Geometry Must Be Vital.

At the beginning of the twentieth century, David Eugene Smith wrote a book about teaching geometry. Although it was written for secondary school geometry teachers, his preface describing why he wrote the book is applicable for all grades. He said that his book "...stands for vitalizing geometry in every legitimate way; for improving the subject matter in such a manner as not to destroy the pupil's interest; for so teaching geometry as to make it appeal to pupils ..." (Smith 1911, p. iv). At the beginning of the twenty-first century, the Principles and Standards for School Mathematics (NCTM 2000) gives us guidance in revitalizing geometry for elementary school students. In this article, we share some of its messages for teachers of prekindergarten through grade 5.

The Geometry Standard

The geometry standard is one of the five content standards in Principles and Standards. This standard is delineated at each grade band with expectations, such as those for prekindergarten through grade 2 and for grade 3 through grade 5 (see fig. 1). As you read the standard and the associated expectations, ask yourself these questions: Does your curriculum contain geometric activities that do not match any of the expectations? Do students in your school miss the opportunity to meet some of these expectations because of lack of experience?

Four Important Messages

One general message from Principles and Standards is that students should study geometry for a purpose, not perform geometric tasks merely because they were asked to do so. For example, the Geometry Standard states that students should use visualization, spatial reasoning, and geometric modeling to solve problems. Solving problems becomes a reason for studying geometry. Similarly, we study geometry to analyze mathematical situations, which may help us learn more about other mathematical topics. For example, we may use geometric models to help us learn about place value or fractions. Principles and Standards includes other messages about geometry. We have chosen four of these, which we present in this article, along with discussions and activities.

Teaching vocabulary appropriately

Vocabulary is important but is not the purpose of studying geometry. Much of the geometry in elementary school curricula is focused on learning vocabulary, often the vocabulary of naming shapes. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) called our attention to vocabulary, saying, "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" (p. 48). Principles and Standards calls for even the youngest children to be introduced to mathematical language but in a natural way that connects with their informal language. The document recommends that children in grades pre-K-2 use their own vocabulary to describe objects while teachers gradually incorporate the conventional terminology. It puts forth the expectation that children in grades 3-5 will develop more precise use of language, not just acquire more words.

We learn vocabulary best by using it, but geometry should go beyond knowing the names of shapes. The expectations point out that children need to describe shapes in terms of the shapes' attributes (e.g., "This quadrilateral is a parallelogram because it has two pairs of parallel sides"), analyze the role of attributes (e.g., how does having two pairs of parallel sides affect the length of the sides?), and make logical arguments to justify conclusions about geometric relationships (e.g., why are the opposite sides equal?).

As we help students build vocabulary, we should always ask ourselves what concepts they are attaching to the words. A second-grade student asked his teacher, "Are all triangles green?" His teacher had not realized until then that she had been using only pattern blocks in her geometry lessons, and the only exemplar of a triangle that her students saw was the green equilateral triangular shape. This student's question was a vivid reminder to incorporate other materials in her lessons. When a class of fifth-grade students chose only the equilateral triangle from many examples and nonexamples of triangles, the teacher realized that an equilateral triangle was the model she consistently gave for a triangle. In both classes, students needed to explore triangles that were not only of different colors but also of different shapes, sizes, and orientations. We need to be aware that vocabulary can limit students' understanding unless it is used correctly and with a wide range of examples to build rich concepts.

Helping students to justify conclusions

Investigating, making conjectures, and developing logical arguments to justify conclusions are important aspects of studying geometry. Throughout grades pre-K-5, the expectations for geometry call for students to investigate subdividing and combining shapes, that is, taking them apart and putting them together. This activity is analogous to developing number sense by looking at a number in terms of its parts (e.g., 23 is 20 + 3 or a little more than 20; 23 is 4 fives and 3 more) or by putting numbers together (e.g., 4 and 6, 3 and 7, and 9 and 1 all make 10). Seeing shapes as made from other shapes and examining how shapes fit together give students a better sense of shapes and make shapes more useful. For example, if children realize that a parallelogram can always be transformed into a rectangle with the same area, then they can find the area of any parallelogram by using their knowledge of the area of a rectangle. Too often in current practice, if children combine shapes at all, they put shapes together to make designs but learn little geometry and make few generalizations. In contrast, Principles and Standards underscores the need to have children put shapes together and take them apart; make predictions about doing so; and in grades 3-6, make logical arguments about their conjectures.

