Gaining Insights into
Children’s
Geometric
Knowledge
W
hile I was working with a third-grade
teacher and his thirty-two students, he
turned to me and asked, “Would you work
with the children on geometry?” Although I eagerly
agreed, several questions raced through my mind:
“I have not had much experience helping children
learn geometry and do not know much about their
thinking. Will I be able to help these children learn
geometry in ways that are meaningful to them?
What should third graders learn about geometry?
What do the children already know about geometry
and the ideas they should learn?” Answers to the
latter two questions were especially important in
helping the children develop a conceptual understanding of geometry by building on their existing
knowledge. This article describes how research on
children’s geometric thinking used in conjunction
with a children’s book provided valuable insights
By Nancy K. Mack
Nancy Mack, [email protected], teaches preservice elementary teachers at
Grand Valley State University, Allendale, MI 49401-9403, and serves as a
volunteer mathematics teacher at Shawmut Hills Elementary School, Grand
Rapids, MI 49504. She is interested in the teaching and learning of fractions
as well as putting research on children’s mathematical thinking into practice
in mathematics teaching in elementary school classrooms.
238
into their prior geometric knowledge of the mathematical names and properties of polygons.
What Should Children Learn
about Geometry?
NCTM’s Principles and Standards for School
Mathematics (2000) suggests that students in
the middle elementary grades should learn about
geometry from a variety of perspectives, including analyzing characteristics and properties of
two-dimensional shapes. In particular, Principles
and Standards states that all students in grades 3–5
should—
• identify, compare, and analyze attributes of
two- and three-dimensional shapes and develop
vocabulary to describe the attributes;
• classify two- and three-dimensional shapes
according to their properties and develop definitions of classes of shapes such as triangles and
pyramids…. (p. 164)
In light of these recommendations, I thought it
important to help the students learn the attributes
and properties of two-dimensional shapes in genTeaching Children Mathematics / November 2007
Copyright © 2007 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.
Photograph by Nancy K. Mack; all rights reserved
eral and polygons in particular. But first I needed to
know what the students already knew about these
concepts.
could help reveal whether the students know the
names of various polygons as well as whether they
understand the concepts of lines and angles and
their relationship to the polygons’ mathematical
names. Further, the book’s examples of tasks that
each shape performs could help reveal connections
the students had made between various polygons
and real-world objects.
Research related to geometric thinking suggests
that when investigating and guiding children’s
learning of geometry, one must consider levels of
thinking. Van Hiele (1999) referred to the lowest level of geometric thinking as the visual level.
A child at this level classifies a two-dimensional
shape only on the basis of its appearance and the
mental visual image he or she has of the shape
rather than the shape’s mathematical properties
(Clements and Sarama 2000; van Hiele 1999). For
example, a child may consider shape 6 a triangle
because it matches her or his mental image of a
triangle; however, the same child may not consider
shapes 9 and 10 triangles because they differ from
her or his mental image of the figure (for numbered
shapes, see template sheet, page 244).
A child begins focusing on the mathematical
properties of two-dimensional shapes at the next
Determining the Students’
Geometric Knowledge
To discreetly determine the students’ existing geometric knowledge and get them excited about learning geometry, I drew on research related to children’s geometric thinking and the picture book The
Greedy Triangle (1994), written by Marilyn Burns
and illustrated by Gordon Silveria. The Greedy
Triangle is a delightful story about a triangle that at
first is happy with its shape and its role in the real
world but gradually becomes dissatisfied, thinking
that if it had one more side and one more angle,
life would be more interesting. The triangle visits
the local Shapeshifter, who transforms the triangle
into a quadrilateral. The pattern of events repeats
several times, and the shape gains more sides and
more angles. Each resulting polygon is referred to
by its mathematical name (e.g., pentagon, hexagon,
heptagon, and octagon).
