<![CDATA[Purpose: This study aims to explore and map the learning trajectories of junior high school students in understanding LEOV through mathematical modeling, comparing the Hypothetical Learning Trajectory (HLT) designed by researchers with the Actual Learning Trajectory (ALT) demonstrated by students during classroom implementation.Method: Employing a design research methodology with a validation study approach, the study was conducted in three phases: Preliminary Design, Experimental Design, and Retrospective Analysis. Participants were seventh-grade students from two parallel classes. Data were collected through classroom observations, student worksheets, interviews, and assessments.Findings: Findings reveal that most students’ learning paths aligned with the anticipated HLT, especially in recognizing variables and constructing equations from contextual problems. However, deviations occurred in more complex tasks, where some students relied on intuition rather than structured reasoning. New, unpredicted informal strategies also emerged, offering insights into students’ natural mathematical thinking processes.Significance: Mathematical modeling proves to be an effective instructional approach for facilitating students' transition from informal reasoning to formal algebraic thinking. The mapping of learning trajectories enables educators to identify specific cognitive developments and potential learning obstacles. These findings suggest that design-based interventions rooted in real-life contexts can significantly enhance conceptual comprehension in algebra. Future research should refine modeling tasks to support deeper abstraction and investigate longitudinal impacts across diverse mathematical domains. The integration of learning trajectory analysis into curriculum design offers a powerful framework to personalize instruction and elevate mathematics education practices globally.]]>
