Forms of Structuring Space by Linear Algebra Students with Video Games and GeoGebra

In this article we analyze how students reason about linear combinations across multiple digital environments. We present the work of three groups of undergraduate students in the Southeast United States (US) who were considered ready to take linear algebra. The students played the game Vector Unknown, reflected upon aspects of their gameplay using GeoGebra, and used that knowledge to design a level for a 3D version of the game under some constraints. We performed a grounded qualitative analysis of each student's activity to identify key episodes of student reasoning about linear combinations using technology. Both authors reviewed the episodes and categorized them according to the type of student activity. We compared their reasoning in 2D and 3D space to understand how they made the transition and finally linked these episodes to the technology used to understand its role in building student understanding. We identify four forms of structuring space: Reasoning with Numeric Sums, Reasoning with Resultant Vectors, Reasoning with Tip-to-Tail Vectors, and Reasoning with Vectors as Points. We found that how the technology represented vectors and linear combinations influenced how students engaged in structuring space.