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5FUZZY LOGIC APPLICATION TO ASSESSMENT OF RESULTS OF ITERATIVE LEARNING
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I. Subbotin, Professor,
National University, Los Angeles, USA,
N.N.Bilotskii, Assocoate Professor, National Pedagogic University, Kiev, UKRAIN
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It is well-known that the iteration is the repeated application of a mathematics procedure where each step is applied to the output of the preceding [HC]. The physiological effectiveness of iterative approach bases on our memory properties. Our brain codes learned information and stores it coded. To retrieve this information the brain needs to decode it. In teaching, our main goal is the establishing as many ways of decoding as possible. One of the main goals of the learning process is long-term retention and transfer [HH]. An effective approach to improving the efficiency of learning is presented in an instructional model called the Iterative Instructional Model [SMB]. In contrast to the traditional consecutive translation along the material with considering polishing of all details before reaching the next state, the iterative approach suggests a holistic approach exploring all sides of a problem (see, for example, [B], [G], [K], [OT], and [T]).
J.Voss [VJ] 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;
d) categorization of this knowledge.
The states a and b could be unified in one state of interpretation the new knowledge. M.Voskogloy in the article [VM] has developed an appealing fuzzy set applications based on the Voss's theory. Created by L.A.Zadeh ([Z1], [Z2]) fuzzy logic has been proven to be extremely productive in many applications (see, for example, [KF], [W], [BE]). There are also some interesting attempts to implement Fuzzy logic ideas in the field of education ([VM], [EO], [PS], [SBB], [SBB1], [SMB]).
We consider in details the approach suggested in [VM] and will employ it for determining relative probabilities of the overall states in the iterative instructional model.
In [VM] the following construction has been developed. 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 At respectively and defines a membership function n
mA by mA (x) = — for each x e U and,
i i n
n
therefore, one can write At = {(x, — ):xe U},
n
where ^ mA (x) = 1, i = 1,2,3.
xeU
A fuzzy relation can be considered here as a fuzzy set of triples, each one of which possess a degree of membership belonging [0, 1]. Consider farther the fuzzy relation
R = {(5, mR (s) : s = (x,y, z) eU3}
where the membership function defined by
mR (s) = mA1 (x) mR2 (y)mR3 (z), for ^ s = (x, y, z) e U3.
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 the article with examples showing straightforwardness of its applications. M.Voskoglou in [VM] described the application of 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 eN, 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 Ai, 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 Ats 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 membership function mAi by mAi(x)=nix/n, for each x in U and therefore we can write Ai={(x,nix/n): x eU}. It becomes clear then that XmA(x)=1, x eU, 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 A(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 Xms(t), while the probability of s(t) is t=l given by p(s)=f (s)/ Sf(s), where fs) 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.
In current article we accommodate the M.Voskogloy's ideas toward measurement of efficiency of iterative learning model.
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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 [9] 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 the Iterative Instructional Model (IIM), we can consider here the following tree important states of the knowledge acquisition process.
State A1. First iteration: Developing of main basic knowledge, first look at the «road map» of the topic (corresponds to Voss's stages of representation and interpretation of the input data).
State A2. Second iteration: Developing of general main knowledge, considering «a main infrastructure map» of the topic (corresponds to Voss's stage of generalization of the new knowledge).
Sate A3. Third iteration, «a detailed map»: Developing of completed detailed knowledge in the topic (corresponds to Voss's stage of categorization of the new knowledge).
At each state, the students knowledge acquisition were observed and labeled corre-
spondingly to above described ideas from [VM]:
c stands for the «below satisfactory»,
b for «satisfactory»,
and a for «above satisfactory» level of acquisition.
Of, course at each state the criteria of labeling were adapted accordingly to students' real progress at predecessor states.
We observed that 4, 18, and 5 students achieved below unsatisfactory, satisfactory, and above satisfactory respectively (n1c =4, nib= 18, nia= 5) at the first state.
Thus An= {(c, 4/27), (b,.18/27), (a,5/27)]}.
At the next state we found that
Ai2= {(c, 3/27), (b, 20/27), (a, 4/27)}.
At the final state we found that
A13 = {(c, 2/27), (b, 22/27), (a, 3/27)}.
Another data set gave us the following results
A2i= {(c,5/25), (b, 18/25), (a,2/25)}.
A22= {(c,4/25), (b,20/25), (a,1/25)}.
A23= {(c, 0/25), (b, 19/25), (a,6/25)}.
Looking at the Ais for both sets, we can see that the higher the state iteration, the enhanced degree of acquisition of knowledge we achieve.
From the table below it turns out that the profile s=(b,b,b) has the highest pseudofre-quency for the two groups of data sets of our experiment
m() =mA}(c)mA2(b)mA3(a) =0.402368, ms(2) =0.43776, and therefore f (s)= 0.840128.
