HARDBACK
\$54.95

### 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.

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 175
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. 15

OCR for page 175
16 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

OCR for page 175
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

OCR for page 175
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).

OCR for page 175
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. Deelopment 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

OCR for page 175
10 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

OCR for page 175
11 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

OCR for page 175
12 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).

OCR for page 175
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

OCR for page 175
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,

OCR for page 175
15 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-

OCR for page 175
212 MATHEMATICS LEARNING IN EARLY CHILDHOOD Boulton-Lewis, G.M. (1987). Recent cognitive theories applied to sequential length measuring knowledge in young children. British Journal of Educational Psychology, 5, 330-342. Boulton-Lewis, G.M., Wilss, L.A., and Mutch, S.L. (1996). An analysis of young children’s strategies and use of devices of length measurement. Journal of Mathematical Behaior, 15, 329-347. Bowerman, M. (1996). Learning how to structure space for language: A cross-linguistic per- spective. In P. Bloom, M.A. Peterson, L. Nadel, and M.F. Garrett (Eds.), Language and Space (pp. 385-436). Cambridge, MA: MIT Press. Bragg, P., and Outhred, L. (2001). So that’s what a centimetre looks like: Students’ under- standings of linear units. In M.V.D. Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology in Mathematics Education (vol. 2, pp. 209-216). Utrecht, The Netherlands: Freudenthal Institute. Bronowski, J. (1947). Mathematics. In D. Thompson and J. Reeves (Eds.), The Quality of Education. London, England: Muller. Brosnan, M.J. (1998). Spatial ability in children’s play with Lego blocks. Perceptual and Mo- tor Skills, , 19-28. Brown, D.L., and Wheatley, G.H. (1989). Relationship between spatial knowledge and math- ematics knowledge. In C.A. Maher, G.A. Goldin, and R.B. Davis (Eds.), Proceedings of the Eleenth Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 143-148). New Brunswick, NJ: Rutgers University. Bryant, P.E. (1982). The role of conflict and of agreement between intellectual strategies in children’s ideas about measurement. British Journal of Psychology, , 242-251. Burger, W.F., and Shaughnessy, J.M. (1986). Characterizing the van Hiele levels of develop- ment in geometry. Journal for Research in Mathematics Education, 1, 31-48. Carpenter, T.P., and Lewis, R. (1976). The development of the concept of a standard unit of measure in young children. Journal for Research in Mathematics Education, , 53-58. Carpenter, T.P., Coburn, T., Reys, R., and Wilson, J. (1975). Notes from national assessment: Basic concepts of area and volume. Arithmetic Teacher, 22, 501-507. Carpenter, T.P., Corbitt, M.K., Kepner, H.S., Lindquist, M.M., and Reys, R.E. (1980). Na- tional assessment. In E. Fennema (Ed.), Mathematics Education Research: Implications for the 10s (pp. 22-38). Alexandria, VA: Association for Supervision and Curriculum Development. Casey, B., Paugh, P., and Ballard, N. (2002). Sneeze Builds a Castle. Bothell, WA: The Wright Group/McGraw-Hill. Casey, B., Kersh, J.E., and Young, J.M. (2004). Storytelling sagas: An effective medium for teaching early childhood mathematics. Early Childhood Research Quarterly, 1, 167-172. Casey, M.B. (2005, April). Ealuation of NSF-Funded Mathematics Materials: Use of Story- telling Contexts to Improe Kindergartners’ Geometry and Block-Building Skills. Paper presented at the National Council of Supervisors of Mathematics, Anaheim, CA. Casey, M.B., and Erkut, S. (2005, April). Early Spatial Interentions Benefit Girls and Boys. Paper presented at the Biennial Meeting of the Society for Research in Child Develop- ment, Atlanta, GA. Casey, M.B., and Erkut, S. (in press). Use of a storytelling context to improve girls’ and boys’ geometry skills in kindergarten. Journal of Applied Deelopmental Psychology. Casey, M.B., Nuttall, R.L., and Pezaris, E. (2001). Spatial-mechanical reasoning skills versus mathematics self-confidence as mediators of gender differences on mathematics subtests using cross-national gender-based items. Journal for Research in Mathematics Educa- tion, 2, 28-57.

