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

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.