<![CDATA[This study addresses students’ persistent difficulties in understanding fractions, particularly their tendency to rely on procedural calculations rather than conceptual reasoning and mathematical representation. The research aimed to explore a Hypothetical Learning Trajectory (HLT) based on Realistic Mathematics Education (RME) for fifth-grade students on fraction topics, including equivalent fractions and operations with different denominators. Using a design research approach, the study was conducted through three phases: preliminary design, teaching experiment, and retrospective analysis, involving six heterogeneous students. Data were collected through observations, interviews, documentation, and student work analysis, and were analyzed by comparing the Hypothetical Learning Trajectory (HLT) with the Actual Learning Trajectory (ALT).The findings reveal that the HLT effectively facilitated students’ progression from informal to formal understanding through contextual activities, visual representations, and collaborative learning. The novelty of this study lies in its in-depth analysis of the alignment between HLT and ALT in fraction learning, which remains underexplored in prior RME-based studies. The results also indicate that deviations between predicted and actual learning trajectories provide valuable insights for refining instructional design.These findings imply that adaptive and well-structured learning trajectories are essential for supporting meaningful conceptual understanding and enhancing students’ mathematical representation skills in fraction learning.]]>
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