OCR for page 175
21 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT Casey, M.B., Andrews, N., Schindler, H., Kersh, J.E., and Samper, A. (in press). The develop- ment of spatial skills through interventions involving block building activities. Cognition and Instruction. Clements, D.H. (1999a). Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60. Clements, D.H. (1999b). Subitizing: What is it? Why teach it? Teaching Children Mathemat- ics, 5, 400-405. Clements, D.H. (1999c). Teaching length measurement: Research challenges. School Science and Mathematics, (1), 5-11. Clements, D.H. (2003). Teaching and learning geometry. In J. Kilpatrick, W.G. Martin, and D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathemat- ics (pp. 151-178). Reston, VA: National Council of Teachers of Mathematics. Clements, D.H., and Barrett, J. (1996). Representing, connecting and restructuring knowledge: A micro-genetic analysis of a child’s learning in an open-ended task involving perimeter, paths and polygons. In E. Jakubowski, D. Watkins, and H. Biske (Eds.), Proceedings of the 1th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 211-216). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Clements, D.H., and Battista, M.T. (1991). The Deelopment of a Logo-Based Elementary School Geometry Curriculum. Final Report, NSF Grant No. MDR 8651668. Buffalo, NY/Kent State, OH: State University of New York and Kent State University Presses. Clements, D.H., and Battista, M.T. (1992). Geometry and spatial reasoning. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 420-464). New York: Macmillan. Clements, D.H., and Battista, M.T. (2001). Length, perimeter, area, and volume. In L.S. Grinstein and S.I. Lipsey (Eds.), Encyclopedia of Mathematics Education (pp. 406-410). New York: RoutledgeFalmer. Clements, D.H., and McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270-279. Clements, D.H., and Sarama, J. (1998). Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Deelopment. NSF Grant No. ESI-9730804. Buffalo: State University of New York. Available: http://www. gse.buffalo.edu/org/buildingblocks/ [accessed July 2009]. Clements, D.H., and Sarama, J. (2003). Young children and technology: What does the re- search say? Young Children, 5(6), 34-40. Clements, D.H., and Sarama, J. (2004). Building blocks for early childhood mathematics. Early Childhood Research Quarterly, 1, 181-189. Clements, D.H., and Sarama, J. (2007a). Building Blocks—SRA Real Math Teacher’s Edition, Grade PreK. Columbus, OH: SRA/McGraw-Hill. Clements, D.H., and Sarama, J. (2007b). Early childhood mathematics learning. In F.K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age. Clements, D.H., and Sarama, J. (2007c). Effects of a preschool mathematics curriculum: Summative research on the building blocks project. Journal for Research in Mathematics Education, , 136-163. Clements, D.H., and Sarama, J. (in press). Experimental evaluation of the effects of a research- based preschool mathematics curriculum. American Educational Research Journal. Clements, D.H., and Stephan, M. (2004). Measurement in preK-2 mathematics. In D.H. Clements, J. Sarama and A.-M. DiBiase (Eds.), Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp. 299-317). Mahwah, NJ: Erlbaum.

