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7FUZZY LOGIC AND THE CONCEPT OF THE ZONE
OF PROXIMATE DEVELOPMENT
(Нечеткая логика и понятие зоны ближайшего развития)
I.Subbotin, Professor, National University, Los Angeles, USA, F.Mossovar-Rahmani, Professor and Chair, National University Academic Headquarters,
LaJolla, USA, N.Bilotskii, Assocoate Professor, National Pedagogic University, Kiev, UKRAIN
У onammi розглядаються деякг додатки гдей нечтког логгки до формал1зацп концепцп зони найближчогорозвитку Л.С.Виготського.
Ключовi слова: навчання i вчення, зона найближчого розвитку, математичне моделю-вання, нечгткг множини, невпевнетсть.
Introduction. Created by L.AZadeh ([18], [19]), fuzzy logic has been successfully developed by many researchers and has been proven to be extremely productive in many applications (see, for example, [2], [4], [6], [7], [17], [1], and others). There are also some interesting attempts to implement Fuzzy logic ideas in the field of education ([13], [14], [9], [3], [11], [12]).
The main goal of every professional instructor is to intensify the learning process, to achieve the highest possible level of knowledge acquisition for entire class and for each individual student. K.Lewin wrote 'There is nothing more practical than a good theory," [8, 1952, p.169]. There are some important developments in the theoretical ground of the learning process allow us to develop some interesting applications useful for instructors in everyday classroom.
Perhaps the central here is a Vygotsky's concept of the Zone of Proximate Development (ZPD). L.Vygotsky [16, p. 86] defines the ZPD as "the distance between the actual
developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers". Vy-gotsky argued that providing the appropriate assistance (scaffolding) gives the student enough of a "boost" to achieve the task. Once the student, with the help of scaffolding masters the task, the scaffolding can be removed and the student will be able to complete the task again on his own. In the learning process it is important to recognize and assess a student's intellectuality capacity. "The concept of scaffolding is a process, wherein an instructor provides only the needed support by providing tasks that will enable a learner to build on prior knowledge, and once the stage of zone of proximal development has been reached, the guidance is gradually removed"
(http://projects.coe.uga.edu/epltt/index.php ?ttle=Vygotsky%27s_constructivism#Vygotsk y.27s_Theories). "The zone of proximal development defines those functions that have
not yet matured but are in the process of maturation, functions that will mature tomorrow but are currently in an embryonic state. These functions could be termed the "buds" or "flowers" of development rather than the "fruits" of development" ([16, p.86]).
Another striking approach has been developed by J.Voss [15] who argued that learning as a specific case of knowledge transfer consists of successive problem-solving activities, in which the input information is represented of existing knowledge with the solution occurring when the input is appropriately represented. This process implements the following states: a) representation of the input data, b) interpretation of this data, c) generalization of the new knowledge, and d) categorization of this knowledge. The states a and b could be unified in one state of interpretation the new knowledge. M.Voskoglou in the articles [13] and [14] has developed an appealing fuzzy set applications based on the Voss's theory. In the following few paragraphs we cite parts of these articles.
Let Ai, i =1,2,3, be the states of interpretation, generalization, and categorization respectively, and a, b, c, d, e the linguistic variables of negligible, low, intermediate, high, and complete acquisition of knowledge respectively of each of the Ai. M.Voskoglou considers the set U = {a, b, c, d, e} and represents the A/s as fuzzy sets in U. He denotes by nia, nib, nic, nid, nie the numbers of the students that have achieved negligible, low, intermediate, high, and complete acquisition of the state Ai respectively and defines a membership function n
mA by mA (x) = —- for each x e U and, i n
therefore, one can write
n
A, = {— ^ ):xe U}, n
where ^ mA (x) = 1, i = 1,2,3. A fuzzy rela-
xeU '
tion can be considered here as a fuzzy set of triples, each one of which possess a degree of membership belonging to [0, 1]. Consider further the fuzzy relation
R = {(5, mR (5): 5 = (x, y, z) e U3} where the membership function is defined by
mR (5) = mA (x) mR2 (y)mR3 (zX
for all s = (x, y, z) e U3 if (x, y, z) is a well ordered triple and zero otherwise.
