<![CDATA[This study investigates students’ learning trajectory in understanding angle measurement within a Pendidikan Matematika Realistik Indonesia (PMRI)-based instructional design through a design research approach. It aims to examine how students progressively construct conceptual understanding of angles, moving from informal contextual experiences toward formal measurement using a protractor, with particular attention to transitions across multiple representations. Data were collected through a sequence of teaching experiments involving contextual tasks based on the traditional Limas house, non-standard circular representations, and formal protractor-based measurement. Students’ written work, classroom observations, and responses were analyzed to compare the actual learning trajectory with the Hypothetical Learning Trajectory (HLT). The findings reveal that students initially interpreted angles through visual perception, emphasizing shape-related features rather than rotational meaning. Although circular representation activities promoted an emerging understanding of angles as parts of a complete rotation, this conceptual development remained fragmented. A critical discontinuity occurred during protractor use, where persistent difficulties in alignment, scale selection, and interpreting angle magnitude hindered conceptual progression. To address these challenges, a bridging phase integrating circular representations, rotational movement, and degree measurement was introduced. This intervention enabled students to reconceptualize angles as measures of rotation, resulting in greater measurement accuracy and conceptual consistency, particularly for benchmark angles of 45°, 90°, and 180°. The study concludes that learning angle measurement is a non-linear process requiring explicit representational mediation, with the bridging phase serving as a crucial instructional mechanism for facilitating students’ transition from informal experiences to formal mathematical reasoning.]]>
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