OCR for page 175
21 MATHEMATICS LEARNING IN EARLY CHILDHOOD Clements, D.H., Sarama, J., Battista, M.T., and Swaminathan, S. (1996). Development of students’ spatial thinking in a curriculum unit on geometric motions and area. In E. Jakubowski, D. Watkins, and H. Biske (Eds.), Proceedings of the 1th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathemat- ics Education (vol. 1, pp. 217-222). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Clements, D H., Battista, M.T., Sarama, J., and Swaminathan, S. (1997a). Development of students’ spatial thinking in a unit on geometric motions and area. The Elementary School Journal, , 171-186. Clements, D.H., Battista, M.T., Sarama, J., Swaminathan, S., and McMillen, S. (1997b). Stu- dents’ development of length measurement concepts in a logo-based unit on geometric paths. Journal for Research in Mathematics Education, 2(1), 70-95. Clements, D.H., Swaminathan, S., Hannibal, M.A.Z., and Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 0, 192-212. Clements, D.H., Battista, M.T., and Sarama, J. (2001). Logo and geometry. Journal for Re- search in Mathematics Education Monograph Series, 10. Clements, D.H., Sarama, J., and Wilson, D.C. (2001). Composition of geometric figures. In M.v.d. Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 273-280). Utrecht, The Netherlands: Freudenthal Institute. Clements, D.H., Wilson, D.C., and Sarama, J. (2004). Young children’s composition of geo- metric figures: A learning trajectory. Mathematical Thinking and Learning, 6, 163-184. Cooper, T.J., and Warren, E. (2007, April). Deeloping Equialence of Expressions in the Early to Middle Elementary Years. Paper presented at the Research Pre-session of the 85th An- nual Meeting of the National Council of Teachers of Mathematics, Atlanta, GA. Cuneo, D. (1980). A general strategy for quantity judgments: The height + width rule. Child Deelopment, 51, 299-301. Curry, M., and Outhred, L. (2005). Conceptual understanding of spatial measurement. In P. Clarkson, A. Downtown, D. Gronn, M. Horne, A. McDonough, R. Pierce and A. Roche (Eds.), Building Connections: Research, Theory, and Practice: Proceedings of the 2th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 265-272). Melbourne, Australia: MERGA. Dehaene, S., Izard, V., Pica, P., and Spelke, E.S. (2006). Core knowledge of geometry in an Amazonian indigene group. Science, 11, 381-384. Del Grande, J.J. (1986). Can grade two children’ spatial perception be improved by inserting a transformation geometry component into their mathematics program? Dissertation Abstracts International, , 3689A. Dewey, J. (1933). How We Think: A Restatement of the Relation of Reflectie Thinking to the Educatie Process. Boston, MA: D.C. Heath. Dougherty, B.J., and Slovin, H. (2004). Generalized diagrams as a tool for young children’s problem solving. In M.J. Høines and A.B. Fuglestad (Eds.), Proceedings of the 2th Conference of the International Group for the Psychology in Mathematics Education (vol. 2, pp. 295-302). Bergen, Norway: Bergen University College. Ebbeck, M. (1984). Equity for boys and girls: Some important issues. Early Child Deelop- ment and Care, 1, 119-131. Ehrlich, S.B., Levine, S.C., and Goldin-Meadow, S. (2005, April). Early Sex Differences in Spatial Skill: The Implications of Spoken and Gestured Strategies. Paper presented at the Biennial Meeting of the Society for Research in Child Development, Atlanta, GA. Ellis, S. (1995). Deelopmental Changes in Children’s Understanding of measurement Proce- dures and Principles. Paper presented at the biennial meetings of the Society for Research in Child Development, Indianapolis, IN.

OCR for page 175
215 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT Enochs, L.G., and Gabel, D.L. (1984). Preservice elementary teaching conceptions of volume. School Science and Mathematics, , 670-680. Eylon, B.-S., and Rosenfeld, S. (1990). The Agam Project: Cultiating Visual Cognition in Young Children. Rehovot, Israel: Department of Science Teaching, Weizmann Institute of Science. Fennema, E.H., and Carpenter, T.P. (1981). Sex-related differences in mathematics: Results from national assessment. Mathematics Teacher, , 554-559. Fennema, E.H., and Sherman, J. (1977). Sex-related differences in mathematics achievement, spatial visualization, and affective factors. American Educational Research Journal, 1, 51-71. Fennema, E.H., and Sherman, J.A. (1978). Sex-related differences in mathematics achievement and related factors. Journal for Research in Mathematics Education, , 189-203. Fey, J., Atchison, W.F., Good, R.A., Heid, M.K., Johnson, J., Kantowski, M.G., et al. (1984). Computing and Mathematics: The Impact on Secondary School Curricula. College Park: University of Maryland. Filippaki, N., and Papamichael, Y. (1997). Tutoring conjunctions and construction of geom- etry concepts in the early childhood education: The case of the angle. European Journal of Psychology of Education, 12(3), 235-247. Fisher, N.D. (1978). Visual influences of figure orientation on concept formation in geometry. Dissertation Abstracts International, , 4639A. Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrecht, The Netherlands: Reidel. Fuson, K.C., and Hall, J.W. (1982). The acquisition of early number word meanings: A con- ceptual analysis and review. In H.P. Ginsburg (Ed.), Children’s Mathematical Thinking (pp. 49-107). New York: Academic Press. Fuson, K.C., and Murray, C. (1978). The haptic-visual perception, construction, and drawing of geometric shapes by children ages two to five: A Piagetian extension. In R. Lesh and D. Mierkiewicz (Eds.), Concerning the Deelopment of Spatial and Geometric Concepts (pp. 49-83). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Envi- ronmental Education. Fuys, D., Geddes, D., and Tischler, R. (1988). The an Hiele Model of Thinking in Geometry Among Adolescents. Reston, VA: National Council of Teachers of Mathematics. Gagatsis, A., and Patronis, T. (1990). Using geometrical models in a process of reflective thinking in learning and teaching mathematics. Educational Studies in Mathematics, 21, 29-54. Gao, F., Levine, S.C., and Huttenlocher, J. (2000). What do infants know about continuous quantity? Journal of Experimental Child Psychology, , 20-29. Gelman, R., and Williams, E.M. (1997). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In D. Kuhn and R. Siegler (Eds.), Cognition, Perception, and Language, Volume 2: Handbook of Child Psychology (5th ed., pp. 575- 630). New York: Wiley. Gerhardt, L.A. (1973). Moing and Knowing: The Young Child Orients Himself in Space. Englewood Cliffs, NJ: Prentice-Hall. Gibson, E.J., Gibson, J.J., Pick, A.D., and Osser, H. (1962). A developmental study of the discrimination of letter-like forms. Journal of Comparatie and Physiological Psychol- ogy, 55, 897-906. Ginsburg, A., Cooke, G., Leinwand, S., Noell, J., and Pollock, E. (2005). Reassessing U.S. International Mathematics Performance: New Findings from the 200 TIMSS and PISA. Washington, DC: American Institutes for Research. Ginsburg, H.P., Inoue, N., and Seo, K.-H. (1999). Young children doing mathematics: Obser- vations of everyday activities. In J.V. Copley (Ed.), Mathematics in the Early Years (pp. 88-99). Reston, VA: National Council of Teachers of Mathematics.

