# Fuzzy Logic as a Tool for Assessing Students’ Knowledge and Skills

## Abstract

**:**

## 1. Introduction

## 2. A Fuzzy Model for Assessing Student Groups’ Performance

_{1}= knowledge of a subject matter, S

_{2}= problem solving related to this subject matter and S

_{3}= ability to adapt properly the already existing knowledge for use in analogous similar cases (analogical reasoning; of course the teacher could choose characteristics different for those mentioned here and may be more than three in total). However, the more are the characteristics chosen for assessment, the more complicated (technically) becomes our model.). Denote by a, b, c, d, and e the linguistic labels (fuzzy expressions) of very low, low, intermediate, high and very high success respectively of a student in each of the S

_{i}s and set U = {a, b, c, d, e}.

_{i}, i = 1, 2, 3, a fuzzy subset, A

_{i}of U. For this, if n

_{ia}, n

_{ib}, n

_{ic}, n

_{id}and n

_{ie}denote the number of students that faced very low, low, intermediate, high and very high success with respect to S

_{i}respectively, we define the membership function m

_{Ai}for each x in U, as follows:

_{ix’}s are obtained with respect to the linguist labels of U, which are fuzzy expressions by themselves. Therefore, the application of a fuzzy approach by using membership degrees instead of probabilities seems to be the most suitable for this case. However, as it is well known, the membership function is usually defined empirically in terms of logical or/and statistical data. In our case the above definition of seems to be compatible with common sense. Then, the fuzzy subset A

_{i}of U corresponding to S

_{i}has the form:

^{3}(i.e., a fuzzy subset of U

^{3}) of the form:

_{R}we give the following definition:

_{R}(s) = (x) (y) (z), if s is well ordered, and 0 otherwise. In fact, if for example the profile (b, a, c) possessed a nonzero membership degree, how it could be possible for a student, who has failed at the problem solving stage, to perform satisfactorily at the stage of analogical reasoning, where he/she has to adapt the existing knowledge for solving problems related to analogous similar cases?

_{s}instead of m

_{R}(s). Then the probability p

_{s}of the profile s is defined in a way analogous to crisp data, i.e., by p

_{s}= . We define also the possibility r

_{s}of s to be r

_{s}= , where max {m

_{s}} denotes the maximal value of m

_{s}for all s in U

^{3}. In other words the possibility of s expresses the “relative membership degree” of s with respect to max {m

_{s}}. From the above two definitions it becomes evident that p

_{s}< r

_{s}for all s in , which is compatible to the common logic. In fact, whatever is probable it is also possible, but whatever is possible need not be very probable.

_{1}(t), A

_{2}(t) and A

_{3}(t) with t = 1, 2,…, k. The values of these variables represent fuzzy subsets of U corresponding to the students’ characteristics under assessment for each of the k groups; e.g., A

_{1}(2) represents the fuzzy subset of U corresponding to the knowledge of a subject matter (characteristic S

_{1}) for the second group (t = 2). Obviously, in order to measure the degree of evidence of the combined results of the k groups, it is necessary to define the probability p(s) and the possibility r(s) of each profile s with respect to the membership degrees of s for all groups. For this reason we introduce the pseudo-frequencies f(s) = m

_{s}(1)+m

_{s}(2)+…. +m

_{s}(k) and we define the probability and possibility of a profile s by p(s) = and r(s) = respectively, where max{f(s)} denotes the maximal pseudo-frequency. The same method could be applied when one wants to study the combined results of k different assessments of the same student group.

## 3. Defuzzification Methods

#### 3.1. The Centroid Method

_{c}, y

_{c}) as the coordinates of the center of gravity, say F

_{c}, of the graph F, which we can calculate using the following well-known (e.g., see [8]) formulas:

_{i}, i = 1,2,3,4,5, having the lengths of their sides on the x axis equal to 1.

_{1}+ y

_{2}+ y

_{3}+ y

_{4}+ y

_{5}= 1. Therefore, we can write:

_{i}= , where x

_{1}= a, x

_{2}= b, x

_{3}= c, x

_{4}= d and x

_{5}= e.

