The Transition from Geometry to Abstraction
Illustration Practices in Linear Algebra Textbooks
A. Abid (TU Delft - Electrical Engineering, Mathematics and Computer Science)
M. Dhume – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
M. Skrodzki – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
R.R. Venkatesha Prasad – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
Students often struggle to abandon geometric intuition when transitioning to formal abstraction in linear algebra. Prior textbook analyses documented a decline in visual support alongside this transition, but the communicative function of the remaining illustrations has not been examined in detail. This qualitative study investigates the role of textbook illustrations across treatments of Euclidean spaces (R2-R3), general Rn, and abstract vector spaces. A thematic analysis was conducted through open coding of four undergraduate textbooks, informed by established typologies. The analysis reveals three shifts in function. First, illustrations systematically shed spatial characteristics to distance readers from geometric intuition. Second, the way illustrations coordinate different mathematical representations shifts from literal geometric mapping in R2-R3 to metaphorical proxies in abstract spaces. Third, illustrations shift from introducing new concepts to reinforcing established theory through repeated examples. These findings emphasize limitations of relying on geometric models for abstraction.