OCR for page 175
216 MATHEMATICS LEARNING IN EARLY CHILDHOOD Ginsburg, H.P., Greenes, C., and Balfanz, R. (2003). Big Math for Little Kids. Parsippany, NJ: Dale Seymour. Gopnik, A., and Meltzoff, A.N. (1986). Words, plans, things, and locations: Interactions between semantic and cognitive development in the one-word stage. In S.A. Kuczaj, II, and M.D. Barrett (Eds.), The Deelopment of Word Meaning (pp. 199-223). Berlin, Germany: Springer-Verlag. Greabell, L.C. (1978). The effect of stimuli input on the acquisition of introductory geomet- ric concepts by elementary school children. School Science and Mathematics, (4), 320-326. Greenes, C., Ginsburg, H.P., and Balfanz, R. (2004). Big math for little kids. Early Childhood Research Quarterly, 1, 159-166. Guay, R.B., and McDaniel, E. (1977). The relationship between mathematics achievement and spatial abilities among elementary school children. Journal for Research in Mathematics Education, , 211-215. Hannibal, M.A.Z., and Clements, D.H. (2008). Young Children’s Understanding of Basic Geometric Shapes. Manuscript submitted. Héraud, B. (1989). A conceptual analysis of the notion of length and its measure. In G. Vergnaud, J. Rogalski, and M. Artique (Eds.), Proceedings of the 1th Conference of the International Group for the Psychology of Mathematics Education (pp. 83-90). Paris, France: City University. Hiebert, J.C. (1981). Cognitive development and learning linear measurement. Journal for Research in Mathematics Education, 12, 197-211. Hiebert, J.C. (1984). Why do some children have trouble learning measurement concepts? Arithmetic Teacher, 1(7), 19-24. Hofmeister, A.M. (1993). Elitism and reform in school mathematics. Remedial and Special Education, 1(6), 8-13. Horvath, J., and Lehrer, R. (2000). The design of a case-based hypermedia teaching tool. International Journal of Computers for Mathematical Learning, 5, 115-141. Howlin, P., Davies, M., and Udwin, U. (1998). Syndrome specific characteristics in Williams syndrome: To what extent do early behavioral patterns persist into adult life? Journal of Applied Research in Intellectual Disabilities, 11, 207-226. Huntley-Fenner, G. (2001). Why count stuff?: Young preschoolers do not use number for measurement in continuous dimensions. Deelopmental Science, , 456-462. Inhelder, B., Sinclair, H., and Bovet, M. (1974). Learning and the Deelopment of Cognition. Cambridge, MA: Harvard University Press. Irwin, K.C., Vistro-Yu, C.P., and Ell, F.R. (2004). Understanding linear measurement: A comparison of Filipino and New Zealand children. Mathematics Education Research Journal, 16(2), 3-24. Johnson-Gentile, K., Clements, D.H., and Battista, M.T. (1994). The effects of computer and noncomputer environments on students’ conceptualizations of geometric motions. Jour- nal of Educational Computing Research, 11(2), 121-140. Jones, S.S., and Smith, L.B. (2002). How children know the relevant properties for generalizing object names. Deelopmental Science, 2, 219-232. Jordan, N.C., Kaplan, D., Oláh, L.N., and Locuniak, M.N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Deelopment, , 153-175. K-13 Geometry Committee. (1967). Geometry: Kindergarten to Grade Thirteen. Toronto, Canada: Ontario Institute for Studies in Education. Kabanova-Meller, E.N. (1970). The role of the diagram in the application of geometric theo- rems. In J. Kilpatrick and I. Wirszup (Eds.), Soiet Studies in the Psychology of Learning and Teaching Mathematics (vol. 4, pp. 7-49). Chicago: University of Chicago Press.

