# Assessing Students’ Mathematical Knowledge with Fuzzy Logic

^{*}

## Abstract

**:**

## 1. Introduction

#### Background

## 2. Review of Literature

#### 2.1. Fuzzy Logic

- triangular function ([26]; a < b < c):

- trapezoidal function ([26]; a < b < c < d):

- Gaussian function [28]:

- fuzzification: We convert clear (crisp) values into fuzzy values via the membership function;
- inference: We use a set of inference rules set by the user, permitting conversion of the fuzzy input values into fuzzy output values; and
- defuzzification: We convert fuzzy output values into clear (crisp) output values using some defuzzification methods.

- Mean of maximum (MOM) is defined as follows:$$MOM\left(A\right)=\frac{1}{\left|T\right|}\cdot {\displaystyle \sum}_{x\in T}x,$$
- Center of Gravity (COG) is defined as follows:$$COG\left(A\right)=\frac{{{\displaystyle \sum}}_{x}{\mu}_{A}\left(x\right)\cdot x}{{{\displaystyle \sum}}_{x}{\mu}_{A}\left(x\right)},$$$$\mathit{COG}\left(A\right)=\frac{{\displaystyle \int}{\mu}_{A}\left(x\right)\cdot x\text{\hspace{0.17em}}\mathrm{dx}}{{\displaystyle \int}{\mu}_{A}\left(x\right)\text{\hspace{0.17em}}\mathit{dx}},$$

#### 2.2. Assessing Students’ Knowledge with Fuzzy Logic

- inference rules are defined based on experience;
- it is impossible to predict the final result; and
- experience is again used to define membership functions.

#### 2.3. The Proposed Model

## 3. Materials and Methods

#### 3.1. Aims of the Research

- RQ1: How are the COG hypothetical grades different from school grades?
- RQ1.1: Are the COG, student grades, and INVALSI scores correlated?
- RQ1.2: Is there a difference between COG hypothetical grades and student grades?
- RQ1.3: Do the COG grades differ between the four high school typologies?

- RQ2: How are the MOM hypothetical grades different from the school grades?
- RQ2.1: Are the MOM, student grades, and INVALSI scores correlated?
- RQ2.2: Is there a difference between MOM hypothetical grades and student grades?
- RQ2.3: Do the MOM grades differ between the four high school typologies?

- RQ3: Is there any difference between the COG and MOM hypothetical grades?

#### 3.2. Methodology

#### 3.3. Sample

#### 3.4. Data Collection

- school typology (i.e., SL, OL, TS, and VS);
- oral and written grades in mathematics; and
- achievement on the INVALSI mathematics test.

#### 3.5. Procedure

#### 3.6. Data Analysis

^{2}-test to check the differences between groups. As a post hoc test, we used the Dwass–Steel–Crichlow–Fligner (DSCF) pairwise comparison. Whenever possible, we present the Cohen’s d measure of effect size [41].

## 4. Results

#### 4.1. Student Grades and Achievements on INVALSI

^{2}= 49.4; p < 0.001; ε

^{2}= 0.0217), oral grade (χ

^{2}= 78.2; p < 0.001; ε

^{2}= 0.0343), and INVALSI score (χ

^{2}= 591.9; p < 0.001; ε

^{2}= 0.2598), as presented in Table 8.

#### 4.2. Center of Gravity Fuzzy Logic and Hypothetical Grades

^{2}= 289; p < 0.001; ε

^{2}= 0.127; see Table 11).

#### 4.3. Mean of Maximum Fuzzy Logic and Hypothetical Grades

^{2}= 354; p < 0.001; ε

^{2}= 0.155; Table 14).

