# An Interval AHP Technique for Classroom Teaching Quality Evaluation

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Interval Number and Interval Judgment Matrix

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Analytic Hierarchy Process

#### 2.3. Hierarchical Evaluation Structure: Reformed Teaching Observation Protocol

## 3. Methodology

#### 3.1. Determining the Evaluation Criteria’ Weights with I-AHP

#### 3.1.1. Constructing Interval Judgment Matrices

#### 3.1.2. Calculating Criteria’ Basic Weights

#### 3.2. Determining the Assessor’ Weight

#### 3.2.1. Calculating the Similarity Coefficient of Assessors’ Evaluation

#### 3.2.2. Calculating the Difference of Assessors’ Evaluation

#### 3.2.3. Calculating the Weight of Evaluation Assessors

#### 3.2.4. Calculating the Criteria’ Final Weights

#### 3.3. Comprehensive Evaluation Model

## 4. Case Study

#### 4.1. Evaluation Object and Subject

#### 4.2. Constructing Interval Judgment Matrices

#### 4.3. Calculation the Weights of Factors ${F}_{1}$–${F}_{5}$

#### 4.3.1. Calculation the Basic Weights of Factors ${F}_{1}$–${F}_{5}$

#### 4.3.2. Calculating the Weights of Factors ${F}_{1}$–${F}_{5}$

#### 4.4. Assessors’ Weights

#### 4.4.1. Calculating the Similarity Coefficient of Assessors’ Evaluation

#### 4.4.2. Calculating the Degree of Difference of Assessors’ Evaluation and the Weight of Evaluation Assessors

#### 4.5. Final Weights for Factors ${F}_{1}$–${F}_{5}$

## 5. Comprehensive Evaluation with Interval Numbers

#### 5.1. Evaluation Standard and Data Collection

#### 5.2. Comprehensive Evaluation

## 6. Results and Analysis

#### 6.1. Results and Analysis of Interval Weights

#### 6.1.1. Ranking for Interval Weights

#### 6.1.2. Analysis for the Ranking Results

#### 6.2. Results and Analysis of Aggregated Scores

#### 6.2.1. Ranking for Aggregated Scores

#### 6.2.2. Analysis of Total Aggregated Scores

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Interval Reciprocal Judgment Matrices for Items

## References

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Value of Importance | Comparative Judgment |
---|---|

1 | ${F}_{i}$ is as important as ${F}_{j}$ |

3 | ${F}_{i}$ is slightly more important than ${F}_{j}$ |

5 | ${F}_{i}$ is strongly more important than ${F}_{j}$ |

7 | ${F}_{i}$ is very strongly more important than ${F}_{j}$ |

9 | ${F}_{i}$ is extremely more important than ${F}_{j}$ |

2,4,6,8 | Represents the median value of the above adjacent judgment |

Reciprocal | If the ratio of the importance of ${F}_{i}$ and ${F}_{j}$ is ${f}_{ij}$, |

then ratio of ${F}_{j}$ and ${F}_{i}$ is ${f}_{ji}=1\left./\phantom{1{f}_{ij}}\right)\phantom{\rule{0.0pt}{0ex}}{f}_{ij}$ |

Target Level | Factor Level (F) | Item Level (I) |
---|---|---|

Teaching quality evaluation | ${F}_{1}$. Lesson Design and Implementation | ${I}_{1}$. Respect student preconceptions and knowledge of mathematics |

${I}_{2}$. Form a math learning group | ||

${I}_{3}$. Explore before formal presentation | ||

${I}_{4}$. Seek alternative approaches different in textbooks | ||

${I}_{5}$. Adopt student ideas in teaching | ||

${F}_{2}$. Content: Propositional Knowledge | ${I}_{6}$. Involve fundamental concepts of mathematics | |

${I}_{7}$. Promote coherent understanding of mathematical concepts | ||

${I}_{8}$. Teacher have a solid grasp of the contents (especially for unrelated questions) | ||

${I}_{9}$. Encourage abstraction (mathematics models or formulas) | ||

${I}_{10}$. Emphasize the connection between mathematics and other disciplines or social life | ||

${F}_{3}$. Content: Procedural Knowledge | ${I}_{11}$. Students use models, formulas, graphics to express their understanding | |

${I}_{12}$. Students make predictions, assumptions or estimates | ||

${I}_{13}$. Make critical inferences or estimates of results | ||

${I}_{14}$. Students reflect on their learning in mathematics class | ||

${I}_{15}$. Students infer or question corresponding conclusions, concepts and formulas | ||

${F}_{4}$. Classroom culture: Communicative Interactions | ${I}_{16}$. Students communicate their understanding and ideas with various ways | |

${I}_{17}$. Teachers’ questions lead to students’ thinking differently about mathematics | ||

