Alexander Shirokov/iStockphoto.com
Paint
bucket
polygons
420 March 2010 • teaching children mathematics Copyright © 2010 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
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Use photo editing
software as a teaching
tool to bring inaccessible
polygon definitions within
reach of your students’
understanding.
By Michael Todd Edwards and Suzanne R. Harper
D
uring a two-week summer professional
d e v e l o p m e n t w o r k s h o p, t e a m s o f
intermediate-level school teachers and
college methods instructors crafted mathematics
learning modules—activities, lesson plans,
worksheets, and technology-oriented tasks—with
the primary goal of strengthening students’
understanding of various geometric concepts. They
recast the paint bucket features of photo editing
software as a teaching tool to foster discussion,
debate, and heightened understanding of polygons
with fifth-grade geometry students. As students
“filled” geometric shapes with the paint bucket, they
explored nuances of polygons too often overlooked
by contemporary elementary-level textbooks.
Read on to see how the paint bucket metaphor
could allow your students to build sophisticated
notions of polygons in ways that are developmentally
appropriate for early-grades learners.
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teaching children mathematics • March 2010 421
Figu r e 1
Typical shape puzzles for young children reinforce shape
prototypes.
Figure 2
Students’ initial task was to identify polygons and
nonpolygons from a collection of prototypical shapes
projected at the front of the classroom.
they resemble door prototypes (Clements and
Sarama 2000). Triangles are reduced to shapes
that resemble pieces of pizza or slices of pie.
When octagons are reduced to stop signs,
students fail to recognize irregular eight-sided
shapes as octagons, too. Children’s puzzles,
picture books, and school texts reinforce the
view of regular shapes as prototypes (e.g., triangle, square, pentagon) while de-emphasizing
essential characteristics that define particular
shapes. The wooden puzzle pieces in figure 1
suggest that squares and rectangles are necessarily distinct; they inadvertently reinforce
the idea that squares are not rectangles and
rectangles are not squares. The notion is lost
that any quadrilateral with four right angles is
a rectangle.
As the teachers in the professional development session constructed first drafts of polygon
lessons, they took into account the importance
of prototypical and nonprototypical shapes
in their students’ geometric development,
designing progressively nonprototypical activities. They aimed to build on previous, intuitive
understanding of polygons while stretching
students to construct increasingly formal
definitions through encounters with unfamiliar
shapes. What follows are details of these initial
lessons as well as the revisions they prompted.
What is a polygon?
Prototypes and nonprototypes
Eduard Härkönen/iStockphoto.com
For the purposes of the study, a prototype is an
early, typical example that serves as a basis or
standard for subsequent stages of investigation (Merriam-Webster 2008). Prototypes are
important in early stages of geometry learning.
For instance, they provide examples that allow
children to associate names with various types
of polygons (e.g., triangle, square, rectangle,
pentagon) and nonpolygons (e.g., crescent,
circle). Unfortunately, as students continue
to study shapes, overreliance on prototypes
ultimately inhibits their recognition of essential and nonessential attributes (Driscoll et al.
2007). When shapes are not sufficiently similar
to prototypes in orientation or proportion,
children fail to identify them correctly (Hoffer
1988). For instance, students may mistakenly
conclude that objects are rectangles only when
422
March 2010 • teaching children mathematics As part of the lesson-study process, a team of
teachers observed the lesson as it was taught in
a colleague’s fifth-grade classroom. The teacher
began with a short warm-up asking students to
identify polygons and nonpolygons from a collection of prototypical shapes on an overhead
projector screen at the front of the classroom
(see fig. 2). Although the shapes were prototypical, identifying the polygons was surprisingly
problematic for the fifth graders. After several
minutes of classroom discussion, a majority of
students agreed that polygons must have six
sides. Although this inaccurate claim initially
startled the team, upon further reflection and
discussion with students, the basis for this
flawed thinking became clear. Students had
observed that a shape with six sides is, in fact, a
polygon. They reasoned that all polygons must
have six sides. The idea that a shape with six
sides is one particular type of polygon was difficult for them to fully comprehend.
