10.57647/ijm2c.2025.150424

A Mamdani Fuzzy–APOS Model for Quantitatively Assessing Students’ Skill in Solving Linear Systems by the Inverse Matrix Method

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  • Hanieh Hashemi1,
  • Mohammad Hassan Behzadi*2,
  • Hamid Rasouli1,
  • Mahdi Azhini1,
  1. Science and Research Branch, Islamic Azad University, Tehran, Iran
  2. Department of Mathematics and Computer Science, Science and Research Branch, Islamic Azad University, Tehran, Iran.

Received: 17-07-2025

Revised: 05-08-2025

Accepted: 12-08-2025

Published in Issue 22-08-2025

Copyright (c) 2025 Hanieh Hashemi, Dr. Mohammad Hassan Behzadi, Dr. Hamid Rasouli, Dr.Mahdi Azhini (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Hashemi, H., Behzadi, M. H., Rasouli, H., & Azhini, M. (2025). A Mamdani Fuzzy–APOS Model for Quantitatively Assessing Students’ Skill in Solving Linear Systems by the Inverse Matrix Method. International Journal of Mathematical Modelling & Computations, 15(4). https://doi.org/10.57647/ijm2c.2025.150424
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Abstract

Accurately diagnosing students’ algebraic problem-solving skills remains a challenge in engineering education, making rigorous mathematical modelling essential. This study introduces and validates a mathematically explicit Mamdani Fuzzy–APOS model that quantitatively evaluates undergraduate engineering students’ ability to solve 3×3 and 4×4 linear systems via the inverse-matrix method. The model formalises each APOS stage (Action, Process, Object, Schema) as a fuzzy submodel and links them through a well-posed rule base of 44 expert-elicited fuzzy rules. For every stage, five observable indicators scored on a 0–5 scale are fuzzified into four linguistic labels (Weak to Excellent) using Gaussian membership functions ( =1.0, =1.8), and the intermediate results are defuzzified into a single cognitive score. Empirical validation with 70 undergraduates demonstrated strong consistency (Spearman’s ρ = 0.906, p < .001; = 0.821; RMSE = 3.75; NRMSE = 0.150) and robustness under ±10% parameter perturbations. While a classical linear-regression baseline achieves higher predictive accuracy ( = 0.978; = 0.956), the Mamdani Fuzzy–APOS framework delivers fully interpretable, stage-wise diagnostics with only a modest decrease in prediction performance. The framework therefore offers precise, interpretable diagnostics that enable instructors to pinpoint specific cognitive weaknesses and design targeted instructional interventions, highlighting its practical value as an educational decision-support tool grounded in mathematical modelling. A key innovation of this work is the novel integration of APOS theory with a Mamdani fuzzy-inference system (APOS-FIS), delivering a mathematically rigorous, cognitively grounded assessment framework that balances interpretability and predictive performance.

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