The Greedy Triangle’s use of mathematical
names and its focus on adding sides and angles
Teaching Children Mathematics / November 2007
239
level, the descriptive level (Clements and Sarama
2000; van Hiele 1999). A child at this level would
view shapes 9 and 10 as triangles because they are
closed figures made up of three straight sides and
three angles. However, the child would not be able
to determine that two of the angles in shape 6 were
equal because two sides were the same length.
Only at the third level, the informal deduction
level, is a child able to deduce that one property
logically follows from another—for example, that a
quadrilateral containing four right angles is a figure
with two pairs of parallel sides. At this level, a child
is also able to use mathematical properties to generate definitions for two-dimensional shapes and
perceive relationships among various shapes, such
as recognizing that squares are both rectangles and
parallelograms (van Hiele 1999).
To determine whether the students had a broad
visual image of various polygons, knew the polygons’ mathematical names and properties, and
perceived relationships among them on the basis
of their mathematical properties, I created a set of
shape cards to use with The Greedy Triangle. The
cards (see template sheet, page 244) depict triangles, quadrilaterals, pentagons, hexagons, octagons, and circles of various sizes and orientations
(Clements and Sarama 2000). For each polygon, I
created five different shapes, at least one of which
attempted to match the children’s visual image of
what that polygon might be. For example, shapes
11 and 21 might likely reflect children’s visual
images of a pentagon and octagon, respectively.
The other depictions were not intended to match
children’s visual images of a particular polygon.
For example, shapes 5 and 14 might likely challenge children’s visual images of a quadrilateral
and pentagon, respectively. Further, the set of cards
include two nonpolygons—shapes 31 and 32—that
resemble triangles when viewed strictly from a
visual perspective but not when considering the
mathematical properties of each shape (Clements
and Sarama 2000; Woleck 2003).
To provide several opportunities to learn about
the students’ geometric knowledge, I developed a
sequence of activities in which the children worked
with the shape cards and interactively read The
Greedy Triangle. The specific sequence is outlined
in figure 1 and described more fully in the following section. The children’s responses to the activities and questions provided many insights into their
knowledge of the mathematical names and properties of polygons, their level of geometric thinking,
and their disposition toward learning geometry.
240
Figure 1
Sequence of activities for gaining insights
into children’s geometric knowledge
1. From a bag, each student blindly draws one
card depicting a two-dimensional shape.
2. The teacher calls out each shape’s mathematical name. Students find others in the
room with the same type of shape and form
a “shape group.”
3. Students in each shape group respond to
questions about their shape. Questions
relate to the shape’s mathematical name, its
properties, and real-world objects having
this shape.
4. The teacher reads The Greedy Triangle
(Burns 1994). At appropriate times during
the reading, each shape group shares its responses to the questions. After reading the
book, the teacher asks the students about
other geometric ideas they are familiar with.
Names of two-dimensional
shapes
To begin, each student drew a card from a paper
bag that contained the set of shape cards (see fig. 2).
The students were not told anything about the cards
before they reached into the bag.
Excitement quickly rose in the classroom as one
student after another selected a card from the bag
and saw the shape depicted. Without any prompting, the students eagerly shared their shapes with
one another, making comments such as, “Mine’s
a square,” “I got a triangle,” and “I don’t know
what mine is.” Their initial comments suggested
that many of them knew the mathematical names
square, rectangle, triangle, and circle but did not
know the names of other polygons, including pentagon, hexagon, and octagon.
Some children looked excited but puzzled as
they rotated their cards and carefully examined the
polygon depicted. Their puzzlement suggested that
their shape did not match their visual image of a
particular polygon.
Visual appearance: The
properties of shapes
Next, I called out each shape’s mathematical name
and asked the students to find all the other students
who had drawn the same shape and form a group
(see fig. 3). This would be their “shape group” for
the remainder of the activities. The students were
intentionally not told how many different shapes
there were or how many of each shape they would
Teaching Children Mathematics / November 2007
find. However, to encourage them to think and stay
involved with the activity, I did tell them that they
would find at least one person who had the same
shape. This approach helped determine whether the
children focused on visual appearance or mathematical properties when classifying polygons and looking
for similarities among polygons as well as circles.