In the activity that follows, we suggest an investigation that could take many turns. You may present the investigation with different scaffolding than that suggested here to make it more appropriate for your students. We suggest that you keep a record of how you approached the investigation.

We will share some conjectures that children made and their reasoning about those conjectures. Before reading about what the children found, try to answer the question and make conjectures yourself.

In a fifth-grade class, the teacher began by asking the question discussed in figure 2 about quadrilaterals only. The students had no difficulty finding different combinations but spent some time discussing whether the two combinations shown below were different or the same.

Lonnie argued that they were different because one joined a vertex to one side and the other joined the same vertex to another side. "In the first one, you get a 3, 4 [the class representation for a combination of a three-sided and a four-sided figure], and [in] the second one you get a 4, 3." The class agreed that the shapes were different in some way but that the many possible ways to get a 3, 4 combination would keep one counting forever. The agreement was reached that only one combination would be counted.

The class agreed that four different combinations were possible for a four-sided shape (see fig. 3 for a summary of the different types). Before beginning the investigation of the five-sided shape, the teacher asked the students whether their answer would have been different if they had begun with a special quadrilateral, such as a square, a rectangle, a parallelogram, or a trapezoid. The students quickly convinced themselves that the number of different combinations would be the same. The teacher asked them to predict how many combinations they would get for a five-sided figure. Most students thought that they would get more; some jumped to the conclusion that if a four-sided figure had four combinations, then a five-sided figure would have five combinations.

Some children completed their investigation of the five-sided figure, finding five combinations and deciding that they had found the pattern. What a surprise when they found seven different combinations for a six-sided polygon! They began to wonder whether one could predict the number of combinations for a seven-sided figure, an eight-sided figure, or one with even more sides. Interested students pursued this investigation independently because the teacher believed that class time for most students would be better spent looking at the data already collected.

The teacher asked, "Are you sure you have all the combinations for the four-sided [figure]? Can you convince us?" This question forced the children to reflect on how they had drawn lines. Lounie had already helped them focus on connecting a vertex to a side, and they saw that other lines went from a vertex to a vertex or from a side to a side. One group of students analyzed the four-sided combinations as follows:

You can only have one "vertex to vertex" because if you begin with one vertex, the two other vertices are on side lines, so you have to go across. You can only have one "vertex to side." We already agreed to that with Lonnie. But you can have two "side to side." You can go to an adjacent side or to an opposite side.

Patterns and conjectures, such as the following three, began to emerge from the data. The students' conjectures follow:

* Monica. There will always be a triangle.

* Mike. If you make a triangle from side to adjacent side, the other figure will have one more side than the original.

* Shawana. The sum of [the] number of sides always varies by 3. For example, in the 4-sided case, the sum goes from 6 to 8; in the 6-sided case, the sums vary from 8 to 10.

Can you find other conjectures? Which of the students' conjectures are clear? Which are reasonable? How could students make a logical argument, such as the one above, for each?

The teacher understood what Monica meant when she said enthusiastically, "There will always be a triangle" and pointed to the side-to-adjacent-side example in each figure. But Susan said, "What do you mean? There is not a triangle in the 4, 4 one." Monica, with some help from Susan, revised her conjecture: "No matter what number of sides you start with, you can always draw a triangle by connecting a side to an adjacent side." She convinced her classmates by drawing what she called a "huge polygon"--one with many sides--and showing the triangle. When Monica drew her example, Mike said, "That helps me justify my conjecture." He showed that when you join a side to an adjacent side, you still have all the original sides in the nontriangular figure, as well as the line. "So you have all the sides plus one." The class was convinced that Shawana's conjecture was correct, but no one could give a convincing argument. The pattern was evident to the students, but they did not have the sophistication to form an argument .

Although we have described this investigation in a fifth-grade setting, parts of this activity could be used at lower grade levels. If you teach young children, you may want to have them investigate only the shapes that they can make beginning with a rectangle and drawing one line inside the rectangle. You may want to ask specific questions, such as "Can you make two triangles? [Yes.] Can you make two rectangles? [Yes.] Can you make a triangle and a rectangle? [No.] Can you make two trapezoids? [Yes.]" You can repeat the investigation several times, using a triangle, parallelogram, hexagon, or other shapes.

