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15FUZZY LOGIC AND LEARNING ASSESSMENT (НЕЧЕТКАЯ ЛОГИКА И ОЦЕНКА РЕЗУЛЬТАТОВ ОБУЧЕНИЯ)
I. Subbotin, Professor,
National University, Los Angeles, USA,
H. Badkoobehi, Associate Professor School of Engineering and Technology,
Los Angeles, USA, N.N.Bilotskii, Assocoate Professor, National Pedagogic University, Kiev, UKRAIN
Обговорюються деякг можливг шляхи використання нечгтко! логгки до оцгнки результат1в навчального процессу. Неч1тка лог1ка, започаткована Ь.Л. 2аёвИ в 1965, успшно розвивалась останмм часом багатьма досл1дниками 7 було показано, що вона може бути дуже пл1дною в багатьох застосуваннях, включаючи теор1ю управл1ння, б1знес, медицину та т. В статт1 можна знайти короткий огляд таких ¡дей 7 новий тдх1д до ощнювання результат1в процессу навчання студент1в. Отримат критерп ( запропонований практичний метод) дуже прост! 7 можуть слугувати ефективним доповненням в1домим методам статистики.
Even though everybody knows what the meaning of learning is, it is very difficult to give a satisfactory definition of this notion. As many things connected to real world, the concept of learning is too diverse and complicated to be framed within some quantitative model. There are some impressive efforts towards the describing and even formalizing the learning process (see, for example, [PG], [VJ]), but there is no way to achieve the complete detailed account of it. By evaluating our students' work on the on-going basis, we assess their knowledge of the subject by assigning grades or pass-non-pass options. But how many times do we have some doubt about adequateness of our assessment? How many times do we need to adjust our judgments based on different real life situations? Do we always feel right when giving our student grade B instead of C or A? What about A- or B+? One can answer mention that everything depends on the correctly organized set of criteria. But this is the real world, not an abstract mathematical model. So we, as our students, could make
mistakes. How can we reduce the impact of these natural errors? First of all, we need to recognize that it is very possible to have a wrong impression about student's performance and try to develop some effective tools of assessment of learning under uncertain conditions. Since not only the process of learning but almost all real life processes have this fuzzy nature, we can look around for the implementation of already existing tools from different human activity areas proven to be effective in such fuzzy situations.
In 1965 L.A.Zadeh ([Z1], [Z2]) introduced the ideas of so called Fuzzy logic as a prospective tool in the control theory for solving some engineering problems that could not be solved with the help of the regular mathematics tools because of their very complicated nature. The main distinction of Fuzzy logic from the conventional Boolean logic is in the following: in the conventional logic all variables have only two values, 1 and 0, which means that an element belongs to the given set or does not; in Fuzzy logic, an element may belong to the set in different degrees (for example, student X gets an A since he accomplished correctly 100% of
®
assignments, and student Y gets the same grade since he accomplished correctly 90% of assignments, so teacher is sure that X deserve A, but not entirely sure about Y). In Fuzzy logic, basic Boolean operations (and, or, not) are redefined in the way that they can take inputs between 0 and 1. In general, there are two ways to do so. Firstly, A and B mean minimum (A, B); C or D means maximum (C, D); and NOT E means 1-E. In the second way, A and B = AB; C or D = C + D - (C-D); and NOT E = 1-E. Note, that these are correct generalizations of standard Boolean operations, since if the standard Boolean variables will be used in any operations above; the result will coincide with the conventional. Fuzzy logic has been successfully developed by many researchers and has been proven to be extremely productive in many applications (see, for example, [D], [JVR], [KF], [W], [B], and others). There are also some interesting attempts to implement Fuzzy logic ideas in the field of education ([VM], [PS]).
As some other researchers (see, for example, [VM]), we will base our consideration on the ideas of Voss [VJ], who developed the argument 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. In the article [VM], the following Fuzzy logic applications have 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. 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 m4 (x) = —- for each x e U and,
n
n
therefore, one can write Ai = {(x, — ):xe U},
n
where ^ mA (x) = 1, i = 1,2,3. A fuzzy
xeU
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 = {(s,mR(s): s = (x,y,z)e U3} where the membership function defined by
mR (s) = mA(x) mR2(y)mR3(z), for all s = (x, y, z) eU3.
This fuzzy relation R represents all the possible profiles of student's behavior during the learning process. Further, Voskoglou develops the procedure of comparing few groups of students based on his ideas and supplies the article with examples showing the simplicity of its applications.
We will try to employ another approach to the assessment of students learning. The main base of this approach has been developed in [SBB]. Our approach is visible, does not implement on the final stage any complicated calculations, and, what is important, can be employed to a single student assessment and to the class assessment as well. Depending on evaluation criteria, this approach could be used for the comparing or just for individual independent assessment.