A triple is said to be well ordered if x corresponds to a degree of acquisition equal or greater than y, and y corresponds to a degree of acquisition equal or greater than z.
In fact, if for example (b, a, c) possessed a non zero membership degree, how it could be possible for a student, who had not generalized at all, to categorize satisfactorily the new knowledge?
This fuzzy relation R represents all the possible profiles of student's behavior during the learning process. Further, M.Voskoglou develops the procedure of comparing few groups of students based on his ideas and supplies his articles with examples showing straightforwardness of its applications. He finally argues [14] that the totalpossibilistic uncertainty (i.e. the sum of strife and non specificity [7, p.28]) on the ordered possibility distribution of a student group during the process of learning can be used as a measure of its capacity in learning a subject matter.
M.Voskoglou in [13] described the developed procedure in the following way:
"Let us consider a group G of n students during the process of learning in the classroom, n e N, n > 2. Obviously, from the point of view of the teacher, there exists an uncertainty about the degree of acquisition of each state of the process from his students, a fact which gave us the hint to introduce the fuzzy sets theory in order to achieve a mathematical representation of the process of learning in the classroom. For this, let us denote by A, i=1, 2, 3, the state of interpretation, generalization and categorization respectively, and by a, b, c, d, e the linguistic labels of negligible, low, intermediate, high and complete acquisition respectively of each of the A/s. Consider the set U = {a,b,c,d,e} , then we are going to represent the A/s as fuzzy sets in U. In fact, if nia, nib, nic, nid and nie denote the numbers of the students that have achieved negligible, low, intermediate, high and complete acquisition of the state Ai respectively, i=1, 2, 3, we can define the
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membership function mAi by mAi(x)=nix/n, for each x in U and therefore we can write Ai={(x,nix/n): x e U}. It becomes clear then that YjmAi(x)=1, x e U, i=l ,2,3 .At this point notice that a fuzzy relationship, like the classical ones, can be considered as a fuzzy set of tuples each one of which possesses a degree of membership included between 0 and 1. Consider now the fuzzy relation R={(s, mr(s)):s=(x, y, z) e U3}, where the membership function mR is defined by mR(s)=mAl(x)mA2(y)mA3(z), for all s=(x,c,z) in U3. This definition satisfies the axioms of aggregation operations in fuzzy sets and further we have that XmR(s)=1. The fuzzy relation R represents all the possible profiles (overall states) of the behavior of a student during the learning process. In the next, and in order to simplify our notation, we shall write ms instead of mR(s). Assume now that one wants to study the behavior of k groups of students during the learning process of the same subject, or the behavior of the same group of students during the learning process of k different subjects, k e N, k > 2. In this case it becomes necessary to introduce the fuzzy variables Ai(t), where i=1, 2, 3 and t=1,2,... k. Then the pseudofrequency f (s) of the overall state s(t) is k given by the sum Yms(t), while the probability of s(t) is t=l given by p(s)=f (s)/ Sf(s), where Xf(s) denotes the sum of all pseudofrequences. But, since X ms=1, it becomes clear that Xf (s)=k and therefore p(s)=f(s)/k. Finally the possibility of s(t) is given by r(s)=f(s)/max f(s), where max f(s) denotes the maximal pseudofrequency. The possibility of s(t) measures the degree of evidence of combined results, i.e. in other words one may say that r(s) gives the "relative probability" of s(t) with respect to the other overall states."
Applications of the Voskoglou's Fuzzy Model to ZPD. One of the main goals of the learning process is long-term retention and transfer [5]. In our era of overwhelming informational pressure and pace instructional approaches, it is very important to evaluate the student abilities to progress in learning acquisition and to measure and determine the needed echelon of scaffolding toward the learning intensification. In the following paragraphs we
will discuss some implementations of the described above fuzzy techniques to evaluation of effectiveness of student learning based on the ZPD concept. We will use the developed in [13] method. We would like to remark, that the following exposition is just an example illustrating efficiency of a Fuzzy set approach.