_{1}− y

_{2})

^{2}= y

_{1}

^{2}+ y

_{2}

^{2}− 2y

_{1}y

_{2}, therefore y

_{1}

^{2}+ y

_{2}

^{2}2y

_{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 2y

_{1}y

_{3}, and so on. Hence it is easy to check that (y

_{1}+ y

_{2}+ y

_{3}+ y

_{4}+ y

_{5})

^{2}5(y

_{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 5(y

_{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}= . In this case the first of Formulas (2) gives that x

_{c}= .

_{c}, or y

_{c}. Therefore, the unique minimum for y

_{c}corresponds to the center of gravity F

_{m}( , ).

_{1}= y

_{2}= y

_{3}= y

_{4}= 0 and y

_{5}= 1. Then from Formulas (2) we get that x

_{c}= and y

_{c}= . Therefore the center of gravity in the ideal case is the point F

_{i}( , ). On the other hand the worst case is when y

_{1}= 1 and y

_{2}= y

_{3}= y

_{4}= y

_{5}= 0. Then for formulas (2) we find that the center of gravity is the point F

_{w}( , ). Thus, the “area” where the center of gravity F

_{c}lies is represented by the triangle F

_{w}F

_{m}F

_{i}of Figure 2.

_{c}2.5 the group having the center of gravity which is situated closer to F

_{i}is the group with the higher y

_{c}; and for two groups with the same x

_{c}< 2.5 the group having the center of gravity which is situated farther to F

_{w}is the group with the lower y

_{c}. Based on the above considerations we formulate our criterion for comparing the groups’ performances as follows:

- Among two or more groups the group with the biggest x
_{c}performs better. - If two or more groups have the same x
_{c}≥ 2.5, then the group with the higher y_{c}performs better. - If two or more groups have the same x
_{c}< 2.5, then the group with the lower y_{c}performs better.

#### 3.2. The Group’s Uncertainty

_{1}= 1 r

_{2}……. r

_{rn}r

_{rn+1}of a group of students defined by

_{ij}denotes the group (i = 1,2) and the second index denotes the corresponding students’ characteristic S

_{j}(j = 1,2,3). We calculated the membership degrees of the 5

^{3}(ordered samples with replacement of 3 objects taken from 5) in total possible students’ profiles as it is described in Section 2 (see column of m

_{s}(1) in Table 1). For example, for the profile s = (c, c, a) one finds that m

_{s}= 0.5 × 0 .5 × 0.25 = 0.06225. From the values of the column of m

_{s}(1) it turns out that the maximal membership degree of students’ profiles is 0.06225. Therefore, the possibility of each s in U

^{3}is given by r

_{s}= . The possibilities of the students’ profiles are presented in column of r

_{s}(1) of Table 1. One could also calculate the probabilities of the students’ profiles using the formula for p

_{s}given in section 2. However, according to Shackle [16] and many other researchers after him, human cognition is better presented by possibility rather than by probability theory. Therefore, adopting this view, we considered that the calculation of the probabilities was not necessary.

A_{1} | A_{2} | A_{3} | m_{s}(1) | r_{s}(1) | m_{s}(2) | r_{s}(2) | f(s) | r(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 |

_{s}(2) and r

_{s}(2) of Table 1 respectively.

_{s}(1) + m

_{s}(2) and the combined possibilities of all profiles presented in the last two columns of Table 1.

_{12}= {(a, 0),(b, 0),(c, 0.67),(d, 0.33),(e, 0)}. Therefore

_{s}(1) in Table 1) one finds that the ordered possibility distribution for the first student group is:

_{s}(2) in Table 1) is:

## 4. Students’ Individual Assessment

_{i}, i = 1, 2, 3, there exists a unique element x of U with membership degree 1, while all the others have membership degree 0. The centroid method is trivially applicable in this marginal case.

_{11}= {(a, 0), (b, 0), (c,0), (d,1), (e,0)} and A

_{21}= {(a, 0), (b, 0), (c,1), (d,0), (e,0)}, then obviously the first student demonstrates a better performance with respect to the knowledge acquisition (characteristic S

_{1}). This is crossed by the centroid method, since x

_{c}

_{ 11}= and x

_{c}

_{ 21}= .

