TRIANGULAR FUZZY LOGIC MODEL FOR LEARNING ASSESSMENT Текст научной статьи по специальности «Математика»

нях, включаючи проектування, теоргяуправлтня, бгзнес, медицина, освта i так дал1. У ц1й cmammi розглядаються деяк застосування нечтко'г логти для ощнки резульmamiв навчання. Ця стаття продовжуе серю статей aвmoрiв, присвячених щй mемi. Тут ми розглядаемо нoвi, бшьш точш трикутш нечтт мoделi для ощнки устшности

Ключовг слова: нечетка логта, нечгткi модел1, оцгнкар1вня знанъ, трикутт модели.

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МЕТОДИЧНА НАУКА - ВЧИТЕЛЮ МАТЕМАТИКИ

TRIANGULAR FUZZY LOGIC MODEL FOR LEARNING ASSESSMENT

(Трикутна модель нечггко! лопки для ощнки усшшносл)

Dr. Igor Ya. Subbotin, Professor, Department of Mathematics, College of Letters and Sciences,

National University, Los Angeles, USA, e-mail: isubboti@nu.edu N. Bilotskii, Associate Professor, Department of Mathematics, Kiev National Pedagogic University, UKRAINE,

e-mail: mikbil@mail.ru

j......

Створенi Л.А. Заде нечтк логти, довели свою користсть в багатьох застосуван-

Introduction. There are some impressive efforts towards the formalizing of the learning process (see, for example, [PG], [VJ]). Evaluating our students' work, we assess their knowledge and skills by assigning grades. Since the process of learning has a fuzzy nature, we can look around for the implementation of already existing proven to be effective in fuzzy situations tools. While 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.

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 standard mathematics tools because of their very complicated nature. This theory lets us handle and process information in a similar way as the human brain does. 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).

This fuzzy logic approach could be realized in the following diagram 1, describing so called membership function, which simply assigns to each of the considered element its degree of belonging to the corresponding sets. Formally, it could be described as a function F = {(x, f(x): x eU}, where U is the universal set of the discourse, and the range E(F) of function F is [0, 1]. It is a very common approach to divide the interval of the specific grades on three parts and assign the corresponding grade using + and - . For example, 80 - 82 = B-, 83 - 86 = B, 87 - 89 = B+. On the diagram 1 student Z has, lets say, 0.25 or 25% degree membership in the set corresponds to the grade C and, therefore, 0.75 or 75% degree membership in the set B, while student X has the 1.0 degree membership in the set corresponds to the grade A. It follows from the simple geometric considerations that in the described cases all degrees of membership for the same element complement each other to 1. It is worthy to note that the same kind of simple

(84)

geometric arguments that this rules are valid for any boundary distributions (for example, 80 - 81 - B - , not 80 - 82 as above,

and so on). Moreover, we do not have to make these boundaries even symmetrical.

C

z B y

Diagram 1

A x

As we already mentioned in the previous papers [SB, SBB, SMB, VS], 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 problemsolving 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 Ai,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 m. (x) = -iL for each x e U and,

n

therefore, one can write Aj ={(x, ):xe U},

n

where ^ m^ ( 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 [0, 1]. Consider farther the fuzzy relation

R = {(s, (s): s = (x, y, z) 6 U3} where the membership function defined by

mR (s) = mA (x) mR2 (y)mRi (z), for al1 S =

=( x, y, z) 6 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 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 [SB and SBB]. This approach is visible, does not implement 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 ap-

proach could be used for the comparing or just for individual independent assessment.

There is a commonly used in the fuzzy logic approach to measure the performance with the pair of numbers (xc,yc) as coordinates of the center of mass (the so-called "centroid method") of the represented figure U, which we can calculate using the following well-known formulas:

(1)

jj xdxdy jj ydxdy

x =

jj dxdy c jj dxdy

It is not a problem to calculate such numbers using the formulas above; however it could take some significant amount of time. So it would be much more useful in

1

У2 У1

Уз

F

(xc,y

everyday life to simplify the situation as described in diagram 1. For this, we use a rectangular diagram 2. This process implements the following states: C) representation of the input data and interpretation of this data (y1), B) generalization of the new knowledge (y2), and A) categorization of this knowledge (y3). In this case, formulas (1) can be easily transformed to the following simple formulas [SBB]:

(2)

1

xc = -c 2

Ус =

yi + 3 y 2 + 5 Уз

^ У1 + У2 + УЗ J

222 У1 + У22 + УЗ2

V У1 + У2 + УЗ .у .

