# Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Basic Definitions

#### 2.2. The Usage of Fuzzy Logic in Education

#### 2.3. Research Aims

## 3. Materials and Methods

#### 3.1. Methodology

#### 3.2. Data Collection

- Teacher-given grades on oral evaluations in mathematics;
- Teacher-given grades on written evaluations in mathematics;
- Scores on the national assessment INVALSI;
- Students’ gender.

#### 3.3. Data Filtering

- Grade 8: 619 students presented missing data on both teacher-given grades and were therefore excluded from the analyses. The remaining sample consisted of students who had at least one teacher-assigned grade (oral or written). We did not make any distinction between oral or written grades, since in Italian middle school educational system, students receive only one grade for mathematics, which refers to both oral and written evaluations;
- Grade 10: 3008 students were excluded due to missing data. The remaining sample consisted of students who had at least one teacher-assigned grade. For the students who had both oral and written grades, only the oral grade was considered, since it includes a broader range of evaluations;
- Grade 13: 6174 students presented missing data on both teacher-given grades. The filtering procedure is the same as for grade 10.

#### 3.3.1. Teacher-Given Grades

#### 3.3.2. The INVALSI Test

#### 3.4. Application of Fuzzy Logic

#### 3.4.1. Fuzzification of Teacher-Given Grades

#### 3.4.2. Fuzzification of Students’ Attainments on the National Assessment INVALSI

#### 3.4.3. Rules and Inference

- 4.
- It is possible to have the “Very high” (VH) level only if both tests are very high (VH), so the VH levels of [7] have been adapted to H;
- 5.
- High performance (H) on the INVALSI tests can only produce at least medium (M) ratings;
- 6.
- The final grade is high (H) only if both ratings are high (H) or one rating is very high (VH) and the other is at least medium (M).

#### 3.4.4. Defuzzification

#### 3.5. Quantitative Analysis

- 7.
- The Spearman’s ρ coefficient to compute the correlation between data [50,51], e.g., to what extent are teacher-given grades and fuzzy final grades correlated. The requirement is that each variable is measured on at least ordinal scale; the usage of this coefficient does not make any assumption regarding the distribution of the variables.
- 8.
- The Wilcoxon W signed-rank test for paired samples to check for differences between two categories [52], e.g., the difference between the traditional and fuzzy final grades. The assumptions of the test are that samples are random samples, which are mutually independent, and that the measurement scale is at least ordinal.
- 9.
- The Kruskal–Wallis χ
^{2}-test to check for differences among three or more categories [53] with the Dwass-Steel-Critchlow-Fligner (DSCF) post-hoc test, e.g., whether fuzzy final grades differ among different teacher-given grades. The assumptions are similar to those for the Wilcoxon test.

^{2}coefficient [54] for the Kruskal–Wallis test and the rank biserial correlation coefficient r

_{rb}[55] for the Wilcoxon test. Effect sizes lower than 0.10 are considered small, those around 0.25 are considered middle, and those greater than 0.40 are considered large [56,57].

## 4. Results

#### 4.1. Differences between Classical and Fuzzy Logic Assessment

#### 4.1.1. Grade 8

^{8}; p < 0.001) with a big effect size (r

_{rb}= 0.922).

^{2}(7) = 15,035; p < 0.001; ε

^{2}= 0.517). The DSCF pairwise comparisons between school grades are all statistically significant (p < 0.05), except for the comparison among students with teacher-given grade 3 and 4 (W = 3.89; p = 0.108).

#### 4.1.2. Grade 10

^{8}; p < 0.001) with a medium effect size (r

_{rb}= 0.600).

^{2}(9) = 13,150; p < 0.001; ε

^{2}= 0.401). The DSCF pairwise comparisons between school grades are all statistically significant (p < 0.05), except for the comparison among students with teacher-given grade 6 and 7 (W = 4.43; p = 0.055).

#### 4.1.3. Grade 13

^{8}; p < 0.001) with a medium effect size (r

_{rb}= 0.690).

^{2}(9) = 11,351; p < 0.001; ε

^{2}= 0.373). The DSCF pairwise comparisons between school grades are all statistically significant (p < 0.05), except for the comparison among students with teacher-given grade 1 and 2 (W = 4.37; p = 0.063).

