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Fuzzy logic, which is based on fuzzy sets theory introduced by Zadeh in 1965, provides a rich and meaningful addition to standard logic. The applications which may be generated from or adapted to fuzzy logic are wide-ranging and provide the opportunity for modeling under conditions which are imprecisely defined. In this article we develop a fuzzy model for assessing student groups’ knowledge and skills. In this model the students’ characteristics under assessment (knowledge of the subject matter, problem solving skills and analogical reasoning abilities) are represented as fuzzy subsets of a set of linguistic labels characterizing their performance, and the possibilities of all student profiles are calculated. In this way, a detailed quantitative/qualitative study of the students’ group performance is obtained. The centroid method and the group’s total possibilistic uncertainty are used as defuzzification methods in converting our fuzzy outputs to a crisp number. According to the centroid method, the coordinates of the center of gravity of the graph of the membership function involved provide a measure of the students’ performance. Techniques of assessing the individual students’ abilities are also studied and examples are presented to illustrate the use of our results in practice.

There used to be a tradition in science and engineering of turning to probability theory when one is faced with a problem in which uncertainty plays a significant role. This transition was justified when there were no alternative tools for dealing with the uncertainty. Today this is no longer the case.

The applications which may be generated from or adapted to fuzzy logic are wide-ranging and provide the opportunity for modeling under conditions which are inherently imprecisely defined, despite the concerns of classical logicians. A real test of the effectiveness of an approach to uncertainty is the capability to solve problems which involve different facets of uncertainty. Fuzzy logic has a much higher problem solving capability than standard probability theory. Most importantly, it opens the door to construction of mathematical solutions of computational problems which are stated in a natural language. In contrast, standard probability theory does not have this capability, a fact which is one of its principal limitations. All the above gave us the impulsion to introduce principles of fuzzy logic to describe in a more effective way a system’s operation

The present paper proposes the use of fuzzy logic in assessing students’ knowledge and skills. The text is organized as follows: In section 2 we use our general fuzzy framework mentioned above as a tool for students groups’ assessment. In section 3 we apply the methods of the “center of gravity” and of the system’s total possibilistic uncertainty as defuzzification methods in converting our fuzzy outputs to a crisp number and we present an example illustrating our results in practice. In section 4 we study techniques of students’ individual assessment and finally in section 5 we state our conclusions and we discuss our plans for future research. For general facts on fuzzy sets we refer freely to the book of Klir and Folger [

One of the problems faced by teachers is the assessment of their students’ knowledge and aptitudes. In fact, our society demands not only to educate, but also to classify the students according to their qualifications as being suitable or unsuitable for carrying out certain tasks or holding certain posts. According to the standard methods of assessment, a mark, expressed either with a numerical value within a given scale (e.g., from 0 to 10) or with a letter (e.g., from A to F) corresponding to the percentage of a student’s success, is assigned in order to characterize his/her performance. However, this crisp characterization, based on principles of the bivalent logic (yes-no), although it is the one usually applied in practice, it is not probably the most suitable to determine a student’s performance. In fact, the teacher can be never absolutely sure about a particular numerical grade characterizing the student’s abilities and skills. In contrast, fuzzy logic, due to its nature of including multiple values, offers a wider and richer field of resources for this purpose. Therefore, the application of fuzzy logic that we shall attempt in this section seems to be a valuable tool for developing a framework for the students’ assessment.