One of the authors taught on-line mathematics classes for prospective elementary school teachers at the National University, California, USA. During first week of this course, the students should learn the vital and intricate for the beginner topic Sets as a basis for whole numbers. Studying this material, students should read the corresponding book chapter, solve a significant amount of problems, answer for the board discussion questions, participate in a live format virtual synchronize classroom sessions, and take tests. Taking into account the following G.Polya's [10] important remark: "For an effective learning the learner discovers alone the biggest possible, under the circumstances, part of the new information", we try to provide the scaffolding in the most efficiently supporting the students' independent and group work way. Based on ZPD, we can consider tree important states of the knowledge acquisition process.
State A1: An independent student introductory work (book reading, lecture reading, to be acquainted with discussion questions and problem sets). This is the phase of "things that can be done on own".
State A2: Light scaffolding (group discussions, on-line chats, individual help, and group work). This is the phase of "needs little support".
Sate A3: Major instructor help (live lecture, discussions, consultations, guided problem solving). This is the "needs much support" phase.
At each state, the students knowledge acquisition were observed and labeled correspondingly to above described ideas from [13]: c stands for the "below satisfactory", b" for "satisfactory", and a for "above satisfactory" level of acquisition.
We observed that 16, 8, and 1 students achieved below unsatisfactory, satisfactory, and above satisfactory respectively (n1c =12,
nib= 8, nia=5) at the first state.
Thus An= {(c,16/25), (b,8/25), (a,1/25)]}.
At the next state we found that A12= {(c,8/25), (b,15/25), (a,2/25)}.
At the final state we found that A13 = {(c,2/25), (b, 18/25), (a,5/25)}.
Another data set gave us the following results.
A21= {(c, 10/20), (b 8/20), (a,2/20)}.
A22= {(c,4/20), (b, 12/20), (a,4/20)}.
A23= {(c,0/20), (b, 16/20), (a,4/20)}.
Looking at the Ais for both sets, we can see that the higher the state of scaffolding, the enhanced degree of acquisition of knowledge we achieve.
Table 1, Part 1: Probabilities and possibilities of profiles
Tuples
Data Set I
Data Set2
A1A2A3 A11 A12 A13 A21 A22 A23
aaa 0.04 0.08 0.2 0.1 0.2 0.2
aab 0.04 0.08 0.72 0.1 0.2 0.8
aac 0.04 0.08 0.08 0.1 0.2 0
aba 0.04 0.6 0.2 0.1 0.6 0.2
abb 0.04 0.6 0.72 0.1 0.6 0.8
abc 0.04 0.6 0.08 0.1 0.6 0
aca 0.04 0.32 0.2 0.1 0.2 0.2
acb 0.04 0.32 0.72 0.1 0.2 0.8
acc 0.04 0.32 0.08 0.1 0.2 0
baa 0.32 0.08 0.2 0.4 0.2 0.2
bab 0.32 0.08 0.72 0.4 0.2 0.8
bac 0.32 0.08 0.08 0.4 0.2 0
bba 0.32 0.6 0.2 0.4 0.6 0.2
bbb 0.32 0.6 0.72 0.4 0.6 0.8
bbc 0.32 0.6 0.08 0.4 0.6 0
bca 0.32 0.32 0.2 0.4 0.2 0.2
bcb 0.32 0.32 0.72 0.4 0.2 0.8
bcc 0.32 0.32 0.08 0.4 0.2 0
caa 0.64 0.08 0.2 0.5 0.2 0.2
cab 0.64 0.08 0.72 0.5 0.2 0.8
cac 0.64 0.08 0.08 0.5 0.2 0
cba 0.64 0.6 0.2 0.5 0.6 0.2
cbb 0.64 0.6 0.72 0.5 0.6 0.8
cbc 0.64 0.6 0.08 0.5 0.6 0
cca 0.64 0.32 0.2 0.5 0.2 0.2
ccb 0.64 0.32 0.72 0.5 0.2 0.8
ccc 0.64 0.32 0.08 0.5 0.2 0