Including all aspects of geometry

Geometry is more than shapes. We often neglect parts of the Geometry Standard, such as the expectations about spatial orientation and location. This omission deprives students of a foundation for understanding everyday applications, such as giving and receiving directions and reading maps, and other mathematical topics, including visualizing two- and three-dimensional objects, using geometric models for numerical and algebraic relationships, working with coordinates, and graphing.

Spatial orientation is having the "spatial sense" of where one is in the world and how to get around. The other expectations about location deal with representing positions and directions mathematically. We record such information mathematically on maps. Even children in preschool and kindergarten can create simple but meaningful maps. For example, preschoolers can use landscape toys, such as houses, cars, and trees, to make maps of a yard. Kindergartners can make models of their classroom that accurately cluster pieces of toy furniture. Primary-grade children are able to sketch recognizable maps of the areas around their homes from memory. Children in preschool and higher grades can locate themselves and other objects along routes that include landmarks. They can also learn from maps. For example, children learn routes more quickly if they first examine maps of those routes.

Still, most students are not competent users of maps. Typical school experiences fail to connect map skills with other curriculum areas, especially mathematics. Developing children's ability to make and use mental maps is important, as is developing geometric ideas from experiences with maps. We should go beyond teaching isolated "map skills" and geography to engage in actual mapping, surveying, drawing, and measuring in local environments. Starting in preschool and continuing through the elementary school years, four basic questions arise: direction (which way?), distance (how far?), location (where?), and representation (what objects?) (NCTM 2000, p. 98). All these questions include important mathematics.

Young children can learn about directions, such as above, over, and behind. From these beginnings, they can develop navigation ideas, such as left, right, and front, and global directions, such as north, east, west, and south. Such ideas, along with distance and measurement concepts, might be developed as children create and read maps of their own environments. For example, children might mark with masking tape a path from a table to the wastebasket, emphasizing the continuity of the path. With the teacher, children could draw a map of this path. Items appearing alongside the path, such as a table or an easel, can be added. Finally, the number of steps or other measures of distance can be recorded on the map.

Marking items on a path records their location. To build on these ideas, children might use cutouts of a tree, swing set, and sandbox and lay them out on a felt board as a simple map of the playground. They can discuss how moving a schoolyard item, such as a table, would change the map of the yard. On the map, locate children sitting in or near the tree, swing set, and sandbox. Scavenger hunts on the playground can help students give and follow directions or clues. Soon, students should determine locations with coordinates. Even young children can use coordinates that adults provide for them. Students throughout the elementary grades need further experiences in structuring and working with two-dimensional grids to develop precise working concepts of grids, grid lines, and points and the overall structure of order and distance relationships in a coordinate grid.

Finally, to answer questions of representation, children also learn the mapping and mathematical processes of abstraction 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. This activity could be followed by using maps that are "miniaturizations" and those that show abstract symbols. Using symbols is beneficial even for young children. Too much reliance on literal pictures and icons may hinder their understanding of maps, leading children to believe, for example, that some physical roads are red. A teacher might have each child pick some object in the room and denote its location with an X on their maps. Children could exchange maps, trying to identify the mystery objects and thereby testing their maps. In a similar vein are symbols of boundaries. Ask children if they ever "mark off" a region for their play , as in a sandbox. Such dividing lines are similar to the closed curves that act as state boundaries in the United States. Discuss the idea that a boundary creates two regions--one inside, and one outside, the curve.

As children work with model buildings or blocks, give them experience with another spatial skill--perspective. For example, students might identify block structures from various viewpoints, matching same-structure views that are portrayed from different perspectives. Students may also try to find the viewpoint from which a photograph was taken. These experiences address such confusions of perspective as preschoolers' "seeing" windows and doors of buildings in vertical aerial photographs. Perspective and direction are particularly important in aligning maps with the environment. Some children of any age will find it difficult to use a map that is not so aligned. Teachers should introduce such situations gradually.

Allowing for growth

The curriculum should allow for growth across the grades. Both the shapes investigation and the mapping activities illustrate how the geometry curriculum can encourage the growth of mathematical ideas in students from preschool to middle school. Let us consider one more example that links shapes and navigation. Young children can abstract and generalize directions and measurement by working with navigation software.