OCR for page 175
21 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT Kamii, C., and Clark, F.B. (1997). Measurement of length: The need for a better approach to teaching. School Science and Mathematics, , 116-121. Kamii, C., Miyakawa, Y., and Kato, Y. (2004). The development of logico-mathematical knowledge in a block-building activity at ages 1-4. Journal of Research in Childhood Education, 1, 13-26. Kay, C.S. (1987). Is a Square a Rectangle? The Development of First-Grade Students’ Un- derstanding of Quadrilaterals with Implications for the van Hiele Theory of the De- velopment of Geometric Thought. Doctoral dissertation, University of Georgia, 1986. Dissertation Abstracts International, (08), 2934A. Kersh, J., Casey, B., and Young, J.M. (2008). Research on spatial skills and block building in girls and boys: The relationship to later mathematics learning. In B. Spodek and O.N. Saracho (Eds.), Contemporary Perspecties on Mathematics, Science, and Technology in Early Childhood Education. Charlotte, NC: Information Age. Klein, A., Starkey, P., and Wakeley, A. (1999). Enhancing Pre-kindergarten Children’s Readi- ness for School Mathematics. Paper presented at the American Educational Research Association. Klein, A., Starkey, P., and Ramirez, A.B. (2002). Pre-K Mathematics Curriculum. Glenview, IL: Scott Foresman. Kosslyn, S.M. (1983). Ghosts in the Mind’s Machine. New York: W.W. Norton. Kouba, V.L., Brown, C.A., Carpenter, T.P., Lindquist, M.M., Silver, E.A., and Swafford, J.O. (1988). Results of the fourth NAEP assessment of mathematics: Measurement, geometry, data interpretation, attitudes, and other topics. Arithmetic Teacher, 5(9), 10-16. Lappan, G. (1999). Geometry: The forgotten strand. NCTM News Bulletin, 6(5), 3. Lean, G., and Clements, M.A. (1981). Spatial ability, visual imagery, and mathematical per- formance. Educational Studies in Mathematics, 12, 267-299. Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W.G. Mar- tin, and D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 179-192). Reston, VA: National Council of Teachers of Mathematics. Lehrer, R., Jenkins, M., and Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer and D. Chazan (Eds.), Designing Learning Enironments for Deeloping Understanding of Geometry and Space (pp. 137-167). Mahwah, NJ: Erlbaum. Lehrer, R., Jacobson, C., Thoyre, G., Kemeny, V., Strom, D., Horvarth, J., et al. (1998). De- veloping understanding of geometry and space in the primary grades. In R. Lehrer and D. Chazan (Eds.), Designing Learning Enironments for Deeloping Understanding of Geometry and Space (pp. 169-200). Mahwah, NJ: Erlbaum. Leushina, A.M. (1974/1991). The Deelopment of Elementary Mathematical Concepts in Preschool Children (vol. 4). Reston, VA: National Council of Teachers of Mathematics. Levine, S.C., Huttenlocher, J., Taylor, A., and Langrock, A. (1999). Early sex differences in spatial skill. Deelopmental Psychology, 5(4), 940-949. Lubinski, C.A., and Thiessen, D. (1996). Exploring measurement through literature. Teaching Children Mathematics, 2, 260-263. Mansfield, H.M., and Scott, J. (1990). Young children solving spatial problems. In G. Booker, P. Cobb and T.N. deMendicuti (Eds.), Proceedings of the 1th Annual Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 275-282). Oaxlepec, Mexico: International Group for the Psychology of Mathematics Education. Maratsos, M.P. (1973). Decrease in the understanding of the word “big” in preschool children. Child Deelopment, , 747-752.