#### 4.2. Differences in Fuzzy Logic Assessment among Students of Grade 8, 10, and 13

^{2}(2) = 51.2; p < 0.001), although the effect size is extremely low (ε

^{2}= 5.55 × 10

^{−4}). The DSCF test has shown that there are statistically significant differences in fuzzy grades between the 8th and 10th grade students (W = −8.572; p < 0.001), as well as between the 8th and 13th grade students (W = −8.983; p < 0.001). However, there are no statistically significant differences between the 10th and 13th grade students (W = −0.824; p = 0.830). Due to the low effect size, results indicate that students’ fuzzy achievements are hardly dependent on students’ classes. Moreover, high school students have similar fuzzy achievements, thus fuzzy grades are almost uniformly distributed in all grades.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**General diagram illustrating the methodology used in the present study. * Data from school year 2018–2019 was used.

**Figure 6.**The output surface for the fuzzy inference system, plotting the output variable “Final Grade” against the first two input variables, i.e., “INVALSI” and “Grade”.

Grade | Original Sample | Final Sample | % of Original Sample |
---|---|---|---|

8 | 29,675 | 29,056 | 97.9% |

10 | 35,802 | 32,794 | 91.6% |

13 | 36,589 | 30,415 | 83.1% |

Grade | Gender | Frequency (f) | Percentage Frequency (%f) |
---|---|---|---|

8 | Male | 14,983 | 51.6% |

Female | 14,073 | 48.4% | |

10 | Male | 15,663 | 47.8% |

Female | 17,131 | 52.2% | |

13 | Male | 14,785 | 48.6% |

Female | 15,630 | 51.4% |

Level | Membership Function |
---|---|

Very low (VL) | $\mathrm{Tri}\left(x,1,1,3\right)$ |

Low (L) | $\mathrm{Tri}\left(x,1,3,5\right)$ |

Medium (M) | $\mathrm{Trap}\left(x,3,5,6,8\right)$ |

High (H) | $\mathrm{Tri}\left(x,6,8,10\right)$ |

Very high (VH) | $\mathrm{Tri}\left(x,8,10,10\right)$ |

Level | Membership Function |
---|---|

Very low (VL) | $\mathrm{Gauss}\left(x,120,40\right)$ |

Low (L) | $\mathrm{Gauss}\left(x,160,40\right)$ |

Medium (M) | $\mathrm{Gauss}\left(x,200,40\right)$ |

High (H) | $\mathrm{Gauss}\left(x,240,40\right)$ |

Very high (VH) | $\mathrm{Gauss}\left(x,280,40\right)$ |

Teacher-Given Grades | ||||||
---|---|---|---|---|---|---|

VL | L | M | H | VH | ||

INVALSI | VL | VL | VL | L | L | M |

L | VL | L | L | M | M | |

M | L | L | M | M | H | |

H | M | M | M | H | H | |

VH | M | M | H | H | VH |

Mathematics Grade | INVALSI | Fuzzy Grade | |
---|---|---|---|

Mean | 6.79 | 201 | 5.62 |

Standard deviation | 1.42 | 38.5 | 1.78 |

Median | 7 | 200 | 6 |

Minimum | 3 | 66.5 | 1 |

Maximum | 10 | 326 | 10 |

Skewness (SE) | 0.160 (0.0144) | 0.247 (0.0144) | −0.0461 (0.0144) |

Kurtosis (SE) | −0.693 (0.0287) | −0.0442 (0.0287) | −0.390 (0.0287) |

Fuzzy Grades | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

School Grade | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |

3 | 3 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 (0.0%) |

4 | 0 | 384 | 2 | 202 | 420 | 7 | 0 | 0 | 0 | 0 | 1015 (3.5%) |

5 | 0 | 0 | 2860 | 0 | 0 | 1792 | 0 | 16 | 0 | 0 | 4668 (16.1%) |

6 | 0 | 0 | 3276 | 0 | 0 | 4274 | 0 | 49 | 0 | 0 | 7599 (26.2%) |

7 | 0 | 0 | 18 | 0 | 2412 | 4064 | 0 | 26 | 0 | 0 | 6520 (22.4%) |

8 | 0 | 0 | 35 | 0 | 0 | 2533 | 0 | 2710 | 0 | 0 | 5278 (18.2%) |

9 | 0 | 0 | 0 | 0 | 11 | 1270 | 717 | 0 | 1353 | 0 | 3351 (11.5%) |

10 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 300 | 0 | 310 | 613 (2.1%) |