Let us consider a class of _{1}_{2}_{3}_{i}s

We are going to attach to each students’ characteristic _{i},_{i}_{ia}, n_{ib}, n_{ic}, n_{id}_{ie}_{i}_{Ai}

In fact, if one wanted to apply probabilistic standards in measuring the degree of the students’ success at each stage of the process, then he/she should use the relative frequencies_{ix’}s_{i}_{i}

In order to represent all possible students’ ^{3}^{3}

For determining properly the membership function _{R}

For example,

We define now the membership degree of a profile s to be m_{R}(s) =

Next, for reasons of brevity, we shall write _{s}_{R}(s)_{s}_{s}_{s}_{s} = _{s}}_{s}^{3}_{s}}_{s} < r_{s}

Assume now that one wants to study the _{1}(t), A_{2}(t)_{3}(t)_{1}(2)_{1}_{s}(1)+m_{s}(2)+…. +m_{s}(k)

The above model gives, through the calculation of probabilities and possibilities of all students’ profiles, a quantitative/qualitative view of their realistic performance.

A common and useful defuzzification technique is the method of the

_{c}, y_{c})_{c}

Subbotin

Here we shall apply the centroid method as a defuzzification technique for the student groups’ assessment model developed in the previous section. For this, we characterize a student’s performance as very low (a) if y _{i}, i = 1,2,3,4,5, having the lengths of their sides on the x axis equal to 1.

Bar graphical data representation.

Here

Therefore, Formulas (1) are transformed into the following form:

Normalizing our fuzzy data by dividing each m(x), x _{1} + y_{2} + y_{3} + y_{4} + y_{5} = 1._{i}_{1} = a, x_{2} = b, x_{3} = c, x_{4} = d and x_{5} = e.

But 0 _{1} − y_{2})^{2} = y_{1}^{2} + y_{2}^{2} − 2y_{1}y_{2}, therefore y_{1}^{2} + y_{2}^{2} _{1}y_{2}, with the equality holding if, and only if, y_{1} = y_{2}. In the same way one finds that y_{1}2+y_{3}2 _{1}y_{3}, and so on. Hence it is easy to check that (y_{1} + y_{2} + y_{3} + y_{4} + y_{5})^{2} _{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2} + y_{5}^{2}), with the equality holding if, and only if y_{1} = y_{2} = y_{3} = y_{4} = y_{5}. However, y_{1} + y_{2} + y_{3} + y_{4} + y_{5} = 1; therefore, 1 _{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2} + y_{5}^{2}) (3), with the equality holding if, and only if y_{1} = y_{2} = y_{3} = y_{4} = y_{5} = _{c}

Further, combining the Inequality (3) with the second of Formulas (2) one finds that _{c}_{c}_{c}_{m} (

The ideal case is when _{1} = y_{2} = y_{3} = y_{4} = 0_{5} = 1_{c} =_{c} =_{i} (_{1} = 1_{2} = y_{3} = y_{4} = y_{5} = 0_{w} (_{c}_{w} F_{m} F_{i} of

Graphical representation of the “area” of the center of gravity.

Then from elementary geometric considerations it follows that for two groups of students with the same _{c}_{i}_{c}_{c} < 2.5_{w}_{c}

_{c} performs better.

_{c} ≥ 2.5, then the group with the higher y_{c} performs better.

_{c} < 2.5, then the group with the lower y_{c} performs better.

It is well known that the amount of information obtained by an action can be measured by the reduction of uncertainty resulting from this action. Accordingly a system’s uncertainty is connected to its capacity in obtaining relevant information. Therefore, a measure of uncertainty could be adopted as an alternative defuzzification technique for the student groups’ assessment model developed in the previous section.

Within the domain of possibility theory uncertainty consists of

Strife is measured by the function

r:r_{1} = 1 _{2} _{rn} _{rn+1} of a group of students defined by

Similarly, non-specificity is measured by the function

The sum

We must emphasize that the two defuzzification methods presented above treat differently the idea of a group’s performance. In fact, the weighted average plays the main role in the centroid method,

There are also other defuzzification techniques in use, such as the calculation of the group’s

Next we give an example illustrating our results in practice.