Table 1, Part II: Probabilities and possibilities of profiles
A1A2A3 ms(1) ms(2) f(s) p(s) r(s)
aaa 0.00064 0.004 0.00464 0.00232 0.008983
aab 0.002304 0.016 0.018304 0.009152 0.035437
aac 0.000256 0 0.000256 0.000128 0.000496
aba 0.0048 0.012 0.0168 0.0084 0.032525
abb 0.01728 0.048 0.06528 0.03264 0.126382
abc 0.00192 0 0.00192 0.00096 0.003717
aca 0.00256 0.004 0.00656 0.00328 0.0127
acb 0.009216 0.016 0.025216 0.012608 0.048818
acc 0.001024 0 0.001024 0.000512 0.001982
baa 0.00512 0.016 0.02112 0.01056 0.040888
bab 0.018432 0.064 0.082432 0.041216 0.159588
bac 0.002048 0 0.002048 0.001024 0.003965
bba 0.0384 0.048 0.0864 0.0432 0.16727
bbb 0.13824 0.192 0.33024 0.16512 0.639345
bbc 0.01536 0 0.01536 0.00768 0.029737
bca 0.02048 0.016 0.03648 0.01824 0.070625
bcb 0.073728 0.064 0.137728 0.068864 0.266641
bcc 0.008192 0 0.008192 0.004096 0.01586
caa 0.01024 0.02 0.03024 0.01512 0.058545
cab 0.036864 0.08 0.116864 0.058432 0.226249
cac 0.004096 0 0.004096 0.002048 0.00793
cba 0.0768 0.06 0.1368 0.0684 0.264845
cbb 0.27648 0.24 0.51648 0.25824 0.999905
cbc 0.03072 0 0.03072 0.01536 0.059474
cca 0.04096 0.02 0.06096 0.03048 0.118019
ccb 0.147456 0.08 0.227456 0.113728 0.440355
ccc 0.016384 0 0.016384 0.008192 0.031719
Remark here that in this case we don't need to distinguish well ordered profiles in the sense of the Voskoglou's definition, because the states Ai, i=1,2,3 are independent to each other.
From this table it turns out that the profile s=(c, b, b) had the highest pseudofrequency for the two groups of data sets of our experiment ms(l) =mA i (c)mA2(b)mA 3(a)=0.64x0.6x0.7 2=0.27648, ms(2) = 0.5x0.6x0.8 = 0.24, and therefore f (s)=0.51648. Thus it had also the highest probability of occurrence p(s) = 0,25824, or 25.8%, while its
possibility was 0.999905 ~ 1.
Fuzzy logic and scaffolding effectiveness.
In [11] and [12] some another approach complementing in some sense the M. Voskoglou's concept is has been introduced. There is a commonly used in fuzzy logic approach to measure performance with the pair of numbers (xc,yc) as the coordinates of the center of mass of the represented figure F, which we can calculate using the following well-known formulas:
(1)
x.
_ F
U xdxdy || ydxdy yc
F
|| dxdy c || dxdy
F F
In Fig. 1 c stands for the "below satisfactory" (y1) , b" for "satisfactory" (y2), and a for "above satisfactory" (y3) level of acquisition for each states A1, A2, A3.
Fig. 1: The center of mass model for measure performance
In this case, formulas (1) can be easily transformed to the following simple formulas [11]:
(2)
xc =
yc =
1 f yi + 3 y2 + 5 y A
V yi + y 2 + y3
2 A
f
yi + y2 + y3
^ y1 + y2 + y3 ) It is easy to see that formulas (2) can be generalized for the case when our figure consists not only from three rectangles, but from n rectangles In this case we will come to the following formulas [11]:
f n \ f n \
(3) xc=2
E (2' -1) y,
1=1
E yi
i=1
1
■ yc =T
E y/
i=1 n
E yi
V 1=1
Base on our data above, we can obtain the following table.
Table 2: State I
Ratio of the class students Set I Set II
reached the following stage of knowledge acquisition at the State I of independent student work (State A1)
C 0.64 0.5
B 0.32 0.4
A 0.04 0.1
y2 +y3 =1. Hence, formulas (2) will look like this:
(4)
xc =1 (y + 3y2 + 5 y3), yc =1 (y12 + y22 + y32).
