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6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement Geometry, spatial thinking, and measurement make up the second area of mathematics we emphasize for young children. In this chapter we pro- vide an overview of childrenâs development in these domains, lay out the teaching-learning paths for children ages 2 through kindergarten in each broad area, and discuss instruction to support their progress through these teaching-learning paths. As in Chapter 5, the discussion of instruction is closely tied to the specific mathematical concepts covered in the chapter. Chapter 7 provides a more general overview of effective instruction. GEOMETRY AND SPATIAL THINKING The Dutch mathematician Hans Freudenthal stated that geometry and spatial thinking are important because âGeometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes, and moves. The space that the child must learn to know, explore, and conquer, in order to live, breath and move better in it. Are we so accustomed to this space that we cannot imagine how important it is for us and for those we are educating?â (Freudenthal, 1973, p. 403). This section describes the two major ways children understand that space, starting with smaller scale perspectives on geometric shape, including com- position and transformation of shapes, and then turning to larger spaces in which they live. Although the research on these topics is far less developed than in number, it does provide guidelines for developing young childrenâs learning of both geometric and spatial abilities. 175

176 MATHEMATICS LEARNING IN EARLY CHILDHOOD Shape Shape is a fundamental idea in mathematics and in development. Be- yond mathematics, shape is the basic way children learn names of objects, and attending to the objectsâ shapes facilitates that learning (Jones and Smith, 2002). Steps in Thinking About Shape Children tend to move through different levels in thinking as they learn about geometric shapes (Clements and Battista, 1992; van Hiele, 1986). They have an innate, implicit ability to recognize and match shapes. But at the earliest, prerecognition level, they are not explicitly able to reliably distinguish circles, triangles, and squares from other shapes. Children at this level are just starting to form unconscious visual schemes for the shapes, drawing on some basic competencies. An example is pattern matching through some type of feature analysis (Anderson, 2000; Gibson et al., 1962) that is conducted after the visual image of the shape is analyzed by the visual system (Palmer, 1989). At the next level, children think visually or holistically about shapes (i.e., syncretic thought, a fusion of differing systems; see Clements, ÂBattista, and Sarama, 2001; Clements and Sarama, 2007b) and have formed schemes, or mental patterns, for shape categories. When first built, such schemes are holistic, unanalyzed, and visual. At this visual/holistic step, children can recognize shapes as wholes but may have difficulty forming separate men- tal images that are not supported by perceptual input. A given figure is a rectangle, for example, because âit looks like a door.â They do not think about shapes in terms of their attributes, or properties. Children at this level of geometric thinking can construct shapes from parts, but they have difficulty integrating those parts into a coherent whole. Next, children learn to describe, then analyze, geometric figures. The culmination of learning at this descriptive/analytic level is the ability to rec- ognize and characterize shapes by their properties. Initially, they learn about the parts of shapesâfor example, the boundaries of two-dimensional (2-D) and three-dimensional (3-D) shapesâand how to combine them to create geometric shapes (initially imprecisely). For example, they may explicitly understand that a closed shape with three straight sides is a triangle. In the teaching-learning path articulated in Table 6-1, this is called the âthinking about partsâ level. Children then increasingly see relationships between parts of shapes, which are properties of the shapes. For instance, a student might think of a parallelogram as a figure that has two pairs of parallel sides and two pairs of equal angles (angle measure is itself a relation between two sides, and

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 177 TABLE 6-1â Space and Shapes in Two Dimensions Goals C. Perceive, Say, A. Perceive, Say, Describe/Discuss, Describe/Discuss, and Construct Steps/Ages and Construct B. Perceive, Say, Describe/ Compositions and (Level of Objects in 2-D Discuss, and Construct Spatial Decompositions in Thinking) Space Relations in 2-D Space 2-D Space Step 1 (Ages 2 and 3) Thinking Recognition and Recognize shapes in many Solve simple puzzles visually/ informal description different orientations and sizes. involving things in holistically (including at least Trial-and-error geometric the world. circles, squares, movements (informal, not Create pictures by then triangles, quantified). representing single rectangles). â¢ se relational language, U objects, each with a including vertical different shape. directionality terms as âupâ and âdown,â referring to a 2-D environment. â¢ nformally recognizes area as I filling 2-D space (e.g., âI need more papers to cover this tableâ). Thinking Shapes by number about of sides (starting parts with restricted cases, e.g., prototypical equilateral triangle, square). Step 2 (Age 4) Thinking Recognition and Recognize shapes (to the left) visually/ informal description in many different orientations, holistically at multiple sizes, and shapes (e.g., âlongâ orientations, and âskinnyâ rectangles and sizes, and shapes triangles). (includes circles â¢ atch shapes by using M and half/quarter geometric motions to circles, squares superimpose them. and rectangles, â¢ se relational words U triangles, and of proximity, such as others [the pattern âbeside,â ânext to,â and block rhombus, âbetween,â referring to a 2-D trapezoids, hexagons environment. regular]). continued

178 MATHEMATICS LEARNING IN EARLY CHILDHOOD TABLE 6-1â Continued Goals C. Perceive, Say, A. Perceive, Say, Describe/Discuss, Describe/Discuss, and Construct Steps/Ages and Construct B. Perceive, Say, Describe/ Compositions and (Level of Objects in 2-D Discuss, and Construct Spatial Decompositions in Thinking) Space Relations in 2-D Space 2-D Space Thinking Describe and name Move shapes using slides, flips, Move shapes using about shapes by number and turns. slides, flips, and turns parts of sides (up to the â¢ se relational language U to combine shapes to number they can involving frames of reference, build pictures. count). such as âto this side of,â For rectangular Describe and name âabove.â spaces shapes by number of â¢ ompare areas by C â¢ opy a design C corners (vertices). superimposition. shown on a grid, For rectangular spaces placing squares â¢ ile a rectangular space with T onto squared-grid physical tiles (squares, right paper. triangles, and rectangles with unit lengths) and guidance. Relating Sides of same/ Predict effects of rigid geometric Combine shapes parts and different length. motions. with intentionality, wholes â¢ ight vs. nonright R recognizing them as angles. new shapes. â¢ n an âequilateral I triangle world,â create pattern block blue rhombus, trapezoid, and hexagons from triangles. Step 3 (Age 5) Thinking Recognition and visually/ informal description, holistically varying orientation, sizes, shapes (includes all above, as well as octagons, parallelograms, convex/concave figures).

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 179 TABLE 6-1â Continued Goals C. Perceive, Say, A. Perceive, Say, Describe/Discuss, Describe/Discuss, and Construct Steps/Ages and Construct B. Perceive, Say, Describe/ Compositions and (Level of Objects in 2-D Discuss, and Construct Spatial Decompositions in Thinking) Space Relations in 2-D Space 2-D Space Thinking Shape by number Create and record original about of sides and corners compositions made using parts (including new squares, right triangles, and shapes). rectangles on grid paper. Extend to equilateral grids and pattern blocks (those with multiples of 60Â° and 120Â° angles). â¢ egin to use relational B language of ârightâ and âleft.â â¢ raw a complete covering D of a rectangle area. Count squares in rectangular arrays correctly and (increasingly) systematically. Relating Measure of sides Compare area using Composition on grids parts and (simple units), gross superimposition. and in puzzles with wholes comparison of angle â¢ or rectangular regions, draw F systematicity and sizes. and count by rows (initially anticipation, using may only count some rows as a variety of shape rows). sets (e.g., pattern â¢ dentify and create symmetric I blocks; rectangular figures using motions (e.g., grids with squares, paper folding; also mirrors as right triangles, and reflections). rectangles; tangrams). NOTE: Most of the time should be spent on 2-D, about 85 percent (there are many beneficial overlapping activities). equality of angles another relation). Owing usually to a lack of good expe- riences, many students do not reach this level until late in their schooling. However, with appropriate learning experiences, even preschoolers can be- gin to develop this level of thinking. In Table 6-1 this is called the ârelating parts and wholesâ level. Development of Shape Concepts What ideas do preschool children form about common shapes? De- cades ago, Fuson and Murray (1978) reported that, by 3 years of age, over

180 MATHEMATICS LEARNING IN EARLY CHILDHOOD 60 percent of children could name a circle, a square, and a triangle. More recently, Klein, Starkey, and Wakeley (1999) reported the shape-naming ac- curacy of 5-year-olds as circle, 85 percent; square, 78 percent; triangle, 80 percent; rectangle, 44 percent. In one study (Clements et al., 1999), children identified circles quite accurately (92, 96, and 99 percent for 4-year-olds, 5-year-olds, and 6-year-olds, respectively), and squares fairly well (82, 86, and 91 percent). Young children were less accurate at recognizing triangles and rectangles, although their averages (e.g., 60 percent for triangles for all ages 4-6) were not remarkably smaller than those of elementary students (64-81 percent). Their visual prototype for a triangle seems to be of an isosceles triangle. Their average for rectangles was a bit lower (just above 50 percent for all ages). Childrenâs prototypical image of a rectangle seems to be a four-sided figure with two long parallel sides and âclose toâ square corners. Thus, young children tended to accept long parallelograms or right trapezoids as rectangles. In a second study (Hannibal and Clements, 2008), children ages 3 to 6 sorted a variety of manipulable forms. Certain mathematically irrelevant characteristics affected childrenâs categorizations: skewness, aspect ratio, and, for certain situations, orientation. With these manipulatives, orienta- tion had the least effect. Most children accepted triangles even if their base was not horizontal, although a few protested. Skewness, or lack of sym- metry, was more important. Many rejected triangles because âthe point on top is not in the middle.â For rectangles, many children accepted nonright parallelograms and right trapezoids. Also important was aspect ratio, the ratio of height to base. Children preferred an aspect ratio near one for triangles; that is, about the same height as width. Children rejected both triangles and rectangles that were âtoo skinnyâ or ânot wide enough.â Spatial Structure and Spatial Thinking Spatial thinking includes two main abilities: spatial orientation and spatial visualization and imagery. Other important competencies include knowing how to represent spatial ideas and how and when to apply such abilities in solving problems. Spatial Orientation Spatial orientation involves knowing where one is and how to get around in the world. As shown in Chapter 3, spatial orientation is, like number, a core cognitive domain, for which competencies, including the ability to actively and selectively seek out information, are present from birth (Gelman and Williams, 1997). Children have cognitive systems that are based on their own position and their movements through space, and