#### 4.4. Comparing the Two Fuzzy Grading Methods

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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

Extremely low (EL) | $\mathrm{Trap}\left(x,1,1,2,4\right)$ |

Low (L) | $\mathrm{Trian}\left(x,2,4,6\right)$ |

Average (A) | $\mathrm{Trian}\left(x,4,6,8\right)$ |

Good (G) | $\mathrm{Trian}\left(x,6,8,10\right)$ |

Very good (VG) | $\mathrm{Trian}\left(x,8,10,10\right)$ |

Written Grade | ||||||
---|---|---|---|---|---|---|

EL | L | A | G | VG | ||

Oral grade | EL | EL | EL | L | L | A |

L | EL | L | L | A | A | |

A | L | L | A | G | G | |

G | L | A | G | G | VG | |

VG | A | G | G | VG | VG |

**Table 3.**Definition of membership functions for student achievements on the INVALSI test (Fuzzy logic 2).

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

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

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

Average (A) | $\mathrm{Gauss}\left(x,200,40\right)$ |

Good (G) | $\mathrm{Gauss}\left(x,240,40\right)$ |

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

Fuzzy Logic 1 | ||||||
---|---|---|---|---|---|---|

EL | L | A | G | VG | ||

INVALSI | EL | EL | EL | L | L | A |

L | EL | L | L | A | A | |

A | L | L | A | G | G | |

G | L | A | G | G | VG | |

VG | A | G | G | VG | VG |

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

Extremely low (EL) | $\mathrm{Trap}\left(x,1,1,2,4\right)$ |

Low (L) | $\mathrm{Trian}\left(x,2,4,6\right)$ |

Average (A) | $\mathrm{Trian}\left(x,4,6,8\right)$ |

Good (G) | $\mathrm{Trian}\left(x,6,8,10\right)$ |

Very good (VG) | $\mathrm{Trian}\left(x,8,10,10\right)$ |

Oral Grade | COG Fuzzy Logic 1 * | COG Fuzzy Logic 2 * | COG Final Grade | MOM Fuzzy 1 * | MOM Fuzzy 2 * | MOM Final Grade |
---|---|---|---|---|---|---|

1 | 3.24 | 5.71 | 6 | 3.00 | 5.00 | 5 |

2 | 3.24 | 5.71 | 6 | 3.00 | 5.00 | 5 |

3 | 3.24 | 5.71 | 6 | 3.00 | 5.00 | 5 |

4 | 4.00 | 5.87 | 6 | 4.02 | 6.04 | 6 |

5 | 5.00 | 5.98 | 6 | 5.01 | 7.00 | 7 |

6 | 5.00 | 5.98 | 6 | 5.01 | 7.00 | 7 |

7 | 6.00 | 6.69 | 7 | 6.00 | 8.02 | 8 ** |

8 | 7.00 | 6.81 | 7 | 6.99 | 7.00 | 7 |

9 | 7.00 | 6.81 | 7 | 6.99 | 7.00 | 7 |

10 | 8.00 | 7.77 | 8 | 7.98 | 8.02 | 8 |

Written Grade | Oral Grade | INVALSI | |
---|---|---|---|

Mean | 6.50 | 6.59 | 178 |

Median | 6 | 7 | 175 |

Standard deviation | 1.43 | 1.41 | 35.4 |

Minimum | 1 | 2 | 90.1 |

Maximum | 10 | 10 | 311 |

Skewness | −0.102 | −0.176 | 0.512 |

Kurtosis | −0.214 | 0.0209 | 0.239 |

Variable | School Typology | Mean | Standard Deviation | Median |
---|---|---|---|---|

Written grade | SL | 6.65 | 1.40 | 7 |

OL | 6.65 | 1.42 | 7 | |

TS | 6.58 | 1.50 | 7 | |

VS | 6.15 | 1.33 | 6 | |

Oral grade | SL | 6.73 | 1.40 | 7 |

OL | 6.74 | 1.31 | 7 | |

TS | 6.75 | 1.53 | 7 | |

VS | 6.16 | 1.30 | 6 | |

INVALSI | SL | 219.35 | 35.25 | - |

OL | 175.47 | 30.44 | - | |

TS | 183.41 | 29.83 | - | |

VS | 155.41 | 26.28 | - |

**Table 9.**Dwass–Steel–Crichlow–Fligner pairwise comparison for school grades and INVALSI scores between school typologies.

