Developing a Qualitative Geometry from the Conceptions of Young Children

More than half a century ago, Piaget concluded from an investigation of children's representational thinking about the nature of space that the development of children's representational thought is topological before it is Euclidean. This conclusion, commonly referred to as the "topological primacy thesis," has essentially been rejected. By giving emphasis to the ideas that develop rather than the order in which they develop, this work set out to develop a new form of non-metric geometry from young children's early and intuitive topological, or at least non-metric, ideas. I conducted an eighteen-week teaching experiment with two children, ages six and seven. I developed a new dynamic geometry environment called Configure that I used in tandem with clinical interviews in each of the episodes of the experiment to elicit these children's non-metric conceptions and subsequently support their development. I found that these children developed significant and authentic forms of geometric reasoning. It is these findings, which I refer to as qualitative geometry, that have implications for the teaching of geometry and for research into students' mathematical reasoning.