). We update the r esulting concept, The concept C1 is included in the updated vector by making the
first coor dinate as 1 in the r esulting vector. Suppose A1 E
A2 then consider A2 E and r epeat the same procedur e. This
pr ocedur e is r epeated till we get a limit cycle or a fixed point.


Using the linguistic questionnair e and the expert’s opinion w e

have taken the follow ing eleven concepts C1, C2 ,..., C10

C1 = Effective delivery of conceptual knowledge
C2 = Effective knowledge tr ansfer (EKT) C3 = Innovative Pedagogical Practices
C4 = Knowledge base
C5 = Instr uctional media and technology (ohp/lcd/videolips
C6 = Brain storming
C7 = Mind mapping
C8 = Case study for illustration of concept
C9 = Modelling for practical application of concept
C10 = Student know ledge testing methods -(GD/QA/MCQ’s)

Now we pr oceed on to apply the effect of combined overlap block. FCM of equal length. Let us consider the eleven concepts C1, C2 ,..., C10 . W e divide these concepts into cyclic way of classes, each having j ust four concepts in the follow ing way.

The dir ected graph and the r elation matrix for the class C =
{C1 , C2 , C4 , C10 }. Given by the expert is as follows:
segregate the student community of a class into sections. C1 C2
Further , the section that would require a different approach of teaching and tutoring


Let C1 , C2 ,…, Cn be the nodes of an FCM, W ith feedback. Let E be the associated adj acency matr ix. Let us find the hidden patter n when C1 is switched on. When an input is given as the

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C4 C10

The directed graph and the relational matrix for the class C =
{C3 , C6 , C 7, C8}. Given by the expert is as follows:

C1 C2 C4 C10

C1 0 1 1 0
C2 1 0 1 1
C4 1 1 0 1
C10 0 1 1 0
The directed graph and the relational matrix for the class C =
{ C3 , C5 , C6 , C7}. Given by the expert is as follows:
C3 C6

C3 C5

C3 C6 C7 C8

C3 0 1 1 1
C6 1 0 1 0
C7 1 1 0 0
C6 C7
C8 1 0 0 0

The directed graph and the relation matrix for the class C ={C3, C7, C8, C9} Given by the expert is as follows:
C3 C5 C6 C7

C3 0 1 1 1
C5 1 0 0 1
C6 1 0 0 1
C7 1 1 1 0
C3 C7

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C8 C9

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C3 C7 C8 C9

C3 0 1 1 1
C7 1 0 0 1
C8 1 0 0 0
C9 1 0 0 0
The combined direct graph and combined overlap block

FCM of equal sizes as follows:


The directed graph and the relation ma trix for the class C
={C1, C3, C8, C10} Given by the expert is as follows: C6

C1 C3 C5
C7 C8
C(m) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10

C1  0 1 1 1 0 0 0 2 0 2 

C8 C10

C2  1 0 0 1 0 0 0 0 0 1 

C3  1 0 0 0 1 2 3 0 0 1 

 

C4  1 1 0 0 0 0 0 0 0 1 

C1 C3 C8 C10

C1 0 1 1 1
C3 1 0 1 0
C8 1 1 0 0
C10 0 0 0 0

C  0 0 1 0 0 0 1 0 0 0 

 

C6  0 0 2 0 0 0 2 0 0 0 

C7  0 0 3 0 1 2 0 0 1 0 

C8  2 0 3 0 0 0 0 0 0 0 

 

C9  0 0 1 0 0 0 0 0 0 0 

C10  0 1 0 1 0 0 0 0 0 0 

Now using the matrix A of the combined overlap block FCM,

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We determine the hidden pattern. Suppose the concept S 1 is in the on state and another nodes are in the off sate. Let the initial input vector be X={ 0 0 0 0 1 0 0 0 0 0 }
X C(m) = {0 0 1 0 0 0 1 0 0 0} =X1
X1 C(m) = {1 0 3 0 2 4 3 2 2 0}  (1 0 1 0 1 1 1 1 1 0) = X2

X2 C(m) = {3 1 1 1 2 4 6 4 2 2} (1 1 1 1 1 1 1 1 1 1) = X3
X3 C(m) = {5 3 1 3 2 4 6 4 2 2} (1 1 1 1 1 1 1 1 1 1) = X3
Where Denotes the resultant vector after thresholding
and updating.
X3 is the hidden pattern which is the fixed point.


