Designing a hypothetical learning trajectory for Lobachevskian geometry through validated cultural artefacts: a design study with explicit analogy boundaries

Abstrak

Hyperbolic geometry remains one of the least researched domains in geometry education, and undergraduate students consistently transfer Euclidean properties into hyperbolic settings without noticing. This design study developed a hypothetical learning trajectory (HLT) for Lobachevskian geometry that used Indonesian cultural artefacts as conceptual bridges. The preparation and design phase of an educational design research cycle was conducted. Three artefacts were selected: the traditional fish trap bubu, the Minangkabau roof gonjong, and the layered cake lupis. Each artefact was mapped onto a target hyperbolic concept through a six-step procedure specifying the community of origin, the selected geometric feature, the hyperbolic model employed, the assumptions imposed, and the boundary of the analogy. A six-activity trajectory was generated for the parallel axiom, the Saccheri quadrilateral, and the angle sum of a triangle. The mapping made explicit that none of the artefacts constitutes empirical evidence of hyperbolic properties, since all three are surfaces embeddable in Euclidean space. The resulting trajectory is presented as a testable conjecture rather than a validated intervention..