As we mentioned in [SBB] assessing our students' knowledge acquisition we are not completely sure about a particular numerical grade, which could belong to the two adjacent groups of grades with different degrees of membership, which we can measure, for instance, in percentages. There is a commonly used in the Fuzzy logic approach to measure this performance with the pair of numbers (xcyc) as coordinates of the center of mass of the represented figure U, which we can calculate using the following well-known formulas:
JJ xdxdy jj ydxdy
JJ dxdy y° JJ dxdy
(1)
It is not a problem to calculate such numbers using the formulas above; however it could take some significant amount of time. As any assessment, our approach is very approximate. So it would be much more useful in everyday life to simplify the situation as
described in diagram 1. This process implements the following states: C)
representation of the input data and ^
J2
yi
Y3
Now consider the situation close to the traditional one. Let's say we would like to compare the performances of two different classes (Class I and Class II) in the same test. Checking the test we obtained the following table of results distribution (in decimals). We join the all scores less or equal to C.
1
Y2 Ti
yi
Obviously in this case Yi + Y2 + Y3 = yi + y2 +y3 =]. Hence, formulas (2) will look like this:
= 2 (y +3 y 2 +5 ys ) yc = 2 (2+y 2+ys2 )•
(4)
Let's analyze the possible positions for the center of mass in this partial case. The ideal case here is the situation when all students get perfect knowledge of the subject, i.e. whenyi = y2 = 0 andy3 =i. In this case Ft has coordinates (2.5, 0.5). The worst scenario appears when all students
interpretation of this data (yi), B) generalization of the new knowledge (y2), and A) categorization of this knowledge (yj).
Ratio ofthe
class
students Class I Class II
reached the
following
stage of knowledge
acquisition
C yi yi
B y2 Y2
A y3 y3
received C, i.e. y1 =1, y2 = y3 = 0; and here Fw (0.5, 0.5). The situation when all students get B for the class: y1 = y3 =0, y2 =1, will be reflected by the center of mass Fo (1.5, 0.5). Clearly, the area that covered by possible positions of Fc is symmetrical with the axis of symmetry y = 1.5. We will find the minimum
for yc = 2 (2 + y22 + y32). Since yi + y2 +y3=1,
F • (Wc)
diagram 1
(wJ • F
diagram 2
(m>
yc = 2(yi2 + y2 + y32 ) = ) (yi + y2 + y3 )2 - 2yiy2- 2y3y2- 2yiy3 =1 -(yi y 2+y3 y 2+yi y3 ) 2-(yi2+y 22 + y32 );
so that,
2 (yi2 + y22 + y32 ) 2,aml yi2 + y22 + y32 ^ ^
i.e.
y = 2 (yi2 + y22 + y32 ) -6.
2^ ,, , 6 The equal sign is possible if and only if yi = y2 = y3.
ii
It is easy to see that this unique minimum is reached in the point Fm (—, — ) when
2 6
i
yi = y2 = y = 3.
That is way the area for Fc is approximately could be represented as the "triangle" on the diagram 3 below.
À
* J . ff.
-►
0 12 3
diagram 3
Base on this consideration it is logical to introduce the following criteria for the class performance comparison:
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.
It is important to note that from elementary geometric (or algebraic) considerations it directly follows that for two classes with the same xc >1.5 the class has the center of mass which is situated closer to Ft if and only if its yc is higher; andfor two classes with the same xc <1.5 the class has the center of mass which is situatedfarther from Ft if and only if its yc is lower.
In the case when it is not clear how to decide which class is better by using the common sense our rule (5) could help. Usually we compare classes' GPAs (Grade Point Averages in the American system's meaning), and the indicators commonly called "the quality of knowledge" - the ratio of the sum of the numbers of all B and A to whole amount of grades. There are some ambiguous cases here. For example, consider the following two classes' grades:
Ratio of the class students reached the following stage of knowledge acquisition Class I Class n
C 10 0
B 0 20
A 50 40
@
For these both classes the GPA is 3.(6). "The quality of knowledge" for the second class is higher then for the first one. The standard deviation for the second class is definitely smaller. So from the common point of view and from the statistical point of view the situation in the second class is
better. However, some instructors could prefer the situation in the first class, since there are much more "perfect" students in this class. Everything is determined by the set of goals preference.
Using the formulas (4) we calculate:
1 ( 1 „0 5 ( 13 1
2 ( 6+3 ■ 0+55 J=163=2?
and
13
x 2 = -l 0 + 3 — + 5 — I = — = 2-,y 2 = -l 02 + - + - I = —. c2 2 + 3 3 J 6 3 2 + 9 9 J 18
-1+02+*=13.
36 36 J 36
1 4
As we can see xc1= xc2= 21 , and yc1 >
3
yc2 . It means that in this case the centers of
mass lie on the same vertical line x = 21, but
3
the first center is a bit higher. So this class performs better by our standards (5).