${I}_{18}$. Students actively discuss mathematics problems | ||

${I}_{19}$. The direction of the class is determined by the discussion of students | ||

${I}_{20}$. Students actively express their views without being ridiculed | ||

${F}_{5}$. Classroom culture: Student/ Teacher Relationships | ${I}_{21}$. Encourage students to actively participate in discussion | |

${I}_{22}$. Encourage students to solve mathematical problems in many ways | ||

${I}_{23}$. Teacher is patient when students think about problems or complete assignments | ||

${I}_{24}$. When students investigate or study, the teacher acts as a resource | ||

${I}_{25}$. Teacher listens carefully when students discuss and express their views |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

RI | 0.00 | 0.00 | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 | 1.49 | 1.51 | 1.48 |

Assessor | $\mathit{\Lambda}$ | ${\mathit{\lambda}}_{{}_{max}}$ | $\mathit{\alpha}/\mathit{\beta}$ | $\mathbf{CR}$ | ${\mathit{w}}_{1\mathit{L}}/{\mathit{w}}_{1\mathit{R}}$ |
---|---|---|---|---|---|

No.1 | ${\mathsf{\Lambda}}_{1}^{-}$ | 5.2669 | 0.9779 | 0.0596 | ${w}_{1L}$=(0.3033, 0.1160, 0.0706, 0.4004, 0.1056) |

${\mathsf{\Lambda}}_{1}^{+}$ | 5.0958 | 0.9958 | 0.0214 | ${w}_{1R}$= (0.4482, 0.1484, 0.0757, 0.2365, 0.0690) | |

No.2 | ${\mathsf{\Lambda}}_{2}^{-}$ | 5.2148 | 0.9896 | 0.0479 | ${w}_{2L}$=(0.1424, 0.3161, 0.2915, 0.1143, 0.1170) |

${\mathsf{\Lambda}}_{2}^{+}$ | 5.3465 | 0.9813 | 0.0773 | ${w}_{2R}$= (0.2607, 0.3275, 0.2450, 0.0890, 0.0674) | |

No.3 | ${\mathsf{\Lambda}}_{3}^{-}$ | 5.3339 | 0.9733 | 0.0745 | ${w}_{3L}$= (0.1674, 0.0814, 0.0993, 0.3766, 0.2527) |

${\mathsf{\Lambda}}_{3}^{+}$ | 5.2508 | 0.9775 | 0.0560 | ${w}_{3R}$= (0.2856, 0.1094, 0.1224, 0.3050, 0.1509) | |

No.4 | ${\mathsf{\Lambda}}_{4}^{-}$ | 5.2592 | 0.9804 | 0.0579 | ${w}_{4L}$= (0.0655, 0.1221, 0.3655, 0.2712, 0.1370) |

${\mathsf{\Lambda}}_{4}+$ | 5.4217 | 0.9615 | 0.0941 | ${w}_{4R}$= (0.1151, 0.1457, 0.3793, 0.2202, 0.1201) | |

No.5 | ${\mathsf{\Lambda}}_{5}^{-}$ | 5.3621 | 0.9582 | 0.0808 | ${w}_{5L}$= (0.2589, 0.2254, 0.0958, 0.1962, 0.1767) |

${\mathsf{\Lambda}}_{5}^{+}$ | 5.4353 | 0.9532 | 0.0972 | ${w}_{5R}$= (0.3487, 0.2395, 0.0895, 0.1629, 0.1177) |

Factor | Lower Weight | Upper Weight |
---|---|---|

${F}_{1}$ | 0.1936 | 0.3030 |

${F}_{2}$ | 0.1747 | 0.1940 |

${F}_{3}$ | 0.1777 | 0.1679 |

${F}_{4}$ | 0.2695 | 0.2043 |

${F}_{5}$ | 0.1577 | 0.1057 |

Factor | Lower Weight | Upper Weight |
---|---|---|

${F}_{1}$ | 0.1990 | 0.3108 |

${F}_{2}$ | 0.1795 | 0.199 |

${F}_{3}$ | 0.1826 | 0.1722 |

${F}_{4}$ | 0.2769 | 0.2095 |

${F}_{5}$ | 0.162 | 0.1085 |

Factor | Lower Weight (${\mathit{w}}_{{\mathit{F}}_{\mathit{i}}}^{-}$) | Upper Weight (${\mathit{w}}_{{\mathit{F}}_{\mathit{i}}}^{+}$) | Items | Lower Weight (${\mathit{w}}_{{\mathit{I}}_{\mathit{i}}}^{-}$) | Upper Weight (${\mathit{w}}_{{\mathit{I}}_{\mathit{i}}}^{+}$) |
---|---|---|---|---|---|