OCR for page 175
21 MATHEMATICS LEARNING IN EARLY CHILDHOOD Miller, K.F. (1984). Child as the measurer of all things: Measurement procedures and the development of quantitative concepts. In C. Sophian (Ed.), Origins of Cognitie Skills: The Eighteenth Annual Carnegie Symposium on Cognition (pp. 193-228). Hillsdale, NJ: Erlbaum. Miller, K.F. (1989). Measurement as a tool of thought: The role of measuring procedures in children’s understanding of quantitative invariance. Deelopmental Psychology, 25, 589-600. Mix, K.S., Huttenlocher, J., and Levine, S.C. (2002). Quantitatie Deelopment in Infancy and Early Childhood. New York: Oxford University Press. Mullet, E., and Paques, P. (1991). The height + width = area of a rectangle rule in five-year- olds: Effects of stimulus distribution and graduation of the response scale. Journal of Experimental Child Psychology, 52(3), 336-343. Mulligan, J., Prescott, A., and Mitchelmore, M.C. (2004). Children’s development of structure in early mathematics. In M.J. Høines and A.B. Fuglestad (Eds.), Proceedings of the 2th Conference of the International Group for the Psychology in Mathematics Education (vol. 3, pp. 393-401). Bergen, Norway: Bergen University College. Mullis, I.V.S., Martin, M.O., Beaton, A.E., Gonzalez, E.J., Kelly, D.L., and Smith, T.A. (1997). Mathematics Achieement in the Primary School Years: IEA’s Third International Math- ematics and Science Study (TIMSS). Chestnut Hill, MA: Center for the Study of Testing, Evaluation, and Educational Policy, Boston College. Murphy, C.M., and Wood, D.J. (1981). Learning from pictures: The use of pictorial informa- tion by young children. Journal of Experimental Child Psychology, 2, 279-297. National Council of Teachers of Mathematics. (1989). Curriculum and Ealuation Standards for School Mathematics. Reston, VA: Author. National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Adisory Panel. Washington, DC: U.S. Department of Educa- tion. Available: http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf [accessed August 2008]. Newcombe, N.S., and Huttenlocher, J. (2000). Making Space: The Deelopment of Spatial Representation and Reasoning. Cambridge, MA: MIT Press. Newcombe, N., and Sanderson, H.L. (1993). The Relation Between Preschoolers’ Eery- day Actiities and Spatial Ability. New Orleans, LA: Society for Research in Child Development. Newcombe, N.S., Huttenlocher, J., and Learmonth, A. (1999). Infants’ coding of location in continuous space. Infant Behaior and Deelopment, 22, 483-510. Nührenbörger, M. (2001). Insights into children’s ruler concepts—Grade-2 students’ con- ceptions and knowledge of length measurement and paths of development. In M.V.D. Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology in Mathematics Education (vol. 3, pp. 447-454). Utrecht, The Neth- erlands: Freudenthal Institute. Nunes, T., and Bryant, P. (1996). Children Doing Mathematics. Cambridge, MA: Blackwell. Nunes, T., Light, P., and Mason, J.H. (1993). Tools for thought: The measurement of length and area. Learning and Instruction, , 39-54. Outhred, L.N., and Mitchelmore, M.C. (1992). Representation of area: A pictorial perspec- tive. In W. Geeslin and K. Graham (Eds.), Proceedings of the Sixteenth Psychology in Mathematics Education Conference (vol. II, pp. 194-201). Durham, NH: Program Com- mittee of the Sixteenth Psychology in Mathematics Education Conference. Palmer, S.E. (1989). Reference frames in the perception of shape and orientation. In B.E. Shepp and S. Ballesteros (Eds.), Object Perception: Structure and Process (pp. 121-163). Hillsdale, NJ: Erlbaum. Petitto, A.L. (1990). Development of number line and measurement concepts. Cognition and Instruction, , 55-78.