EXAMPLE: The following data was obtained by assessing the mathematical skills of two groups of students of the Technological Educational Institute of Patras, Greece being at their first term of studies:

According to the above notation the first index of _{ij}_{j}^{3} (ordered samples with replacement of 3 objects taken from 5) in total possible students’ profiles as it is described in _{s}(1) in _{s} = 0.5 × 0 .5 × 0.25 = 0.06225._{s}(1) it turns out that the maximal membership degree of students’ profiles is ^{3}_{s} = _{s}(1) of _{s}

Profiles with non zero membership degrees.

A_{1} |
A_{2} |
A_{3} |
_{s} |
_{s} |
_{s} |
_{s} |
||
---|---|---|---|---|---|---|---|---|

b | b | b | 0 | 0 | 0.016 | 0.258 | 0.016 | 0.129 |

b | b | a | 0 | 0 | 0.016 | 0.258 | 0.016 | 0.129 |

b | a | a | 0 | 0 | 0.016 | 0.258 | 0.016 | 0.129 |

c | c | c | 0.062 | 1 | 0.062 | 1 | 0.124 | 1 |

c | c | a | 0.062 | 1 | 0.062 | 1 | 0.124 | 1 |

c | c | b | 0 | 0 | 0.031 | 0.5 | 0.031 | 0.25 |

c | a | a | 0 | 0 | 0.031 | 0.5 | 0.031 | 0.25 |

c | b | a | 0 | 0 | 0.031 | 0.5 | 0.031 | 0.25 |

c | b | b | 0 | 0 | 0.031 | 0.5 | 0.031 | 0.25 |

d | d | a | 0.016 | 0.258 | 0 | 0 | 0.016 | 0.129 |

d | d | b | 0.016 | 0.258 | 0 | 0 | 0.016 | 0.129 |

d | d | c | 0.016 | 0.258 | 0 | 0 | 0.016 | 0.129 |

d | a | a | 0 | 0 | 0.016 | 0.258 | 0.016 | 0.129 |

d | b | a | 0 | 0 | 0.016 | 0.258 | 0.016 | 0.129 |

d | b | b | 0 | 0 | 0.016 | 0.258 | 0.016 | 0.129 |

d | c | a | 0.031 | 0.5 | 0.031 | 0.5 | 0.062 | 0.5 |

d | c | b | 0.031 | 0.5 | 0.031 | 0.5 | 0.062 | 0.5 |

d | c | c | 0.031 | 0.5 | 0.031 | 0.5 | 0.062 | 0.5 |

e | c | a | 0.031 | 0.5 | 0 | 0 | 0.031 | 0.25 |

e | c | b | 0.031 | 0.5 | 0 | 0 | 0.031 | 0.25 |

e | c | c | 0.031 | 0.5 | 0 | 0 | 0.031 | 0.25 |

e | d | a | 0.016 | 0.258 | 0 | 0 | 0.016 | 0.129 |

e | d | b | 0.016 | 0.258 | 0 | 0 | 0.016 | 0.129 |

e | d | c | 0.016 | 0.258 | 0 | 0 | 0.016 | 0.129 |

The outcomes of

The membership degrees and the possibilities of students’ profiles are presented in columns of m_{s}(2) and r_{s}(2) of

In order to study the combined results of the two groups’ performance we also calculated the pseudo-frequencies _{s}(1) + m_{s}(2)

We compare now the two groups’ performance by applying the centroid method. For the first characteristic (knowledge of the subject matter) we have:

Thus, by our criterion the first group demonstrates better performance.

For the second characteristic (problem solving abilities) we have:

Normalizing the membership degrees in the first of the above fuzzy subsets of U (0.5:0.75 ≈ 0.67 and 0.25:0.75 ≈ 0.33) we get A_{12} = {(a, 0),(b, 0),(c, 0.67),(d, 0.33),(e, 0)}. Therefore

By our criterion, the first group again demonstrates a significantly better performance.

Finally, for the third characteristic (analogical reasoning) we have

Based on our calculations we can conclude that the first group demonstrated a significantly better performance concerning the knowledge of the subject matter and problem solving, but performed identically with the second one concerning analogical reasoning.

