Developing Understanding of Geometry and Space in the Primary Grades
Richard Lehrer, Cathy Jacobson, Greg Thoyre, Vera Kemeny, Dolores Strom, Jeffrey Horvath, Stephen Gance, and Matthew Koehler University of Wisconsin-Madison
Our approach to geometry with young children begins with students' in- formal knowledge about situations, followed by progressive mathemati- cal reinterpretation of these experiences, an approach consistent with the Dutch approach to "realistic mathematics education" (see Gravemeijer, chap. 2, this volume). Young children's everyday activities--looking, walking, drawing, building, and manipulating objects--are a rich source of intuitions about spatial structure ( Freudenthal, 1983; Piaget & Inhelder, 1948/ 1956; Streefland, 1991; van Hiele, 1986). By looking at pattern and form in the world, children develop informal knowledge about geometric constructs like perspective, symmetry, and similarity. For example, preschoolers pretend that miniatures are small-scale versions of familiar things, and even infants distinguish contour and symmetry ( Fantz, 1958; Gravemeijer, chap. 2, this volume; Haith, 1980). By walking in their neigh- borhoods, children learn to reason about landmarks, routes, and other el- ements of large-scale space ( Piaget, Inhelder, & Szeminska, 1960; Siegel & White, 1975). By drawing what they see, children represent form ( Good- now , 1977). By building structures with blocks, toothpicks, or Tinkertoys, children experience first-hand how shape and form play roles in function (e.g., objects that roll vs. those that do not) and structure (e.g., sturdiness; see Middleton & Corbett, chap. 10, this volume).
Everyday experiences like these, and the informal knowledge chil- dren develop over time by participating in them, constitute a springboard into geometry. For example, the ideas that children develop about position and direction while walking in their neighborhood can be elaborated mathematically in a variety of ways--as coordinate systems, as compass