${F}_{1}$ | 0.1990 | 0.3108 | ${I}_{1}$ | 0.1243 | 0.1717 |

${I}_{2}$ | 0.2183 | 0.2497 | |||

${I}_{3}$ | 0.2604 | 0.2516 | |||

${I}_{4}$ | 0.2487 | 0.2128 | |||

${I}_{5}$ | 0.1482 | 0.1142 | |||

${F}_{2}$ | 0.1795 | 0.1990 | ${I}_{6}$ | 0.1282 | 0.1914 |

${I}_{7}$ | 0.1439 | 0.1768 | |||

${I}_{8}$ | 0.3149 | 0.3012 | |||

${I}_{9}$ | 0.2101 | 0.1684 | |||

${I}_{10}$ | 0.2029 | 0.1623 | |||

${F}_{3}$ | 0.1826 | 0.1722 | ${I}_{11}$ | 0.0878 | 0.1460 |

${I}_{12}$ | 0.2063 | 0.2454 | |||

${I}_{13}$ | 0.2499 | 0.2462 | |||

${I}_{14}$ | 0.3182 | 0.2549 | |||

${I}_{15}$ | 0.1378 | 0.1075 | |||

${F}_{4}$ | 0.2769 | 0.2095 | ${I}_{16}$ | 0.2008 | 0.2646 |

${I}_{17}$ | 0.1628 | 0.1938 | |||

${I}_{18}$ | 0.1574 | 0.1475 | |||

${I}_{19}$ | 0.2054 | 0.1758 | |||

${I}_{20}$ | 0.2737 | 0.2183 | |||

${F}_{5}$ | 0.1620 | 0.1085 | ${I}_{21}$ | 0.2002 | 0.2694 |

${I}_{22}$ | 0.2300 | 0.2444 | |||

${I}_{23}$ | 0.1802 | 0.1816 | |||

${I}_{24}$ | 0.2209 | 0.1892 | |||

${I}_{25}$ | 0.1687 | 0.1155 |

Factors | Items | Evaluation Value (${\mathit{x}}_{\mathbf{ij}},\mathit{i}=1,2,\cdots ,25,\mathit{j}=1,2,\cdots ,5$) | ||||
---|---|---|---|---|---|---|

