In the geometry lessons that are presented throughout this book, the issues at hand are not just related to shape or aspects of shape (vertices, edges,
faces) and their features (straight, curved). Rather, the metric aspects of geometry become characteristic features of the interaction between participants. They become issues, as we see in Chapter 5, precisely when teachers or peers ask students to render accountable this or that position they are taking. For example, in the preceding chapter, we observe repeated instances where others (Sylvia, Jane, Mrs. Turner, and Lilian) challenge Melissa to make her position on the nature of the mystery object-which she has articulated to be a cube in her model and in her talk-visible, rational, and reportable for the purposes at hand. Much of the three students’ activity consists in producing ways of accounting for the process of transformation between what they feel when they touch the mystery object and the different models they build thereof. Because they are asked to produce only one model-i.e., that all three girls featured in the preceding chapter have to have the same model-the activity structure encourages them to produce rational accounts for the transformations that they are produce.