Young children designing and defining space: Supporting the co-development of mathematical practices and foundational understandings of space and symmetry

Abstract

Visualizing and reasoning about space are foundational to the epistemic enterprise of mathematics. However, current K-12 math curriculum largely neglects cultivating children’s mathematical understandings of and visualizing of space. The three papers of this dissertation explore how to begin cultivating young children’s mathematical considerations of space at the onset of schooling, beginning with the defining of concepts of shape and symmetry. Each paper addresses how to design learning ecologies where children participate in generating mathematical conceptual systems that open new considerations of and relations between 3-D structures and their symmetries. The first paper is a conceptual paper that demonstrates how symmetries and their physical instantiations as informal and enacted motions have served as epistemic tools across historical and individual scales of development. Drawing together these two scales of development, I conclude with principals to guide the design of learning ecologies where children are afforded opportunities to engage with symmetries as tools to author mathematical ideas. The second and third papers present the first two iterations of a design study conducted in first-grade classrooms. Finding from this empirical work demonstrate that 3-D isometry and symmetry transformations are firmly within the grasp of young children. However, beyond their accessibility, introducing these forms of transformation into children’s existing activities of constructing and analyzing 3-D structures supported children to develop new design and mathematical defining practices and conceptual understandings of mathematical objects (e.g., polyhedra). Together these papers begin to articulate a child’s geometry that invites children to use their experience with and visualization of space as resources to generate powerful mathematical ideas.