OCR for page 175
21 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT Piaget, J., and Inhelder, B. (1967). The Child’s Conception of Space. (F.J. Langdon and J.L. Lunzer, Trans.). New York: W.W. Norton. Piaget, J., and Inhelder, B. (1971). Mental Imagery in the Child. London, England: Routledge and Kegan Paul. Piaget, J., Inhelder, B., and Szeminska, A. (1960). The Child’s Conception of Geometry. Lon- don, England: Routledge and Kegan Paul. Plumert, J.M., and Nichols-Whitehead, P. (1996). Parental scaffolding of young children’s spatial communication. Deelopmental Psychology, 2(3), 523-532. Prigge, G.R. (1978). The differential effects of the use of manipulative aids on the learning of geometric concepts by elementary school children. Journal for Research in Mathematics Education, , 361-367. Raven, K.E., and Gelman, S.A. (1984). Rule usage in children’s understanding of “big” and “little.” Child Deelopment, 55, 2141-2150. Razel, M., and Eylon, B.-S. (1990). Development of visual cognition: Transfer effects of the Agam program. Journal of Applied Deelopmental Psychology, 11, 459-485. Reynolds, A., and Wheatley, G.H. (1996). Elementary students’ construction and coordina- tion of units in an area setting. Journal for Research in Mathematics Education, 2(5), 564-581. Rosch, E. (1975). Cognitive representations of semantic categories. Journal of Experimental Psychology: General, 10, 192-233. Rosser, R.A., Horan, P.F., Mattson, S.L., and Mazzeo, J. (1984). Comprehension of Euclidean space in young children: The early emergence of understanding and its limits. Genetic Psychology Monographs, 110, 21-41. Rulence-Paques, P., and Mullet, E. (1998). Area judgment from width and height information: The case of the rectangle. Journal of Experimental Child Psychology, 6(1), 22-48. Russell, J. (1975). The interpretation of conservation instructions by five-year-old children. Journal of Child Psychology and Psychiatry, 16, 233-244. Sales, C. (1994). A Constructiist Instructional Project on Deeloping Geometric Problem Soling Abilities Using Pattern Blocks and Tangrams with Young Children. Unpublished Masters, University of Northern Iowa, Cedar Falls. Sarama, J., and Clements, D.H. (2002). Building blocks for young children’s mathematical development. Journal of Educational Computing Research, 2(1 and 2), 93-110. Sarama, J., Clements, D.H., and Vukelic, E.B. (1996). The role of a computer manipulative in fostering specific psychological/mathematical processes. In E. Jakubowski, D. Watkins, and H. Biske (Eds.), Proceedings of the 1th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 567-572). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Semrud-Clikeman, M., and Hynd, G.W. (1990). Right hemispheric dysfunction in nonverbal learning disabilities: Social, academic, and adaptive functioning in adults and children. Psychological Bulletin, 10, 196-209. Sena, R., and Smith, L.B. (1990). New evidence on the development of the word big. Child Deelopment, 61, 1034-1052. Seo, K.-H., and Ginsburg, H.P. (2004). What is developmentally appropriate in early child- hood mathematics education? In D.H. Clements, J. Sarama, and A.-M. DiBiase (Eds.), Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp. 91-104). Mahwah, NJ: Erlbaum. Share, D.L., Moffitt, T.E., and Silva, P.A. (1988). Factors associated with arithmetic and reading disabilities and specific arithmetic disability. Journal of Learning Disabilities, 21, 313-320.

OCR for page 175
220 MATHEMATICS LEARNING IN EARLY CHILDHOOD Shepard, R.N. (1978). Externalization of mental images and the act of creation. In B.S. Randhawa and W.E. Coffman (Eds.), Visual Learning, Thinking and Communication. New York: Academic Press. Shepard, R.N., and Cooper, L.A. (1982). Mental Images and Their Transformations. Cam- bridge, MA: MIT Press. Shusterman, A., and Spelke, E. (2004). Investigations in the development of spatial reasoning: Core knowledge and adult competence. In P. Carruthers, S. Laurence, and S. Stich (Eds.), The Innate Mind: Structure and Contents. New York: Oxford University Press. Silverman, I.W., York, K., and Zuidema, N. (1984). Area-matching strategies used by young children. Journal of Experimental Child Psychology, , 464-474. Smith, I. (1964). Spatial Ability. San Diego, CA: Knapp. Sophian, C. (2002). Learning about what fits: Preschool children’s reasoning about effects of object size. Journal for Research in Mathematics Education, , 290-302. Sophian, C., and Kailihiwa, C. (1998). Units of counting: Developmental changes. Cognitie Deelopment, 1, 561-585. Sowell, E.J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20, 498-505. Spiers, P.A. (1987). Alcalculia revisited: Current issues. In G. Deloche and X. Seron (Eds.), Mathematical Disabilities: A Cognitie Neuropyschological Perspectie. Hillsdale, NJ: Erlbaum. Spitler, M.E., Sarama, J., and Clements, D.H. (2003). A Preschooler’s Understanding of a Triangle: A Case Study. Paper presented at the 81st Annual Meeting of the National Council of Teachers of Mathematics. Starkey, P., Klein, A., Chang, I., Qi, D., Lijuan, P., and Yang, Z. (1999, April). Enironmental Supports for Young Children’s Mathematical Deelopment in China and the United States. Paper presented at the Society for Research in Child Development, Albuquerque, NM. Starkey, P., Klein, A., and Wakeley, A. (2004). Enhancing young children’s mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Re- search Quarterly, 1, 99-120. Starkey, P., Klein, A., Sarama, J., and Clements, D.H. (2006). Preschool Curriculum Ealua- tion Research. Paper presented at the American Educational Research Association. Steffe, L.P. (1991). Operations that generate quantity. Learning and Indiidual Differences, , 61-82. Steffe, L.P., and Cobb, P. (1988). Construction of Arithmetical Meanings and Strategies. New York: Springer-Verlag. Stephan, M., Bowers, J., Cobb, P., and Gravemeijer, K.P.E. (2003). Supporting Students’ De- elopment of Measuring Conceptions: Analyzing Students’ Learning in Social Context (vol. 12). Reston, VA: National Council of Teachers of Mathematics. Stevenson, H.W., and McBee, G. (1958). The learning of object and pattern discrimination by children. Journal of Comparatie and Psychological Psychology, 51, 752-754. Stevenson, H.W., Lee, S.-Y., and Stigler, J.W. (1986). Mathematics achievement of Chinese, Japanese, and American children. Science, 21, 693-699. Stewart, R., Leeson, N., and Wright, R.J. (1997). Links between early arithmetical knowledge and early space and measurement knowledge: An exploratory study. In F. Biddulph and K. Carr (Eds.), Proceedings of the Twentieth Annual Conference of the Mathematics Education Research Group of Australasia (vol. 2, pp. 477-484). Hamilton, New Zea- land: MERGA. Stigler, J.W., Lee, S.-Y., and Stevenson, H.W. (1990). Mathematical Knowledge of Japanese, Chinese, and American Elementary School Children. Reston, VA: National Council of Teachers of Mathematics.