Total | 3 (0.0%) | 384 (1.3%) | 6200 (21.3%) | 202 (.7%) | 2843 (9.8%) | 12943 (48.0%) | 717 (2.5%) | 3101 (10.7%) | 1353 (4.7%) | 310 (1.1%) | 29056 (100.0%) |

Fuzzy Grade | |||||
---|---|---|---|---|---|

Teacher-Given Grade | Mean | Standard Deviation | Median | Minimum | Maximum |

3 | 2.50 | 0.905 | 3 | 1 | 3 |

4 | 3.67 | 1.36 | 4 | 2 | 6 |

5 | 4.17 | 1.47 | 3 | 3 | 8 |

6 | 4.72 | 1.51 | 6 | 3 | 8 |

7 | 5.63 | 0.523 | 6 | 3 | 8 |

8 | 7.01 | 1.05 | 8 | 3 | 8 |

9 | 7.42 | 1.35 | 7 | 5 | 9 |

10 | 9.00 | 1.02 | 10 | 6 | 10 |

Mathematics Grade | INVALSI | Fuzzy Grade | |
---|---|---|---|

Mean | 6.15 | 206 | 5.54 |

Standard deviation | 1.45 | 39.1 | 1.60 |

Median | 6 | 203 | 6 |

Minimum | 1 | 72.3 | 1 |

Maximum | 10 | 314 | 10 |

Skewness (SE) | 0.004 (0.0135) | 0.226 (0.0135) | −0.293 (0.0135) |

Kurtosis (SE) | −0.233 (0.0271) | −0.053(0.0271) | 0.009 (0.0271) |

Fuzzy Grades | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

School Grade | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |

1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 (0.0%) |

2 | 3 | 89 | 0 | 14 | 11 | 1 | 0 | 0 | 0 | 0 | 118 (0.4%) |

3 | 102 | 0 | 683 | 0 | 0 | 51 | 0 | 0 | 0 | 0 | 835 (2.5%) |

4 | 0 | 519 | 0 | 536 | 1951 | 227 | 0 | 0 | 0 | 0 | 3233 (9.9%) |

5 | 0 | 0 | 2282 | 0 | 0 | 3652 | 0 | 209 | 0 | 0 | 6143 (18.7%) |

6 | 0 | 0 | 2550 | 0 | 0 | 6560 | 566 | 0 | 0 | 9676 (29.5%) | |

7 | 0 | 0 | 17 | 0 | 2008 | 4654 | 0 | 180 | 0 | 0 | 6859 (20.9%) |

8 | 0 | 0 | 17 | 0 | 0 | 1745 | 0 | 2362 | 0 | 0 | 4124 (12.6%) |

9 | 0 | 0 | 0 | 0 | 8 | 634 | 202 | 0 | 785 | 0 | 1629 (5.0%) |

10 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 98 | 0 | 69 | 173 (0.5%) |

Total | 108 (0.3%) | 608 (1.9%) | 5549 (16.9%) | 550 (1.7%) | 3978 (12.1%) | 17539 (53%) | 202 (0.6%) | 3415 (10.4%) | 785 (2.4%) | 69 (0.2%) | 32794 (100.0%) |

