In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 2). Palmerston North, NZ: MERGA. © MERGA Inc. 2009 Linear Algebra Snapshots through APOS and Embodied, Symbolic and Formal Worlds of Mathematical Thinking Sepideh Stewart Mike Thomas The University of Auckland The University of Auckland <stewart@math.auckland.ac.nz> <m.thomas@math.auckland.ac.nz> Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here is as part of the first named author's recent PhD thesis where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the learning and teaching of some linear algebra concepts. This paper highlights some of the overall findings of this research and suggests applications for learning and teaching in undergraduate mathematics classrooms. Introduction In recent years many mathematics education researchers have been concerned with students’ difficulties related to the undergraduate linear algebra courses. There is agreement that teaching and learning this course is a frustrating experience for both teachers and students, and despite all the efforts to improve the curriculum the learning of linear algebra remains challenging for many students (Dorier & Sierpinska, 2001). Students may cope with the procedural aspects of the course, solving linear systems and manipulating matrices, but struggle to understand the crucial conceptual ideas underpinning them. The concepts may be presented through a definition in natural language, which may be linked to a symbolic presentation. These definitions are considered to be fundamental as a starting point for concept formation and deductive reasoning in advanced mathematics (Vinner, 1991; Zaslavsky & Shir, 2005). Interestingly enough, at the end of the course many students do reasonably well in their final examinations, since the questions are mainly set on using techniques and following certain procedures, rather than understanding the concepts (Dorier, 1990). This, as Sierpinska, et al. (2002, p. 2) describe is a “waste of students intellectual possibilities”. They believe “linear algebra, with its axiomatic definitions of vector space and linear transformation, is a highly theoretical knowledge, and its learning cannot be reduced to practicing and mastering a set of computational procedures” (ibid, p. 1). The action-process-object-schema (APOS) development in learning proposed by Dubinsky and others (Dubinsky & McDonald, 2001) suggests an approach different from the definition-theorem-proof that often characterises university courses. Instead mathematical concepts are described in terms of a genetic decomposition into their constituent actions, process and objects in the order these should be experienced by the learner. In more recent years Tall has introduced the idea of three worlds of mathematics, the embodied, symbolic and formal (Tall, 2004). The worlds describe a hierarchy of qualitatively different ways of thinking that individuals develop as new conceptions are compressed into more thinkable concepts (Tall & Mejia-Ramos, 2006). The embodied world, containing embodied objects (Gray & Tall, 2001), is where we think about the things around us in the physical world, and it “includes not only our mental perceptions of real-world objects, but also our internal conceptions that involve visuo-spatial imagery.” (Tall, 2004, p. 30). The symbolic world is the world of procepts, where actions, processes and their corresponding objects are realized and symbolized. The formal world of thinking