As the students searched for matching shapes,
most of them initially focused only on the visual
appearance of the polygon, circle, or nonpolygon.
When this approach did not quickly produce a
match, some students counted the polygon’s number of sides in their attempt to find a matching
shape. Other children persisted in focusing on the
visual appearance by concentrating on a particular
feature, such as concavity.
Maddie and Mike, who had drawn the two
nonpolygon cards, did not think their shapes were
legitimate. On their own, they realized that the twodimensional shapes, the primary focus of this activity, were closed figures consisting of straight sides.
As Mike explained in reference to shape 31, “I don’t
have a real shape. The lines aren’t touching.” Mad-
die added that shape 32 was “not a shape because
this line is curved. It can only be a shape if the lines
are straight.” Maddie and Mike intuitively deduced
these attributes of a polygon and recognized that the
shapes they had drawn were counterexamples.
The shape search resulted in the formation of
several shape groups (see fig. 4). These included two
groups of triangles. One triangle group consisted
of shapes 6, 7, and 8, and the other consisted of
shapes 9 and 10. We also had one group for foursided shapes (shapes 1, 2, 3, and 4), one group for
five-sided shapes (shapes 11, 12, and 15), one group
for six-sided shapes (shapes 16, 17, 18), and one
group for eight-sided shapes (shapes 21, 22, 23, 24).
Further, there were two mixed-shape groups, one
consisting of four- and five-sided irregular polygons
(shapes 5, 13, and 14) and another consisting of sixand eight-sided irregular polygons (shapes 19, 20,
and 25). In addition, there was one group of circles
(shapes 26–30). Maddie and Mike formed their own
group, which they called “Not a Shape. Nothing”
(the nonpolygons, shapes 31 and 32).
The students in the triangle group consisting of
shapes 6, 7, and 8 explained that their shapes were
triangles because they had three sides and looked
like triangles. This group also explained that shapes
9 and 10 “sort of look like triangles, but not really.
We don’t think they’re triangles, but maybe they
are.” The students’ varied explanations illustrated
how, when a shape appeared to match their visual
image of a triangle, they focused only on the number of sides. Similarly, students in the group for
four-sided shapes mentioned the number of sides
of their polygons. Students in other shape groups,
however, only occasionally mentioned the number
of sides. Their primary focus was the polygon’s
visual appearance, and they explained their grouping as “We sort of look alike” and “We don’t look
like them [any of the other shape groups].” Students
in the mixed-shape groups commented that they
were not sure they actually belonged together, but,
because they did not look like the other shapes,
“we made a group that kinda looks the same, but
not really.”
The students’ comments and actions during the
shape search provided additional evidence that
their geometric thinking focused primarily on the
polygons’ visual appearance and that their visual
image of a particular polygon was likely limited.
Further, their explanations for their groupings
reflected Clements and Sarama’s (2000) suggestion that children may talk about such concepts as
the number of sides of a geometric shape without
Figure 2
Photograph by Nancy K. Mack; all rights reserved
A student randomly selecting a shape
Teaching Children Mathematics / November 2007
241
1. What is the name of your shape? The name
needs to be the same for everyone in your
group.
2. What are two ways that all the shapes in your
group are the same?
3. Where are two places that you find this shape
outside your classroom?
Figure 3
Photograph by Nancy K. Mack; all rights reserved
Students seeking their shape group
Figure 4
Student-initiated shape groups (after further discussion, some
­students voluntarily changed groups)
Shapes
Corresponding shapes in template
Polygons
Triangles (group 1)
6, 7, 8
Triangles (group 2)
9, 10
Four-sided shapes
1, 2, 3, 4
Five-sided shapes
11, 12, 15
Six-sided shapes
16, 17, 18
Eight-sided shapes
21, 22, 23, 24
Mixed shapes (4 and 5 sides)
5, 13, 14
Mixed shapes (6 and 8 sides)
19, 20, 25
Circles
26, 27, 28, 29, 30
Nonpolygons
31, 32
fully understanding how the concept relates to the
shape’s mathematical name.
