Hypothetical Learning Trajectory Using Straws to Teach Subtraction to Elementary School Students - Pinasthi, Titah; Amir, Mohammad Faizal (Department of Mathematics Education University of Muhammadiyah Malang, 2025)
This study developed a context-based Hypothetical Learning Trajectory (HLT) using straws as concrete manipulatives to support elementary students’ understanding of two-digit addition and subtraction. Employing a Design Research approach, the study consisted of an initial design, two cycles of teaching experiments, and a retrospective analysis. The HLT was constructed through sequential activities that guided students from concrete representations to symbolic reasoning related to place value, addition with and without carrying, and subtraction with and without borrowing. The findings indicate that straw-based manipulatives effectively supported students in developing the concept of regrouping, enhancing calculation accuracy, and strengthening conceptual understanding, particularly during carrying and borrowing processes. Across Activities A–D, the actual learning trajectory aligned with the hypothetical pathway, with notable benefits for students with lower initial abilities. Overall, the results confirm that bundling-based concrete media such as straws offer a meaningful and feasible instructional alternative for early-grade mathematics learning.