Calculating the possibilities of all profiles (column of r_{s}(1) in

Thus, with the help of a calculator one finds that

Also the group’s non-specificity is

≈3.32(0.5

The ordered possibility distribution for the second student group (column of r_{s}(2) in

Therefore, since

The outputs of our fuzzy model developed above can be used not only for assessing the performance of student groups’, but also for the students’ individual assessment. In fact, if _{i}, i = 1, 2, 3, there exists a unique element

For example, if _{11}_{21}_{1}_{c}_{ 11} = _{c}_{ 21} =

As a consequence of the above situation (_{s} = 1_{2}

A. Jones developed a fuzzy model to the field of Education involving several theoretical constructs related to assessment, amongst which is a technique for assessing the deviation of a student’s knowledge with respect to the teacher’s knowledge, which is taken as a reference [

Let _{1}, S_{2}, S_{3}}_{1}, m(S_{1})), (S_{2}, m(S_{2})), (S_{3}, m(S_{3})_{1}, 1), (S_{2}, 1), (S_{3}, 1)

Then the _{i}_{1},1-m(S_{1})), (S_{2}, 1-m(S_{2})), (S_{3},1-m(S_{3})

This assessment by reference to the teacher provides us with the ideal student as the one with nil deviation in all his/her components and it defines a relationship of partial order among students’. The following example illustrates this theoretical framework in practice.

EXAMPLE: We reconsider the group of 35 students of the School of Technological Applications of the Technological Educational Institute of Patras, Greece of our example of section 2. In assessing the students’ individual performance by applying the A. Jones technique we found the following types of deviations with respect to the teacher:

_{1}_{1}, 0.75), (S_{2}, 0.75), (S_{3}, 1)

_{2}_{1}, 0.5), (S_{2}, 1), (S_{3}, 1)

_{3}_{1}, 0.5), (S_{2}, 0.75), (S_{3}, 1)

_{4}_{1}, 0.5), (S_{2}, 0.75), (S_{3}, 0.75)

_{5}_{1}, 0.25), (S_{2}, 0.5), (S_{3}, 0.75)

_{6}_{1}, 0.25), (S_{2}, 0.25), (S_{3}, 0.5)

_{7}_{1}, 0), (S_{2}, 0.5), (S_{3}, 0.75)

_{8}_{1}, 0), (S_{2}, 0.5), (S_{3}, 0.5)

_{9}_{1}, 0), (S_{2}, 0.25), (S_{3}, 0.5)

_{10}_{1}, 0), (S_{2}, 0.25), (S_{3}, 025)

_{11}_{1}, 0), (S_{2}, 0), (S_{3}, 0.25)

On comparing the above types of students’ deviations it becomes evident that the students possessing the type _{3}_{1}_{4}_{3}_{1}_{2}_{6}_{7}

Notice that the teacher may put a target for his/her class and may establish didactic strategies in order to achieve it. For example he/she may ask for the deviation, say d, to be

The following conclusions can be drawn from those presented in this paper:

Fuzzy logic, due to its nature of including multiple values, offers a wider and richer field of resources for assessing the students’ performance than the classical crisp characterization does by assigning a mark to each student, expressed either with a numerical value within a given scale or with a letter corresponding to the percentage of the student’s success.

In this article we developed a fuzzy model for assessing student groups’ knowledge and skills, in which the students’ characteristics under assessment are represented as fuzzy subsets of a set of linguistic labels characterizing their performance.

The group’s total possibilistic uncertainty and the coordinates of the center of gravity of the graph of the membership function involved were used as defuzzification methods in converting our fuzzy outputs to a crisp number.

Techniques of assessing the students’ performance individually were also discussed and examples were presented illustrating the use of our results in practice.

Our model is actually a proper adaptation of a more general fuzzy model developed in earlier papers to represent in an effective way a system’s operation

The author declares no conflict of interest.