Longitudinal Study of Children's Reasoning About Space and Geometry
Richard Lehrer, Michael Jenkins, Helen Osana University of Wisconsin-Madison
The development of children's ideas about qualities of space is of longstanding interest to researchers and educators alike. For the former, reasoning about space provides a window to issues of mind, such as how children represent images or process configural information ( Anderson, 1983; Eilan, McCarthy, & Brewer, 1993; Kosslyn, 1980). For the latter, because effective instructional design ideally begins with children's prior knowledge ( Bruer, 1994; Freudenthal, 1973), children's reasoning provides the foundation for instruction about the mathematics of space.
Although the teaching and learning of geometry in the United States is traditionally reserved for high-school curricula, recent recommendations of national educational teaching organizations suggest that geometry instruction should begin in the primary grades (e.g., Goldenberg , Cuoco, & Mark, chap. 1, this volume; National Council of Teachers of Mathematics, 1991). The pioneering efforts of Piaget ( Piaget & Inhelder, 1948/ 1956; Piaget, Inhelder, & Szeminska, 1960) and van Hiele ( 1959, 1986) remain the most extensive sources of information about school-age children's initial conceptions about space and corresponding trajectories of change. Much of this work, however, was conducted several decades ago, and more contemporary efforts have not embraced the wide range of mathematical concepts and mental skills characteristic of the earlier work. Consequently, our task was to develop a contemporary and widespread portrait of children's emerging skills in reasoning about space. Our purpose was to describe the development of prototypical forms of spatial reasoning in ways that teachers would find accessible and useful for guiding instruction about geometry in the primary grades so that teachers could use students' knowledge of mathematics as the building blocks of instruction ( Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). In chapter 7,