1 2 з

Diagram 2

It is easy to see that the 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 [SBB]:

(3)

1

xc = — c 2

f n \

Z (2i -1) y

i=1

1

,yc = 2

Z y2

i=1 n

Z Уг

V >=1 does

However, this consideration does not reflect the commonly used situation when the teacher is not sure about the grading and assessing the performance of the students whose performance could be assess as marginal between and close to two adjacent levels.

Y

For example, it is something like between 81 and 79 percents. The proposed below 'triangular model" fits this situation. In general, this model looks more precise.

Instead of rectangles, in the triangular model we use triangles. But the most important advantage here is that we allow these triangles have intersections. Namely, we allow to any two adjacent triangles have 25% of their bases belongs to both of them. This way, we cover the situation of uncertainty of assessment of marginal grades described above.

4 \ 7

Diagram 3

10

)

0

(86)

Note, that the triangular form not only better realizes the case of marginal grades. It also correlates the marginal grades through the shape of the triangles in the intersections and this way makes needed correlations.

For the center of mass coordinates we will use the following formulas from the commonly used definition:

(4) xc= Yc= M^miy\

where mi is the mass of the /'-triangle, and (x^y) is the coordinates of its center of mass, M is the mass of the considered figure. In our specific case, we can assume that Zi n = the center of mass of a triangle is the point of intersection of its medians, and since this point divides the median in the proportion 2:1 from the vertex, we can conclude that yi=1/3Yi. Without loss of generality, for the sake of easy calculations, we assume first that the triangles in the Diagram 3 are isosceles, and their bases are 4 units each. Then Xi=3i-1. At the end of calculation, we will easily return to the on - unit base by simple dividing by 4.Taking into account, that we can allow the density of the figure to be 1, we can equal the mass of a figure to its area. So, mx =Si=2Yi. Considering any pair of adjacent triangles and taking into account that Y1 +Y2+Y3 =1, after some simple elementary geometric reasoning we find that

Xc=m~(2Yj +5Y2+8Ys);

YC=3M

0? + YÏ + Yf);

Y-.Y?

Y-Y-<

M=2 +12 ) MYZ+Y3y In the general case of n triangles, we have the following formulas

(6)

-2?w —IZYi

M

YiY,

M=2 ^i+SUl)

In order to transform our formulas to

the case of the one -unit base for x, we just

need to divide the results by 4. So finally,

we obtaine i

Xc=2m(2Yj +5Y2+8Y3);

Yc=6m

(Yl+Yi +Yf);

Y7Y,

M=0.5 (7)

16(Yi +Y2 ) 16(y2+r3)"

X,

i

Zn y 2

1

Yc=6M

-zr

M=0.5 i6(yt )

In our previous article [SBB] we have considered some examples of comparing the performances of two classes in some marginal cases. In the case when it is not clear how to decide which class performance is better we usually 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

Ratio of the class stu- Class Class

dents reached the fol- I II

lowing stage of knowledge acquisition

C 10 0

B 0 20

A 50 40

For these both classes the GPA is 3.7. "The quality of knowledge" for the second class is higher than 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.

As we can see xc1= xc2= 2- =2.33 and

3

yc1 > yc2. It means that in this case the centers of mass lie on the same vertical line x =2.33< 2.5, and the first center is a bit higher. So the second class performs better by the following standards from [SBB].

Among two or more classes the class with the biggest xc performs better;

If two or more classes have the same xc > 2.5, then the class with the higher ycper-forms better. If two or more classes have the same xc < 2.5, then the class with the lower yc performs better.

Consider the same example using our

Y Class I Class II

Yi 0.17 0

®

Y2 0 0.33

Уз 0.83 0.67

We obtain: Xci= 1.75 XC2= 1.81

So, our triangular model in one step also shows that the second class performs slightly better than the first one. It means that the triangular model is more sensitive even in marginal cases.

