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7FUZZY LOGIC AND ITERATIVE ASSESSMENT
(НЕЧЕТКАЯ ЛОГИКА И ПОВТОРНАЯ ОЦЕНКА РЕЗУЛЬТАТОВ ОБУЧЕНИЯ)
I. Subbotin, Professor,
National University, Los Angeles, USA, F. Mossavar-Rahmani, Associate Professor, National University, Los Angeles, USA,
N.N.Bilotskii, Assocoate Professor, National Pedagogic University, Kiev, UKRAIN
Стаття e завершальною роботою циклу застосування нечтког логики до оцтки результатов навчального процесу. Нечткалоггка, яка введенаL.A.Zadeh в 1965, устшнорозвивалася останшм часом багатьма дошдниками i було показано, що вона може бути надзвичайно пл1дною в багатьох додатках, включаючи iнжинiринговi розробки, теорт управлтня, бiзнес, медицину i тлн. У статтi розглядаеться повторна оцткарезультатiв навчання студентiв.
One of the main goals of the learning process is long-term retention and transfer [H]. An effective approach to improve the efficiency of learning is presented in the instructional model called Iterative Instructional Model. In contrast to the traditional consecutive translation along the material, with polishing of all details before reaching the next step, the iterative approach suggests a holistic approach exploring all sides of a problem using analogies of well-known iteration process (see, for example, [B], [G], [K], [OT], and [T]).
Recall 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 a learned information and store it coded. To retrieve this information the brain need to decode it. The speed and efficiency of decoding (retrieving) depend on the amount of ways (connections) of the information decoding [H]. The right way of learning requires creating and developing as many ways of decoding as possible. We can reach this goal by practicing, active discussions, and
other activities, using all possible kinds of memory (visual, motor, audio, and so on), employing the holistic approach, raising vertical and horizontal connections between main parts and details of the material. One of the very important components is the developing of strong connections with the already learned material.
Among the main principles of the iterative learning are the following.
• A holistic approach. At every stages we considers whole theme, from the beginning to the end, as the one whole thing.
• An expanding pace. Every stage bases on its predecessor and brings a new level of understanding and clarity, adds more details and connections
• A multi-repetitive character. Each stage requires repeating of the whole material of the theme. The final stage repeats all details and connections.
• A uniform level of knowledge acquisition. Each stage covers the material at the same deep level.
Now we illustrate the iterative approach with a simple example from Algebra.
The conventional way of teaching of this theme in American schools usually includes the following steps, which we organize in the traditional order of instruction.
• Main Definitions.
• Solving quadratic equations by factoring.
• Completing to a Square.
• Solving quadratic equations by using the quadratic formula.
• Applications.
Briefly it looks like the following.
□ Main Definition
Quadratic equations in one variable are equations of the form ax + bx + c =0, where a # 0, b, c are real numbers that we will call the coefficients, and x is a variable. Quadratic equations are very important because of its applications to many parts of sciences. To solve the equation ax2 + bx + c =0 means find all the numbers x, for which the equation is a correct equality (the numbers, which we can substitute to the left part to get 0).
□ Solving quadratic equations by factoring.
This partial method is based on one of the main properties of the real number set - The Principle of Zero Products:
If ab = 0, then a = 0 or b = 0.
The main idea here is to factor the trinomial ax + bx + c in some way in order to get a product of the form a(x - x1)(x - x2). Thus a(x - xi)(x - x2) = 0, and by using The Principle of Zero Products we can reduce the solution to solutions of two linear equations x - x1 = 0, and x - x2 = 0 (remember that a ^0).
□ Completing to a Square
Completing to a square is another method
of solving of quadratic equations.
For the equation ax2 + bx + c =0 using the condition a #0 we can write:
2 i /2 b c.
ax + bx + c = a( x + — x + —) =
a a
= a ( x2 + — x +
4a2
b2 c)
■"TT+" ) = 4a a
a[( x + — ) 2a
b x2 b2
-4ac
4a2
□ Solving quadratic equations by using the quadratic formula.
