Navigation as a source of geometric knowledge: young children's use of length, angle, distance, and direction in a reorientation task

Abstract

Geometry is one of the highest achievements of our species, but its foundations are obscure. Consistent with longstanding suggestions that geometrical knowledge is rooted in processes guiding navigation, the present study examines potential sources of geometrical knowledge in the navigation processes by which young children establish their sense of orientation. Past research reveals that children reorient both by the shape of the surface layout and the shapes of distinctive landmarks, but it fails to clarify what shape properties children use. The present study explores 2-year-old children's sensitivity to angle, length, distance and direction by testing disoriented children's search in a variety of fragmented rhombic and rectangular environments. Children reoriented themselves in accord with surface distances and directions, but they failed to use surface lengths or corner angles either for directional reorientation or as local landmarks. Thus, navigating children navigate by some but not all of the abstract properties captured by formal Euclidean geometry. While navigation systems may contribute to children's developing geometric understanding, they likely are not the sole source of abstract geometric intuitions.

Figure 1

Illustrations of possible ways which distance, angle, and direction can be used for reorientation in a rhombic environment in either egocentric (left) or allocentric (right) coordinates. Given four locations (A, B, C, D) that are at the center of each wall and therefore equidistant from the center of the room (array tested in Experiment 2), the left diagram shows distance differences from the center position from which location A (and its geometric twin C), for instance, can be encoded using distance and direction (i.e., “left of the farthest point of the surrounding layout (d2)" or "right of the closest point of the surrounding layout (d1)"). The right diagram shows distance differences between the surfaces themselves from which A (or C) can be encoded (i.e., “left of the narrower space between surfaces (d1)" or "right of the wider space between surfaces (d2)”) Note that distances could be measured either from the centers of walls or as the average of the distances of each point on a wall. Note that in principle, measurements of angle sizes could substitute for measurement of distances, in either reference frame (e.g., “left of the narrower angle," or "right of the wider angle”).

Figure 2

Schematic diagrams of the geometric arrays in Experiments 1 (left) and 2 (right). Proportions of searches are given for the four hiding locations (represented by the letters C, R, A, and B): the correct target location (C), with the rotationally opposite geometric twin across from it (R), and the other incorrect hiding places clockwise (A) and counter-clockwise (B) from the correct location. Data are rotated so that all locations for a given condition are aligned.

Figure 3

Schematic diagrams of the geometric arrays in Experiments 3 (left) and 4 (right). Proportions of searches are given for the four hiding locations (represented by the letters C, R, A, and B): the correct target location (C), with the rotationally opposite geometric twin across from it (R), and the other incorrect hiding places clockwise (A) and counter-clockwise (B) from the correct location. Data are rotated so that all locations for a given condition are aligned.

Figure 4

Schematic diagrams of the geometric arrays in Experiment 5 (left) and Experiment 6 (right), in which the hiding places were located between the sides/corners (and required directional reorientation) (top) or directly at the sides/corners (bottom). Proportions of searches are given for the four hiding locations (represented by letters C, R, A, B): the correct target location (C), the rotationally opposite location (R), and the geometrically incorrect hiding places clockwise (A) and counter-clockwise (B) from the correct location. Data are rotated so that all locations for a given condition are aligned.

Figure 5

Schematic diagrams of the geometric arrays in Experiment 7 (left) and Experiment 8 (right), in which the hiding places were located between the side surfaces (and required directional reorientation) (top) or directly at the surfaces (bottom). Proportions of searches are given for the four hiding locations (represented by letters C, R, A, B): the correct target location (C), the rotationally opposite location (R), and the geometrically incorrect hiding places clockwise (A) and counter-clockwise (B) from the correct location. Data are rotated so that all locations for a given condition are aligned.