Modes of Description and the Problem of Representation in Linear Algebra

By J. Hillel. 2000

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Examining individual and collective level mathematical progress (2014, background): “Other studies analyzed student difficulties with the notions of basis (Hillel, 2000), linear transformation (Dreyfus, Hillel, & Sierpinska, 1999), rank (Dorier, Robert, Robinet, & Rogalski, 2000), linear independence (Bogomolny, 2007; Harel, 1997; Trigueros & Possani, 2013), and span (Stewart & Thom”
Eigenvalues and eigenvectors: embodied, symbolic and formal thinking (2011, background): “It has been recognised by Hillel (2000) and Sierpinska (2000) that conceptual difficulties in linear algebra are often linked to its three kinds of description or representation: the general theory; the specific theory of R n ; and the geometry of”
A hypothetical learning trajectory for conceptualizing matrices as linear transformations (2017, background): “the symbolic notation of linear algebra [24,25]; working with formal definitions in linear algebra [15]; and coordinating across the different modes or representations involved in linear algebra [17,18].”
Subspace in linear algebra: investigating students’ concept images and interactions with the formal definition (2011, background): “Such a view is algebraic in the sense of Hillel’s “algebraic language of linear algebra” (Hillel, 2000).”