We considered a some kind of "unusual" case when xc1= xc2 . Most of the time we deal with the cases when xc1# xc2. Since y1 + y2 +y3 =1,
xc = "2 ( + 3y2 + 5y3 ) = -2 ( + 3 - 3y1 - 3y3 + 5y3 )
1 3
=2 (3+2 y3- 2 y1 )=2+(y - y1)-
It follows that for two classes the class with the bigger difference y3 - y1 performs
better. So, in the most amount of cases we just need to look on this difference for the making our decision.
If we will use our traditional GPA calculation, then for our class we will get GPA = 2y1 + 3y2 + 4y3 . Taking in the account the
equality xc = 2 ( + 3 y2 + 5 y3) we have 2xc
- GPA= y3 - y,. So xc = 2 (GPA + (y3 - y )).
In the regular case, when we measure the student performance using the letter grades F = 0, D= 1, C = 2, B = 3, and A = 4, the formulas (3) will be transformed in the following:
1 ' y + 3 y2 + 5 y3 + 7 y4 + 9 y. (
y1 + y 2 + y3 + y4 + y5
(
. y, =■
y + y 2 + y3 + y4 + y 5 y1 + y 2 + y3 + y4 + y 5
2
Since
y1 + y2 + y3 + y4 + y5 = we can write
xc =1 (y1 + 3 y2 + 5 y3 + 7yA + 9ys),
yc = 1 (y12 + y22 + y32 + y42 + y52 ). (6)
Taking in account that GPA = 0y + 1y2 + 2y3 + 3 y4 + 4y5, we can easily
obtain that in this case xc = GPA + 2 (1 - yj).
The ideal case, when all students received the grade A provides us with Ft (4.5, 0.5). In the worst scenario when all students received the grade F we have Fw (0.5, 0.5). With the help of the elementary inequalities it is not difficult to establish that the unique
minimum is reached in the point Fm (2.5, — )
10
when y1 = y2 = y3 = y4 = y5 = 5.
Now we can reformulate our criterion (5) in the following form.
Among two or more classes the class with the biggest xc performs better;
(7) If two or more classes have the same xc > 2.5, then the class with the higher yc performs better. If two or more classes have the same xc < 2.5, then the class with the lower yc performs better.
Again we can note that from elementary geometric (or algebraic) considerations it directly follows that for two classes with the same xc >2.5 the class has the center of mass which is situated closer to Ft if and only if its yc is higher; and for two classes with the same xc <2.5 the class has the center of mass which is situated farther from Fi if and only if its yc is lower.
For example, in the following case
2
Ratio of the class students reached the following grade Class I Class П
F 2 out of 17 1 out of 17
D 3 out of 17 3 out of 17
C 2 out of 17 3 out of 17
B 4 out of 17 6 out of 17
A 6 out of 17 4 out of 17
GPA's for these two classes is the same 43
and is equal to — ~ 2.529. However, xc for the 17
second class xc is bigger, and according to our criterion (7) the second class performs better.
In the Ukrainian grading system the following five numerical grades 1, 2, 3, 4, and 5 are used. The average grade AG (an analogy of the GPA) is an average of the numerical grades received by students. For this system it is easy to obtain the following equality: AG =
2 + Xc. So we can compare two classes with
the same AG (xc) by using the second part of the criterion (7).
At the end we would like to admit that this article, as a previous one, raises more questions than gives answers. However, this is another argument supporting our strong feeling that the implementation of Fuzzy logic to the study of the process of learning has a very prosper future.
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Резюме. Subbotin I., Badkoobehi H., Bilotskii N.N. НЕЧЕТКАЯ ЛОГИКА И ОЦЕНКА РЕЗУЛЬТАТОВ ОБУЧЕНИЯ. Статья обсуждает некоторые предполагаемые пути применения нечеткой логики к оценке результатов учебного процесса. Нечеткая логика, введенная L.A. Zadeh в 1965, успешно развивалась в последнее время многими исследователями и было показано, что она может быть чрезвычайно плодотворной во многих приложениях, включая инжиниринговые разработки, теорию управления, бизнес, медицину и т.д. В статье можно найти короткий обзор таких идей и новый подход к оцениванию результатов процесса обучения студентов. Полученные критерии (и предложенный практический метод) очень просты и могут служить эффективным дополнением к известным методам статистики.
Summary. Subbotin I., Badkoobehi H., Bilotskii N.N. FUZZY LOGIC AND LEARNING ASSESSMENT. The article discusses some prospective ways of application of Fuzzy logic to assessment of learning process. Fuzzy logic, introduced by L.A. Zadeh in 1965, has been successfully developed lately by many researchers and has been proven to be extremely productive in many applications, including engineering, control theory, business, medicine, and so on. In the article one can find a short review ofsuch ideas and a new approach to the assessment of student's learning. The obtained criteria (and suggested for everyday practices) are very simple and could serve as an effective complement to the well-known common statistics methods.
Надшшла доредакцп 28.10.2005р.