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 181 external references. They can learn to represent spatial relations and move- ment through space using both of these systems, eventually mathematizing their knowledge. Children as young as age 2 can implicitly use knowledge of multiple landmarks and distances between them to determine or remember loca- tions. By about age 5, they can explicitly represent that information, even interpreting or creating simple models of spaces, such as their classroom. Similarly, they can implicitly use distance and direction when they move at age 1-2. They do so more reliably when they move themselves, another justification for providing children of all ages with opportunities to explore large spaces in which they can navigate safely. By age 4, children explicitly use distance and direction and reason about their locations. For example, they can point to one location from another, even though they never walked a path that connected the two (Uttal and Wellman, 1989). Language for spatial relationships is acquired in a consistent order, even across different languages (Bowerman, 1996). The first terms acquired are in, on, and under, along with such vertical directionality terms as up and down. These initially refer to transformations (e.g., âonâ not as a smaller object on top of another, but only as making an object become physically attached to another; Gopnik and Meltzoff, 1986). Children then learn words of proximity, such as âbesideâ and âbetween.â Later, they learn words referring to frames of reference, such as âin front of,â âbehind.â The words âleftâ and ârightâ are learned much later, and are the source of confusion for several years. In these early years, children also can learn to analyze what others need to hear in order to follow a route through a space. Such learning is dependent on relevant experiences, including language. Learning and us- ing spatial terminology can affect spatial competence (Wang and Spelke, 2002). For example, teaching preschoolers the spatial terms âleftâ and ârightâ helped them reorient themselves more successfully (Shusterman and Spelke, 2004). However, language provides better support for simpler rep- resentations, and more complex spatial relationships are difficult to capture verbally. In such cases, children benefit from learning to interpret and use external representations, such as models or drawings. Young children can begin to build mental representations of their spatial environments and can model spatial relationships of these environ- ments. When very young children tutor others in guided environments, they build geometrical concepts (Filippaki and Papamichael, 1997). Such environments might include interesting layouts inside and outside class- rooms, incidental and planned experiences with landmarks and routes, and frequent discussion about spatial relations on all scales, including distinguishing parts of their bodies (Leushina, 1974/1991), describing spa- tial movements (forward, back), finding a missing object (âunder the table

182 MATHEMATICS LEARNING IN EARLY CHILDHOOD thatâs next to the doorâ), putting objects away, and finding the way back home from an excursion. As for many areas of mathematics, verbal inter- action is important. For example, parental scaffolding of spatial commu- nication helped both 3- and 4-year-olds perform direction-giving tasks, in which they had to clarify the directions (disambiguate) by using a second landmark (âitâs in the bag on the tableâ), which children are more likely to do the older they are. Both age groups benefited from directive prompts, but 4-year-olds benefited more quickly than younger children from nondirective prompts (Plumert and Nichols-Whitehead, 1996). Children who received no prompts never disambiguated, showing that interaction and feedback from others is critical to certain spatial communication tasks. Children as young as 3Â½ to 5 years of age can build simple but mean- ingful models of spatial relationships with toys, such as houses, cars, and trees (Blaut and Stea, 1974), although this ability is limited until about age 6 (Blades et al., 2004). Thus, younger children create relational, geometric correspondences between elements, which may still vary in scale and per- spective (Newcombe and Huttenlocher, 2000). As an example, children might use cutout shapes of a tree, a swing set, and a sandbox in the playground and lay them out on a felt board as a simple map. These are good beginnings, but models and maps should eventually move beyond overly simple iconic picture maps and challenge children to use geometric correspondences. Four questions arise: direction (which way?), distance (how far?), location (where?), and identification (what objects?). To answer these questions, children need to develop a variety of skills. They must learn to deal with mapping processes of abstrac- tion, generalization, and symbolization. Some map symbols are icons, such as an airplane for an airport, but others are more abstract, such as circles for cities. Children might first build with objects, such as model buildings, then draw pictures of the objectsâ arrangements, then use maps that are miniaturizations and those that use abstract symbols. Teachers need to con- sistently help children connect the real-world objects to the representational meanings of map symbols. As noted in Chapter 4, equity in the education of spatial thinking is an important issue. Preschool teachers spend more time with boys than girls and usually interact with boys in the block, construction, sand play, and climbing areas and with girls in the dramatic play area (Ebbeck, 1984). Boys engage in spatial activities more than girls at home, both alone and with caretakers (Newcombe and Sanderson, 1993). Such differences may interact with biology to account for early spatial skill advantages for boys (note that some studies find no gender differences (e.g., Brosnan, 1998, Chapter 15; Ehrlich, Levine, and Goldin-Meadow, 2006; Jordan et al., 2006; Levine et al., 1999; Rosser et al., 1984).

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 183 Spatial Visualization and Imagery Spatial images are internally experienced, holistic representations of objects that are to a degree isomorphic to their referents (Kosslyn, 1983). Spatial visualization is understanding and performing imagined movements of 2-D and 3-D objects. To do this, you need to be able to create a mental image and manipulate it, showing the close relationship between these two cognitive abilities. An image is not a âpicture in the head.â It is more abstract, more malleable, and less crisp than a picture. It is often segmented into parts. Some images can cause difficulties, especially if they are too inflexible, vague, or filled with irrelevant details. Peopleâs first images are static. They can be mentally recreated, and even examined, but not transformed. For example, one might attempt to think of a group of people around a table. In contrast, dynamic images can be transformed. For example, you might mentally âmoveâ the image of one shape (such as a book) to another place (such as a bookcase, to see if it will fit). In mathematics, you might mentally move (slide) and rotate an image of one shape to compare that shape to another one. Piaget argued that most children cannot perform full dynamic motions of images until the primary grades (Piaget and Inhelder, 1967, 1971). However, preschool children show initial transformational abilities (Clements et al., 1997a; Del Grande, 1986; Ehrlich et al., 2005; Levine et al., 1999). With guidance, 4-year-olds and some younger children can generate strategies for verifying congruence for some tasks, moving from more primitive strategies, such as edge matching (Beilin, 1984; Beilin, Klein, and ÂWhitehurst, 1982) to the use of geometric transformations and super- position. Interventions can improve the spatial skills of young children, especially when embedded in a story context (Casey, 2005). Computers are especially helpful, as the screen tools make motions more accessible to reflection and thus bring them to an explicit level of awareness for children (Clements and Sarama, 2003; Sarama et al., 1996). Similarly, other types of imagery can be developed. Manipulative work with shapes, such as tangrams (a puzzle consisting of seven flat shapes, called tans, which are put together in different ways to form distinct geo- metric shapes), pattern blocks, and other shape sets, provides a valuable foundation (Bishop, 1980). After such explorations, it is useful to engage children in puzzles in which they see only the outline of several pieces and have them find ways to fill in that outline with their own set of tangrams. Similarly, children can begin to develop a foundation for spatial structur- ing by forming arrays with square tiles and cubes (this is discussed in more detail in the section on measurement). Also challenging to spatial visualization and imagery are âsnapshotâ activities (Clements, 1999b; Yackel and Wheatley, 1990). Children briefly

184 MATHEMATICS LEARNING IN EARLY CHILDHOOD see a simple arrangement of pattern blocks, then try to reproduce it. The configuration is shown again for a couple of seconds as many times as necessary. Older children can be shown a line drawing and try to draw it themselves (Yackel and Wheatley, 1990). This often creates interesting discussions revolving around âwhat I saw.â Spatial visualization and imagery have been positively affected by in- terventions that emphasize building and composing with 3-D shapes (Casey et al., in press). Another series of activities described above that develops imagery is the sequence of tactile-kinesthetic exploration of shapes. Achievable and Foundational Geometry and Spatial Thinking Although longitudinal research is needed, extant research provides guidance about which geometric and spatial experiences are appropriate for and achievable by young children and will contribute to their math- ematical development. First, of the mathematics children engage in spon- taneously in child-centered school activities, the most frequent deals with shape and pattern. Second, each of the recently developed, research-based preschool mathematics curricula includes geometric and spatial activities (Casey, Paugh, and Ballard, 2002; Clements and Sarama, 2004; Ginsburg, Greenes, and Balfanz, 2003; Klein, Starkey, and Ramirez, 2002), with some of these featuring such a focus in 40 percent or more of the activities. Third, pilot-testing has shown that these activities were achievable and motivating to young children (Casey, Kersh, and Young, 2004; Clements and Sarama, 2004; Greenes, Ginsburg, and Balfanz, 2004; Starkey, Klein, and Wakeley, 2004), and formal evaluations have revealed that they con- tributed to childrenâs development of both numerical and spatial/geometric concepts (Casey and Erkut, 2005, in press; Casey et al., in press; Clements and Sarama, 2007c, in press; Starkey et al., 2004, 2006). Fourth, previous work has shown that well-designed activities can effec- tively build geometric and spatial skills and general reasoning abilities (e.g., Kamii, Miyakawa, and Kato, 2004). Fifth, results with curricula in Israel that involved only spatial and geometric activities (Eylon and Â Rosenfeld, 1990) are remarkably positive. Children gained in geometric and spatial skills and showed pronounced benefits in the areas of arithmetic and writ- ing readiness (Razel and Eylon, 1990). Similar results have been found in the United States (Swaminathan, Clements, and Schrier, 1995). Children are better prepared for all school tasks when they gain the thinking tools and representational competence of geometric and spatial sense. In this section, we describe teaching-learning paths for spatial and geometric thinking in 2-D and 3-D contexts. For each area outlined below, children should be engaged in activities that cover a range of difficulty, in- cluding perceive, say, describe/discuss, and construct (measurement in one,