2W1 "2 c 2.....+
In this case, for the Set 1 we have xc1 = 0.5(0.64 + 3 ■ 0.32 + 5■ .0.04) = 0.9; yc1= 0.5(0.642 + 0.322 + 0.042) = 0.2563. For the set II we have xc2 = 0.5(0.5 + 3■ 0.4 + 5- .0.1) = 1.1; yc1=
0.5(0.52 + 0.42 + 0.12) = 0.21 By the following rule developed in [12] Among two or more classes the class with the biggest xc performs better;
(5) If two or more classes have the same xc > 1.5, then the class with the higher yc performs better. If two or more classes have the same xc <1.5, then the class with the lower yc performs better; we conclude that the Set II shows a better performance rate at the State I.
Consider now the State II of light scaffolding.
Table 3: State II
Ratio of the class students Set I Set II
reached the following stage of knowledge acquisition at the State II of light scaffold-
ing (State A2)
C 0.32 0.2
B 0.6 0.6
A 0.08 0.2
In this case, for the Set 1 we have
xc] = 0.5(0.32 + 3 ■ 0.6 + 5 .0.08) = 0.45; yc1= 0.5(0.322 + 0.62 + 0.082) = 0.1362. For the set II we have xc2 = 0.5(0.2 + 3■ 0.6 + 5- .0.2) = 1.5; yc1=
0.5(0.22 + 0.62 + 0.22) = 0.22. Base on the rule (5) we conclude that the Set II demonstrates a better performance rate at the State II.
Consider now the State III of major scaffolding.
Obviously in this case Y1 + Y2 + Y3 = y1 +
Table 4: State III
Ratio of the class stu- Set I Set II
dents reached the fol-
lowing stage of knowledge acquisition at the State III of major scaffolding (State A3)
C 0.08 0
B 0.72 0.8
A 0.2 0.2
In this case, for the Set 1 we have
xc1 = 0.5(0.08 + 30.72 + 5.0.2) = 1.64; yc1= 0.5(0.08 + 0.722 + 0.22) = 0.2644.
For the set II we have xc2 = 0.5(0.0 + 3-0.8 + 5-.0.2) = 1.7; yc1= 0.5(0.02 + 0.82 + 0.22) = 0.34.
Base on the rule (5) we conclude that the Set II again shows a slightly better performance rate as the State III.
Analyzing the data at all three states we can conclude that the performance of both sets significantly improved at each following state. At the first state of independent work the numbers showing the performance of each sets are close; the similar picture we observe at the final state of major scaffolding (actually this state is the state of direct explicit instructions). However, we monitor a significant difference in the performance of these sets at the Stage II of light scaffolding: for the second group of students this light scaffolding is much more efficient that for the first group.
Conclusions. The Zone of Proximate Development is one of the central ideas in the nowadays education. Its applications are especially efficient in mathematics and sciences teaching. Applying fuzzy logic to formalization of the process of learning helps us in obtaining quantitative information for this process (possibilities and probabilities of profiles, comparing student performances, and so on), as well as a qualitative view on the degree of acquisition of the successive steps of the learning process. The described in the article fuzzy models help the instructor to get specific concrete information regarding the students' cognitive status and to choose the appropriate scaffolding.
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Резюме. Субботин И.Я., Моссовар-Рахмани Ф., Билоцкий Н.Н. НЕЧЕТКАЯ ЛОГИКА И ПОНЯТИЕ ЗОНЫ БЛИЖАЙШЕГО РАЗВИТИЯ. В статье рассматриваются некоторые приложения идей нечеткой логики к формализации концепции зоны ближайшего развития Л. С.Выготского.
Ключевые слова: обучение и учение, зона ближайшего развития, математическое моделирование, нечеткие множества, неуверенность.
Abstract. Subbotin I., Mossovar-Rahmani F., Bilotskii N. FUZZY LOGIC AND THE CONCEPT OF THE ZONE OF PROXIMATE DEVELOPMENT. The article discusses some applications of fuzzy logic ideas to formalizing of the L. Vygotsky's concept of the Zone of Proximate Development and to student learning assessment.
Key words: learning and transfer, zone of proximate development, mathematical modeling, fuzzy sets, uncertainty.
Стаття надшшла доредакцп 18.01.2011 р.
