Designing Learning Trajectories for Nets of Polyhedrons through Simulation-Based PBL

Abstract

Students frequently experience difficulties in learning nets of polyhedrons because they must coordinate two-dimensional representations with three-dimensional objects and mentally anticipate folding processes. This study aimed to develop and refine an empirically grounded learning trajectory for nets of polyhedrons by integrating Problem-Based Learning (PBL), contextual simulation, animated videos, and student worksheets. A design research approach was employed through three phases: preliminary design, teaching experiment, and retrospective analysis. The pilot teaching experiment involved six ninth-grade students representing high, moderate, and low levels of mathematical achievement, while the large-scale implementation involved 28 ninth-grade students in a regular classroom. Data were collected through classroom observations, semi-structured interviews, students’ written work, worksheets, learning outcome tasks, and instructional documentation, and were analyzed by comparing predicted responses in the Hypothetical Learning Trajectory with students’ actual learning processes. The findings showed a progressive development from recognizing familiar packaging patterns to visualizing folding processes, distinguishing valid and invalid nets, analyzing relationships among faces and edges, constructing alternative nets, and communicating mathematical justifications. Animated visualization and contextual simulation helped students connect informal experiences with formal geometric concepts, while collaborative investigation and reflection supported the transition from perceptual judgments to structural reasoning. Retrospective analysis produced a refined eight-stage learning trajectory: gift-shop simulation, activation of prior knowledge, animated visualization, identification of valid and invalid nets, collaborative investigation, alternative net construction, presentation and mathematical justification, and reflection and generalization. The trajectory offers a practical and theoretically informed design for supporting conceptual understanding and spatial visualization, although further scaffolding is needed to strengthen students’ written mathematical communication.