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 185 two, and three dimensions is described in the following section). Tables 6-1 and 6-2 summarize development of spatial and geometric thinking, as well as measurement, in two and three dimensions. Ages are grouped in the same was as in the previous chapter in order to illustrate how childrenâs engagement with mathematics should build and develop over the prekin- dergarten years. In the tables, childrenâs competence within each band is described on the basis of the level of sophistication in their thinking. These levels are called thinking visually/holistically, thinking about parts, and relating parts and wholes. Step 1 (Ages 2 and 3) 2-D and 3-D Objects Very young children match shapes implicitly in their play. Working at the visual/holistic level (see Table 6-1), they can describe pictures of objects of all sorts, using the shape implicitly in their recognition. By age 2 to 3, they also learn to name shapes, with 2-D shapes being more familiar in most cultures, beginning with the familiar and symmetric circle and square and extending to at least prototypical triangles. Although they may name 3-D shapes by the name of one of its faces (calling a cube a square), their ability to match 2-D to corresponding 2-D (and similar for 3-D) indicates their intuitive differentiation of 2-D and 3-D shapes. Children also learn to recognize and name additional shapes, such as triangles and rectanglesâat least in their prototypical formsâand can be- gin to describe them in their own words. With appropriate knowledge of number, they can begin to describe these shapes by the number of sides they have, just starting to learn the concepts and terminology of the thinking about parts level of geometric thinking. Spatial Relations From the first year of life, children develop an implicit ability to move objects. They also learn relationship language, such as âupâ and âdownâ and similar vocabulary. They learn to apply that vocabulary in both 3-D contexts and in 2-D situations, such as the âbottomâ of a picture that they are drawing on a horizontal surface. Compositions and Decompositions At the visual/holistic level, children can solve simple puzzles involving things in the world (e.g., wooden puzzles with insets for each separate ob-

186 MATHEMATICS LEARNING IN EARLY CHILDHOOD TABLE 6-2â Space and Shapes in Three Dimensions Goals C. Perceive, Say, Describe/ A. Perceive, Say, B. Perceive, Say, Discuss, and Construct Steps/Ages Describe/Discuss, and Describe/Discuss, and Compositions and (Levels of Construct Objects in Construct Spatial Decompositions in 3-D Thinking) 3-D Space Relations in 3-D Space Space Step 1 (Ages 2 and 3) Thinking See and describe Understand and use Represent real-world visually/ pictures of objects of all relational language, objects with blocks that holistically sorts (3-D to 2-D).* including âin,â have a similar shape. âout,â âon,â âoff,â â¢ ombine unit blocks by C and âunder,â along stacking. with such vertical directionality terms as âupâ and âdown. Thinking Discriminate between about 2-D and 3-D shapes parts intuitively, marked by accurate matching or naming. Step 2 (Age 4) Thinking Describe the difference Match 3-D shapes. visually/ between 2-D and 3-D â¢ ses relational words U holistically shapes, and names of proximity, such common 3-D shapes as âbeside,â ânext informally and with toâ and âbetween,â mathematical names âabove,â âbelow,â (âballâ/sphere; âboxâ âover,â and âunder.â or rectangular prism, ârectangular block,â or âtriangular blockâ; âcanâ/cylinder). Thinking Identify faces of 3-D Identify (matches) the Combine building blocks, about objects as 2-D shapes faces of 3-D shapes to using multiple spatial parts and name those shapes. (congruent) 2-D shapes, relations. â¢ se relational U and match faces of language involving congruent 2-D shapes, frames of reference naming the 2-D shapes. such as âin front â¢ epresent 2-D and R of,â âin back of,â 3-D relationships with âbehind,â âbefore.â objects.

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 187 TABLE 6-2â Continued Goals C. Perceive, Say, Describe/ A. Perceive, Say, B. Perceive, Say, Discuss, and Construct Steps/Ages Describe/Discuss, and Describe/Discuss, and Compositions and (Levels of Construct Objects in Construct Spatial Decompositions in 3-D Thinking) 3-D Space Relations in 3-D Space Space Relating Informally describe why Compose building blocks parts and some blocks âstack to produce composite wholes wellâ and others do shapes. Produce arches, not. enclosures, corners, and crosses systematically. Step 3 (Age 5) Thinking Name common visually/ 3-D shapes with holistically mathematical terms (spheres, cylinder, rectangle, prism, pyramid). Thinking Begin to use relational Fill rectangular about language of ârightâ and containers with cubes, parts âleft.â filling one layer at a time. Relating Describe congruent Understand and can Substitution of shapes. parts and faces and, in context replicate the perspective Build complex structures. wholes (e.g., block building), of a different viewer. â¢ uild structures from B parallel faces of blocks. pictured models. NOTE: Less time on 3-D than on 2-D, about 10 percent of the time on 3-D. *Research indicates that very young children mainly use shape for object identification. Research says children with lower socioeconomic status have difficulty with describing objects and need to learn the vocabulary to do so. ject pictured). They create pictures with geometric shapes (circles, circle sec- tions, polygons), often representing single objects with different shapes, but eventually combining shapes to make, for example, the body of a vehicle or an animal. That is, initially children manipulate shapes individually, but they are unable to combine them to compose a larger shape. For example, they might use a single shape for a sun, a separate shape for a tree, and another separate shape for a person. Initially, they cannot accurately match shapes to even simple frames. Later, children learn to place 2-D shapes contiguously to form pictures. In free-form âmake a pictureâ tasks, for example, each shape used repre-

188 MATHEMATICS LEARNING IN EARLY CHILDHOOD sents a unique role or function in the picture (e.g., one shape for one leg). Children can fill simple frame-based shapes puzzles using trial and error, but they may have limited ability to use turns or flips to do so; they cannot use motions to see shapes from different perspectives. Thus, children view shapes only as wholes and see few geometric relationships between shapes or between parts of shapes (i.e., a property of the shape). Composition with 3-D shapes usually begins with stacking blocks. Children then learn to stack congruent blocks and make horizontal âlines.â Next they build a vertical and horizontal structure, such as a floor or a simple wall. Later, some 3-year-olds begin to extend their buildings in mul- tiple directions, possibly creating arches, enclosures, corners, and crosses, but often using unsystematic trial and error and simple addition of pieces. Step 2 (Age 4) 2-D and 3-D Objects Beginning at the visual/holistic level, preschoolers learn to recognize a wide variety of shapes, including shapes that are different sizes and are presented at different orientations. They also begin to recognize that geometric figures can belong to the same shape class, but have different measures and proportions. Similarly, preschoolers learn to describe the dif- ferences between 2-D and 3-D shapes informally. They also learn to name common 3-D shapes informally and with mathematical names (ball/sphere, box/rectangular prism, rectangular block, triangular block, can/cylinder). They name and describe these shapes, first using their own descriptions and increasingly adopting mathematical language. For example, âdiamondâ gives way to ârhombusâ and âcornersâ become âanglesâ (or vertices). Eventually, they adopt the terminology of the thinking about parts level, such as identifying shapes as triangles because they have three sides. Faces of 3-D shapes are identified as specific 2-D shapes. Such descriptions build geometric concepts, as well as reasoning skills and language. They encourage children to view shapes analytically. Chil- dren begin to describe some shapes in terms of their properties, such as saying that squares have four sides of equal length, and thus make initial forays into thinking at the relating parts and wholes step. They informally describe the properties of blocks in functional contexts, such as that some blocks roll and others do not. Spatial Relations Also beginning at the visual/holistic level, preschool children learn to extend their vocabulary of spatial relations with such terms as âbeside,â

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 189 ânext to,â and âbetween,â which they can apply in 3-D and 2-D spaces. Later, they extend this to terms that involve frames of reference, such as âto the side of,â âabove,â and âbelow.â Later, at the thinking about parts level, preschoolers recognize âmatch- ingâ shapes at different orientations. They can learn to check if pairs of 2-D shapes are congruent by using geometry motions intuitively, moving from less accurate strategies, such as side-matching, or using lengths, to the use of superimposition (placing one shape on top of the other). They begin to use the geometric motions of slides, flips, and turns explicitly and intention- ally, in discussing their solutions to puzzles or in applying such motions in computer environments to manipulate shapes. They learn to predict the effects of geometric motions, thus laying the foundation for thinking at the relating parts and wholes level. Children also begin to be able to cover a rectangular space with physi- cal tiles and represent their tilings with simple drawings, although they may initially leave gaps in each and may not align all the squares. This is mainly a competence of spatial structuring but it has close connections to the ability to construct compositions in 2-D space. Preschoolers also learn about the parts of 3-D shapes, using motions to match the faces of 3-D shapes to 2-D shapes and representing 2-D and 3-D relationships with objects. For example, they may make a simple model of the classroom, using a rectangular block for the teacherâs desk, small cubes for chairs, and so forth. Compositions and Decompositions At the thinking about parts level, preschoolers can place shapes con- tiguously to form pictures in which several shapes play a single role (e.g., a leg might be created from three contiguous squares), but they use trial and error and do not anticipate creation of new geometric shapes. When filling in a frame or picture outline, children use gestalt configuration or one component, such as side length (Sarama et al., 1996). For example, if several sides of the existing arrangement form a partial boundary of a shape (instantiating a schema for it), children can find and place that shape. If such cues are not present, they match by a side length. Children may at- tempt to match corners but do not understand angle as a quantitative entity, so they try to match shapes into corners of existing arrangements in which their angles do not fit. Rotating and flipping are used, usually by trial and error, to try different arrangements (a âpicking and discardingâ strategy). Thus, they can complete a frame that suggests placement of the individual shapes but in which several shapes together may play a single semantic role in the picture. Later, preschoolers begin to develop relating parts and wholes thinking.