This formula allows us to solve any quadratic equation.
Here is the formula:
- b ±y[D , ^ ,2 ,
x =-, where D = b - 4ac.
2a
The last number D is called the discriminant and this number plays crucial role in the problem of determining of number of solutions of a quadratic equation.
□ Applications.
Usually the instructor finishes the theme by solving some word problems, which employs quadratic equations.
The Iterative Model suggests totally different order of steps.
First iteration. The key question: WHAT?
The main goal here is to introduce students to the main topics of the theme, main relations between them and connections to the previously learned materials. We should not worry about very surface student understanding on this stage. The way of realizing is survey, lecture, reading, and so on.
At this stage we start with the definitions as above. Here the instructor reminds what is a solution of an equation in general and introduces students to the brief history of equations, including the problems of solution of the equations of the degree higher than 2.
Now we immediately jump almost to the end: namely, teach the quadratic
formula x :
b ±4d
2a
where D = b - 4ac,
This method naturally leads to the next step.
and its implementations.
Considering examples the instructor discusses the possible number of real solutions. Summarizing, the instructor mentions that number D plays crucial role in the problem of determining of number of solutions of a quadratic equation. Thus,
if D > 0, our equation has exactly two different real solutions;
if D = 0, the equation has only one solution ( or two equal solutions);
if D < 0, there is no real solution (however, we have two complex solutions, which we can find using the same formula).
It would be very appropriate to tell to the students that we already reach the final front of the theme and there will be no more complications. It will reduce their anxiety and build students' self-confidence.
By the same reasons it is very important to make sure that everybody in the class on this stage can apply the quadratic formula for solving equations. At this stage we do not need to worry about logical understanding how we come to this formula.
The same method can be used in particular in business courses. The concept should be defined first and then underlying theories need to be discussed. After that faculty must make sure that students are able to relate theories discussed with what they have learned before or to the theories in other related disciplines.
Second iteration. The key question: HOW?
Focus on the structure and main details of the theme studied at the first step, their connections, exploring some second line details and relations, their functioning in the whole system, making this system working. On this stage we start to build the infrastructural system of the material. The best ways of realizing are: active discussions, projects, solving easy (non-creative) problems. The discussion method helps the students to elaborate the key concepts. This method is especially useful in helping students to learn to evaluate the logic of, and evidence for, their own and others' position and gives them the opportunities to formulate applications of principles. We have seen a very positive result in using this method in both learning and achieving set goals. In [MCMSW] the authors have cited the following advantages for using discussion method. They argued that discussion method will:
• Help students learn to think in terms of the subject matter by giving them practice in thinking.
• Help students become aware of and formulate problems using information gained from readings or lecture.
• Use the resources of members of the group.
• Gain acceptance for information or theories counter to folklore o pervious beliefs of students.
• Develop motivation for further learning.
• Get prompt feedback on how well objectives are being attained.
At this stage it is appropriate to introduce the solving equations by using factoring and using the formula, to discuss the difference between these methods.
At the end every student supposes to solve any equations using the formula. Moreover, everybody suppose to be able to explain HOW this formula works, and HOW to apply the factoring method.
Discussion methods also have been widely used in business courses. Through this exercise, students will learn how to define their positions and learn from others.
Third iteration. The key question: WHY?
Observe the relations of each main part with the system as a whole and the role which this part plays in the process. Trying to select and learn the main reasons for the existence and functioning of each part, exploring detours and shortcuts between parts. On this stage we develop the infrastructure system of the theme. Best way to realize: team projects, working in groups, practicing with the problems requiring non-ordinal approach. For example, solve: we cannot factor every trinomial. WHY? After one completes the review we come to the questions WHY the quadratic formula works? In other words, WHY the solutions could be always found in this way?
