Core foundations of abstract geometry

Abstract

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 later-developing 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.

Keywords: map reading; mathematical cognition; spatial cognition.

Fig. 1.

Two nonsymbolic geometry tasks. (A) Schematics of the three rectangular enclosures that were used in the navigation task. (B) All 16 displays used in the visual form analysis task, which required children to locate the geometric deviant in a group of shapes. Children performed above chance in 11 of the 16 trials and at chance in the five trials outlined in red (binomial test: ***P < 0.001; **P < 0.01; *P < 0.05). (C) Proportion of correct responses in each condition of the navigation task. Children performed above chance in both the 6:9 and 6:8 conditions. They used the enclosures’ relative wall distances with greater difficulty as their aspect ratio approached 1 (***P < 0.001; **P < 0.01).

Fig. 2.

Maps and schematics of the 3D environments in the two map tasks. (A) Six maps used in both map tasks, which depicted intact triangles at a 0.13:1 scale. Each map was presented at a constant orientation relative to the child, who faced a different direction relative to the array on each trial (0°, 60°, 120°, 180°, 240°, or 300°). (B) Overhead view of the triangular array for the distance map task. Three boards of white foam core (25 cm × 92 cm) were arranged as the sides of a 30-60-90 triangle (102 cm × 176.67 cm × 204 cm). (C) Overhead view of the triangular array for the angle map task. Three corners of white foam core (25 cm high) were arranged as the corners of a 30-60-90 triangle with two 46-cm segments defining each corner. Proportion of correct responses at each target location in the distance map task (D) and the angle map task (E). The gray horizontal line indicates chance-level (0.33) performance (binomial test: ***P < 0.001; **P < 0.01; *P < 0.05). In both tasks, more correct responses occurred at the most geometrically distinct locations, indicating that children were using this geometric information when searching for targets (McNemar test: ***P < 0.001; **P < 0.01; *P < 0.05).

Fig. 3.

Partial regression plots controlling for the effects of age and verbal intelligence and showing that reorientation performance predicted performance on the distance map task (A) and visual form analysis performance predicted performance on the angle map task (B).