190 MATHEMATICS LEARNING IN EARLY CHILDHOOD For example, they might combine pattern block shapes (angles that are mul- tiples of 30Â°) to make composites that they recognize as new shapes and to fill puzzles with growing intentionality and anticipation (âI know what will fitâ). Shapes are chosen using angles as well as side lengths. The equilateral triangle world of pattern blocks provides a microworld, in which matching by sides (all of which are equal in length or double the unit length), fitting angles (multiples of 30Â°), and composing (two equilateral triangles can âmakeâ the blue rhombus, a rhombus and a triangle make a trapezoid, etc.) are facilitating at this beginning step. Eventually, children consider several alternative shapes with angles equal to the existing arrangement. Rotation and flipping are used intentionally (and mentally, i.e., with anticipation) to select and place shapes (Sarama et al., 1996). Children can fill complex frames (Sales, 1994) or cover regions (Mansfield and Scott, 1990). Related to their ability to tile the rectangular section of a plane, chil- dren can copy designs made from squares (and, for some, also isosceles right triangles) and place these shapes onto squared-grid paper. This square- based microworld is simple and not only facilitates composition, but also develops the foundations of much of mathematics (spatial structuring, multiplication, area, volume, coordinates, etc.). Using 3-D shapes, preschoolers combine building blocks using multiple spatial relations, extending in multiple directions and with multiple points of contact among components, showing flexibility in integrating parts of the structure. Thus, they can reliably produce arches, enclosures, corners, and crosses, including enclosures that are several blocks in height. Later, they can learn to compose building blocks with anticipation, understanding what 3-D shape will be produced with a composition of 2 or more other (simple, familiar) 3-D shapes. Step 3 (Age 5) 2-D and 3-D Objects Kindergartners learn to recognize additional shapes, such as paral- lelograms, and, more importantly, learn to describe why a certain figure is classified into a given class of shapes (at the relating parts and wholes level). They may therefore discuss that parallelograms have two pairs of sides that are equal in length and two pairs of angles of equal size. This remains just the beginning of this type of thinking, as concepts of parallelism, perpen- dicularity, and angle measure develop over many years thereafter. Kindergartners also learn the names of more 3-D shapes, such as spheres, cylinders, prisms, and pyramids. They describe congruent faces of such shapes and begin to understand and discuss such properties as parallel faces in some contexts (e.g., building with blocks).

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 191 Spatial Relations Kindergartners begin to use relational terms ârightâ and âleftâ in both 3-D and 2-D contexts, using scaffolds and other guidance as needed. They can also continue to develop the ability to tile a plane with square tiles without gaps and begin to represent such a tiling by drawing. They can learn to count the squares in their tiling, using more systematic strategies for keeping track, such as counting one row at a time. Finally, kindergart- ners can understand and can replicate the perspectives of different viewers. These competencies reflect an initial development of thinking at the relating parts and wholes level. Compositions and Decompositions Kindergartners continue to develop the ability to intentionally and systematically combine shapes to make new shapes and complete puzzles. They do so with increasing anticipation, based on the shapesâ attributes, indicating development of mental images of the component shapes. A sig- nificant advance is that they can combine shapes with different properties, extending the pattern block (30Â°) shapes common at early steps to such shapes as tangrams (angles multiples of 45Â°), and with sets of various shapes that include angles that are multiples of 15Â° as well as sections of circles. Using 3-D shapes, kindergartners can substitute a composite shape for a congruent whole shape. They learn to build complex structures, such as bridges with multiple arches, with ramps and stairs at the ends. They can build structures with cubes or building blocks from 2-D pictures of these structures. Children of this age also can learn to move squares and right triangles on grids to create original designs. They can also record these designs on squared-grid paper. Instruction to Support the Teaching-Learning Paths Learning and Teaching About Shape Without good experiencesââeducativeâ rather than âmis-Âeducativeâ (Dewey, 1933)âstudents often rely on impoverished visual prototypes that they develop based on limited examples and limited experiences with language. In contrast, good experiences include providing a variety of e Â xamplesâfor example, with triangles, not all equilateral or isosceles, and not all with a horizontal base, as well as discussions about triangles and their attributes that go beyond simple memorized definitions. Most children in the United States do not have these good experiences. Teachers and curriculum

192 MATHEMATICS LEARNING IN EARLY CHILDHOOD writers assume that children in early childhood classrooms have little or no knowledge of geometric figures. And teachers have had few experiences with geometry in their own education or in their professional development. Thus, it is unsurprising that most classrooms exhibit limited geometry in- struction. One early study found that kindergarten children had a great deal of knowledge about shapes and matching shapes before instruction began. Their teacher tended to elicit and verify this prior knowledge but did not add content or develop new knowledge. That is, about two-thirds of the interactions had children repeat what they already knew (Thomas, 1982). Furthermore, many of their attempts to add content were mathematically inaccurate (âevery time you cut a square, you get two trianglesâ). Such neglect is reflected in student achievement. U.S. students are not prepared for learning more sophisticated geometry, especially when com- pared with students of other nations (Carpenter et al., 1980; Fey et al., 1984; Kouba et al., 1988; Starkey et al., 1999; Stevenson, Lee, and Stigler, 1986; Stigler, Lee, and Stevenson, 1990). In some international studies, they score at or near the bottom in every geometry task (Beaton et al., 1997; Lappan, 1999). The research reviewed to this point suggests that development of geo- metric knowledge is fueled by experience and education, not just matura- tion. If the shape categories children experience are limited, so will be their concepts of shapes. If the examples and nonexamples children experience are rigid, so will be their mental prototypes. Many children learn to accept only isosceles triangles, for example. Others learn richer concepts, even at a young age. Such children are likely to have had good experiences with shapes, including rich, varied examples and nonexamples and discussions about shapes and their characteristics. Good experiences should begin early. Children need to experience varied examples and nonexamples and understand the attributes of shapes that are mathematically relevant as well as those (orientation, size) that are not. So, examples of triangles and rectangles should include a wide variety of shapes, including âlong,â âskinny,â and âfatâ examples. Direct empiri- cal support for this finding is strongest for 4-year-olds, who are motivated to explore shape (Seo and Ginsburg, 2004) and have achieved substantial gains in geometric knowledge through curricular interventions, often sur- passing the concepts of much older students in business-as-usual curricula (Casey and Erkut, in press; Casey et al., in press; Clements and Sarama, 2007c, in press; Starkey et al., 2006; Starkey, Klein, and Wakeley, 2004). Beyond perceiving and naming shapes, children can and should discuss the parts and attributes of shapes. Again, there are several reasons for this recommendation. First, such descriptive activity encourages children to move beyond visual prototypes to the use of mathematical criteria. Second, discussions redirect attention and build strong concepts, mutually affect-

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 193 ing and benefiting mental images (Clements and Sarama, 2007b). Third, these types of discussions are interesting to, and beneficial for, children as young as ages 3 and 4 (as the evaluations of the research-based curricula show; see also Spitler, Sarama, and Clements, 2003). Instructional activities that promote such reflection and discussion include building shapes from components. For example, children might build squares and other polygons with toothpicks and marshmallows. They might also form shapes with their bodies, either singly or with their friends. Another sequence of activities involves tactile-kinesthetic exploration of shapes (feeling shapes hidden in a box). Such nonvisual exploration of shapes does not allow simple matching to prototypes. Instead, they force children to carefully put the parts of the shape into relationship with each other. First, teachers place a small number of shapes on the table and hide a shape congruent to one of these in the box (Clements and Sarama, 2007a). Children feel the shape and point to the matching shape, then pull out the hidden shape to check. Later, children do not have the shapes on the table. Instead, they have to name the shape they are feeling. Even later, they have to describe the shape without using its name, so that their friends could name the shape. In this way, children learn the properties of the shape, moving from intuitive to explicit knowledge. The sequence in Table 6-1 indicates that 3-year-olds may begin to associate certain shapes with a known small number, even if only at an intuitive level. In comparison, 4-year-olds can explicitly adopt terminol- ogy of the thinking about parts step, illustrated by a preschooler stating that an obtuse triangle âmust be a triangle, because it has three sides.â As 4-year-olds start to see that some shapes have four sides that are the same length, they begin a long journey into the relating parts and wholes level of geometric thinking. Kindergartners can explicitly discuss why they call a certain shape a rectangle. Teachers might start by having children gather rectangles and have them describe why their shapes are rectangles in their own words. They could also show children a variety of shapes and have them decide whether they were or were not rectangles and why. Another useful instructional task is to challenge children to use sticks or straws of varying lengths to make triangles. Older children could draw a series of rectangles, increasing in size. Some children increase the lengths in both dimensions (e.g., length and width), some in only one dimension, leading to rich discussions. Early childhood curricula traditionally introduce shapes in four basic categories: circle, square, triangle, and rectangle. The separation of the square and the rectangle sets up a misconception that violates the math- ematical relationship between these shapes: A square is a rectangle; it is a special kind of rectangle in which all sides are the same length. The idea that a square is not a rectangle, however, is rooted by age 5 (Clements

