When it's on zero, the lines become parallel: Preservice elementary teachers' diagrammatic encounters with division by zero.

• Pairs of preservice teachers explored division with a transformable diagram. • Participants' diagrammatic encounters with division by zero were analyzed. • We show that dynamic diagrams can provide material contexts for investigating arithmetic concepts. We investigated preservice elementary teachers' diagrammatic encounters with division by zero. Pairs of preservice teachers explored a transformable diagram where the locations of points on the x and y axes could be continuously varied. Quotients were defined in the diagram as the intersection of a line with the y-axis. For zero divisors, the quotient line was parallel to the y-axis, and there was no point of intersection. We report our analysis of two episodes where the transformability of the diagram spurred encounters with division by zero. In each episode, pairs of preservice teachers used repeated movements of the points in the diagram to explore the conditions under which the quotient line would become parallel to the y-axis. Our analysis shows how these movement-based material experiments gave rise to different conceptions of division by zero. We discuss how transformable diagrams create new material contexts for exploring arithmetic concepts. [ABSTRACT FROM AUTHOR]