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shapes 3, 5, and 6 in large part because the
working definition was sufficiently rigorous to
detect these shapes as polygons.
After spending the better part of a fifty-minute class session identifying polygons and nonpolygons with fifth graders, the lesson-study
team decided to spend the afternoon revising
the module.
Troublesome definitions
After lunch, the team met to review video of
the morning’s lesson and discuss possible follow-up activities for the fifth graders. Watching
the video made it apparent that a more mathematically rigorous definition of polygons was
needed to engage students in investigations
of nonprototypical shapes. A quick search of
the Web and several textbooks soon revealed
a surprising number of different definitions
for polygon. (In an examination of eightyfive school mathematics texts, Usiskin and
Griffin [2008, p. 2] found twenty-one essentially different definitions for polygon). The
current team’s research uncovered definitions
of two general types:
• Simplistic—Although arguably accessible to
elementary school students, the definition
fails to discriminate between polygons and
nonpolygons for examples beyond prototypical shapes.
F igure 3
After considerable discussion, a classroom
teacher proposed that the class search the
Internet for polygon. Quickly accessing an
online dictionary, students noted that the word
polygon is derived from two Greek roots: poly,
meaning many, and gon, meaning angle (Page
2008). From these roots, the group initially
defined a polygon as a geometric shape consisting of many angles. The fifth graders and classroom teachers considered this definition carefully. Several children objected to the apparent
ambiguity of the word many in this definition.
Ultimately, after several minutes of discussion,
the definition that was deemed acceptable by
all teachers and students was a geometric shape
consisting of at least three angles.
With this revised definition in hand, the
lead teacher asked the fifth graders to revisit
the shapes in figure 2. After less than a minute
of discussion, small groups of students agreed
unanimously (and correctly) that shapes 1, 2, 3,
and 5 are polygons because each of these shapes
has at least three angles. Shape 4, a circle, is not
a polygon because it has no angles. At this point,
the teachers seemed relieved. Although no one
had anticipated students’ difficulties with the
first warm-up task, constructing a working definition for polygon with students was clearly an
appropriate use of instructional time. Slightly
behind schedule, the lead instructor proceeded
to a second warm-up task.
The teacher asked students to identify
polygons and nonpolygons from a collection
of nonprototypical shapes (see fig. 3). Once
again, shapes were projected at the front of the
classroom for students to consider. This time,
students were given index cards and were asked
to jot down numbers corresponding to polygonal shapes.
Much to the chagrin of all the teachers in the
room, the initial polygon definition (namely,
a geometric shape consisting of at least three
angles) proved inadequate when considering
nonprototypical shapes. Shapes 2 and 4 generated considerable discussion and disagreement among students. Although shape 2 is not
closed, approximately half the class identified it
as a polygon because the shape comprises three
angles. Similarly, despite the fact that shape 4 is
not simple, more than half the class identified
it as a polygon because it is composed of three
polygons. Most students correctly identified
After an Internet search to revise their definition, the fifth
graders revisited nonprototypical shapes.
teaching children mathematics • March 2010 423
Figu r e 4
This nonprototypical shape is
composed of line segments. It is
not a polygon, because two sides
intersect.
• Inaccessible—Detailed enough to discriminate between polygons and nonpolygons,
the definition nevertheless contains terminology too abstract and too technical for
­intermediate-level students.
Figure 5
Below are examples from each category along
with a short commentary regarding the inappropriateness of the definition for a follow-up
lesson with fifth graders.
The lesson-study team used photo editing software to
develop another series of nonprototypical shapes for
students to explore their working definition.
A well-known fourth-grade math text defines
a polygon as a closed, two-dimensional figure
that has straight sides (Macmillan/McGrawHill 2002, p. 412). Although the definition is
relatively accessible, it lacks the rigor required
to distinguish polygons from nonpolygons. The
shape in figure 4 is composed of line segments;
but it is not a polygon, because two sides intersect (i.e., it is not simple). As an example of an
inaccessible definition, the popular Web site
SparkNotes.com claims that polygons are one
type of a simple closed curve, defining them as
the union of three or more line segments whose
endpoints meet. Although arguably more rigorous, this definition is problematic for use with
young students because it assumes that they
know the mathematical meanings of the terms
simple and closed, not to mention union, endpoints, and so on.