194 MATHEMATICS LEARNING IN EARLY CHILDHOOD et al., 1999; Hannibal and Clements, 2008). It is time to change the pre- sentation of squares as an isolated set. Instead, recent approaches present many Âexamples of squares and rectangles, varying orientation, size, and so forth, including squares as examples of rectangles. If children say âthatâs a square,â teachers respond that it is a square that is a special type of rectangle, using double-naming (âitâs a square-rectangleâ). This approach has been shown to be successful with preschoolers and kindergartners ( Â Clements and Sarama, 2007c, in press; Clements, Sarama, and Wilson, 2001; Sarama and Â Clements, 2002). Kindergarten and first graders can discuss general categories, such as quadrilaterals and triangles, counting the sides of various figures to choose their category. They can then build hierar- chical relationships of subsets of these general categories (Kay, 1987). Children should also learn about composing and decomposing shapes from other shapes. This competence is significant in that the concepts and actions of creating and then iterating units and higher order units in the context of constructing patterns, measuring, and computing are established bases for mathematical understanding and analysis (Clements et al., 1997b; Reynolds and Wheatley, 1996; Steffe and Cobb, 1988). In addition, there is empirical support that this type of composition corresponds with, and supports, childrenâs ability to compose and decompose numbers (Clements et al., 1996). The sequence in Table 6-1 is based on a series of developmental stud- ies describing childrenâs capabilities (Clements, Sarama, and Wilson, 2001; M Â ansfield and Scott, 1990; Sales, 1994; Sarama, Clements, and Vukelic, 1996). These studies were synthesized into an empirically verified develop- mental progression that identified skills that are achievable for children at different ages, especially if provided opportunities to learn (Clements, Wilson, and Sarama, 2004). Starting with a lack of competence in composing geo- metric shapes, they gain abilities to combine shapes into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes). As further evidence, interventions at the preschool level have shown notable gains in this ability for 2-D shapes (Casey and Erkut, in press). Intentional interventions with 3-D shape construction (i.e., building with unit blocks) have also resulted in statistically significant gains (Casey et al., in press). Many activities develop these abilities. With a variety of groups of shapes, such as pattern blocks, tangrams, or groups with a greater variety of shapes, children can be encouraged to combine shapes creatively to create pictures and designs. Noting childrenâs developmental level, teachers can make suggestions and pose challenges that will facilitate their learning of more sophisticated thinking. Outline puzzles that can be filled with those same groups of shapes are also motivating and particularly useful because they can be designed to pro- mote a particular level of thinking. Teachers can then view childrenâs active

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 195 problem solving and provide them with puzzles that will be attainable but challengingâthat is, that will promote their development to the next level of thinking for 2-D geometric composition. Similar teaching strategies can develop composition of 3-D shapes. Discussions about childrenâs own creative constructions may make explicit ideas about length and symmetry, among others. Also, problems can be designed to encourage spatial and mathematical thinking and sequenced to match developmental progressions (Casey et al., in press; Kersh, Casey, and Young, 2008) early problem for children might be to build an enclo- sure with walls that are at least two blocks high and include an arch. This introduces the problem of bridging, which involves balance, measurement, and estimation. A second problem might be to build more complex bridges, such as ones with multiple arches and ramps or stairs at the end. This introduces planning and seriation. The third problem might be to build a complex tower with at least two floors, or stories. Children could be pro- vided with cardboard ceilings, so they to make the walls fit the constraints of the cardboardâs dimensions. The recommended approaches and activities in this section have been performed successfully with 3- and 4-year-olds in classrooms serving low- and middle-income children, with strong positive results on child outcomes (Clements and Sarama, 2007c, in press; Starkey et al., 2006; Starkey, Klein, and Wakeley, 2004). Use of Manipulatives, Pictures, and Computers Research suggests that the use of manipulatives can help young children develop geometric and spatial thinking (Clements and McMillen, 1996). Using a greater variety of manipulatives is beneficial (Greabell, 1978). Such tactile-kinesthetic experiences as body movement and manipulating geometric solids help young children learn geometric concepts (Gerhardt, 1973; Prigge, 1978). Children also fare better with solid cutouts than with printed forms, the former encouraging the use of more senses (Stevenson and McBee, 1958). However, such benefits are not straightforward or certain (Clements, 1999a; National Mathematics Advisory Panel, 2008). These materials must be used in the context of a complete mathematics program to intentionally develop specific skills and concepts. Also, from the beginning, manipulatives should be used to help childrenâeven young childrenâdevelop mental representations that are increasingly abstract. Pictures can also support learning. Children as young as 5 or 6 (but not most younger children) can use information in pictures to build a pyramid, for example (Murphy and Wood, 1981). Thus, pictures can give students an immediate, intuitive grasp of certain geometric ideas. Instructionally, pictures need to be sufficiently varied so the ideas that students form are

196 MATHEMATICS LEARNING IN EARLY CHILDHOOD not too limited. With experience, children can become sophisticated in interpreting geometric relationships in pictures. Diagrams are also useful tools for visualizing numerical and arithmetic problems, and the more e Â xperience children have with the geometric and measurement attributes of pictures and shapes, the more competence they will have in constructing and interpreting such diagrams. However, research indicates that it is rare for pictures to be superior to manipulatives. In fact, in some cases, pictures may not differ in effectiveness from instruction with symbols (Sowell, 1989). The reason may lie not so much in the nonconcrete nature of the pictures as in their nonmanipulabilityâthat is, that children cannot act on them as flexibly and extensively. This is one reason that manipulatives on c Â omputersâeven though 2-Dâcan benefit learning and teaching. In fact, computers may have some specific advantages (Clements and McMillen, 1996). For example, some computer manipulatives offer more flexibility than their noncomputer counterparts. Computer-based pattern blocks, for example, can be composed and decomposed in more ways than physical pattern blocks. As another example, children and teachers can save and later retrieve any arrangement of computer manipulatives. Similarly, computers allow storage and replay of sequences of actions on manipula- tives. Computers can also be used to carry out mathematical processes that are difficult or impossible to perform with physical manipulatives. For example, a computer environment might automatically draw shapes sym- metrical to anything the child constructs or draws. As a final illustration, computers can help children become aware of, and mathematize, their actions. For example, very young children can move puzzle pieces into place, but they do not think about their actions. Using the computer, however, helps children become aware of and describe these motions (Clements and Battista, 1991; Johnson-Gentile, Clements, and Battista, 1994). Manipulativesâphysical or computerâare one tool that can assist children in constructing mathematical meaning. They do not always do that, however, and the point of using them lies not in their use in promoting manipulations or random play, but to develop abstract ideas. In this view, manipulatives are successful to the extent that they become unnecessary because children have built mental images and concepts that they use for mathematical thinking (Clements, 1999a). MEASUREMENT Geometric measurement connects and enriches the two critical domains of geometry and number. Childrenâs understanding of measurement has its roots in infancy and the preschool years, and it grows over many years, as the research described in Chapter 3 shows. Even preschoolers can be guided

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 197 to learn important concepts if provided appropriate measurement experi- ences. They naturally encounter and discuss quantities (Seo and Ginsburg, 2004). They initially learn to use words that represent quantity or mag- nitude of a certain attribute. Then they compare two objects directly and recognize equality or inequality (Boulton-Lewis, Wilss, and Mutch, 1996). At age 4-5, most children can learn to overcome perceptual cues and make progress in reasoning about and measuring quantities. They are ready to learn to measure, connecting number to the quantity, yet the average child in the United States, with limited measurement experience, exhibits limited understanding of measurement until the end of the primary grades. We examine this development in more detail for the attribute of length. Length Measurement Length is a characteristic of an object found by quantifying how far it is between the end points of the object. Distance is often used similarly to quantify how far it is between any two points in space. Measuring length or distance consists of two aspects: (1) identifying a unit of measure and subdividing (mentally and physically) the object by that unit; (2) placing that unit end to end (iterating) alongside the object. Subdividing and unit iteration are complex mental accomplishments that are too often ignored in traditional measurement curriculum materials and instruction. Many researchers therefore go beyond the physical act of measuring to investigate childrenâs understandings of measuring as covering space and quantifying that covering. Appendix B describes concepts that are basic to understand- ing length measurement. Before kindergarten, many children lack understanding of measurement ideas and procedures, such as lining up end points when comparing the lengths of two objects. Even 5- to 6-year-olds, given a demarcated ruler, write in numerals haphazardly, with little regard to the size of the spaces. Few use zero as a starting point, showing a lack of understanding of the origin concept. At age 4-5, however, many children can, with opportunities to learn, become less dependent on perceptual cues and thus make progress in reasoning about or measuring quantities. From kindergarten to Grade 2, children can significantly improve in measurement knowledge (Ellis, 1995). They learn to represent length with a third object, using transitivity to com- pare the length of two objects that are not compared directly in a wider variety of contexts (Hiebert, 1981). They can also use given units to find the length of objects and associate higher counts with longer objects (Hiebert, 1981, 1984). Some 5-year-olds and most 7-year-olds can use the concept of unit to make inferences about the relative size of objects; for example, if the numbers of units are the same, but the units are different, the total size is different (Nunes and Bryant, 1996).

