Journal of Mathematical Behavior 36 (2014) 73–94 Contents lists available at ScienceDirect The Journal of Mathematical Behavior journal h om epa ge: ww w.elsevier.com/locate/jmathb Making sense of qualitative geometry: The case of Amanda Steven Greenstein ∗ Montclair State University, Department of Mathematical Sciences, 1 Normal Avenue, Montclair, NJ 07043, United States a r t i c l e i n f o Keywords: Geometric reasoning Topological reasoning Teaching experiment Student thinking a b s t r a c t This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry envi- ronment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implica- tions for the teaching and learning of geometry and for research into students’ mathematical reasoning. © 2014 Elsevier Inc. All rights reserved. 1. Introduction Geometry in traditional elementary school classrooms has been principally about identifying canonical shapes and match- ing those shapes to their given names (Clements, 2004). These picture-driven geometric experiences have done little to move students beyond the stage in which they identify shapes not by their properties but by their appearance. They have argued, “That’s a triangle, because it looks like a triangle.” Inevitably, little conceptual change in geometry has occurred throughout the elementary grades (Lehrer & Chazan, 1998; Lehrer, Jenkins, & Osana, 1998). No new geometric knowledge is developed beyond what children already know (Thomas, 1982, as cited in Clements, 2004). What makes learning geometry in this kind of environment especially detrimental to young children’s development is that concepts of shape are stabilizing as early as age 6 (Clements, 2004). This means that if young children’s engagement is not expanded beyond a set of conventional, rigid shapes, these shapes develop into a set of visual prototypes that could rule their thinking throughout their lives (Burger & Shaughnessy, 1986; Clements, 2004). For instance, most children require that triangles have horizontal bases, all triangles are acute, and one dimension of a rectangle is twice as long as the other (Clements, 2004). Geometry does, of course, possess a significant visual component. However, as children often experience it, geometric thinking is restricted to passive observation of static images. The problem is, children don’t only see shapes that way. They see them as malleable and often provide “morphing explanations” (Lehrer et al., 1998, p. 142) for shapes they identify as similar. When geometry is about static images on paper, then engagement with, and understanding of, geometry is inevitably constrained to holistic representations of those shapes. Furthermore, arbitrary attributes of shape emerge as fundamental properties when shapes are static. For example, young children distinguish between a square and a regular diamond, because they see rotation as a property of shape. A study by Lehrer et al. (1998) found that “half of the first- and ∗ Tel.: +1 973 655 4287 E-mail address: greensteins@mail.montclair.edu http://dx.doi.org/10.1016/j.jmathb.2014.08.004 0732-3123/© 2014 Elsevier Inc. All rights reserved.