Linear Algebra Proofs and Tall\u27s Worlds of Mathematics

Proofs are notoriously difficult. While the challenges students face when working on proofs are well-documented, more research is needed on what students do when working on proofs, especially in the context of linear algebra. My research focuses on student work on proofs in linear algebra through the lens of Tall’s worlds of mathematics: the embodied world, the symbolic world, and the formal world. The embodied world consists of graphs, diagrams, and their properties. The symbolic world contains operations, formulas, and calculations. The formal world consists of axioms, formal definitions, and formal proofs. I conducted taskbased interviews with linear algebra students in which they determined if given proofs were valid and then constructed their own proofs for different statements. In different interviews, I encouraged participants to use different worlds of mathematics. Through this study, I hoped to gain some understanding of how approaches to proof constructions and validations within the different worlds of mathematics affect students’ personal proof constructions and validations. I also sought to understand what participants viewed as challenging or helpful about each world of mathematics with regards to proofs. I found that participants often chose a world of mathematics based on the given topic rather than their preferred world or my encouragement to use a given world. Encouraging participants to use the embodied world resulted in their using generic examples. Encouraging participants to use the symbolic or formal worlds had little effect, likely due to participants’ views of the symbolic and formal worlds which differed from my views of the symbolic and formal worlds. Participants said the embodied world was helpful for developing understanding, but felt limited by its specificity. A challenge of the symbolic world was the level of precision needed and the large number of variables. Participants viewed the formal world as helpful for proofs and its generality, but logic was a challenge. Reflecting on the study, participants said that the three-world framework was helpful for organizing their thoughts when working on problems