Fuzzy Grade | |||||
---|---|---|---|---|---|

Teacher-Given Grade | Mean | Standard Deviation | Median | Minimum | Maximum |

1 | 1.00 | .000 | 1 | 1 | 1 |

2 | 2.53 | 1.10 | 2 | 1 | 6 |

3 | 2.94 | 1.02 | 3 | 1 | 6 |

4 | 4.42 | 1.16 | 5 | 2 | 6 |

5 | 4.95 | 1.54 | 6 | 3 | 8 |

6 | 5.33 | 1.47 | 6 | 3 | 8 |

7 | 5.75 | .599 | 6 | 3 | 8 |

8 | 7.13 | 1.02 | 8 | 3 | 8 |

9 | 7.56 | 1.42 | 7 | 5 | 9 |

10 | 8.73 | 1.10 | 8 | 6 | 10 |

Mathematics Grade | INVALSI | Fuzzy Grade | |
---|---|---|---|

Mean | 6.33 | 204 | 5.54 |

Standard deviation | 1.46 | 39.8 | 1.62 |

Median | 6 | 201 | 6 |

Minimum | 1 | 69.5 | 1 |

Maximum | 10 | 341 | 10 |

Skewness (SE) | −0.005 (0.014) | 0.211 (0.014) | −0.205 (0.014) |

Kurtosis (SE) | −0.215 (0.0281) | −0.177(0.0281) | −0.053 (0.0281) |

Fuzzy Grades | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

School Grade | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |

1 | 4 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 (0.0%) |

2 | 1 | 50 | 0 | 13 | 12 | 2 | 0 | 0 | 0 | 0 | 78 (0.3%) |

3 | 59 | 0 | 534 | 0 | 0 | 68 | 0 | 0 | 0 | 0 | 661 (2.4%) |

4 | 0 | 449 | 0 | 320 | 1458 | 228 | 0 | 0 | 0 | 0 | 2455 (10.5%) |

5 | 0 | 0 | 2024 | 0 | 0 | 2907 | 0 | 178 | 0 | 0 | 5109 (16.8%) |

6 | 0 | 0 | 2889 | 0 | 0 | 5500 | 0 | 506 | 0 | 0 | 8895 (29.2%) |

7 | 0 | 0 | 22 | 0 | 2379 | 4141 | 0 | 416 | 0 | 0 | 6688 (22.0%) |

8 | 0 | 0 | 42 | 0 | 0 | 2062 | 0 | 2141 | 0 | 0 | 4245 (14.0%) |

9 | 0 | 0 | 0 | 0 | 18 | 877 | 298 | 13 | 783 | 0 | 1989 (6.5% |

10 | 0 | 0 | 0 | 0 | 0 | 5 | 0 | 134 | 10 | 141 | 290 (1.0%) |

Total | 64 (.2%) | 499 (1.6%) | 5512 (18.1%) | 333 (1.1%) | 3867 (12.7%) | 15790 (51.9%) | 298 (1.0%) | 3118 (10.3%) | 793 (2.6%) | 141 (0.5%) | 30415 (100.0%) |

Fuzzy Grade | |||||
---|---|---|---|---|---|

Teacher-Given Grade | Mean | Standard Deviation | Median | Minimum | Maximum |

1 | 1.40 | 0.894 | 1 | 1 | 3 |

2 | 2.88 | 1.31 | 2 | 1 | 6 |

3 | 3.13 | 1.13 | 3 | 1 | 6 |

4 | 4.41 | 1.24 | 5 | 2 | 6 |

5 | 4.88 | 1.57 | 6 | 3 | 8 |

6 | 5.14 | 1.55 | 6 | 3 | 8 |

7 | 5.68 | 0.608 | 6 | 3 | 8 |

8 | 6.98 | 1.07 | 8 | 3 | 8 |

9 | 7.33 | 1.40 | 7 | 5 | 9 |

10 | 8.97 | 1.05 | 9 | 6 | 10 |

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## Share and Cite

**MDPI and ACS Style**

Doz, D.; Felda, D.; Cotič, M. Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach. *Axioms* **2022**, *11*, 359.
https://doi.org/10.3390/axioms11080359

**AMA Style**

Doz D, Felda D, Cotič M. Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach. *Axioms*. 2022; 11(8):359.
https://doi.org/10.3390/axioms11080359

**Chicago/Turabian Style**

Doz, Daniel, Darjo Felda, and Mara Cotič. 2022. "Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach" *Axioms* 11, no. 8: 359.
https://doi.org/10.3390/axioms11080359