# Using Dynamic Geometry Software to Explore Eigenvectors: The Emergence of Dynamic-Synthetic-Geometric Thinking

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## Abstract

This article analyses students’ thinking as they interacted with a dynamic geometric sketch designed to explore eigenvectors and eigenvalues. We draw on the theory of instrumental genesis and, in particular, attend to the different dragging modalities used by the students throughout their explorations. Given the kinaesthetic and dynamic features of the environment, we also draw on theories of embodied cognition to analyse students’ emergent visual and kinaesthetic understandings. Our analysis suggests that, in contrast with the predominantly analytic-arithmetic mode of thinking (and the consequent procedural knowledge) reported in the literature, our students developed a synthetic-geometric mode of thinking, which researchers have pointed to as being essential to the understanding of linear algebra, but absent for most students. We also found that their synthetic-geometry mode of thinking strongly featured *motion*-*based* conceptions of eigenvectors and eigenvalues, thus leading us to characterise their thinking as *dynamic*-*synthetic*-*geometric*.

## Keywords

Linear algebra Dynamic geometric environments Instrumental genesis Dynamic and kinaesthetic imagery Embodiment Gestures## References

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