Abstract 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.

中文翻译：

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当它为零时，线变得平行：职前小学教师与除以零的图解相遇

摘要 我们调查了职前小学教师与除以零的图解遭遇。一对职前教师探索了一个可转换的图表，其中 x 和 y 轴上的点的位置可以连续变化。商在图中定义为一条线与 y 轴的交点。对于零除数，商线平行于 y 轴，没有交点。我们报告了我们对两个情节的分析，其中图表的可转换性刺激了与除以零的相遇。在每一集中，成对的职前教师使用图中点的重复移动来探索商线平行于 y 轴的条件。我们的分析显示了这些基于运动的材料实验是如何产生不同的除以零概念的。我们讨论可转换图如何为探索算术概念创建新的材料上下文。