For example, preschoolers and kindergartners might drive an on-screen "car" around a street map by specifying how far it should move forward and when it should turn. Primary-grade students might give the Logo turtle the directions to move through a maze. Students learn concepts of orientation, direction, perspective, and measurement when they give computer commands, such as "forward ten steps, right turn, forward five steps." One first grader explained how she turned the turtle 45 degrees, rotating her hand as she counted: "I went five, ten, fifteen, twenty ... forty-five! It's like a car speedometer. You go up by fives!" This child is mathematizing the act of turning; she is applying a unit to an instance of turning and using her counting abilities to determine a measurement.

These experiences provide a solid foundation for such difficult geometric ideas as shape. For example, intermediate-grade students can use the Logo turtle to draw regular polygons. Combining navigation and shape ideas, they can first walk the paths of equilateral triangles and squares, then command the turtle to "walk" similar paths. They realize that the forward (distance) commands must have the same input because the sides of each regular polygon must be the same length. The turns for polygons with five or more sides are not obvious, but students can be guided to realize that the turtle always "turns all the way around," that is, 360 degrees. By reasoning that each turn is equal, because all the angles must be equal in a regular polygon, students can be guided to figure Out that each turn of a pentagon is 360 [divided by] 5, or 72, degrees. This "total turtle trip" theorem is a powerful idea with many applications.

The process standards--problem solving, reasoning and proof, communication, connections, and representations--can be especially powerful in shaping your role in the classroom as you include geometry Each process standard discusses the role of the teacher, and the examples are often drawn from geometry. Geometry activities are particularly fruitful for developing mathematical processes. We encourage you to read those standards for your grade level. Here, we emphasize two central ways to revitalize geometry in your classroom.

Providing opportunities to use geometry

Provide opportunities for your students to learn and use geometry. Geometry is often neglected in our curriculum, yet we know that geometry allows students to become involved in mathematics and show their talents and thoughts. "Some students' capabilities with geometric and spatial concepts exceed their numerical skills. Building on these strengths fosters enthusiasm for mathematics and provides a context in which to develop number and other mathematical concepts" (NCTM 2000, p. 97).

As you choose geometry activities, return to the main messages of this article and ask yourself how well the activities and messages are aligned. Children may have fun making pictures from geometric shapes in kindergarten, in second grade, and even in fifth grade. However, we need to ask what new concepts students will learn, what reasoning skills we will help them develop, and what connections we are making with other areas of mathematics.

Encourage reasoning

Foster reasoning through geometry. Geometry is a wonderful area in which to encourage reasoning, as the shape investigation in this article illustrates. Students often find talking about shapes and properties easy because models are available. Making and testing conjectures become more interesting because many relationships are not as obvious or as well known in geometry as in number.

"The teacher must establish the expectation that the class as a mathematical community is continually developing, testing, and applying conjectures about mathematical relationships" (NCTM 2000, p. 191). The discussion of the role of the teacher in grades 3-5 in the Reasoning and Proof Standard continues from this statement to emphasize the importance of the students' taking responsibility for articulating their own reasoning and working hard to understand the reasoning of others. Such responsibility does not begin in grade 3, however. The pre-k-2 discussion of the teacher's role in developing reasoning emphasizes the role of questioning and that of justifying answers. It also reminds us that students' conclusions often seem odd to adults. If we take the time to listen carefully to children's explanations of their thinking, we may see that their reasoning is not faulty but that their assumptions, based on their experiences, are different.

Conclusion

Our world becomes more complex almost daily. Advanced technology requires us to further develop visual and spatial skills; the flood of information that we receive forces us to think and justify our thoughts; and the increase in our interaction with mathematics prompts us to appreciate the beauty of geometry and enhance our own humanity.

Doug Clements, clements@buffalo.edu, was previously a preschool and kindergarten teacher and is now a professor of early childhood, mathematics, and computer education at the State University of New York at Buffalo, Buffalo, NY 14260. He conducts research in computer applications in education, early development of mathematical ideas, and the learning and teaching of geometry. He was also a member of the Pre-K-2 Writing Group for the Standards 2000 Project. Mary Lindquist, lindquist_mary@colstate.edu, NCTM president from 1992 to 1994, is on the faculty of Columbus State University, Columbus, GA 31907. She chaired the Commission on the Future of the Standards.

