Different pathways shared conceptual goal: A readiness-based hypothetical learning trajectory for sphere volume

Abstrak

Learning the volume of a sphere requires students to coordinate concrete experiences, representations, quantitative relationships, and symbolic generalization, making it a context for examining understanding. However, differentiated learning trajectories have rarely been designed to account for students’ readiness while tracing how different pathways support reflective abstraction; thus, how readiness-responsive differentiation can be embedded within a Hypothetical Learning Trajectory (HLT) remains insufficiently understood. This study aimed to design a readiness-based differentiated HLT and investigate how it facilitates students’ reflective abstraction in learning sphere volume. Employing design research comprising preliminary design, teaching experiment, and retrospective analysis, the study involved 25 ninth-grade students, and data were collected through interviews, observations, document analysis, and diagnostic assessments. Findings revealed readiness-responsive pathways within the same HLT. Low-readiness students relied on concrete experiences and empirical reasoning, exhibiting recognition-based thinking; medium-readiness students identified patterns and organized quantitative relationships, progressing from representation toward structural abstraction; and high-readiness students generalized and justified mathematical relationships, demonstrating structural awareness. The differentiated HLT accommodated these differences through representations and scaffolding ranging from manipulatives and visual models to relational reasoning and symbolic generalization. Despite divergent pathways, all students converged toward conceptual understanding of the sphere volume formula. These findings indicate that readiness-based differentiation concerns not divergent learning goals but differentiated representational, reasoning, and scaffolding pathways, offering a design principle for embedding branching and convergent HLTs that preserve a conceptual goal while supporting reflective abstraction.