t 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.
A
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 geome-
try. 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
Principles and
Standards
Geometry Must
Be Vital
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
Mary M. Lindquist and
School Mathematics (1989) called our
Douglas H. Clements
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
Doug Clements, [email protected], 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, [email protected], 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, [email protected], Department of Public Instruction, Raleigh,
NC 27601, and Barbara Reys, [email protected], 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.
MARCH 2001
Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
409
FIGURE 1
The geometry standard (NCTM 2000, p. 396)
Standards
Instructional programs from
prekindergarten through grade 12
should enable all students to—
Pre-K–2 Expectations
In prekindergarten through grade 2 all
students should—
Grade 3–5 Expectations
In grades 3–5, all students should—
specify locations and describe spatial
relationships using coordinate geometry
and other representational systems
• describe, name, and interpret relative
• describe location and movement using
positions in space and apply ideas
common language;
about relative position;
• make and use coordinate systems to
• describe, name, and interpret direction
specify locations and to describe
and distance in navigating space and
paths;
apply ideas about direction and distance; • find the distance between points along
• find and name locations with simple
horizontal and vertical lines of a
relationships, such as “near to” and in
coordinate system.
coordinate systems such as maps.
apply transformations and use
symmetry to analyze mathematical
situations
• recognize and apply slides, flips, and
turns;
• recognize and create shapes that
have symmetry.
• predict and describe the results of
sliding, flipping, and turning twodimensional 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 threedimensional shapes and designs.
use visualization, spatial reasoning,
and geometric modeling to solve
problems
• create mental images of geometric
shapes using spatial 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.
• build and draw geometric objects;
• create and describe mental images of
objects, patterns, and paths;
• identify and build a three-dimensional
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.
• identify, compare, and analyze
• recognize, name, build, draw, compare,
attributes of two- and threeanalyze characteristics and properties
and sort two- and three-dimensional
dimensional shapes and develop
of two- and three-dimensional geometric
shapes;
vocabulary to describe the attributes;
shapes and develop mathematical
• describe attributes and parts of two• classify two- and three-dimensional
arguments about geometric relationships
and three-dimensional shapes;
shapes according to their properties
• investigate and predict the results of
and develop definitions of classes
putting together and taking apart twoof shapes, such as triangles and
and three-dimensional shapes.
pyramids;
• investigate, describe, and reason about
the results of subdividing, combining,
and transforming shapes;
• explore congruence and similarity;
• make and test conjectures about
geometric properties and relationships
and develop logical arguments to justify
conclusions
410
TEACHING CHILDREN MATHEMATICS
FIGURE 2
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.
This activity began as a question about quadrilaterals.
Investigation: Given a convex shape, how many different combinations of two types of polygons can be made by cutting the shape
with one line? (A type of polygon is identified by the number of sides; for example, a three-sided or a four-sided polygon.)
Convex four-sided shapes
Concave shape
3
4
Save for another
investigation
MARCH 2001
Line makes a three-sided and
a four-sided polygon
4
4
Line makes two
four-sided polygons
411
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.
What are your conjectures?
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 combina412
tions 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. Lonnie 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-adjacentside 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
TEACHING CHILDREN MATHEMATICS
FIGURE 3
Results of investigating four-, five-, and six-sided figures
3, 3
(a) Vertex to vertex
3, 5
3, 4
3, 5
3, 4
(b) Vertex to side
4, 4
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
MARCH 2001
4, 5
4, 5
3, 6
4, 4
3, 6
4, 4
(d) Side to side
3, 5
(c) Side to side
3, 7
4, 6
5, 5
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
413
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 meaThe E-Standards include an Illumisurement concepts, might be develnations section designed to “illumioped as children create and read maps
nate” the NCTM’s Principles and
of their own environments. For examStandards for School Mathematics
ple, children might mark with mask(2000) by providing resources for
ing tape a path from a table to the
teaching and learning. One of the
wastebasket, emphasizing the continuresources is i-Math, which is comity of the path. With the teacher, chilposed of online investigations that
dren could draw a map of this path.
are ready-to-use, interactive-multiItems appearing alongside the path,
media-mathematics lessons. The
such as a table or an easel, can be
three-part ladybug example presents
added. Finally, the number of steps or
a rich computer environment in
other measures of distance can be
which students can use their knowlrecorded on the map.
edge of number, measurement, and
Marking items on a path records
geometry to solve interesting probtheir location. To build on these ideas,
lems. Planning and visualizing, estichildren might use cutouts of a tree,
mating and measuring, and testing
swing set, and sandbox and lay them
and revising are components of the
out on a felt board as a simple map of
ladybug activities. These interactive
the playground. They can discuss how
figures can help students build ideas
moving a schoolyard item, such as a
about navigation and location, as
table, would change the map of the
described in the Geometry Standard,
yard. On the map, locate children sitand use these ideas to solve probting in or near the tree, swing set, and
lems, as described in the Problem
sandbox. Scavenger hunts on the playSolving Standards. See the Illuminaground can help students give and foltions Web site at standards.nctm.
low directions or clues. Soon, students
org/document/eexamples/chap4/4.
should determine locations with coor3/index.htm.
dinates. 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 cir414
cles 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
TEACHING CHILDREN MATHEMATICS
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 ÷ 5, or 72, degrees.
This “total turtle trip” theorem is a powerful idea
with many applications.
Your Role in the
Classroom
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
MARCH 2001
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.
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. ▲
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