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 provide an overview of children’s 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 “Geometry 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?” (Freudenthal, 1973, p. 403). This section describes the two major ways children understand that space, starting with smaller scale perspectives on geometric shape, including composition 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 children’s learning of both geometric and spatial abilities.



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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 children’s 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 “Geometry 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?” (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 children’s learning of both geometric and spatial abilities. 15

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16 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 objects’ 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, Battista, 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 “it looks like a door.” 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 shapes—for example, the boundaries of two-dimensional (2-D) and three-dimensional (3-D) shapes—and 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 “thinking about parts” 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

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1 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT TABLE 6-1 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). • Use relational language, objects, each with a including vertical different shape. directionality terms as “up” and “down,” referring to a 2-D environment. • Informally recognizes area as filling 2-D space (e.g., “I need more papers to cover this table”). 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., “long” orientations, and “skinny” rectangles and sizes, and shapes triangles). (includes circles • Match shapes by using and half/quarter geometric motions to circles, squares superimpose them. and rectangles, • Use relational words triangles, and of proximity, such as others [the pattern “beside,” “next to,” and block rhombus, “between,” referring to a 2-D trapezoids, hexagons environment. regular]). continued

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1 MATHEMATICS LEARNING IN EARLY CHILDHOOD TABLE 6-1 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 • Use relational language to combine shapes to number they can involving frames of reference, build pictures. count). such as “to this side of,” For rectangular Describe and name “above.” spaces shapes by number of • Compare areas by • Copy a design corners (vertices). superimposition. shown on a grid, For rectangular spaces placing squares • Tile a rectangular space with 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 • Right vs. nonright recognizing them as angles. new shapes. • In an “equilateral triangle world,” 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).

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1 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT TABLE 6-1 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 60° and 120° angles). • Begin to use relational language of “right” and “left.” • Draw a complete covering 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 • For rectangular regions, draw 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 • Identify and create symmetric 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 “relating parts and wholes” level. Deelopment 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

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10 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). Children’s prototypical image of a rectangle seems to be a four-sided figure with two long parallel sides and “close to” 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 children’s 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 “the point on top is not in the middle.” 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 “too skinny” or “not wide enough.” 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

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11 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 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., “on” 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 “beside” and “between.” Later, they learn words referring to frames of reference, such as “in front of,” “behind.” The words “left” and “right” 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 “left” and “right” 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 (“under the table

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12 MATHEMATICS LEARNING IN EARLY CHILDHOOD that’s next to the door”), 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 (“it’s in the bag on the table”), 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 3½ 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 objects’ 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).

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1 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 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 “picture in the head.” 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. People’s 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 “move” 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 Whitehurst, 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 “snapshot” activities (Clements, 1999b; Yackel and Wheatley, 1990). Children briefly

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1 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 “what I saw.” 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 children’s 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 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,

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15 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT 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 children’s engagement with mathematics should build and develop over the prekin- dergarten years. In the tables, children’s competence within each band is described on the basis of the level of sophistication in their thinking. These levels are called thinking isually/holistically, thinking about parts, and relating parts and wholes. Step 1 (Ages 2 and 3) 2-D and -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 rectangles—at least in their prototypical forms—and 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 “up” and “down” and similar vocabulary. They learn to apply that vocabulary in both 3-D contexts and in 2-D situations, such as the “bottom” 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-

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