198 MATHEMATICS LEARNING IN EARLY CHILDHOOD Children as young as kindergartners may be proficient with a conven- tional ruler and understand quantification in limited measurement contexts. However, their skill decreases when features of the ruler deviate from the convention. Thus, measurement is supported by characteristics of mea- surement tools, but children still need to develop understanding of key measurement concepts. For example, they may initially iterate a unit leav- ing gaps between subsequent units or overlapping adjacent units (Horvath and Lehrer, 2000; Lehrer, 2003). These children may think of measuring as the physical activity of placing units along a path in some manner, rather than the activity of covering the space/length of the object with no gaps. Furthermore, children often begin counting at the numeral 1 on a ruler (Lehrer, 2003) or, when counting paces heel-to-toe, start their count with the movement of the first foot, missing the first foot and counting the sec- ond foot as one (Lehrer, 2003; Stephan et al., 2003). Again, children may not be thinking about measuring as covering space. Rather, the numerals on a ruler (or the placement of a foot) signify when to start counting, not an amount of space that has already been covered (i.e., 1 is the space from the beginning of the ruler to the hash mark, not the hash mark itself). Many children initially find it necessary to iterate the unit until it âfills upâ the length of the object and will not extend the unit past the end point of the object they are measuring (Stephan et al., 2003). Finally, many children do not understand that units must be of equal size. They will even measure with tools subdivided into different size units and conclude that quantities with more units are larger (Ellis et al., 2000). This may be a deleterious side effect of counting, in which children learn that the size of objects does not affect the result of counting (Mix, Huttenlocher, and Levine, 2002). However, the researchers base this interpretation on the assumption that units are always âgivenâ in counting contexts. In fact, there are counting contexts in which this is not the case, such as counting whole toy people constructed in two parts, top and bottom, when some are fastened and some are separated (Sophian and Kailihiwa, 1998). Thus, significant development occurs in the early childhood years. However, the foundational ideas about length are usually not integrated, even by the primary grades. For example, children may still not understand the importance of, or be able to create, equal size units (Clements et al., 1997a; Lehrer, Jenkins, and Osana, 1998; Miller, 1984). This indicates that children have not necessarily differentiated fully between counting discrete objects and measuring. Even if they show competence with rulers and are given identical units, children may not spontaneously iterate those they have if they do not have a sufficient number to measure an object (Lehrer, Jenkins, and Osana, 1998)âeven when the units are rulers themselves ( Â Clements, 1999c). Some children can or do not mentally partition the object to be measured.

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 199 Many recent curricula or other instructional guides advise a sequence of instruction in which children compare lengths, measure with nonstan- dard units, incorporate the use of manipulative standard units, and measure with a ruler (Clements, 1999c; Kamii and Clark, 1997). The basis for this sequence is, explicitly or implicitly, the theory of measurement of Piaget et al. (1960). The argument is that this approach motivates children to see the need for a standard measuring unit. Although such an approach has been shown to be effective, it may not be necessary to follow a nonstandard-to-standard units approach. For example, Boulton-Lewis et al. (1996) found that children used nonstandard units unsuccessfully but were successful at an earlier age with standard units and measuring instruments. The researchers concluded that nonstan- dard units are not a good way to initially help children understand the need for standardized conventional units in the length measuring process. Just as interesting were childrenâs strategy preferences. Children of every age pre- ferred to use standard rulers, even though their teachers were encouraging them to use nonstandard units. Furthermore, children measured correctly with a ruler before they could devise a measurement strategy using nonstandard units. To realize that ar- bitrary units are not reliable, a child must reconcile the varying lengths and numbers of arbitrary units. Emphasizing nonstandard units too early may defeat the purpose it is intended to achieve. That is, early emphasis on various nonstandard units may interfere with childrenâs development of the basic measurement concepts required to understand the need for standard units. In contrast, using manipulative standard units, or even standard rul- ers, is less demanding and appears to be a more interesting and meaningful real-world activity for young children (Boulton-Lewis et al., 1996). These findings have been supported by additional research (Boulton-Lewis, 1987; Clements and Battista, 2001; Clements et al., 1997b; HÃ©raud, 1989). Thus, early experience measuring with different units may be exactly the wrong thing to do. Another study (Nunes, Light, and Mason, 1993) suggests that children can meaningfully use rulers before they reinvent such ideas as units and iteration. In it, children ages 6 to 8 communicated about lengths using string, centimeter rulers, or one ruler and one broken ruler starting at 4 cm. The traditional ruler supported the childrenâs rea- soning more effectively than the string; their accurate performance almost doubled. Their strategies and language (it is as long as the âlittle line just after threeâ) indicated that children gave âcorrect responses based on rigorous procedures, clearly profiting from the numerical representation available through the rulerâ (p. 46). They did even better with the broken ruler than the string, showing that they were not just reading numbers off the ruler. The unusual context confused children only 20 percent of the time. The researchers concluded that conventional units already chosen

200 MATHEMATICS LEARNING IN EARLY CHILDHOOD and built into the ruler do not make measurement more difficult. Indeed, children benefited from the numerical representation provided by even the broken ruler. Such research has led several authors to argue that early rule use should be encouraged, not avoided or delayed (Clements, 1999c; NÃ¼hrenbÃ¶rger, 2001; Nunes et al., 1993). Rulers allow children to connect instruction to their previous measurement experiences with conventional tools. In con- trast, dealing with informal, 3-D units deemphasizes the one-dimensional (1-D) nature of length and focuses on the counting of discrete objects. In this way, it deemphasizes both the zero point and the iteration of line seg- ment lengths as units (Bragg and Outhred, 2001). The Piagetian-based argument, that children must conserve length be- fore they can make sense of ready-made systems, such as rulers (or com- puter tools, such as those discussed in the following section), may be an overstatement. Findings of these studies support a Vygotskian perspective (Ellis et al., 2000; Miller, 1989), in which rulers are viewed as cultural instruments children can appropriate. That is, children can use rulers, ap- propriate them, and so build new mental tools. Not only do children prefer using rulers, but also they can use them meaningfully and in combination with manipulable units to develop understanding of length measurement. In general, measurement procedures can serve as cognitive tools (Miller, 1989) developed to solve certain practical problems and organize the way children think about amount. Measurement concepts may originally be organized in terms of the contexts and procedures used to judge, compare, or measure specific attributes (Miller, 1989). If so, transformations that do not change length but do change number, such as cutting, may be particularly difficult for children, more so than traditional conservation questions. Children need to learn to distinguish the different attributes (e.g. length, number) and learn which transformations affect which attributes. Another Piagetian idea, from the field of social cognition, is that con- flict is the genesis of cognitive growth. One series of studies, however, indicates that this is not always so. If two strategies, measurement and direct comparison, were in conflict, children learned little and benefited little from verbal instruction. However, if children saw that the results of measurement and direct comparison agreed, then they were more likely to use measurement later than were children who observed both procedures but did not have the opportunity to compare their results (Bryant, 1982). This is a case in which presenting children with conflicting information (between strategies or between results of measuring with different units) too soon is unhelpful or deleterious. Whatever the specific instructional approach taken, research demon- strates several implications. Measurement should not be taught as a simple skill. It is a complex combination of concepts and skills that develops over years. Teachers who understand the foundational concepts of measurement

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 201 will be better able to interpret childrenâs understanding and ask questions that will lead them to construct these ideas. Both research with children and interviews with teachers support the claims that (a) the principles of measurement are difficult for children, (b) they require more attention in school than is usually given, (c) time needs to first be spent in informal measurement, in which the use of measurement principles is evident, and (d) transition from informal to formal measurement needs much more time and care, with instruction in formal measure always returning to basic principles (see Irwin, Vistro-Yu, and Ell, 2004). The sequence in Table 6-3 summarizes achievable goals in linear mea- surement that have been employed in pilot-testing of research-based curri- cula (Casey et al., 2004; Clements and Sarama, 2004; Greenes et al., 2004; Starkey et al., 2004). Again, evaluations confirm the appropriateness of the sequencing (Clements and Sarama, 2007c, in press; Starkey et al., 2004, 2006). Area Measurement and Spatial Structuring Area is an amount of 2-D surface that is contained within a boundary. Area measurement assumes that a suitable 2-D region is chosen as a unit, congruent regions have equal areas, regions do not overlap, and the area of the union of two regions that do not overlap (disjoint union) is the sum of their areas (Reynolds and Wheatley, 1996). Thus, finding the area of a region can be thought of as tiling (or equal partitioning) a region with a 2-D unit of measure. Such understandings are complex, and children develop them over time. These area understandings do not develop well in traditional U.S. instruction (Carpenter et al., 1975), not only for young children, but also for preservice teachers (Enochs and Gabel, 1984). A study of children from Grades 1, 2, and 3 revealed little understanding of area measurement (Lehrer, Jenkins, and Osana, 1998). Asked how much space a square (and a triangle) cover, 41 percent of children used a ruler to measure length. Although area measurement is typically emphasized in the intermediate grades, the literature suggests that some less formal aspects of area measurement can be introduced in earlier years. Concepts that are essential to understanding and learning area measurement are described in Appendix B. One especially important one, spatial structuring, is discussed next. Nascent awareness of area is often noticed in informal observations, such as when a child asks for pieces of colored paper to cover their table. A way to more formally assess childrenâs understanding of area is through comparison tasks. Some researchers report that preschoolers use only one dimension or one salient aspect of the stimulus to compare the area of two surfaces (Bausano and Jeffrey, 1975; Maratsos, 1973; Mullet and Paques, 1991; Piaget et al., 1960; Raven and Gelman, 1984; Russell, 1975; Sena