_{s}= 1, while all the others have membership degree 0. In other words, each student is characterized in this case by a unique profile, which gives us the requested information about his/her performance. For example, if (c, b, a) and (c, b, b) are the characteristic profiles for students x and y respectively, then clearly y demonstrates a better performance than x. In contrast, if (d, b, b) and (c, c, b) are the corresponding profiles, then x demonstrates a better performance than y concerning the knowledge acquisition, but y demonstrates a better performance than x concerning the problem solving skills (characteristic S

_{2}). Mathematically speaking this means that the students’ characteristic profiles define a relationship of partial order among students’ with respect to their performance.

_{1}, S

_{2}, S

_{3}} be the set of the students’ characteristics under assessment that we have considered in section 2. Then a fuzzy subset of X of the form {(S

_{1}, m(S

_{1})), (S

_{2}, m(S

_{2})), (S

_{3}, m(S

_{3})}can be assigned to each student, where the membership function m takes the values 0, 0.25, 0.5, 0.75, 1 according to the level of the student’s performance. The teacher’s fuzzy measurement is always equal to 1, which means that the fuzzy subset of X corresponding to the teacher is {(S

_{1}, 1), (S

_{2}, 1), (S

_{3}, 1)}.

_{i}= {(S

_{1},1-m(S

_{1})), (S

_{2}, 1-m(S

_{2})), (S

_{3},1-m(S

_{3})} of X.

- D
_{1}= {(S_{1}, 0.75), (S_{2}, 0.75), (S_{3}, 1)} (this type of deviation was related with 2 students) - D
_{2}= {(S_{1}, 0.5), (S_{2}, 1), (S_{3}, 1)} (related with 7 students) - D
_{3}= {(S_{1}, 0.5), (S_{2}, 0.75), (S_{3}, 1)} (related with 5 students) - D
_{4}= {(S_{1}, 0.5), (S_{2}, 0.75), (S_{3}, 0.75)} (related with 4 students) - D
_{5}= {(S_{1}, 0.25), (S_{2}, 0.5), (S_{3}, 0.75)} (related with 3 students) - D
_{6}= {(S_{1}, 0.25), (S_{2}, 0.25), (S_{3}, 0.5)} (related with 6 students) - D
_{7}= {(S_{1}, 0), (S_{2}, 0.5), (S_{3}, 0.75)} (related with 1 student) - D
_{8}= {(S_{1}, 0), (S_{2}, 0.5), (S_{3}, 0.5)} (related with 2 students) - D
_{9}= {(S_{1}, 0), (S_{2}, 0.25), (S_{3}, 0.5)} (related with 1 student) - D
_{10}= {(S_{1}, 0), (S_{2}, 0.25), (S_{3}, 025)} (related with 3 students) - D
_{11}= {(S_{1}, 0), (S_{2}, 0), (S_{3}, 0.25)} (related with 1 student)

_{3}of deviation demonstrate a better performance than those possessing the type D

_{1}, the students possessing the type D

_{4}demonstrate a better performance than those possessing the type D

_{3}and so on. However, the students possessing the type D

_{1}demonstrate a better performance with respect to problem solving than those possessing the type D

_{2}, who demonstrate a better performance with respect to the knowledge acquisition. Similarly, the students possessing the type D

_{6}demonstrate a better performance with respect to problem solving and analogical reasoning than those possessing the type D

_{7}, who demonstrate a better performance with respect to the knowledge acquisition. In other words, this type of assessment by reference to the teacher defines a relationship of partial order among students’ with respect to their performance.

## 5. Conclusions and Discussion

- 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.

## Conflicts of Interest

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Voskoglou, M.G. Fuzzy Logic as a Tool for Assessing Students’ Knowledge and Skills. *Educ. Sci.* **2013**, *3*, 208-221.
https://doi.org/10.3390/educsci3020208

**AMA Style**

Voskoglou MG. Fuzzy Logic as a Tool for Assessing Students’ Knowledge and Skills. *Education Sciences*. 2013; 3(2):208-221.
https://doi.org/10.3390/educsci3020208

**Chicago/Turabian Style**

Voskoglou, Michael Gr. 2013. "Fuzzy Logic as a Tool for Assessing Students’ Knowledge and Skills" *Education Sciences* 3, no. 2: 208-221.
https://doi.org/10.3390/educsci3020208