It is very appropriate to work now on the completing to the square methods. This method naturally leads to the next step - the quadratic formula we already studied.
At this stage we introduce students to the most complicated parts of the theme. But they are well prepared to this and have a significant experience to work with the content.
Answering "why" is very crucial in practically all disciplines. In economic or finance courses for example, asking this question force students to do more analysis and come up with a rational justification to elaborate further the relationships between theories and the concepts that have been explained. By going
through this exercise, students will better understand the concept and learn the subject.
Forth iteration. The key question: WHY NOT?
Attention to details. Work neat with each small part. Try to change the order of particles in the maze answering on the question: what happen to the system if we will make this or that permutation? At this stage we seek for the complete understanding of the material. Accompany activities: high-level problems solving requiring creativity and independent research. The best way to accomplish: whole class work, discussion of each problem, special projects and problems requiring whole material involving.
A complete review performed by students is very appropriate. Concluding with the applications. It is very suitable here to explain how to factor any trinomial knowing the solutions of the corresponding equations. At this stage is very efficient to ask the student to write a paper concerned one topic of study. Instructor helps students to find the topic and its place in the system.
The process of writing a paper should be broke into a series of easy iterations such as:
• Finding a topic
• Gathering sources, data, or references
• Developing an outline
• Writing a first draft
• Rewriting
Fifth iteration. Assessment.
Iteration is very useful not only in instruction, but in assessment as well.
There are few levels of the iteration process here:
1. Iteration of the whole process as a chain of steps of assessments.
2. Iterative assessment of a specific theme acquisition.
3. Iterative structure of the specific test.
The following example is a fragment of
an iteratively structured test for the assessing of this theme. It is very important that in such a test a student could start the next question only providing a correct answer for the previous question (this is a very good opportunities for the computer applications!).
Stage 1. Knowledge interpretation.
1. Circle the equations among the following expressions.
2. What is the root of an equation?
3. How many solutions does a quadratic equation have?
4. What the ways of solving quadratic equation you know?
5. Write the formula for the solution of a quadratic equation.
Stage 2. Knowledge generalization.
6. Describe in your own words how to solve a quadratic equation.
7. Solve a specific given quadratic equation.
8. Graphical interpretations.
Stage 3. Knowledge categorization.
9. Application to word problems.
10. Solving problems with analyzing parameters (the highest level of knowledge categorization).
In the standard test we do not usually come to all these details.
Among the main benefits of the iterative instructional approach are the following:
1. a high uniform level of learning;
2. a significantly reduced level of the subject anxiety;
3. an easy and fast decoding of the information (retention);
1. a complete understanding of the theme.
Created by L.A. Zadeh ([Z1], [Z2]) Fuzzy logic turns out in a very efficient instrument of formalizing the mentioned above iterative assessment. 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], [BE], and others). There are also some interesting attempts to implement Fuzzy logic ideas in the field of education ([VM], [PS], [SBB], [SBB1]).
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], some prospective Fuzzy logic applications have been developed. We will try to employ another approach to the assessment of students learning. The main base of this approach has been introduced in [SBB] and [SBB1]. This approach is visible, does not employ on the final stage any complicated calculations, and, what is important, can be implemented 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. In the Fuzzy logic there is a commonly used approach consisting in the presenting of a system performance with the pair of numbers (xc,yc) 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
= F
-, yc = ^
JJ dxdy c JJ dxdy
(1)
As any assessment, this approach is very approximate. So it would useful in everyday life to illustrate the situation presented in diagram 1. 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).
Using our three-step iterative assessment we can build the diagram 1 above [SBB].
V2 Yi
yi
F • Owe)
0 12 3
diagram 1
In this case, formulas (1) can be easily transformed to the following simple formulas [SBB]:
xc = — c 2
1 ( yi + 3 y2 + 5 y. ^
yc =-
yi + y2 + y3
( yi2 + y22 + y32 ^
(2)
V yi + y 2 + y3 J 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]:
f n \
Z (2i -1) y,
i=1
Z y,
i=1
f
•yc
Z y,:
A
(3)
Z y,
V ,=1
Formulas (3) could be useful in the case if one would like to study more than three states of the learning process.
