Application of Grey Numbers to Assessment of the Understanding the Graphical Representation of the Derivative

Grey numbers, which are defined with the help of the real intervals, are very useful in the everyday life for handling approximate data. In the present paper grey numbers are used as a tool for assessing, with linguistic expressions, the student understanding of the graphical representation of the derivative. Although the proposed new assessment method is proved to be equivalent with an analogous method using Triangular Fuzzy Numbers developed in earlier works, the required computational burden is significantly reduced. A classroom application is also presented illustrating our results.


Introduction
Mathematical education researchers report that, although student routine performance on differentiation items is usually adequate, most of them have little intuitive or conceptual understanding of the derivative concept [1][2][3][4][5][6]. Many calculus students are, for example, proficient at differentiating a function and finding its critical values. However, they face difficulties to conceptualize these actions and to work with them if they are not presented in equation form.
It is recalled that the formal definition of the derivative f΄(a) of a function y = f(x) at a point a of its domain is given by However, the concept of the limit is difficult to be understood by students. For example, in calculating the limit appearing in definition (1), while we initially accept that x tends to α with values , x a ≠ to calculate it we finally set x = a. Although this can be easily justified by the continuity of y = f(x) at α, it is frequently confusing the novices. It is characteristic that 150 whole years were required from the time that Newton (1642-1727) and Leibniz (1646-1716) initiated the Differential Calculus until the complete understanding of the concept of the limit.
It seems that many students perform poorly due to their weakness to deal successfully with information about abstract concepts (e.g. functions, limits etc.) given in symbolic form, but also because they luck of the necessary cognitive schemas that could allow them to organize their knowledge on those matters [7]. Consequently, the use by the instructor of various representations of the concept of the derivative is recommended as well as the student orientation to analogous activities [8,9,10,11]. Such representations involve for example the rate of change of the function f(x) with respect to x, the physical nature of the derivative connected to the speed and to the acceleration at a moment of time of a moving object under the action of a steady force and its geometric representation as the slope of the tangent at the point (a,f(a)) of the graph of y = f(x).
In the present paper grey numbers (GNs) will be used as tools for assessing student difficulties for the graphical understanding of the derivative. The rest of the paper is formulated as follows: In Section 2 we discuss the assessment methods that we have used in earlier works. In Section 3 we introduce the necessary for the purposes of the present paper background from the theory of GNs. In Section 4 we develop the new assessment method, while in Section 5 we present a classroom experiment on assessing student understanding of the graphical representation of the derivative. The paper closes with our conclusion and hints for further research.

Assesment Methods
The assessment of a system's effectiveness, i.e. of the degree of attainment of its targets) with respect to an activity performed within the system (e.g. problemsolving, decision making, learning, etc) is a very important task, which enables the correction of the system's weaknesses resulting to the improvement of its general performance. The traditional assessment methods that are usually applied in practice are based on principles of the traditional bivalent logic (yes-no). However, assessment situations appear frequently in real life characterized by a degree of uncertainty and/or ambiguity. For example, a teacher is frequently not sure for a particular numerical grade characterizing a student's performance. In such cases fuzzy logic, due to its property of characterizing the ambiguous/uncertain situations with multiple values, offers rich resources for the assessment purposes.
The traditional method for assessing a group's mean performance is the calculation of the mean value of its members' numerical scores. However, frequently in practice the individual performance is evaluated not by numerical scores, but by qualitative characterizations (grades), like excellent, very good, good, fair, unsatisfactory, etc. In such cases, in which the calculation of the mean value is not possible, one could use the also traditional method of calculating the Grade Point Average (GPA) index (e.g. see [12], p. 125) for assessing the group's performance. However GPA actually measures not the mean, but the group's quality performance, by assigning greater coefficients to the higher scores.
In earlier works we have used the measurement of a fuzzy system's uncertainty as a tool for assessing a group's mean performance (e.g. see [12], Chapter 5). Nevertheless, this method, apart of requiring laborious calculations, can be used to compare the mean performance of two different groups only under the assumption that the existing uncertainty in the two groups before the relative activity in which they are participating is the same, a condition which is not always true.
More recently we have also used fuzzy numbers as tools for evaluating a group's mean performance (e.g. see [12], Chapter 7, [13], etc.), which appears to be a more general and accurate method than the measurement of the uncertainty. The method that will be developed in this paper by using GNs as assessment tools is actually equivalent to the method using the special form of the triangular fuzzy numbers (TFNS). (see Remark of Section 3), but it reduces significantly the required computational burden.

