Embodied, symbolic and formal thinking in linear algebra

Abstract

Students often find their first university linear algebra experience very challenging. While coping with procedural aspects of the subject, solving linear systems and manipulating matrices, they may struggle with crucial conceptual ideas underpinning them, making it very difficult to progress in more advanced courses. This research has sought to apply Tall's three worlds of mathematics, embodied, symbolic and formal, to the linear algebra concepts of linear combination and linear independence by groups of second year university students. The results suggest that the students struggled to understand the concepts through definitions and primarily process conceptions, but that embodied, visual ideas proved a valuable adjunct to their thinking.