A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results
Background: Consistent evaluation methods foster fairness, reduce bias, and enhance student understanding and motivation. Notably, mathematical inconsistencies, such as improper weighting, flawed averaging, and unsound scaling, can undermine the accuracy and reliability of assigned grades. This pape...
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2025-05-01
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| author | Cecilia Leal-Ramírez Héctor Alonso Echavarría-Heras Enrique Villa-Diharce Horacio Haro-Avalos |
| author_facet | Cecilia Leal-Ramírez Héctor Alonso Echavarría-Heras Enrique Villa-Diharce Horacio Haro-Avalos |
| author_sort | Cecilia Leal-Ramírez |
| collection | DOAJ |
| description | Background: Consistent evaluation methods foster fairness, reduce bias, and enhance student understanding and motivation. Notably, mathematical inconsistencies, such as improper weighting, flawed averaging, and unsound scaling, can undermine the accuracy and reliability of assigned grades. This paper addresses the critical need for consistent student evaluation methods, with a primary focus on ensuring mathematical consistency within grading systems. Methods: We propose a scheme aimed at identifying inconsistencies in student evaluation related to the mathematical framework of the used grading method. To explain the functioning of our construct, we provide mathematical representation of conventional grading methods, including summative assessments, rubrics, and the Systematic Task-Based Assessment Method (STBAM) that we have recently developed, which incorporates both traditional and fuzzy logic-based grading modules. We introduce a Consistency Index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mrow><mi>C</mi><mi>I</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> depending on the Mean Absolute Deviation of assigned scores <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>M</mi><mi>A</mi><mi>D</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> and a method’s Complexity Pointer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>C</mi><mi>P</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></semantics></math></inline-formula>). We also propose a method’s Efficiency Index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>β</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> expressed in terms of the Consistency Index and the latter indicator. We compared the mathematical consistency and efficiency of the methods addressed in this study through simulation runs. Results: We demonstrated how the proposed indices can reveal the strengths and weaknesses of each grading scheme analysed. Conclusions: The fuzzy logic-based modulus of the STBAM yielded the highest values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>C</mi><mi>I</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></semantics></math></inline-formula>. However, performing a pending analysis of scalability, teacher training, and cultural adaptability would be essential to strengthen the potential of the STBAM to be adopted as a reliable grading alternative to conventional grading approaches. In the meantime, our approach could provide a clear, logical, and defensible framework for testing the mathematical consistency of student assessment methods. |
| format | Article |
| id | doaj-art-b59c2a172f3547b18bd95f5e04e103d4 |
| institution | DOAJ |
| issn | 2076-3417 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
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| series | Applied Sciences |
| spelling | doaj-art-b59c2a172f3547b18bd95f5e04e103d42025-08-20T03:11:21ZengMDPI AGApplied Sciences2076-34172025-05-011511601410.3390/app15116014A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation ResultsCecilia Leal-Ramírez0Héctor Alonso Echavarría-Heras1Enrique Villa-Diharce2Horacio Haro-Avalos3Centro de Investigación Científica y de Estudios Superiores de Ensenada, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, Ensenada 22860, Baja California, MexicoCentro de Investigación Científica y de Estudios Superiores de Ensenada, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, Ensenada 22860, Baja California, MexicoCentro de Investigación en Matemáticas, A.C. Jalisco s/n, Mineral Valenciana, Guanajuato 36240, Guanajuato, MexicoInstituto Mexicano de Investigación en Pesca y Acuacultura Sustentables (IMIPAS), Km 97.5 Carretera Tijuana-Ensenada, s/n, El Sauzal de Rodríguez, Ensenada 22760, Baja California, MexicoBackground: Consistent evaluation methods foster fairness, reduce bias, and enhance student understanding and motivation. Notably, mathematical inconsistencies, such as improper weighting, flawed averaging, and unsound scaling, can undermine the accuracy and reliability of assigned grades. This paper addresses the critical need for consistent student evaluation methods, with a primary focus on ensuring mathematical consistency within grading systems. Methods: We propose a scheme aimed at identifying inconsistencies in student evaluation related to the mathematical framework of the used grading method. To explain the functioning of our construct, we provide mathematical representation of conventional grading methods, including summative assessments, rubrics, and the Systematic Task-Based Assessment Method (STBAM) that we have recently developed, which incorporates both traditional and fuzzy logic-based grading modules. We introduce a Consistency Index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mrow><mi>C</mi><mi>I</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> depending on the Mean Absolute Deviation of assigned scores <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>M</mi><mi>A</mi><mi>D</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> and a method’s Complexity Pointer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>C</mi><mi>P</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></semantics></math></inline-formula>). We also propose a method’s Efficiency Index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>β</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> expressed in terms of the Consistency Index and the latter indicator. We compared the mathematical consistency and efficiency of the methods addressed in this study through simulation runs. Results: We demonstrated how the proposed indices can reveal the strengths and weaknesses of each grading scheme analysed. Conclusions: The fuzzy logic-based modulus of the STBAM yielded the highest values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>C</mi><mi>I</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></semantics></math></inline-formula>. However, performing a pending analysis of scalability, teacher training, and cultural adaptability would be essential to strengthen the potential of the STBAM to be adopted as a reliable grading alternative to conventional grading approaches. In the meantime, our approach could provide a clear, logical, and defensible framework for testing the mathematical consistency of student assessment methods.https://www.mdpi.com/2076-3417/15/11/6014student gradingmathematical set upconventional scoring methodsFuzzy Inference System approachMean Absolute DeviationConsistency Index |
| spellingShingle | Cecilia Leal-Ramírez Héctor Alonso Echavarría-Heras Enrique Villa-Diharce Horacio Haro-Avalos A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results Applied Sciences student grading mathematical set up conventional scoring methods Fuzzy Inference System approach Mean Absolute Deviation Consistency Index |
| title | A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results |
| title_full | A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results |
| title_fullStr | A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results |
| title_full_unstemmed | A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results |
| title_short | A Study on the Consistency and Efficiency of Student Performance Evaluation Methods: A Mathematical Framework and Comparative Simulation Results |
| title_sort | study on the consistency and efficiency of student performance evaluation methods a mathematical framework and comparative simulation results |
| topic | student grading mathematical set up conventional scoring methods Fuzzy Inference System approach Mean Absolute Deviation Consistency Index |
| url | https://www.mdpi.com/2076-3417/15/11/6014 |
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