202 MATHEMATICS LEARNING IN EARLY CHILDHOOD TABLE 6-3â Linear Measurement (Space in One Dimension) Goals C. Perceive, Say, Steps/ A. Perceive, Say, B. Perceive, Say, Describe/Discuss, and Ages Describe/Discuss, and Describe/Discuss, and Construct Compositions (Levels of Construct Objects in Construct Spatial and Decompositions in Thinking) 1-D Space Relations in 1-D Space 1-D Space Step 1 (Ages 2 and 3) Thinking Informally recognize length as extent of 1-D space. Informally combine visually/ Compare 2 objects directly, noting equality or objects in linear extent. holistically inequality. Step 2 (Age 4) Thinking Compare the length of two objects by representing Understand that lengths about them with a third object. can be concatenated. parts â¢ nitial measurement by laying units end to end, I often with units that are notably square or cubical (to facilitate physical concatenation). Relating Seriate up to six objects by length (e.g., connecting parts and cube towers). wholes Step 3 (Age 5) Thinking Measure by repeated use of a unit, moving from about units that are notably square or cubical to those parts that more closely embody one dimension (e.g., sticks or stirrers). Relating Seriate any number of objects by length, even if Add two lengths to parts and differences between consecutive lengths are not obtain the length of a wholes palpable perceptually. whole. â¢ nitial measurement with simple unit rulers, I including sticks with unit lengths marked off and other unit rulers. â¢ xplore the relationship between the size and E number of units. Interpret bar graphs to answer questions such as âmore,â âless,â as well as simple trends, using length of the bars. NOTE: Less time on 1-D than on 2-D; about 5 percent of the time on 1-D.

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 203 and Smith, 1990). For example, 4- and 5-year-olds may match only one side of figures when attempting to compare their areas (Silverman, York, and Zuidema, 1984). Others claim that children can integrate more than one feature of a region but judge areas with additive combination, for example, making implicit area judgments based on the longest single di- mension (Mullet and Paques, 1991) or height + width rules (Cuneo, 1980; Rulence-Paques and Mullet, 1998). Children ages 6 to 8 use a linear ex- tent rule, such as the diagonal of a rectangle. Only after this age do most children move to explicit use of spatial structuring of multiplicative rules to solve those studiesâ tasks. Note that this does not imply formal use of multiplication, but only that their estimates are best approximated by the area formula. In most of these studies, children did not interact with the materials. Doing so often changes their strategies and improves their estimates. Chil- dren as young as age 3 are more likely to make estimates consistent with multiplicative rules when using manipulatives than when just asked to make a perceptual estimation. For example, they are more accurate when they are asked to count out the right number of square tiles to cover a floor and put them in a cup (Miller, 1984). Similarly, children ages 5 to 6 were more likely to use strategies consistent with multiplicative rules after playing with the stimulus materials (Wolf, 1995). A more accurate strategy for comparing areas than visual estimation is superimposition. Children as young as age 3 have a rudimentary concept of area based on placing regions on top of one another, but it is not until age 5 or 6 that their strategy is accurate and efficient. As an illustration, when asked to manipulate regions, preschoolers in one study used superimposi- tion instead of the less precise strategies of laying objects side-by-side or comparing single sides, both of which use one dimension at best in estimat- ing the area (Yuzawa, Bart, and Yuzawa, 2000). Again, the facilitative effect of manipulation is shown. Children were given target squares or rectangles and asked to choose one that was equal to two standard rectangles in area. They performed better when they placed the standard figures on the targets than when they made perceptual judgments. They also performed better when one target could be overlapped completely with the standard figures (even in the perceptual condition, which suggests that they performed a mental superposition). Higher steps in thinking about area may have their roots in the in- ternalization of such procedures as placing figures on one another, which may be aided by cultural tools (manipulatives) or scaffolding by adults (see Â Vygotsky, 1934/1986). For example, kindergartners who were given practice with origami (paper folding) increased the spontaneous use of the procedure of placing one figure on another for comparing sizes (Yuzawa et al., 1999). Because origami practice includes the repeated procedure

204 MATHEMATICS LEARNING IN EARLY CHILDHOOD of folding one sheet into two halves, origami practice might facilitate the development of an area concept, which is related to the spontaneous use of the procedure. To measure, a unit must be established. Teachers often assume that the product of two lengths structures a region into an area of 2-D units for students. However, the construction of a 2-D array from linear units is nontrivial. Young children often cannot partition and conserve area and instead use counting as a basis for comparing. For example, when it was determined that one share of pieces of paper cookie was too little, preschoolers cut one of that shareâs pieces into two and handed them both back, apparently believing that the share was now âmoreâ (Miller, 1984). As with length measurement, children often cover space, but they do not initially do so without gaps or overlapping (i.e., they do not tile the region with units). They also initially do not extend units over the bound- aries when a subdivision of that unit is needed to fill the surface (Stephan et al., 2003). Even more limiting, children often choose units that physically resemble the region they are covering; for example, choosing bricks to cover a rectangular region and beans to cover an outline of their hands (Lehrer, 2003; Lehrer, Jenkins, and Osana, 1998; Nunes et al., 1993). They also mix different shapes (and areas), such as rectangular and triangular, to cover the same region and accept a measure of â7â even if the seven covering shapes are of different sizes (84 percent of primary grade children; Lehrer, Jenkins, and Osana, 1998). These concepts have to be developed before children can use iteration of equal units to measure area with understanding. Once these problems have been solved, children need to structure 2-D space into an organized array of units to achieve multiplicative thinking in determining volume, a concept to which we now turn. Volume Measurement Volume introduces even more complexity, not only in adding a third dimension and thus presenting a significant challenge to studentsâ spatial structuring, but also in the very nature of the materials that are measured using volume. This leads to two ways to measure volume, illustrated by âpackingâ a space, such as a 3-D array with cubic units, and âfillingâ with iterations of a fluid unit that takes the shape of the container. For the latter, the unit structure may be psychologically 1-D for some children (i.e., simple iterative counting that is not processed as geometric 3-D), especially, for example, in filling a cylindrical jar in which the (linear) height corresponds to the volume (Curry and Outhred, 2005). Given the possible complexities, is either of these more or less appropriate for young children, beyond, say, informal experiences? For children in Grades 1-4, competence in filling volume (e.g., estimat- ing and measuring the number of cups of rice that filled a container) was

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 205 about equivalent to their competence in corresponding length tasks (Curry and Outhred, 2005). The relationship is consistent with the notion that the structure of the task is 1-D, exemplified by some studentsâ treating the height of the rice in the container as if it were a unit length and iterating it, either mentally or using their fingers, up the side of the container. Some students performed better on length, others on filling volume, giving no evidence of a relationship between the two. The task contained some extra demands, such as creating equal measurements; even many first graders made sure that the cup was not over- or underfilled for each iteration. In another study, 3- and 4-year-olds understood that unit size affects the mea- surement of the objectâs volume (Sophian, 2002). Thus, simple experience with filling volume may be appropriate for young children. On the other hand, packing volume is more difficult than length and area (Curry and Outhred, 2005). Most children had little idea of how to estiÂmate or measure on packing tasks. There were substantial increases from Grades 2 to 4, but even the older studentsâ scores were below the corresponding scores for the area task. Furthermore, there was a sugges- tion that understanding of area is a prerequisite to understanding pack- ing volume. Therefore, children should have many experiences building with blocks and filling boxes with cubes. A developmental progression is provided in Table 6-2. A full conceptual understanding of 3-D space will develop only over several years for most children. Achievable and Foundational Measurement in One, Two, and Three Dimensions In this section, we describe childrenâs development of measurement in one, two, and three dimensions. We do not consider measurement of nongeometric attributes, such as weight/mass, capacity, time, and color, because these are more appropriately considered in science and social studies curricula. Again, for each area outlined below, children should be engaged in activities that cover a range of difficulty, including perceive, say, describe/discuss, and construct. Table 6-3 outlines the path for measure- ment of length. Step 1 (Ages 2 and 3) Objects and Spatial Relations Young children naturally encounter and discuss quantities in their play (Ginsburg, Inoue, and Seo, 1999). They first learn to use words that represent quantity or magnitude of a certain attribute. Facilitating this language is important not only to develop communication abilities, but for the development of mathematical concepts. Simply using labels such as

206 MATHEMATICS LEARNING IN EARLY CHILDHOOD âDaddy/Mommy/Babyâ and âbig/little/tinyâ helped children as young as 3 years to represent and apply higher order seriation abilities, even in the face of distracting visual factors, an improvement equivalent to a 2-year gain. At the visual/holistic level (see Table 6-3), children begin by informally recognizing length as extent of 1-D space. For example, they may remark of a road made with building blocks, âThis is long.â They can then compare two objects directly and recognize and describe their equality (e.g., âYou are just as tall as I am!â) or inequality (e.g., âMy pencil is longer than yoursâ) in length. Compositions and Decompositions At the visual/holistic level, children compose lengths intuitively. For ex- ample, they may lay building blocks along a path to âmake a long road.â Step 2 (Age 4) Objects and Spatial Relations At the thinking about parts level, preschool children learn to compare the length of two objects by representing them with a third object and us- ing transitive reasoning (i.e., indirect comparison) (Boulton-Lewis et al., 1996). Again, language, such as the differences between counting-based terms (e.g., a toy, two trucks) and mass terms (e.g., some sand), can help children form relationships between counting and continuous measurement (Huntley-Fenner, 2001). Preschoolers also begin actual measurement by laying physical units end to end and counting them to measure a length. However, they may not recognize the need for equal-length units and initially may make errors, such as leaving gaps between units. One way to engage in discussions of such concepts is to apply the resulting measures to comparison situations. These concepts and skills develop in parallel with competencies in seriating lengths, which emerge last and mark the first level of thinking about relat- ing parts and wholes. Preschoolers also begin to be able to cover a rectangular space with physical tiles and represent their tilings with simple drawings, although they may leave gaps in each and may not align all the squares. Compositions and Decompositions At the thinking about parts level, preschoolers understand that lengths can be concatenated in this way. This understanding, initially implicit, is revealed as children operate on objects.