Grey Numbers
Frequently in the everyday life, as well as in many applications of science and engineering including medicine diagnostics, psychology, sociology, control systems, economy price indices, opinion polls, etc., the data can not be easily determined precisely and in practice estimates of them are used. Apart from fuzzy logic, another effective tool for handling the approximate data is the use of the GNs, which are introduced with the help of the real intervals.
A GN is an indeterminate number whose probable range is known, but which has unknown position within its boundaries. Therefore, if R: denotes the set of real numbers, a GN, say A, can be expressed mathematically by  [14].
From the definition of the GNs it becomes evident that the well known arithmetic of the real intervals [15] can be used to define the basic arithmetic operations among the GNs. More explicitly, if are given GNs, then we define: Addition by: [ ] a a a b b a b b  A A  a a a b b a

The Assesment Method
The assessment method presented in this section was initially developed in [16].
Let G be a group of n objects participating in a certain activity. Assume that one wants to assess the mean performance of G in terms of the fuzzy linguistic expressions (grades) A = Excellent, B = Very good, C = Good, D = Fair and F = Unsatisfactory (Failed).
We introduce a numerical scale of scores from 0 to 100 and we correspond these scores to the linguistic grades as Let n A , n B , n C , n D and n F denote the numbers of the objects of G whose performance was characterized by the grades A, B, C, D and F respectively. Assigning to each object of G the corresponding GN we define the mean value of all those GNs to be the GN:

A Classroom Application
In the Calculus courses the emphasis is usually given on analytic representations and less effort is spent on presenting the concepts graphically. However, it has been reported that most students have no deep conceptual understanding of the derivative's graphical representation and its correlation to its analytic definition [1,3,17].
The following classroom application took place recently at the Graduate Technological Educational Institute of Western Greece in the city of Patras with subjects the students of two different Engineering Departments of the School of Technological Applications. The instructor, who was the same for the two Departments, in order to help students to have a better understanding of the graphical representation of the derivative gave emphasis to the following points during the teaching procedure: 1. Connecting two points  Calculating the slopes of the secant lines of the graph of y = f(x) defined by the point (2,4) and the points οf Table 1 one finds that f ΄(2) = 4.  Although most students of both Departments were aware of the points where the function was not differentiable (x = -4 and x = 0), many of them did not give any explanations. Question 4: The graph of a function y = f(x) is presented in the below figure. Observing that its tangent at the point (a, f(a)) is horizontal and its tangent at (b, f(b)) is vertical with respect to the x-axis, sketch the graph of the derivative function f΄(x). Since the tangent of the given graph at (a, f(a)) is parallel to the x-axis, its slope is equal to zero, which means that f ΄(α) = 0. Consequently, the graph of f ΄(x) intersects the x-axis at a. Also, from Figure 3 one observes that f(x) is strictly decreasing in the interval (-∞ , a), which means that f ΄(x)<0, for all x in (-∞ , a). Therefore, the graph of f ΄(x) in (-∞ , a) lies under the x-axis. Further, the concavity of f(x) in (-∞ , a), is upwards, which means that f ΄΄(x)>0. Consequently, the derivative function f΄(x) is strictly increasing in (-∞ , a).
Working in the same way for the other intervals one obtains the draft design of the graph of f ΄(x) presented in Figure 4. The above questions are included in an unpublished work on teaching the derivatives done jointly by the present author, V. Borji, H. Alamolhodaei and F. Radmehr.
The results of the test are depicted in Table 2: In concluding, the emphasis given by the instructor during the teaching procedure to the points 1-5 presented in the beginning of this section help the students of both Departments to have a better understanding of the graphical representation of the derivative.
Remark: Consider the two extreme cases, where the maximal or the minimal possible numerical score corresponds to each student for each linguistic grade, e.g. for the first Department the 20 scores corresponding to A, are 100 or 85 respectively, etc. Then, the mean values of the student scores for the first Department are 79.93 and 62.42 respectively, while the value of w(M) is equal to the mean value of these two values. It is easy to check that this result is always true. Consequently, the assessment method with the GNs developed in this paper gives a good approximation of a group's mean performance and therefore it is useful when no numerical scores are used and the group's performance is assessed by qualitative grades.

Conclusion
In the paper at hand a method was developed with the help of GNs for the assessment of a group's mean performance, which is useful when qualitative assessment grades and not numerical scores are used for this purpose. Although this method was proved to be equivalent with an analogous method using TFNs developed in earlier works, the required computational burden was significantly reduced.