10 Geometric and Spatial Thinking in Early Childhood Education Douglas H. Clements University at Buffalo, State University of New York Geometry and spatial reasoning are inherently important because they involve" grasping... that space in which the child lives, breathes and ...

Although geometry and spatial reasoning are important as a way to interpret and reflect on the physical environment and also form the foundation for learning mathematics and other subjects, many early childhood and primary school teachers spend little time instructing their students in these areas. This paper examines how young children learn about space and geometry, discusses how they think about specific concepts in this area, and presents activities and teaching approaches that early childhood educators can use to help them develop. Section 1 of the paper examines how children learn about space and geometry and begins with an examination of Piaget's belief that children have constructed "perceptual space" by infancy but develop ideas about space through action; this is followed by a discussion of children's exploration of shapes by touch, drawing of shapes, and the development of perspective taking. This section also describes levels of geometric thinking-from a holistic, unanalyzed visual beginning through description to an analysis of geometric figures-and discusses the important role of education in this development. Section 2 discusses how children of different ages think about salient mathematical concepts: shape, spatial orientation, and spatial visualization and imagery. Section 3 presents suggestions for instructing young children, including use of manipulatives and pictures, computer manipulatives, and the Agam program to develop the visual language of young children. The paper concludes by noting that it is essential that geometry and spatial sense receive greater attention in instruction and research. (Contains approximately 60 references and 11 figures). (KB) Reproductions supplied by EDRS are the best that can be made from the original document.

For over a century, views of young children's mathematics have differed widely. The recent turn of the century has seen a dramatic increase in attention to the mathematics education of young children. We begin with a brief consideration of the history of mathematics in early childhood and then turn to the question of what children know before entering school. The remainder of the chapter examines what children can and should learn about the five major mathematical topics: Number and arithmetic, geometry, measurement, patterning and algebraic thinking, and data and graphing.

We discuss research on both physical manipulatives and virtual manipulatives to provide a framework for understanding, creating, implementing, and evaluating efficacious manipulatives—physical, virtual, and a combination of these two. We provide a theoretical framework and a discussion of empirical evidence supporting that framework, for the use of manipulatives in learning and teaching mathematics, from early childhood through the elementary years. From this reformulation, we reconsider the role virtual manipulatives may play in helping students learn mathematics. We conclude that manipulatives are meaningful for learning only with respect to learners' activities and thinking and that both physical and virtual manipulatives can be useful. When used in comprehensive , well planned, instructional settings, both physical and virtual manipulatives can encourage students to make their knowledge explicit, which helps them build Integrated-Concrete knowledge.

We define and describe how subitizing activity develops and relates to early quantifiers in mathematics. Subitizing is the direct perceptual apprehension and identification of the numerosity of a small group of items. Although subitizing is too often a neglected quantifier in educational practice, it has been extensively studied as a critical cognitive process. We believe that subitizing also helps explain early cognitive processes that relate to early number development and thus deserves more instructional attention. We also contend that integrating developmental/cognitive psychology and mathematics education research affords opportunities to develop learning trajectories for subitizing. A complete learning trajectory includes three components: goal, developmental progression, or learning path through which children move through levels of thinking, and instruction. Such a learning trajectory thus helps establish goals for educational purposes and frames instructional tasks and/or teaching practices. Through this chapter, it is our hope that early childhood educators and researchers begin to understand how to develop critical educational tools for early childhood mathematics instruction. Through this instruction, we believe that children will be able to use subitizing to discover critical properties of number and build on subitizing to develop capabilities such as unitizing, cardinality, and arithmetic capabilities.