10.57647/ijm2c.2026.1602.10

Designing a Smart Flipped Collective Learning Model for Reducing Conceptual Errors Based on APOS Theory: A Case Study on Linear Equations with Female Students in District 2 of Tehran

PDF
  • Somayeh Babaeisadr1,
  • Esmaeil Yousefi*1,
  • Abolfazl Tehranian1,
  • Ahmad Shahvarani Semnani1,
  • Somaeyh Khalashi Ghezelahmad1,
  1. Department of Mathematics‎, ‎SR.C.‎, ‎Islamic Azad University‎, ‎Tehran‎, ‎Iran

Received: 10-08-2025

Revised: 10-10-2025

Accepted: 30-10-2025

Published in Issue 30-06-2026

Published Online: 12-11-2025

Copyright (c) 2025 Somayeh Babaeisadr, ٍEmaeil Yousefi, Abolfazl Tehranian, Ahmad Shahvarani Semnani, Somaeyh Khalashi Ghezelahmad (Author)

Creative Commons License

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

How to Cite

Babaeisadr, S., Yousefi, E., Tehranian, A., Shahvarani Semnani, A., & Khalashi Ghezelahmad, S. (2026). Designing a Smart Flipped Collective Learning Model for Reducing Conceptual Errors Based on APOS Theory: A Case Study on Linear Equations with Female Students in District 2 of Tehran. International Journal of Mathematical Modelling & Computations, 16(2). https://doi.org/10.57647/ijm2c.2026.1602.10
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

This research aims to design a novel hybrid learning model called Smart Flipped Collective Learning Model. This model integrates artificial intelligence within a flipped learning environment with in-person collaborative learning based on APOS(Action-Process-Object-Schema) theory. The study focused on reducing students' conceptual errors in the topic of linear equations. In this novel approach, educational content related to linear equations was provided to students using artificial intelligence, and class time was dedicated to collaborative and interactive activities to deepen learning. The research considered how students' learning processes changed, how their understanding levels improved based on the stages of APOS theory (Action, Process, Object, Schema), and how conceptual errors were reduced through the integration of in-person collaborative learning with flipped instruction. The research method was quasi-experimental with a pre-test and post-test design and a control group. The statistical population included all ninth-grade female students in District 2 of Tehran. Random cluster sampling was performed, and 120 students were selected, with 60 in the experimental group and 60 in the control group. The results showed that the smart flipped collective model is a novel integrative framework with statistically significant and cognitively profound effects on teaching mathematical concepts. The smart flipped collective model led to a significant improvement in mathematical performance and a reduction in conceptual errors. Traditional instruction largely remained at the Action and Process levels, while the smart flipped collective model significantly facilitated the transition to the Object and Schema levels. The effect size at the Schema level was reported to be over 0.23 and at the Object level over 0.25, which are considered very strong effects in educational studies. Overall, the proposed model explained approximately 45% of the variance in complex APOS cognitive components.

Keywords

  • In-Person Collaborative Learning,
  • Flipped Learning,
  • Artificial Intelligence,
  • Smart Flipped Collective Learning Model,
  • Conceptual Errors,
  • APOS Theory,
  • Linear Equations
PDF