A partial solution: the paint bucket
To have deep comprehension of polygon
definitions, one that will serve them well in
their mathematical futures, understanding
the terms simple and closed will help students.
However, these terms are abstract for a number of fifth graders; they must experience the
shapes within familiar, everyday contexts.
Because many elementary school students
have used computer software to color images,
424
March 2010 • teaching children mathematics www.nctm.org
Figu r e 6
Figure 7
Students predicted that the color would stay inside shape 1,
which it did.
Beata Becla/iStockphoto.com
Most upper-grades elementary school students are familiar
with the paint bucket tool (see shape 1) in photo editing
computer programs; when clicked, it shades a closed region
with a user-specified color or pattern.
the team introduced the concept of digital coloring to help students understand simple and
closed in the more rigorous polygon definition. Now more attuned to polygon characteristics, the team used photo editing software to
develop another series of shapes for students
to explore with their working definitions (see
fig. 5).
That afternoon, the team presented a
revised version of the lesson with another
group of fifth graders. Once again, shapes were
projected at the front of the classroom, this
time in photo editing software. When the paint
bucket—a tool common to such software—is
clicked, it shades a closed region with a userspecified color or pattern.
As students identified potential polygons,
their teacher asked them to predict the effect
of clicking inside shape 1 with the paint
bucket (see shape 1 in fig. 6). Already familiar
with the technology, students predicted that
the color would stay inside shape 1, an irregular octagon (see fig. 7). They quickly identified
several more polygons, which the teacher used
the paint bucket tool to color.
Shapes 3 and 7 were designed to help
students refine their working definition of
polygons. When asked if shape 3 is a polygon,
students discussed its angles and straight
sides. The lead teacher asked what would
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teaching children mathematics • March 2010 425
Andrew Johnson/iStockphoto.com
Figure 8
426
When the paint bucket tool was clicked and the color went
beyond shape 3, it prompted a discussion of the polygon
characteristic closed.
March 2010 • teaching children mathematics happen if she used the paint bucket to color
shape 3. One student explained that if you
click on shape 3, the color “will all spread out,”
meaning that the color would go beyond the
shape’s outline, which it did (see fig. 8).
Shape 3 differs from the previously identified polygons; the paint bucket tool did not
shade it in the same manner, prompting a
discussion of closed figures. To the students,
a shape is closed if clicking it with the paint
bucket tool results in a contained painted
region. They concluded that shape 3 is not a
polygon because the color was not contained.
Similarly, they questioned whether shape 7 is a
polygon. On the basis of their previous experience with digital coloring, they predicted that
only part of the shape—namely, one of the
triangles—would be colored.
The teacher tested their conjecture (see
fig. 9). Although the paint bucket tool did not
color shape 7 in the same manner as other
identified polygons, students were not as
quick to identify this shape as a nonpolygon.
To help them do so, the teacher asked them
to count the shape’s sides. Students arrived at
two different answers, seven and nine. Despite
small-group discussions of the merits of each
answer, students were unable to agree on the
number of sides.
The teacher brought the class back together
and asked, “Why can’t you agree? What do you
think this means in terms of shape 7?”
Students agreed that shape 7 is different
from the others and concluded that it must
not be a polygon. Their conclusion prompted
a discussion of simple figures. Shape 7 is selfintersecting, so it is not simple.
The class noted that each shaded polygon
in figure 9 is both closed and simple, terms
that became more mathematically meaningful
as students worked with the paint bucket. The
inaccessible definition of polygon was now
within reach.
Although pedagogically useful, the paint
bucket notion does not replace rigorous mathematical definitions. Some shapes that the
paint bucket will color are not polygons—for
example, shapes with curved versus straight
sides (see shape 6). However, experience
shows that students have less difficulty with
the aspect of the polygon definition having
only line segments as sides than they do with
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Conclusion
Too often, mathematics is erroneously portrayed as a static body of knowledge discovered
long ago. In truth, mathematics is ever changing. Students should know that math is a human
activity that is negotiated through social interaction. The study of definition enables teachers
to foster this view with students. Without question, definition is essential to mathematics.