Names, properties, and
real-world connections
Next, the students worked with their shape group
to respond to three specific questions about having that shape: the shape’s mathematical name, the
shape’s properties, and real-world objects having
that shape:
242
The students enthusiastically discussed the questions. Any uncertainty they had about the shape
groups did not seem to have negatively influenced
their excitement about learning geometry.
Maddie and Mike were initially unsure about
how to respond to the questions because their
shapes were nonpolygons. As Maddie explained,
“We don’t have shapes. The questions are about
shapes. We don’t know what to do.” After some discussion, they decided to respond to the questions by
thinking about a triangle. However, they did not do
so willingly. They kept insisting they did not have
shapes and were unable to set their nonpolygons
aside and think about or draw a triangle. Maddie
and Mike’s responses suggested that their thinking
was strongly influenced by the visual appearance of
their nonpolygons.
We then read The Greedy Triangle in a way that
actively involved the students, who shared their
responses to the three questions at appropriate
times in the story. For example, before we began
reading the book, I asked if we had a triangle group.
The two triangle groups identified themselves, held
up their shape cards for others to see, and shared
their shape’s mathematical name and two properties that all shapes in their group had in common.
Finally, they shared two real-world examples of triangles. When we got to the part in the book where
the triangle changes its shape by gaining one more
side and one more angle, I stopped and asked for
the new shape’s mathematical name. The group
with the shape that matched this new polygon then
shared their shape cards and their responses to the
three questions. We continued reading the book in
this manner, while also frequently making predictions about what would happen next according to
the pattern the students saw emerging in the story.
The students’ responses while reading The
Greedy Triangle further suggested that their knowledge of the mathematical names of two-dimensional
shapes was largely limited to triangle, rectangle,
square, and circle. Only two students knew the
terms pentagon, hexagon, and octagon, but they
appeared to be uncertain about the relationships
between the polygon’s mathematical name and its
Teaching Children Mathematics / November 2007
Figure 5
Photograph by Nancy K. Mack; all rights reserved
Comparing four-sided shapes
number of sides. None of the students was initially
familiar with term quadrilateral, but most were
fascinated with this word and delighted in saying it
over and over. Joel asked if they could learn more
about quadrilaterals, and other students eagerly
seconded his request.
The students’ responses also suggested that their
visual images of various polygons were strong but
limited. With the exception of the triangle and circle
groups, the students responded to the three questions
by focusing on only one of their polygons—the one
thought to match most closely their visual image of
the shape. For example, the group having four-sided
figures responded to the three questions only with
respect to shape 1, a rectangle with typical dimensions and orientation (see fig. 5).
Although the students’ visual image of particular
polygons was strong, it did not necessarily prevent
them from focusing on the shapes’ mathematical
properties. As we read about the greedy triangle
changing its shape, many students revised their
responses to the three questions, realized that they
were in the wrong shape group, and, without any
prompting or guidance from me, found their appropriate group. First, Alex said he thought the two
triangle groups should be one group “because we
Teaching Children Mathematics / November 2007
Figure 6
Students’ misconceptions of a “right angle” and a “left angle”
a. A “right
angle” opens
to the right.
b. A “left
angle” opens
to the left.
all have three sides and three angles,” and the other
children concurred. The two groups then moved
their chairs and formed one large group of triangles.
Later, at varying points in the story, students from
the mixed-shape groups moved to other groups on
their own initiative. When I asked Kayla why she
was changing groups, she explained, “I should be
with the hexagon group. I have six sides and six
angles.” These actions suggested that the students
were learning about relationships between the
243
Template sheet for shape cards (shapes 1–32) (Note: Enlarge for classroom use.)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
244
Teaching Children Mathematics / November 2007
polygons’ mathematical names and their properties.