Edited by Jeane Joyner, jjoyner@dpi.state.nc.us. Department of Public Instruction, Raleigh. NC 27601, and Barbara Reys, reysb@missouri.edu, University of Missouri, Columbia, MO 65211. This department is designed to give teachers information and ideas for using the NCTM's Principles and Standards for School Mathematics (2000). Readers are encouraged to share their experiences related to the Standards with Teaching Children Mathematics. Please send manuscripts to "Principles and Standards," TCM, 1906 Association Drive, Reston, VA 20191-9988.

References

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.

-----. Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

Smith, David Eugene. The Teaching of Geometry. Boston, Mass.: Ginn & Co., 1911.
```                 The geometry standard (NCTM 2000, p. 396)
Standards                        Pre-K-2 Expectations
Instructional programs from      In prekindergarten through grade
prekindergarten through grade 12 2 all students should--
should enable all students to--
* recognize, name, build, draw,
analyze characteristics and        compare, and sort two- and
properties of two- and three-      three-dimensional shapes;
dimensional geometric shapes and
develop mathematical arguments   * describe attributes
about geometric relationships      and parts of two- and
three-dimensional shapes;
* investigate and predict the
results of putting together
and taking apart two- and
three-dimensional shapes.
specify locations and describe   * describe, name, and interpret
spatial relationships using        relative positions in space
coordinate geometry and other      and apply ideas about
representational systems           relative position;
* describe, name, and interpret
direction and distance in navi-
gating space and apply ideas
* find and name locations with
simple relationships, such as
"near to" and in coordinate
systems such as maps.
apply transformations and use    * recognize and apply slides,
symmetry to analyze mathematical   flips, and turns;
situations                       * recognize and create shapes
that have symmetry.
use visualization, spatial       * create mental images of
reasoning, and geometric           geometric shapes using spatial
modeling to solve problems         memory and spatial visualization;
* recognize and represent shapes
from different perspectives;
* relate ideas in geometry to
ideas in number and measurement;
* recognize geometry shapes and
structures in the environment
and specify their location.
Instructional programs from      In grades 3-5, all
prekindergarten through grade 12 students should--
should enable all students to--  * identify, compare, and
analyze attributes of two-
analyze characteristics and        and three- dimensional
properties of two- and three-      shapes and develop vocabulary
dimensional geometric shapes and   to describe the attributes;
develop mathematical arguments   * classify two- and three-
according to their properties
and develop definitions
of classes of shapes, such
as triangles and pyramids;
* investigate, describe, and reason
combining, and transforming shapes;
* explore congruence and similarity;
* make and test conjectures about
geometric properties and
relationships and develop logical
arguments to justify  conclusions
specify locations and describe   * describe location and movement
spatial relationships using        using common language;
coordinate geometry and other    * make and use coordinate
representational systems           systems to specify locations
and to describe paths;
* find the distance between
points along horizontal and
vertical lines of a coordinate
system.
apply transformations and use    * predict and describe the results
symmetry to analyze mathematical   of sliding, flipping, and
situations                         turning two- dimensional shapes;
* describe a motion or series
of motions that will show that
two shapes are congruent;
* identify and describe line and
rotational symmetry in two-
and three- dimensional
shapes and designs.
use visualization, spatial       * build and draw geometric objects;
reasoning, and geometric         * create and describe mental images
modeling to solve problems         of objects, patterns, and paths;
* identify and build a three-dimen-
sional object from two-dimensional
representations of that object;
* identify and build a
two-dimensional representation
of a three-dimensional object;
* use geometric models to solve
problems in other areas of
mathematics, such as number
and measurement;
* recognize geometric ideas and
relationships and apply them to
other disciplines and to problems
that arise in the classroom
or in everyday life.
```

The E-Standards include an Illuminations section designed to "illuminate" the NCTM's Principles and Standards for School Mathematics (2000) by providing resources for teaching and learning. One of the resources is i-Math, which is composed of online investigations that are ready-to-use, interactive-multimedia-mathematics lessons. The three-part ladybug example presents a rich computer environment in which students can use their knowledge of number, measurement, and geometry to solve interesting problems. Planning and visualizing, estimating and measuring, and testing and revising are components of the ladybug activities. These interactive figures can help students build ideas about navigation and location, as described in the Geometry Standard, and use these ideas to solve problems, as described in the Problem Solving Standards. See the illuminations Web site at standards.nctm.org/document/eexamples/chap4/4.3/index.htm.
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