References

  1. Abdolmaleki, J., Khaledian, R., Shirzadi, R., & Abdolmaleki, Z. (2024). A study of the extent of applying collaborative learning strategies in teaching and learning math in elementary schools. The First National Conference on Teaching Skills. [In Persian] https://civilica.com/doc/2192938
  2. Ahmad, M., & Dogar, A. H. (2023). Effect of collaborative learning on conceptual understanding ability in mathematics among 5th grade neglected children. Annals of Human and Social Sciences, 4(2). https://doi.org/10.35484/ahss.2023(4-II)19
  3. Aminifar, E., Karimikia, Kh., & Kamalikhamseh, F. (2014). Identification of conceptual errors of first-grade middle school students in first-degree equations. Iranian Curriculum Studies Quarterly, 9(34), 71-92. [In Persian] https://www.jcsicsa.ir/article_60898.html
  4. Aminifar, E., Saleh Sedghpour, B., & Valinejad Torkaman, F. (2011). The role of technology in math learning. Educational Technology, 6(2), 111-124. [In Persian] https://jte.sru.ac.ir/article_234_bf3fd38476ef3458028d47d455c81347.pdf
  5. Azadi, N., Reyhani, A., Bahrami Samani, E., & Komijani, A. (2024). The effectiveness of using conceptual maps in identifying the misconceptions of pre-service math teachers in graphical literacy. Journal of Mathematics and Society, 8(1), 77-102. [In Persian] https://civilica.com/doc/1947564
  6. Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS Theory: A framework for research and curriculum development in mathematics education. Springer. https://doi.org/10.1007/978-1-4614-7966-6
  7. Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory.
  8. Bergmann, J., & Sams, A. (2012). Flip Your Classroom: Reach Every Student in Every Class Every Day. International Society for Technology in Education.
  9. Chen, X., Xie, H., & Hwang, G.-J. (2020). A multi-perspective study on Artificial Intelligence in Education: grants, conferences, journals, software tools, institutions, and researchers. Computers and Education: Artificial Intelligence, 1, 100005. https://doi.org/10.1016/j.caeai.2020.100005
  10. Delastri, L., & Lolang, E. (2023). Students' conceptual error and procedural error in solving algebraic problems. Multicultural Education, 9(1). https://doi.org/10.5281/zenodo.7508092
  11. Egara, F. O., & Mosimege, M. (2024). Effect of flipped classroom learning approach on mathematics achievement and interest among secondary school students. Education and Information Technologies, 29, 8131–8150. https://doi.org/10.1007/s10639-023-12145-1
  12. Haghjou, S., & Reyhani, A. (2021). Using the APOS-Slope framework to study the understanding of the slope concept. Roshd Mathematics Education Quarterly, 37(1), 42-50. https://www.magiran.com/p2111635
  13. Haghjou, S., & Reyhani, A. (2020). An analysis of the APOS and SOLO theories of math learning. Paper presented at the 5th National Conference on New Approaches in Education and Research, Mazandaran, Mahmoudabad. [In Persian] https://civilica.com/doc/1152685
  14. Holmes, W., Southwell, D. R., & Brown, R. C. (2020). Artificial intelligence in education: Promises and implications for teaching and learning. Routledge.
  15. Jaeger, W. (1943). Paideia: The Ideals of Greek Culture. Oxford University Press.
  16. Johnson, D. W., Johnson, R. T., & Holubec, E. J. (1988). Cooperation in the Classroom. Interaction Book Company.
  17. Jafari Dastjerd, F. (2024). The effect of collaborative learning instruction on the academic performance of female students in math. Paper presented at the International Conference on Recent Research in Science and Technology, Kerman. [In Persian] https://civilica.com/doc/2082961
  18. Karamian, M. M. A., Barkhordari Ahmadi, M., & Mal Khalifeh, M. R. (2025). Design and evaluation of a new teaching method based on factors to reduce math anxiety using the combined Delphi-bwm-topsis technique. International Journal of Mathematical Modelling & Computations, 15(4). https://doi.org/10.57647/ijm2c.2025.150420
  19. Kovari, A. (2025). A systematic review of AI-powered collaborative learning in higher education: Trends and outcomes from the last decade. Social Sciences & Humanities Open, 11, 101335. https://doi.org/10.1016/j.ssaho.2025.101335
  20. Lawan Ahmad, A. (2023). Modified flipped learning as an approach to mitigate the adverse effects of generative artificial intelligence on education. Education Journal, 12(4), 136–143. https://doi.org/10.11648/j.edu.20231204.14
  21. Lo, C. K., & Hew, K. F. (2023). A review of integrating AI-based chatbots into flipped learning: new possibilities and challenges. Frontiers in Education, 8, 1175715. https://doi.org/10.3389/feduc.2023.1175715
  22. Mazur, E. (1997). Peer instruction: Getting students to think in class. In E. F. Redish &
  23. J. S. Rigden (Eds.), AIP Conference Proceedings (pp. 981–988). American Institute of Physics. https://doi.org/10.1063/1.53199
  24. Momeni Rad, A., Khakbaz, A. S., & Hakanloo, M. (2024). A qualitative study of the teaching-learning process of computer application courses using the flipped learning method. Journal of Trends and Achievements in Learning Technology, 1(2), 25-49. [In Persian] https://doi.org/10.22034/jlt.2025.2025303.1007
  25. Naderi, R. (2024). A study of the role of artificial intelligence in flipped learning and its effect on students' academic achievement. The 26th National Conference on Psychology, Educational Sciences, and Social Sciences, Babol. [In Persian] https://civilica.com/doc/2256295
  26. Norouzi, S., Mehrmohammadi, M., Reyhani, A., Fardanesh, H., & Imani Naeini, M. (2017). Correction of some knowledge and belief misconceptions of third-grade elementary students based on Schoenfeld and Gagne's problem-solving frameworks. Research in Curriculum Planning, 14(26), 18-26. [In Persian] https://civilica.com/doc/1014122
  27. Piaget, J., & Inhelder, B. (1969). The psychology of the child. Roozbahani, S. (2024). The effect of metacognitive strategy and flipped learning on students' self-actualization; a case study of Sheikh Mortaza Ansari Elementary School. Journal of Research in Art and Humanities, 9(8), (71). [In Persian] https://civilica.com/doc/2233870
  28. Shahi, R., Eyvazi, M., & Heidari Kolejeh, N. (2021). The effectiveness of the flipped classroom teaching method on the components of academic optimism in math. Education and Evaluation, 15(59), 182-198. [In Persian] https://doi.org/10.30495/JINEV.2023.1971357.2792
  29. Siller, H.-S., & Ahmad, S. (2024). Analyzing the impact of collaborative learning approach on grade six students' mathematics achievement and attitude towards mathematics. EURASIA Journal of Mathematics, Science and Technology Education, 20(2), em2395. https://doi.org/10.29333/ejmste/14153
  30. Tsafe, A. K. (2024). Effective mathematics learning through APOS theory by dint of cognitive abilities. Journal of Mathematics and Science Teacher, 4(1). https://doi.org/10.29333/mathsciteacher/14308
  31. Vygotsky, L. S. (1986). Thought and language (A. Kozulin, Trans.).
  32. Zainuddin, Z., & Halili, S. H. (2016). Flipped Classroom Research and Trends from Different Fields of Study. The International Review of Research in Open and Distributed Learning, 17(3). https://doi.org/10.19173/irrodl.v17i3.2274
  33. Yousefi, E., Azhini, M., & Zamanzadeh Nasrabadi, M. (2025). The effect of flipped instruction on the math performance of ninth-grade students using APOS theory. Journal of Trends and Achievements in Learning Technology. [In Persian]
  34. Zaimbashi, A., Aminifar, E., & Zohrevand, Sh. (2013). Conceptual errors of students in applying variables. [In Persian] https://www.magiran.com/p1710843
  35. Zimmerman, B. J. (2002). Becoming a self-regulated learner: An overview. Theory into Practice, 41(2), 64–70. https://doi.org/10.1207/s15430421tip4102_2