Symbolizing lines and planes as linear combinations in a dynamic geometry environment

In this paper, we focus on students' symbolizing activity and mathematization relating to linear combination and span in the context of a task sequence designed with digital tools. Considering tools and functions of a digital environment (specifically of GeoGebra) with design heuristics of Realistic Mathematics Education, we introduce four tasks (in R-2 and R-3) and results of pilot studies. Data sources come from six individual task-based interviews with two linear algebra students. We analyzed these with a lens of mathematization, with a particular focus on tool use. We found students' use of sliders supported their reasoning about individual linear combinations; the trace function and slider animation supported their intuition for all possible linear combinations of vectors. After shifting from two to three dimensions, students symbolized all possible linear combinations with a parametric vector equation and reasoned about how this could represent particular points on the corresponding lines or planes.