VS | TS | SL | |||
---|---|---|---|---|---|

Written grades | OL | W | −9.25 | −1.73 | −0.189 |

p-value | <0.001 | 0.614 | 0.999 | ||

VS | W | - | 6.93 | 7.08 | |

p-value | - | <0.001 | <0.001 | ||

TS | W | - | 1.25 | ||

p-value | - | 0.815 | |||

Oral grades | OL | W | −11.35 | 0.0152 | 0.180 |

p-value | <0.001 | 1.000 | 0.999 | ||

VS | W | - | 9.85 | 8.45 | |

p-value | - | <0.001 | <0.001 | ||

TS | W | - | 0.089 | ||

p-value | - | 1.000 | |||

INVALSI | OL | W | −16.87 | 7.02 | 23.17 |

p-value | <0.001 | <0.001 | <0.001 | ||

VS | W | - | 22.04 | 29.39 | |

p-value | - | <0.001 | <0.001 | ||

TS | W | - | 19.15 | ||

p-value | - | <0.001 |

Hypothetical Grades | |
---|---|

Mean | 5.73 |

Median | 6 |

Standard deviation | 1.19 |

Minimum | 3 |

Maximum | 9 |

Skewness | −0.329 |

Kurtosis | −0.0879 |

Variable | School Typology | Mean | Standard Deviation | Median |
---|---|---|---|---|

COG hypothetical grade | SL | 6.53 | 1.15 | 6 |

OL | 5.77 | 1.11 | 6 | |

TS | 5.90 | 1.10 | 6 | |

VS | 5.14 | 1.11 | 5 |

**Table 12.**Dwass–Steel–Crichlow–Fligner pairwise comparison of center of gravity hypothetical grades between school typologies.

VS | TS | SL | ||
---|---|---|---|---|

OL | W | −14.62 | 2.57 | 13.14 |

p-value | <0.001 | 0.264 | <0.001 | |

VS | W | - | 16.11 | 21.25 |

p-value | - | <0.001 | <0.001 | |

TS | W | - | 10.89 | |

p-value | - | <0.001 |

Hypothetical Grades | |
---|---|

Mean | 5.56 |

Median | 5 |

Standard deviation | 1.79 |

Minimum | 1 |

Maximum | 10 |

Skewness | 0.314 |

Kurtosis | −0.225 |

Variable | School Typology | Mean | Standard Deviation | Median |
---|---|---|---|---|

MOM hypothetical grade | SL | 6.96 | 1.73 | 7 |

OL | 5.57 | 1.64 | 5 | |

TS | 5.80 | 1.70 | 6 | |

VS | 4.61 | 1.38 | 4 |

**Table 15.**Dwass–Steel–Crichlow–Fligner pairwise comparison for mean of maximum hypothetical grades between school typologies.

VS | TS | SL | ||
---|---|---|---|---|

OL | W | −15.76 | 3.00 | 15.15 |

p-value | <0.001 | 0.147 | <0.001 | |

VS | W | — | 17.34 | 24.43 |

p-value | — | <0.001 | <0.001 | |

TS | W | — | 12.48 | |

p-value | — | <0.001 |

**Table 16.**Center of gravity (COG) and mean of maximum (MOM) hypothetical grades between four school typologies.

School Typology | Higher Grade | W | p | Cohen’s d |
---|---|---|---|---|

SL | MOM | 2513 | <0.001 | 0.481 |

OL | COG | 68,382 | <0.001 | 0.211 |

TS | COG | 39,709 | <0.001 | 0.103 |

VS | COG | 54,143 | <0.001 | 0.617 |

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**MDPI and ACS Style**

Doz, D.; Felda, D.; Cotič, M. Assessing Students’ Mathematical Knowledge with Fuzzy Logic. *Educ. Sci.* **2022**, *12*, 266.
https://doi.org/10.3390/educsci12040266

**AMA Style**

Doz D, Felda D, Cotič M. Assessing Students’ Mathematical Knowledge with Fuzzy Logic. *Education Sciences*. 2022; 12(4):266.
https://doi.org/10.3390/educsci12040266

**Chicago/Turabian Style**

Doz, Daniel, Darjo Felda, and Mara Cotič. 2022. "Assessing Students’ Mathematical Knowledge with Fuzzy Logic" *Education Sciences* 12, no. 4: 266.
https://doi.org/10.3390/educsci12040266