In grades K-4, the mathematics curriculum should include two- and three-dimensional geometry so that students can--

• describe, model, draw, and classify shapes;
• investigate and predict the results of combining, subdividing, and changing shapes;
• develop spatial sense;
• relate geometric ideas to number and measurement ideas;
• recognize and appreciate geometry in their world.

Focus

Geometry is an important component of the K-4 mathematics curriculum because geometric knowledge, relationships, and insights are useful in everyday situations and are connected to other mathematical topics and school subjects. Geometry helps us represent and describe in an orderly manner the world in which we live. Children are naturally interested in geometry and find it intriguing and motivating; their spatial capabilities frequently exceed their numerical skills, and tapping these strengths can foster an interest in mathematics and improve number understandings and skills.

Spatial understandings are necessary for interpreting, understanding, and appreciating our inherently geometric world. Insights and intuitions about two- and three-dimensional shapes and their characteristics, the interrelationships of shapes, and the effects of changes to shapes are important aspects of spatial sense. Children who develop a strong sense of spatial relationships and who master the concepts and language of geometry are better prepared to learn number and measurement ideas, as well as other advanced mathematical topics.

In learning geometry, children need to investigate, experiment, and explore with everyday objects and other physical materials. Exercises that ask children to visualize, draw, and compare shapes in various positions will help develop their spatial sense. Although a facility with the language of geometry is important, it should not be the focus of the geometry program but rather should grow naturally from exploration and experience. Explorations can range from simple activities to challenging problem-solving situations that develop useful mathematical thinking skills.

Evidence suggests that the development of geometric ideas progresses through a hierarchy of levels. Students first learn to recognize whole shapes and then to analyze the relevant properties of a shape. Later they can see relationships between shapes and make simple deductions. Curriculum development and instruction must consider this hierarchy because although learning can occur at several levels simultaneously, the learning of more complex concepts and strategies requires a firm foundation of basic skills.

Discussion

Geometry gives children a different view of mathematics. As they explore patterns and relationships with models, blocks, geoboards, and graph paper, they learn about the properties of shapes and sharpen their intuitions and awareness of spatial concepts. Children's geometric ideas can be developed by having them sort and classify models of plane and solid figures, construct models from straws, make drawings, and create and manipulate shapes on a computer screen. Folding paper cutouts or using mirrors to investigate lines of symmetry are other ways for children to observe figures in a variety of positions, become aware of their important properties, and compare and contrast them. Related experiences help children avoid simplistic and misleading ideas about shapes, such as that implied by one child's observation, "This is an upside-down triangle."

Children can be taught to internalize characteristics of shapes and then translate those internal ideas into descriptions and models by feeling an object inside a paper bag, identifying the object, creating the shape on a geoboard, and naming the object's shape.

Children can also follow verbal directions, such as "Draw [or put on a geoboard] a shape that has four sides and two right angles." Follow-up discussions help children understand the conditions necessary to define a shape. These experiences allow children to develop more complete understandings about shapes and their properties and to build the vocabulary of geometry in a natural manner.

Spatial sense is an intuitive feel for one's surroundings and the objects in them. To develop spatial sense, children must have many experiences that focus on geometric relationships; the direction, orientation, and perspectives of objects in space; the relative shapes and sizes of figures and objects; and how a change in shape relates to a change in size. These experiences depend on a child's ability to follow directions that use words like above, below, and behind and to progress to such activities as using a computer to reproduce a pattern-block design. When children examine the result of combining two shapes to form a new shape, predict the effect of changing the number of sides of a shape, draw a shape after it has been rotated a quarter or half turn, or explore what happens when the dimensions of a shape are changed, they acquire a deeper understanding of shapes and their properties. Such activities promote spatial sense.

Drawing and sketching shapes is an important part of developing spatial sense. The following spatial-visualization tasks illustrate one productive activity. A figure is displayed on an overhead projector for two to three seconds (fig. 9.1) and then children try to draw the figure. The original figure is again briefly displayed, and children make a second attempt at drawing it. Discussion about what the children saw is also important.

Fig. 9.1

Children can cut paper shapes and make new shapes from the parts (fig. 9.2).

Fig. 9.2

When children hold a long loop of yarn so that each hand serves as a vertex, they can explore the effect of changing the size of an angle, or increasing the number of sides while the perimeter is unchanged. See figure 9.3.

Fig. 9.3

Another activity that promotes spatial sense is to have children decide which two-dimensional patterns can be folded to produce a three-dimensional shape. See figure 9.4.

Fig. 9.4

Geometry contributes to the development of number and measurement concepts. Two-dimensional regions are subdivided into congruent parts to teach fraction and decimal concepts. Geometric ideas and a number line are useful models for teaching rounding. For example, 438 is between 400 and 500; it is closer to 400 because it is less than halfway.

Many geometric skills and concepts are essential to the process of problem solving. For example, a primary problem-solving strategy is drawing a picture or diagram, which is, in many situations, a geometric representation of the problem. Three sample problems illustrate this point. See figures 9.5 and 9.6.

Fig. 9.5

Fig. 9.6

In summary, children should have many opportunities to explore geometry in two and three dimensions, to develop their sense of space and relationships in space, and to solve problems that involve geometry and its application to other topics in mathematics or to other fields.