The following syntheses which are based on the Curriculum and Evaluation Standards for School Mathematics, outline some basic aspects of teaching geometry. The accompanying activities, "Which Container Holds the Most?" and "Building Houses," provide practical examples for translating the Standards recommendations into classroom instruction.

### Geometry for the Early and Middle Grades

eometry is an orderly way to describe and represent our inherently geometric world. Basic to the understanding of geometry is the development of spatial sense-an intuitive feel for our surroundings and the objects in them. Spatial capabilities appear early in life, and tapping into these strengths can foster an interest in mathematics. Children who develop a strong sense of spatial relationships and master the concepts of geometry are better prepared to learn number and measurement ideas as well as other advanced mathematical topics.

Classroom experiences that focus on geometric relationships will develop children's spatial sense. They should examine the direction, orientation, and perspective 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. Children can begin with activities that use words like above, below, and behind and progress to using a computer to reproduce a pattern-block design.

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

For middle school students the informal exploration of geometry can be mathematically productive. Geometry at this level links the informal explorations begun in grades KŠ4 to the more formal processes of grades 9-12. Students draw inferences and make logical deductions from geometric problem situations. They can also analyze their thought processes and explanations. Allow sufficient time to discuss the quality of their answers and to think about such questions as: Could it be another way? Students should learn to use correct vocabulary, including such common reasoning terms as and, or, all, some, always, never, and if...then, as well as such descriptors as parallel, perpendicular, and similar. Geometry has its own vocabulary including terms like rhombus, trapezoid, and dodecahedron, and students need ample time to develop confidence in their use of this new and unique language. Definitions should evolve from experiences in constructing, visualizing, drawing, measuring, contrasting, and classifying figures according to their properties. Students who memorize a definition and a textbook example or two are less likely to remember the term or its application.

Computer software allows students to construct two- and three-dimensional shapes on a screen and then flip, turn, or slide them to view from a new perspective. Explorations of flips, slides, turns, stretchers, and shrinkers will illuminate the concepts of congruence and similarity. Observing and learning to represent two- and three-dimensional figures in various positions by drawing and constructing also helps students develop spatial sense.

This synthesis is based on the chapters:"Geometry and Spatial Sense" and "Geometry for Grades 5-8" from Curriculum and Evaluation Standards for School Mathematics. Order from NCTM, 1900 Association Drive, Reston, VA 22091. Telephone: 1-800-235-7566.

Geometry in grades 9-12 encompasses algebraic and synthetic (elementary euclidean) perspectives. The NCTM Standards recommend that topics be integrated across all grade levels.The Standards emphasize

• transformation and coordinate approaches,
• the development of short sequences of theorems,
• deductive arguments expressed orally and in sentence or paragraph form,
• computer-based explorations of 2-D and 3-D figures,
• three-dimensional geometry,
• real-world applications and modeling.

In the study of geometry of two and three dimensions from an algebraic point of view, students deduce properties of figures using transformations and coordinates. They should be able to identify congruent and similar figures using transformations, analyze properties of euclidean transformations, and relate translations to vectors. Students headed for college should also be able to deduce properties of figures using vectors and apply transformations, coordinates, and vectors in problem solving.

The student who understands the interplay between geometry and algebra has more power to construct and analyze problems. Objects and relations in geometry correspond directly to objects and relations in algebra. For example, a point in geometry corresponds to an ordered pair (x,y) of numbers in algebra. A line corresponds to a set of ordered pairs satisfying the equation ax + by = c. The intersection of two lines corresponds to the set of ordered pairs that satisfies the corresponding equations. Connections like these allow translation between the two "languages" and permit concepts in one to clarify and reinforce concepts in the other.

The synthetic component of 9-12 geometry allows students to interpret and draw three-dimensional objects, represent problem situations with geometric models, and apply properties of figures. Students classify figures in terms of congruence and similarity and apply these relationships. Geometry instruction should deepen their understanding of shapes, their properties, and everyday applications. Examples from such activities as recreation (billiards and sailing), practical tasks (purchasing paint for a room), or the arts (perspective in drawing) should be evident throughout the curriculum.

Give students the opportunity to work with three-dimensional figures so they can develop spatial skills that are basic to everyday life. Visualization also includes plane figures. Computer graphics software that allows students to create and manipulate shapes makes conjecturing and testing their attempts at two-dimensional visualization easier.

Computer microworlds such as Logo turtle graphics provide opportunities for a great deal of student involvement.

Of course there are many opportunities for visualization that do not use a computer. Exercises that require the student to draw a diagram provide opportunities for reading mathematics and problem translation.

This synthesis is based on the chapter "Geometry from a Synthetic Perspective" and "Geometry from an Algebraic Perspective" from Curriculum and Evaluation Standards for School Mathematics. Order from NCTM, 1900 Association Drive, Reston, VA 22091. Telephone: 1-800-235-7566.

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