Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support laterdeveloping Euclidean intuitions? To approach these questions, we investigated the relations among young children's use of geometry in tasks assessing: navigation; visual form analysis; and the interpretation of symbolic, purely geometric maps. Children's navigation depended on the distance and directional relations of the surface layout and predicted their use of a symbolic map with targets designated by surface distances. In contrast, children's analysis of visual forms depended on the size-invariant shape relations of objects and predicted their use of the same map but with targets designated by corner angles. Even though the two map tasks used identical instructions and map displays, children's performance on these tasks showed no evidence of integrated representations of distance and angle. Instead, young children flexibly recruited geometric representations of either navigable layouts or objects to interpret the same spatial symbols. These findings reveal a link between the early-arising geometric representations that humans share with diverse animals and the flexible geometric intuitions that give rise to human knowledge at its highest reaches. Although young children do not appear to integrate core geometric representations, children's use of the abstract geometry in spatial symbols such as maps may provide the earliest clues to the later construction of Euclidean geometry.spatial cognition | mathematical cognition | map reading A bstract concepts of formal geometry underlie a wide range of human achievements, but their source has been debated for millennia (1). Human abilities to navigate the environment and to recognize objects develop early and are shared across diverse animal species. In recent years, intensive study at levels from neurons to cognition (2-5) has illuminated the geometric information guiding these abilities in animals from insects to vertebrates (6-8) and in humans from infants to adults (9-14). When navigating, humans and animals represent their position by encoding the distances and directions of extended surfaces in the terrain rather than the angles at which surfaces meet (15, 16). In contrast, humans and animals represent objects by encoding the angles and relative lengths defining 3D part structures or 2D shapes rather than their absolute sizes or the directional relations that distinguish a form from its mirror image (17, 18). Despite the pervasiveness and power of these core geometric representations, neither in isolation is adequate to support abstract geometric intuitions, which require an integrated representation of distance and angle (13,19,20). Still, these two sets of core representations together may ...

Research in developmental cognitive science reveals that human infants perceive shape changes in 2D visual forms that are repeatedly presented over long durations. Nevertheless, infants’ sensitivity to shape under the brief conditions of natural viewing has been little studied. Three experiments tested for this sensitivity by presenting 128 seven‐month‐old infants with shapes for the briefer durations under which they might see them in dynamic scenes. The experiments probed infants’ sensitivity to two fundamental geometric properties of scale‐ and orientation‐invariant shape: relative length and angle. Infants detected shape changes in closed figures, which presented changes in both geometric properties. Infants also detected shape changes in open figures differing in angle when figures were presented at limited orientations. In contrast, when open figures were presented at unlimited orientations, infants detected changes in relative length but not in angle. The present research therefore suggests that, as infants look around at the cluttered and changing visual world, relative length is the primary geometric property by which they perceive scale‐ and orientation‐invariant shape.

Many poor children are underprepared for demanding primary school curricula. Research in cognitive science suggests that school achievement could be improved by preschool pedagogy in which numerate adults engage children's spontaneous, nonsymbolic mathematical concepts. To test this suggestion, we designed and evaluated a game-based preschool curriculum intended to exercise children's emerging skills in number and geometry. In a randomized field experiment with 1540 children (average age 4.9 years) in 214 Indian preschools, 4 months of math game play yielded marked and enduring improvement on the exercised intuitive abilities, relative to no-treatment and active control conditions. Math-trained children also showed immediate gains on symbolic mathematical skills but displayed no advantage in subsequent learning of the language and concepts of school mathematics.