Assessor-1 | Assessor-2 | Assessor-3 | Assessor-4 | Assessor-5 | ||

${F}_{1}$ | ${I}_{1}$ | [0.6,0.7] | [0.65,0.75] | [0.6,0.8] | [0.7,0.8] | [0.8,0.9] |

${I}_{2}$ | [0.7,0.8] | [0.8,0.85] | [0.75,0.85] | [0.7,0.8] | [0.85,0.9] | |

${I}_{3}$ | [0.65,0.75] | [0.7,0.8] | [0.55,0.6] | [0.5,0.6] | [0.6,0.8] | |

${I}_{4}$ | [0.65,0.75] | [0.7,0.8] | [0.55,0.6] | [0.5,0.6] | [0.6,0.8] | |

${I}_{5}$ | [0.4,0.5] | [0.5,0.55] | [0.5,0.6] | [0.6,0.7] | [0.6,0.65] | |

${F}_{2}$ | ${I}_{6}$ | [0.8,0.9] | [0.75,0.8] | [0.8,0.85] | [0.75,0.85] | [0.8,0.9] |

${I}_{7}$ | [0.6,0.8] | [0.7,0.8] | [0.8,0.85] | [0.8,0.9] | [0.75,0.8] | |

${I}_{8}$ | [0.8,0.9] | [0.8,0.85] | [0.8,0.9] | [0.9,0.9] | [0.75,0.85] | |

${I}_{9}$ | [0.5,0.6] | [0.75,0.8] | [0.6,0.7] | [0.65,0.7] | [0.785,0.85] | |

${I}_{10}$ | [0.8.0.9] | [0.8,0.85] | [0.8,0.9] | [0.75,0.8] | [0.75,0.85] | |

${F}_{3}$ | ${I}_{11}$ | [0.6,0.6] | [0.55,0.7] | [0.6,0.65] | [0.5,0.6] | [0.7,0.75] |

${I}_{12}$ | [0.65,0.7] | [0.65,0.75] | [0.7,0.75] | [0.8,0.9] | [0.75,0.8] | |

${I}_{13}$ | [0.75,0.8] | [0.7,0.75] | [0.7,0.8] | [0.6,0.8] | [0.75,0.85] | |

${I}_{14}$ | [0.5,0.6] | [0.4,0.5] | [0.6,0.65] | [0.5,0.55] | [0.6,0.7] | |

${I}_{15}$ | [0.7,0.8] | [0.6,0.65] | [0.6,0.7] | [0.7,0.8] | [0.75,0.85] | |

${F}_{4}$ | ${I}_{16}$ | [0.75,0.9] | [0.7,0.8] | [0.7,0.8] | [0.6,0.8] | [0.7,0.75] |

${I}_{17}$ | [0.75,0.8] | [0.7,0.8] | [0.75,0.85] | [0.65,0.8] | [0.8,0.85] | |

${I}_{18}$ | [0.5,0.6] | [0.55,0.6] | [0.45,0.5] | [0.6,0.7] | [0.6,0.7] | |

${I}_{19}$ | [0.35,0.4] | [0.5,0.6] | [0.55,0.6] | [0.5,0.6] | [0.5,0.6] | |

${I}_{20}$ | [0.75,0.8] | [0.7,0.8] | [0.65,0.7] | [0.6,0.7] | [0.75,0.8] | |

${F}_{5}$ | ${I}_{21}$ | [0.8,0.9] | [0.75,0.8] | [0.75,0.85] | [0.8,0.85] | [0.75,0.85] |

${I}_{22}$ | [0.6,0.7] | [0.7,0.75] | [0.6,0.65] | [0.7,0.8] | [0.7,0.8] | |

${I}_{23}$ | [0.8,0.9] | [0.85,0.9] | [0.75,0.8] | [0.8,0.9] | [0.7,0.8] | |

${I}_{24}$ | [0.6,0.8] | [0.6,0.7] | [0.65,0.7] | [0.75,0.8] | [0.7,0.8] | |

${I}_{25}$ | [0.6,0.7] | [0.5,0.6] | [0.45,0.5] | [0.5,0.6] | [0.6,0.7] |

Factor | Weight (${\mathit{w}}_{{\mathit{F}}_{\mathit{i}}}$) | Average Evaluation Value (${\mathit{y}}_{\mathit{i}}$) | Aggregated Score (${\mathit{S}}_{{\mathit{F}}_{\mathit{i}}}={\mathit{w}}_{{\mathit{F}}_{\mathit{i}}}{\mathit{y}}_{\mathit{i}}$) | Total Aggregated Score (y) |
---|---|---|---|---|

${F}_{1}$ | [0.1990,0.3108] | [0.6135,0.721] | [0.122,0.2241] | |

${F}_{2}$ | [0.1795,0.1990] | [0.7549,0.8372] | [0.1355,0.1666] | [0.6565,0.7510] |

${F}_{3}$ | [0.1826,0.1722] | [0.6310,0.7193] | [0.1152,0.1239] | |

${F}_{4}$ | [0.2769,0.2095] | [0.6257,0.7238] | [0.1733,0.1517] | |

${F}_{5}$ | [0.1620,0.1085] | [0.6817,0.7816] | [0.1104,0.0847] |

Interval Scale | Evaluation Level | Description |
---|---|---|

(0,0.2] | Very poor | The behavior never occurred, the performance is very poor |

(0.2,0.4] | Poor | The behavior occurred at least once, the performance is poor to describe the lesson |

(0.4,0.6] | Medium | The behavior occurred more than once, the performance very loosely describes the lesson |

(0.6,0.8] | Good | The behavior occurred more than two times, the performance fairly descriptive of the lesson |

(0.8,1] | Very good | The performance extremely descriptive of the lesson |

Factor | Weight | Ranking | Items of Minimum/Maximum Weight |
---|---|---|---|

${F}_{1}$ | [0.1990,0.3108] | 1 | ${I}_{5}/{I}_{3}$ |

${F}_{2}$ | [0.1795,0.1990] | 3 | ${I}_{6}/{I}_{8}$ |

${F}_{3}$ | [0.1826,0.1722] | 4 | ${I}_{11}/{I}_{14}$ |

${F}_{4}$ | [0.2769,0.2095] | 2 | ${I}_{18}/{I}_{20}$ |

${F}_{5}$ | [0.1620,0.1085] | 5 | ${I}_{25}/{I}_{22}$ |

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## Share and Cite

**MDPI and ACS Style**

Qin, Y.; Hashim, S.R.M.; Sulaiman, J.
An Interval AHP Technique for Classroom Teaching Quality Evaluation. *Educ. Sci.* **2022**, *12*, 736.
https://doi.org/10.3390/educsci12110736

**AMA Style**

Qin Y, Hashim SRM, Sulaiman J.
An Interval AHP Technique for Classroom Teaching Quality Evaluation. *Education Sciences*. 2022; 12(11):736.
https://doi.org/10.3390/educsci12110736

**Chicago/Turabian Style**

Qin, Ya, Siti Rahayu Mohd. Hashim, and Jumat Sulaiman.
2022. "An Interval AHP Technique for Classroom Teaching Quality Evaluation" *Education Sciences* 12, no. 11: 736.
https://doi.org/10.3390/educsci12110736