Thus it had also the highest probability of occurrencep(s) = 0.420064, or 42%, while its possibility was 0.999996 ~ 1.
Probabilities and possibilities ofprofiles
Table
A1A2A3 A11 A12 A13 A21 A22 A23 ms(1) ms(2) f(s) p(s) r(s)
aaa 0.1852 0.1481 0.1111 0.08 0.04 0.24 0.003047 0.000768 0.003815 0.001908 0.004541
aab 0.1852 0.1481 0.8148 0.08 0.04 0.76 0.022348 0.002432 0.02478 0.01239 0.029496
aac 0.1852 0.1481 0.074 0.08 0.04 0 0.00203 0 0.00203 0.001015 0.002416
aba 0.1852 0.7407 0.1111 0.08 0.8 0.24 0.01524 0.01536 0.0306 0.0153 0.036423
abb 0.1852 0.7407 0.8148 0.08 0.8 0.76 0.111772 0.04864 0.160412 0.080206 0.190937
<9D
abc 0.1852 0.7407 0.074 0.08 0.8 0 0.010151 0 0.010151 0.005076 0.012083
aca 0.1852 0.1111 0.1111 0.08 0.16 0.24 0.002286 0.003072 0.005358 0.002679 0.006378
acb 0.1852 0.1111 0.8148 0.08 0.16 0.76 0.016765 0.009728 0.026493 0.013247 0.031534
acc 0.1852 0.1111 0.074 0.08 0.16 0 0.001523 0 0.001523 0.000761 0.001812
baa 0.6667 0.1481 0.1111 0.72 0.04 0.24 0.01097 0.006912 0.017882 0.008941 0.021285
bab 0.6667 0.1481 0.8148 0.72 0.04 0.76 0.080452 0.021888 0.10234 0.05117 0.121814
bac 0.6667 0.1481 0.074 0.72 0.04 0 0.00730 7 0 0.00730 7 0.003653 0.008697
bba 0.6667 0.7407 0.1111 0.72 0.8 0.24 0.054864 0.13824 0.193104 0.096552 0.22985
bbb 0.6667 0.7407 0.8148 0.72 0.8 0.76 0.402368 0.43776 0.840128 0.420064 0.999996
bbc 0.6667 0.7407 0.074 0.72 0.8 0 0.036543 0 0.036543 0.018272 0.043497
bca 0.6667 0.1111 0.1111 0.72 0.16 0.24 0.008229 0.027648 0.035877 0.017939 0.042704
bcb 0.6667 0.1111 0.8148 0.72 0.16 0.76 0.060353 0.087552 0.147905 0.073952 0.176049
bcc 0.6667 0.1111 0.074 0.72 0.16 0 0.005481 0 0.005481 0.002741 0.006524
caa 0.1481 0.1481 0.1111 0.2 0.04 0.24 0.002437 0.00192 0.004357 0.002178 0.005186
cab 0.1481 0.1481 0.8148 0.2 0.04 0.76 0.017872 0.00608 0.023952 0.011976 0.028509
cac 0.1481 0.1481 0.074 0.2 0.04 0 0.001623 0 0.001623 0.000812 0.001932
cba 0.1481 0.7407 0.1111 0.2 0.8 0.24 0.012187 0.0384 0.050587 0.025294 0.060214
cbb 0.1481 0.7407 0.8148 0.2 0.8 0.76 0.089382 0.1216 0.210982 0.105491 0.251129
cbc 0.1481 0.7407 0.074 0.2 0.8 0 0.008118 0 0.008118 0.004059 0.009662
cca 0.1481 0.1111 0.1111 0.2 0.16 0.24 0.001828 0.00768 0.009508 0.004754 0.011317
ccb 0.1481 0.1111 0.8148 0.2 0.16 0.76 0.013407 0.02432 0.037727 0.018863 0.044906
ccc 0.1481 0.1111 0.074 0.2 0.16 0 0.001218 0 0.001218 0.000609 0.001449
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Резюме. Subbotin I., Bilotskii N. ПРИМЕНЕНИЕ НЕЧЕТКОЙ ЛОГИКИ К ОЦЕНКЕ РЕЗУЛЬТАТОВ ПРОЦЕССА ИТЕРАЦИОННОГО ОБУЧЕНИЯ. В статье продолжается обсуждение применений нечеткой логики к формализации процесса оценки итерационного обучения.
Ключевые слова: итерационное обучение, нечеткая логика, высшее образование.
Abstract. Subbotin I., Bilotskii N. We continue a discussion on some applications of fuzzy logic to formalization of the assessment of iterative students learning.
Key words: iterative learning, fuzzy logic, higher education.
Стаття надшшла доредакци 29.10.2011 р.
