Mathematics Learning in Early Childhood Paths Toward Excellence and Equity (2009) / Chapter Skim
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6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement
6 The Teaching-Learning Paths for Geometry, Spatial Thinking, and Measurement
Pages 175-222
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From page 175... ...
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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From page 176... ...
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.
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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.
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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 • se relational language U to combine shapes to number they can involving frames of reference, build pictures.
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From page 179... ...
Development of Shape Concepts What ideas do preschool children form about common shapes? Decades ago, Fuson and Murray (1978)
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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.
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From page 181... ...
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 environments.
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Children as young as 3½ to 5 years of age can build simple but meaningful 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)
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From page 183... ...
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 geometric shapes) , pattern blocks, and other shape sets, provides a valuable foundation (Bishop, 1980)
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From page 184... ...
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.
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From page 185... ...
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.
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From page 186... ...
Perceive, Say, B Perceive, Say, Discuss, and Construct Steps/Ages Describe/Discuss, and Describe/Discuss, and Compositions and (Levels of Construct Objects in Construct Spatial Decompositions in 3-D Thinking)
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From page 187... ...
Later, children learn to place 2-D shapes contiguously to form pictures. In free-form "make a picture" tasks, for example, each shape used repre
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From page 188... ...
They informally describe the properties of blocks in functional contexts, such as that some blocks roll and others do not. Spatial Relations Also beginning at the visual/holistic level, preschool children learn to extend their vocabulary of spatial relations with such terms as "beside,"
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From page 189... ...
They learn to predict the effects of geometric motions, thus laying the foundation for thinking at the relating parts and wholes level. Children also begin to be able to cover a rectangular space with physical tiles and represent their tilings with simple drawings, although they may initially leave gaps in each and may not align all the squares.
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From page 190... ...
. Using 3-D shapes, preschoolers combine building blocks using multiple spatial relations, extending in multiple directions and with multiple points of contact among components, showing flexibility in integrating parts of the structure.
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They do so with increasing anticipation, based on the shapes' attributes, indicating development of mental images of the component shapes. A significant advance is that they can combine shapes with different properties, extending the pattern block (30°)
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Many children learn to accept only isosceles triangles, for example. Others learn richer concepts, even at a young age.
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Even later, they have to describe the shape without using its name, so that their friends could name the shape. In this way, children learn the properties of the shape, moving from intuitive to explicit knowledge.
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With a variety of groups of shapes, such as pattern blocks, tangrams, or groups with a greater variety of shapes, children can be encouraged to combine shapes creatively to create pictures and designs. Noting children's developmental level, teachers can make suggestions and pose challenges that will facilitate their learning of more sophisticated thinking.
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From page 195... ...
. Such tactile-kinesthetic experiences as body movement and manipulating geometric solids help young children learn geometric concepts (Gerhardt, 1973; Prigge, 1978)
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Computer-based pattern blocks, for example, can be composed and decomposed in more ways than physical pattern blocks. As another example, children and teachers can save and later retrieve any arrangement of computer manipulatives.
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Many researchers therefore go beyond the physical act of measuring to investigate children's understandings of measuring as covering space and quantifying that covering. Appendix B describes concepts that are basic to understanding length measurement.
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. This may be a deleterious side effect of counting, in which children learn that the size of objects does not affect the result of counting (Mix, Huttenlocher, and Levine, 2002)
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The researchers concluded that nonstandard units are not a good way to initially help children understand the need for standardized conventional units in the length measuring process. Just as interesting were children's strategy preferences.
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One series of studies, however, indicates that this is not always so. If two strategies, measurement and direct comparison, were in conflict, children learned little and benefited little from verbal instruction.
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Concepts that are essential to understanding and learning area measurement are described in Appendix B One especially important one, spatial structuring, is discussed next.
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Perceive, Say, Describe/Discuss, and Ages Describe/Discuss, and Describe/Discuss, and Construct Compositions (Levels of Construct Objects in Construct Spatial and Decompositions in Thinking) 1-D Space Relations in 1-D Space 1-D Space Step 1 (Ages 2 and 3)
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Only after this age do most children move to explicit use of spatial structuring of multiplicative rules to solve those studies' tasks. Note that this does not imply formal use of multiplication, but only that their estimates are best approximated by the area formula.
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From page 204... ...
Once these problems have been solved, children need to structure 2-D space into an organized array of units to achieve multiplicative thinking in determining volume, a concept to which we now turn. Volume Measurement Volume introduces even more complexity, not only in adding a third dimension and thus presenting a significant challenge to students' spatial structuring, but also in the very nature of the materials that are measured using volume.
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Step 1 (Ages 2 and 3) Objects and Spatial Relations Young children naturally encounter and discuss quantities in their play (Ginsburg, Inoue, and Seo, 1999)
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For example, they may lay building blocks along a path to "make a long road." Step 2 (Age 4) Objects and Spatial Relations At the thinking about parts level, preschool children learn to compare the length of two objects by representing them with a third object and using transitive reasoning (i.e., indirect comparison)
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From page 207... ...
, along with related use of rulers and consistent discussion, will help children learn both the concepts and procedures of linear measurement. Kindergartners also can learn to fill containers with cubes, filling one layer at a time, intentionally, all of which involves relationships at the thinking about parts level of thinking.
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From page 208... ...
Comparing results of measuring the same object with manipulatives and with rulers and using manipulative length-units to make their own rulers help children connect their experiences and ideas. In second or third grade, teachers might introduce the need for standard length-units and the relation between the size and number of length-units.
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From page 209... ...
. Play with structured materials, such as unit blocks, pattern blocks, and tiles, can lay the groundwork for children's spatial structuring, although achieving the conceptual benchmark will not be achieved until a fter the primary grades for most children, even with high-quality instruction.
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From page 210... ...
Children's early competency in measurement is facilitated by play with structured materials, such as unit blocks, pattern blocks, and tiles and strengthened through opportunities to reflect on and discuss their experiences. It is important to note that the potential of young children's learning in geometry and measurement if a conscientious, sequenced development of spatial thinking and geometry were provided to them throughout their earliest years is not yet known.
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From page 211... ...
Journal for Research in Mathematics Education, 27, 258-292. Battista, M.T., Clements, D.H., Arnoff, J., Battista, K., and Borrow, C.V.A.
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, Mathematics Education Research: Implications for the 1980s (pp.
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, Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp.
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, Building Connections: Research, Theory, and Practice: Proceedings of the 28th Annual Conference of the Mathematics Education Research Group of Australasia (pp.
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Journal for Research in Mathematics Education, 9, 189-203. Fey, J., Atchison, W.F., Good, R.A., Heid, M.K., Johnson, J., Kantowski, M.G., et al.
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Mathematics Education Research Journal, 16(2)
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. Research on spatial skills and block building in girls and boys: The relationship to later mathematics learning.
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Graham (Eds.) , Proceedings of the Sixteenth Psychology in Mathematics Education Conference (vol.
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, Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp.
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Carr (Eds.) , Proceedings of the Twentieth Annual Conference of the Mathematics Education Research Group of Australasia (vol.
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, Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol.
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