RUME 28
2026
Alexandria, Virginia
Shore to Shore: Students’ Progressive Mathematization of Null Spaces
Page: 286
This study explores how undergraduate students develop reasoning about the relationship between a matrix and its null space through progressive mathematization within an inquiry-oriented linear algebra context. Using a playful hallway map task, students engaged in a paired teaching experiment to investigate how modifying the number of rooms and hallways affects the dimension of the null space. We drew on a river journey metaphor to examine students’ movement along and between shores of contextual or formal symbolic reasoning. Findings reveal a dynamic interplay between contextual reasoning and formal symbolic representations, as students moved fluidly between the realistic hallway context and matrix-based interpretations. Through successive horizontal and vertical mathematization, students gradually developed reasoning that approached a reinvention of the rank-nullity theorem. The results highlight the complexity of this journey and suggest further research into providing supports for instruction.