The students’ responses further suggested that
their knowledge of the polygons’ mathematical
properties was largely limited to the number of sides
and angles. All groups described the similarities
among their shapes in terms of the number of sides,
while two groups also referred to angles. No groups
referred to any other mathematical properties.
After finishing the book, I asked the students if
they knew other properties of geometric shapes.
They said they knew about right angles, and some
of them quickly pointed out appropriate examples
in objects in the room. However, Jonny also said
there are “left angles,” which he identified by pointing to an example and saying, “See, it opens to the
left.” Further questioning revealed that the students
had seen examples of right angles that opened
only to the right (see fig. 6a). When the angle was
rotated or flipped to open toward the left (see fig.
6b), Jonny and several other students called it a
“left angle.” Such comments suggested that some
of the students’ conceptions of right angles were
based on visual appearance rather than a conceptual understanding of 90-degree angles. When I
asked the students if they had heard of parallel and
perpendicular lines, the response was thirty-two
blank stares.
The students’ responses also suggested that
they had made connections between various twodimensional shapes and real-world objects. The
students quickly provided numerous real-world
examples of triangles, quadrilaterals, and circles
and, after a little time to think, also a few appropriate examples of pentagons, hexagons, and octagons.
Whenever the book mentioned the same real-world
example they themselves had suggested for a particular polygon, such as a house for a pentagon, the
students cheered and applauded. On their own, they
eagerly looked for additional examples of polygons
on their clothing and in the room. Maria was quite
observant and noticed that the bottom edge of her
open umbrella formed an octagon!
The students’ actions and responses suggested
that they were excited about learning geometry and
eager to learn more about polygons. At the end of
our activities, Alex commented, “That was fun. Can
we do more of this next time?” And Maddie quickly
added, “Yea, I want to learn about quadrilaterals.”
I could build on during instruction. For many
students, their knowledge was initially tied to the
polygons’ visual appearance. Some were able to
begin to shift their focus from visual appearance
to mathematical properties as they considered concepts such as the number of lines and angles with
respect to a polygon’s mathematical name. Further,
the children enjoyed learning about polygons.
As teachers attempt to determine their students’
geometric knowledge, they may find it helpful to
consider questions such as these:
• Can the students identify various shapes by their
mathematical names and properties, or is their
identification based on the visual appearance of
shapes?
• Are the students able to analyze various shapes
to determine their mathematical properties?
• Do the students see various shapes as being
related through the mathematical properties
they share, or are the relationships they perceive
based on visual appearance?
• How can I gain needed insights into the children’s geometric knowledge in a way that will
excite them about learning geometry?
By considering such questions, teachers may
find new ideas to share regarding how to help students learn geometry in ways that are meaningful
to them.
References
Burns, Marilyn. Illustrated by Gordon Silveria. The
Greedy Triangle. New York: Scholastic Inc., 1994.
Clements, Douglas H., and Julie Sarama. “Young Children’s Ideas about Geometric Shapes.” Teaching Children Mathematics 6 (April 2000): 482–88.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, VA: NCTM, 2000.
Van Hiele, Pierre M. “Developing Geometric Thinking
through Activities That Begin with Play.” Teaching
Children Mathematics 5 (February 1999): 310–16.
Woleck, Kristine Reed. “Tricky Triangles: A Tale of One,
Two, Three Researchers.” Teaching Children Mathematics 10 (September 2003): 40–44.
The author would like to thank the students at
Shawmut Hills Elementary School, Grand Rapids,
Michigan, for participating in this lesson and the
teachers and administrators for providing her with
an opportunity to serve as a volunteer mathematics
teacher at the school. s
Closing Comments
My first goal was to learn something about the
students’ geometric knowledge of polygons that
Teaching Children Mathematics / November 2007
245