While analyzing FCM, w hen the concept C5, ‚ Instructional media and technology (ohp/lcd/videolips/e-learning) is in the on state, the other concepts C2 , C2 , C3 , C4 , C5 , C6 , C7 , C8 , C9 , C10 ar e in the on state. The educational pr ocess needs to focus on the equipping and satisfying the needs of the students for ongoing learning, for social and personal competence to meet complex, r eal-life challenges. For this the Teaching learning Pr ocess has t impact the young mind sufficiently to leave a lasting impr ession w ith foundations r ooted str ongly in the individual’s know ledge b ase. This would involve enr iching the classr oom exper ience which can only be done with the effective implementation of the available instr uctional media( teaching tools) and technology.
The teaching pr ocess equipped w ith the tools mentioned in this paper would r ejuvenate not only the teach er but w ould be successful in instigating the young mind to think beyond – as commonly put ‚ thinking out of the b ox‛. As is w ell known, visual perception often leaves a lasting impr ession while practical implementation of concepts w ould for ever for ge the concepts in the thinking faction of the mind. Such lectur es would invite participation fr om the students clarifying doubts and causing them to ventur e in to new dimensions. Ther e w ill be an immense satisfaction for the teacher in terms of effective delivery of know ledge using all the available r esour ces besides the ultimate satisfaction that the student has learnt a new concept.
As technology gives us the tools to dispense infor mation in the best possible way, teacher s can continue on with their efforts to make information available inter estingly and in r elation to the curr ent scenario.


The authors wish to thank Dr. W.B. Vasantha Kandasamy.


[1] Axelrod ,R.(1976).Struc ture o f decision :Th e c ongn itive
maps of political elites.Princeton Unive rsity.
[2] Vasantha Kandasamy W .B and Victor Devadoss A.
“Some N ew Fuzzy Techniques” , Jour.of inst.o f Math
& Co mpu ter science.
[3] Kosko ,B., “Fuzzy Co gnitive Maps”, In ternational
Journal of man-machine studies ,jan(1986)
[4] Kosko , B .Neural N etw orks and Fuzzy System
Prentice Hall of India,1997 .
[5] Kosko ,B. Hidden patterns in Co mbin ed and adaptive Knowled ge Networ ks,International Conference of Neural Ne twor ks(ICNN-86)1988 377-393.
[6] Vasantha Kandasamy W.B and M.Ram Kishore Symptom-Disease Model in Children using FCM, Ultra sci.11(1999)318-324.
[7] Ref: Effective Know ledge tr ansfer and exchange for non- pr ofit or ganisations – A Fr amewor k - by Fataneh Zarinpoush, Shir ley Von Sychowski, Julie Sper ling , Imagine Canada 2007
[8] Ref: Banks F, Leach J and Moon B (1999) New

understandings of teachers’ pedagogic knowledge. In Leach J

and Moon B (Eds) Learne rs and Pedagogy. London: PCP [9] Ref: W ikipedia encyclopedia
[10] Ref: Integrating know ledge of instr uctional media and
technology in pr e-ser vice science teacher education Savittr ee Rochanasmita Arnold1*, Michael J. Padilla2 and Bupphachart Tunhikorn, The Pr ogr am to Pr epar e Resear ch and Development Per sonnel for Science Education,Department of Education, Faculty of Education, Kasetsart University, Bangkok 10900, Thailand,The Eugene T. Moor e School of Education and Educational Collaborations, Clemson University, South Car olina 340702, USA, Department of Education, Faculty of Education, Kasetsart University, Bangkok

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