A detailed account of the discussion on applications of these formulas on can find in [SBB1]. As we mentioned there the final decision about the class performance depends of the main preferences and goals. For instance, sometime the instructor main goal is to achieve the biggest as possible number of students accommodated in the highest level of knowledge categorization. Sometime the instructor goal is to achieve a basic level of knowledge by each student. In other words the "density" of knowledge distribution by our three groups is vary and depends of the main preferences. One can formalizes this introducing some coefficients of density a, b, and c for the areas of a) representation of the input data and interpretation of this data, b) generalization of the new knowledge, and c) categorization of this knowledge. In this way our final formulas (2) will look like this
xc = — c 2
1 ( ayx + 3by2 + 5cy3 ^
ayi + by2 + cy3
It 2 2 , / 2 2 ■ 2 y = _ a yi +b y2 +c y3
s c
2
2
ayi + by 2 + cy3
(4)
As in the case considered in [SBB1] we can assume that y1 + y2 + y3= 1 and
ay1 + by2 + cy3 = 1. Using these equations we can express our variables yj, y2, y3 through one of them, say through ys:
yi =
a -1
a-
y 2 =
a - b a - b 1 - b c - b
Уз,
(5)
a - b a - b
y3
Substituting (5) in (4) we obtain the following simple and useful formula
1 b(a -1) ( a(c - b) ^
xc = 2+1 cIy3 (6)
2 a - b V a - b J
This formula can serve us in the majority of cases discussed in the article [SBB 1]. The formula foryc through y3 is much more complicated. However, the value yc is seldom involves in the process of assessment [SBB1].
1/2 2, 7.2 2, 2 2 \
yc = 2(a y1 +b y2 +c y3 ) =
-(a2 2
+b2
a -1 - b
1 - b -a - b
U - c ^ a - b
y3
V a - b J J
\2
+
c - b a - b
\ \
y3 J J
+ c2 y32) (7)
It is obvious, that for two different distributions (y1, y2, ys) and (z1, z2, z3), z>0, y, > 0, i = 1, 2, S, there is a unique triple (a, b, c) transforming the first one to the second.
To generalize the idea above we can consider the "density function" as a liner function D(x,y) = ax + b. In this case the formulas (1) will look as following
x„
yc
JJ D( x, y) xdxdy
_F_
JJ D(x, y)dxdy '
F
JJ D( x, y) ydxdy
F
JJ D( x, y)dxdy
(8)
After all simplifications we come to the following formulas
xc =-c 3
1 ( (2a + 3b)y1 + (14a + 9b)y2 + (34a +15b)y. ^
(a + 2b) y1 + (3 a + 2b) y2 + (5a + 2b) y3
yc = -
1 ( (a + 2b) y12 + (3a + 2b) y22 + (5a + 2b) y3:
2 [ (a + 2b)y1 + (3a + 2b)y2 + (5a + 2b)y3
Such kinds of formulas are very easy to use with some technological tools including graphing calculators. It is clear that formulas (2) are the partial case of (9) when a = 0 and b = 1.
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Резюме. Subbotin I., Mossavar-Rahmani F., Bilotskii N.N. НЕЧЕТКАЯ ЛОГИКА И ПОВТОРНАЯ ОЦЕНКА РЕЗУЛЬТАТОВ ОБУЧЕНИЯ. Статья является завершающей работой цикла 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 рассматривается повторная оценка результатов обучения студентов.
Summary. Subbotin I., Mossavar-Rahmani F., Bilotskii N.N. FUZZY LOGIC AND ITERATIVE ASSESSMENT. The article continues the discussion of 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 education. We discuss some applications of fuzzy logic ideas to formalization of the iterative assessment of students learning.
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