Students ought to recognize the necessity for
clarity of definition in all matters where precise
thinking is essential. To understand how the
vagueness and ambiguity of ordinary words
leads to serious errors in reflective thought is
to appreciate the importance of clearly defined
concepts in any technical vocabulary (Fawcett
1938, p. 30).
The teachers on this lesson-study team
thought that their students completely understood the definition of polygon; however, this
was the case for only the most basic of shapes.
When students were confronted with unfamiliar shapes, the overly simplistic definitions
that had served them well in the past were no
longer sufficient. Although this insufficiency
caught the team off-guard, it supplied a context for a rich activity for students to further
explore prototypes and nonprototypes of
polygons from a different perspective, using
familiar coloring tools.
As students used the software, they physically interacted with the polygons, obtaining
immediate feedback while strengthening their
understanding of what a polygon is. The paint
bucket metaphor allowed them to build a more
sophisticated notion of polygon that better
served their current needs. More important, it
offered an opportunity to view mathematics as
the dynamic area of study that it is.
bibliography
Clements, Douglas H., and Julie Sarama. “Young
Children’s Ideas about Geometric Shapes.”
Teaching Children Mathematics 6, no. 8 (April
2000): 482–87. Driscoll, Mark, Rachel W. DiMatteo, Johannah
Nikula, and Michael Egan. Fostering Geometric
Thinking: A Guide for Teachers, Grades 5–10.
Portsmouth, NH: Heinemann, 2007.
www.nctm.org
Figu r e 9
the mathematical concepts of closed and
simple.
Some shapes that the paint bucket will color are not
polygons—for example, shape 7.
Fawcett, Harold P. The Nature of Proof: A Description and Evaluation of Certain Procedures Used
in a Senior High School to Develop an Understanding of the Nature of Proof (Thirteenth
Yearbook). New York: Teachers’ College, Columbia University, 1938.
“Geometry: Polygons.” SparkNotes. 2009. http://
www.sparknotes.com/math/geometry1/
polygons/section1.html.
Hannibal, Mary A. “Young Children’s Developing
Understanding of Geometric Shapes.” Teaching
Children Mathematics 5, no. 6 (February 1999):
353–57. Hoffer, Alan R. “Geometry and Visual Thinking.” In
Teaching Mathematics in Grades K–8: ResearchBased Methods, edited by T. R. Post, pp. 232–
61. Boston: Allyn and Bacon, 1988. Lewis, Catherine C. Lesson Study: A Handbook of
Teacher-Led Instructional Change. Philadephia:
Research for Better Schools, 2002.
Mathematics. New York: Macmillan/McGraw-Hill
School, 2002.
Merriam-Webster Online. 2009. www.merriamwebster.com.
teaching children mathematics • March 2010 427
Page, John. “Polygon.” Mathematics Open Reference Project. 2009. www.mathopenref.com/
polygon.html.
Usiskin, Zalman, and Jennifer Griffin. The Classification of Quadrilaterals: A Study in Definition.
Charlotte, NC: Information Age Publishing,
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Call for Manuscripts
The authors thank classroom teachers Gail
Banks, Susie Mabry, and Jaime Hutchins for
the opportunity to explore mathematics with
their students, as well as Kathy Jones, the
Department of Learning Mathematics Coordinator for the Middletown City School District
in Ohio. The authors also thank the anony-
428
mous reviewers for their thoughtful comments
on an earlier draft of this article. The work
reported herein received support from an Ohio
Department of Education Mathematics and
Science Partnership Grant, Miami University
Partnership for Enhancing the Teaching of
Mathematics.
Michael Todd Edwards,
edwardm2 @muohio
.edu, and Suzanne
Harper, harpersr@
muohio.edu, are mathematics education colleagues at Miami University,
Oxford, Ohio, where they teach preservice and inservice mathematics teachers. Their professional interests include the development of technology-oriented
instructional materials that foster rich mathematical
explorations in the early grades.
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