OCR for page 175
221 PATHS FOR GEOMETRY, SPATIAL THINKING, AND MEASUREMENT Swaminathan, S., Clements, D.H., and Schrier, D. (1995). The Agam Curriculum in Kinder- garten Classes: Effects and Processes. Buffalo: University of Buffalo, State University of New York. Tasuoka, K., Corter, J.E., and Tatsuoka, C. (2004). Patterns of diagnosed mathematical content and process skills in TIMSS-R across a sample of 20 countries. American Edu- cational Research Journal, 1, 901-926. Thomas, B. (1982). An Abstract of Kindergarten Teachers’ Elicitation and Utilization of Children’s Prior Knowledge in the Teaching of Shape Concepts: Unpublished manuscript, School of Education, Health, Nursing, and Arts Professions, New York University. Usiskin, Z. (1997, October). The implications of geometry for all. Journal of Mathematics Education Leadership, 1(3), 5-14. Available: http://ncsmonline.org/NCSMPublications/ 1997journals.html#oct97mel [accessed October 2008]. Uttal, D.H., and Wellman, H.M. (1989). Young children’s representation of spatial informa- tion acquired from maps. Deelopmental Psychology, 25, 128-138. van Hiele, P.M. (1986). Structure and Insight: A Theory of Mathematics Education. Orlando, FL: Academic Press. Vinner, S., and Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometrical concepts. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 177-184). Berkeley: Lawrence Hall of Science, University of California. Vygotsky, L.S. (1934/1986). Thought and Language. Cambridge, MA: MIT Press. Wang, R.F., and Spelke, E.S. (2002). Human spatial representation: Insights from animals. Trends in Cognitie Sciences, 6, 376-382. Wheatley, G.H. (1990). Spatial sense and mathematics learning. Arithmetic Teacher, (6), 10-11. Wheatley, G.H., Brown, D.L., and Solano, A. (1994). Long-term relationship between spatial ability and mathematical knowledge. In D. Kirshner (Ed.), Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 225-231). Baton Rouge: Louisiana State University. Wolf, Y. (1995). Estimation of Euclidian quantity by 5- and 6-year-old children: Facilitating a multiplication rule. Journal of Experimental Child Psychology, 5, 49-75. Yackel, E., and Wheatley, G.H. (1990). Promoting visual imagery in young pupils. Arithmetic Teacher, (6), 52-58. Yuzawa, M., Bart, W.M., Kinne, L.J., Sukemune, S., and Kataoka, M. (1999). The effects of “origami” practice on size comparison strategy among young Japanese and American children. Journal of Research in Childhood Education, 1(2), 133-143. Yuzawa, M., Bart, W.M., and Yuzawa, M. (2000). Development of the ability to judge relative areas: Role of the procedure of placing one object on another. Cognitie Deelopment, 15, 135-152. Zykova, V.I. (1969). Operating with concepts when solving geometry problems. In J. Kilpatrick and I. Wirszup (Eds.), Soiet Studies in the Psychology of Learning and Teaching Math- ematics (vol. 1, pp. 93-148). Chicago: University of Chicago.

OCR for page 175