Supporting students’ understanding of quadratic functions as patterns of change through representational transitions

This study develops and analyzes a learning design supporting students’ understanding of quadratic functions as patterns of change through representational transitions, using design research with qualitative retrospective analysis at two public senior high schools in Riau Province, Indonesia, within the Kurikulum Merdeka curriculum. Preliminary data came from 80 tenth-grade students through observations, an initial representation test, and interviews with 12 selected students. A separate teaching experiment with 80 tenth-grade students was implemented across two cycles, integrating a walking activity, the PhET Moving Man simulation, and GeoGebra, with worksheets classified using a three-level representational coding scheme. Students’ initial understanding was predominantly symbolic and procedural, with combined achievement across three representation indicators of only 34.53%. The learning sequence supported movement from contextual situations to numerical patterns, symbolic generalization, and graphical interpretation, though recognizing second differences and generalizing them symbolically remained the central bottleneck. Cycle revisions, notably a structured walking procedure and an added generalization row, were associated with more groups reaching representational coordination (4 of 10 to 7 of 10) and higher quiz scores (63.13% to 84.13%), though this is not a controlled comparison, since cycles occurred at different schools. This study offers a literature-grounded design moving students from procedural recognition toward conceptual understanding of nonlinear change, and identifies second differences as a representational bridge between numerical and symbolic forms. Three design principles are proposed: beginning with contrastive situations, using second differences as a bridge, and formalizing the general form after students recognize the structure of change.