Whitney, Kartal & Zawojewski ICME-12, 2012 abcde+2 ON A SPHERE “IT’S NOT REALLY PARALLEL, IT’S JUST KIND OF PARALLEL” Stephanie R. Whitney Illinois Institute of Technology Swhitne1@iit.edu Ozgul Kartal Illinois Institute of Technology okartal@iit.edu Judith S. Zawojewski Illinois Institute of Technology zawojewski@iit.edu Students in a post-secondary geometry course were engaged in creating an axiomatic system for spherical geometry over the course of a semester. While using Lenárt Spheres they engaged in mathematical activities which included expressing, testing and refining definitions. In a spontaneous discussion, the students began to create a definition for their shared axiomatic system for “parallel” on a sphere. Their conversations are partitioned into segments, or sites, in which there is evidence of students using concept images and definitions to articulate their mathematical activity. Each of these sites are characterized developmentally using the Defining as a Mathematical Activity Frameworks (Zandieh & Rasmussen, 2010). Major mathematical themes in the students’ understanding of parallel on a sphere are also categorized and discussed. Key words: concept image, concept definition, parallel transports, axiomatic system INTRODUCTION This study was situated in a class session in which post-secondary students were transporting their concept images and definitions about parallel lines from plane geometry to the surface of a sphere—a prime site for creating new definitions. In particular, parallel lines in a plane are straight, equidistant at all perpendicular distances, do not intersect, and when cut by a transversal result in corresponding angles are of equivalent size. However, on the surface of a sphere students encountered a “new geometry” where a straight line is a great circle, all distinct great circles intersect, and the notions of equidistance, transversals, and corresponding angles all need to be reconceptualised. In this rich context for definition creation and development, the purpose of this study was to identify the