“I'm trying to remember the distance formula, but I can’t” secondary students’ conceptual understanding of the distance between two points using visual representations

In the United States, secondary mathematics curricula often prioritize procedural skills over conceptual understanding, which has been shown to hinder long-term retention and the development of foundational knowledge for future mathematical thinking. This study explores how secondary students conceptualize and recall the distance between two points when instruction includes visual representations. Through qualitative methods, including task-based interviews with 15 students from Algebra I, Geometry, and Algebra II, this research examines how students reason through distance tasks. The most common reasoning strategies used by students included "Count Diagonally," "Slope," "Count 1-Dimension," "Rotate," and "Count Legs." After participating in the task-based interview using the instructional activities of an initial Hypothetical Learning Trajectory (HLT), participants demonstrated a narrowing of reasoning strategies when tackling distance tasks, leading to improved success rates in identifying the correct solutions. A refined HLT is presented with modifications based on the findings. These findings also contribute to the literature on students’ development of anticipatory schemes for finding the distance between two points and provide valuable insights for educators and curriculum designers.