[B] BINAGHI E. A Fuzzy Logic Inference Model for a Rule-based System in Medical Diagnosis. Expert Systems, Vol 7, No. 3, pp. 134141, 1990.

[D] DOWLING E.T., Mathematics for Economists, Schaum's Outime Series, McGraw - Hill, New York, 1980.

[JVR] JAMSHIDI M., VADIEE N. and ROSS T. (eds.). Fuzzy logic and Control, Prentice-Hall, 1993.

[KF] KLIR G.J. - FOLGER T.A., Fuzzy sets: Uncertainty and Information, Prentice -Hall Int., London, 1988.

[SB] SUBBOTIN, I., BILOTSKIY, N., Fuzzy logic application to assessment of results of iterative learning. Didactics of Mathematics: Problems and Investigations, 2012. v.37. p.89-93.

[SBB] SUBBOTIN I., BADKOOBEHI H., BILOTSKIY, N.: Application of Fuzzy logic to learning assessment. Didactics of Mathematics: Problems and Investigations: 22. -Doneck: Company TEAN, 2004. - 136 p., pp.

38-41.

[SMB] SUBBOTIN, I., F.MOSSOVAR-RAHMANI, F.,N. BILOTSKII, N Fuzzy logic and the concept of the Zone of Proximate Development. Contemporary trends in the development of mathematics and its application aspects 2012: Proceeding of the first International Scientific Internet-Conference (May 17, 2012). Doneck, 2012, p. 301-302.

[VM] VOSKOGLOUM.G., The process of learning mathematics: a fuzzy set approach, Heuristics and Didactics of Exact Sciences, V 10, p.9 - 13, 1999.

[VS] VOSKOGLOU, M., SUBBOTIN, I., Fuzzy measures for students' analogical reasoning skills. Contemporary trends in the development of mathematics and its application aspects 2012: Proceeding of the first International Scientific Internet-Conference (May 17, 2012). Doneck, 2012, p. 219-220.

[VJ] VOSS J. F., Learning and transfer in subject-matter learning: A problem solving model, Int. J. Educ. Research,11, 607-622, 1987.

[W] WILLIAMS T. Fuzzy Logic Simplifies Complex Control Problems. Computer Design, pp. 90-102, March, 1991

[Z1] ZADEH, L. A. Fuzzy sets. Information and Control, 8, 338-353, 1965.

[Z2] ZADEH, L.A. Outline of a New Approach to the Analysis of Complex Systems and Decision processes. IEEE Trans. Systems, Man and Cybernatics, SMC-3, pp. 28-44, 1973.

Резюме. Субботин И.Я., Билоцкий Н.Н. ТРЕУГОЛЬНАЯ МОДЕЛЬ НЕЧЁТКОЙ ЛОГИКИ ДЛЯ ОЦЕНКИ УСПЕВАЕМОСТИ. Созданные Л.А. Заде нечеткие логики, доказали свою полезность во многих приложениях, включая проектирование, теория управления, бизнес, медицина, образование и так далее. В настоящей статье рассматриваются некоторые приложения нечеткой логики для оценки результатов обучения. Эта статья продолжает серию статей авторов, посвященных этой теме. Здесь мы рассматриваем новые, более точные треугольные нечеткие модели для оценки успеваемости.

Ключевые слова: нечеткая логика, нечеткие модели, оценка уровня знаний, треугольные модели.

Abstract. Subbotin Ya., Bilotskii N. TRIANGULAR FUZZY LOGIC MODEL FOR LEARNING ASSESSMENT. Created by L.A. Zadeh, fuzzy logic has been proven to be extremely feasible in many applications, including engineering, control theory, business, medicine, education, and so on. The current article discusses some applications offuzzy logic to assessment of learning. This article is a continuation of the series of the authors' papers dedicated to this theme. We consider here a new, more precised triangular fuzzy model for learning assessment.

Key words: Fuzzy Logic, Fuzzy models, Learning Assessment, Triangular Model.

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