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 207 Step 3 (Age 5) Objects and Spatial Relations Kindergartners move to more sophisticated understanding at the think- ing about parts level by measuring via the repeated use of a unit. However, they initially may not be precise in such iterations. Beginning to develop aspects of thinking at the level of relating parts and wholes, they can ex- plore the concept of the inverse relationship between the size of the unit of length and the number of units required to cover a specific length or distance, recognizing it at least at an intuitive level. However, they may not appreciate the need for identical units. Work with manipulative units of standard measure (e.g., 1 inch or 1 cm), along with related use of rulers and consistent discussion, will help children learn both the concepts and procedures of linear measurement. Kindergartners also can learn to fill containers with cubes, filling one layer at a time, intentionally, all of which involves relationships at the thinking about parts level of thinking. In a similar vein, they can learn to ac- curately count the number of squares in a rectangular array, using increas- ingly systematic strategies, including counting in rows or columns. They represent a complete covering of a rectangleâs area (although initially there may be some inaccuracies, such as in the alignment of drawn shapes). Compositions and Decompositions Kindergartners understand length composition explicitly. For example, they can add to lengths to obtain the length of the whole. They can use a simple ruler (or put a length of connecting cubes together) to measure one plastic snake and measure the length of another snake to find the total of their lengths. Or, more practically, they can measure all sides of a table with unmarked (foot) rulers to measure how much ribbon they would need to decorate the perimeter of the table. Their use of rows or columns in cover- ing a rectangular area also implies at least an implicit composition of units into a composite unit. Instruction to Support the Teaching-Learning Path Length To move children through the teaching-learning path, teachers of the youngest children should observe children in their play, because they en- counter and discuss measurable quantities frequently (Ginsburg, Inoue, and Seo, 1999). Using such words as âbigger/larger/smaller,â and, as soon as possible, âlonger/shorterâ and âtaller/shorterâ directs childrenâs attention

208 MATHEMATICS LEARNING IN EARLY CHILDHOOD to these attributes and also helps them apply seriation abilities. Teachers should listen carefully to see how they are interpreting and using language (e.g., length as the distance between end points or as âone end sticking outâ). Children should be given a variety of experiences comparing the size of objects. Once they can do so by direct comparison, they should compare several items to a single item, such as finding all the objects in the class- room longer than their forearm. Ideas of transitivity can then be explicitly discussed. Next, children should engage in experiences that allow them to connect number to length. Teachers should provide children with both con- ventional rulers and manipulative units using standard units of length, such as centimeter cubes (specifically labeled âlength-unitsâ; from Dougherty and Slovin, 2004). As they explore with these tools, the ideas of length-unit iteration (e.g., not leaving space between successive length-units), correct alignment (with a ruler), and the zero-point concept can be developed. Having older (or more advanced) children draw, cut out, and use their own rulers can be used to discuss these aspects explicitly. In all activities, teachers should focus on the meaning that the numer- als on the ruler have for children, such as enumerating lengths rather than discrete numbers. In other words, classroom discussions should focus on âWhat are you counting?â with the answer in length-units. Given that counting discrete items often correctly teaches children that the length-unit size does not matter, teachers should plan experiences and reflections on the nature of properties of the length-unit in various discrete counting and measurement contexts. Comparing results of measuring the same object with manipulatives and with rulers and using manipulative length-units to make their own rulers help children connect their experiences and ideas. In second or third grade, teachers might introduce the need for standard length-units and the relation between the size and number of length-units. The relationship between the size and number of length-units, the need for standardization of length-units, and additional measuring devices can be explored at this time. The early use of multiple nonstandard length-units would not be used until this point (see Carpenter and Lewis, 1976). Instruc- tion focusing on childrenâs interpretations of their measuring activity can enable them to use flexible starting points on a ruler to indicate measures successfully (Lubinski and Thiessen, 1996). Without such attention, chil- dren are just reading off whatever ruler number aligns with the end of the object into the intermediate grades (Lehrer, Jenkins, and Osana, 1998). By kindergarten, length is used in other areas, such as understanding addition and graphing. For example, bar graphs use length to represent counts or measures. Kindergartners can answer such questions as âmoreâ and âless,â as well as simple trends, using length of the bars. Emphasis on childrenâs solving real measurement problems and, in so

PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 209 doing, building and iterating units, as well as units of units, helps them de- velop strong concepts and skills. Teachers should help children closely con- nect the use of manipulative units and rulers. When conducted in this way, measurement tools and procedures become tools for mathematics and tools for thinking about mathematics (Clements, 1999c; Miller, 1984, 1989). Well before first grade, children have begun the journey toward that end. Area Children need to structure an array to understand area as truly 2-D (see Appendix B). Play with structured materials, such as unit blocks, pattern blocks, and tiles, can lay the groundwork for childrenâs spatial structuring, although achieving the conceptual benchmark will not be achieved until a Â fter the primary grades for most children, even with high-quality instruc- tion. In brief, the too-frequent practice of simple counting of units to find area (achievable by preschoolers) leading directly to teaching formulas may not build the requisite foundational concepts (Lehrer, 2003). Instead, educators should build on young childrenâs initial spatial intuitions and ap- preciate their need to construct the idea of measurement unitsâincluding development of a measurement sense for standard units, for example, find- ing common objects in the environment that have a unit measure. Children need to have many experiences covering quantities with appropriate mea- surement units, counting those units, and spatially structuring the object they are to measure, in order to build a firm foundation for eventual use for formulas. For example, children might build rectangular arrays with square tiles and learn to count the number of manipulatives used in each. Eventually, they need to link counting by groups to reflect the structure of rectangular arrays, for example, counting the squares in an array by skip- counting the number in each row. This long developmental process usually only begins in the years before first grade. However, we should also appreciate the importance of these early conceptualizations. For example, 3- and 4-year-oldsâ use of a linear rating scale to judge area, even if using an additive rule, indicates an im- pressive level of quantitative ability and, according to some, nascent mental structures for algebra at an early age (Cuneo, 1980). Competencies in the major realms of geometry/spatial thinking and number are connected throughout development. The earliest competen- cies may share common perceptual and representational origins (Mix, H Â uttenlocher, and Levine, 2002). Infants are sensitive to both the amount of liquid in a container (Gao, Levine, and Huttenlocher, 2000) and the distance away a toy is hidden in a long sandbox (Newcombe, Â Huttenlocher, and Learmonth, 1999). Visual-spatial deficits in early childhood are Âdetrimental to childrenâs development of numerical competencies (Semrud-Clikeman

210 MATHEMATICS LEARNING IN EARLY CHILDHOOD and Hynd, 1990; Spiers, 1987). Other evidence shows specific spatially re- lated learning disabilities in arithmetic, possibly more so for boys than girls (Share, Moffitt, and Silva, 1988). Primary school childrenâs thinking about units and units of units was found to be consistent in both spatial and nu- merical problems (Clements et al., 1997a). In this and other ways, specific spatial abilities appear to be related to other mathematical competencies (Brown and Wheatley, 1989; Clements and Battista, 1992; Fennema and Carpenter, 1981; Wheatley, Brown, and Solano, 1994). Geometric measure- ment connects the spatial and numeric realms explicitly. SUMMARY This chapter describes geometry and spatial thinking and measure- ment, which comprise the second essential domain for young childrenâs mathematical development. The research in this domain is less developed than for number, but it does provide guidance for educators regarding what young children can and should do to develop competence in these areas. The teaching-learning path for geometry and spatial relations demonstrates how young children move through levels of thinking as they learn about 2-D and 3-D objects. The use of manipulatives, pictures, and computers play an important role in facilitating childrenâs progress along this path. Early childhood teachers should help children extend their thinking by building on simple conventional models (e.g., child represents classroom with cut out pictures) and challenge them by asking them to use geometric correspondences (e.g., directionâwhich way?, identificationâwhich ob- ject?) to solve problems. Measurement, the second major area covered in this chapter, con- nects and enriches the two crucial domains of geometry and number. The teaching-learning path for measurement describes childrenâs developing competence in linear measurement and initial steps toward understanding areas and volume. The teaching-learning path outlined for length em- phasizes the need to provide experiences that allow children to compare the size of objects and to connect number to length. Children also need opportunities to solve real measurement problems which can help build their understanding of units, length-unit iteration, correct alignment and the zero-point concept. Childrenâs early competency in measurement is facilitated by play with structured materials, such as unit blocks, pattern blocks, and tiles and strengthened through opportunities to reflect on and discuss their experiences. It is important to note that the potential of young childrenâs learning in geometry and measurement if a conscientious, sequenced development of spatial thinking and geometry were provided to them throughout their earliest years is not yet known. Research on the learning of shapes and

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