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Math in Motion: Origami Math for Students
Who Are Deaf and Hard of Hearing
Kaili ChenNational Institute of Education, Singapore
This article aims to provide an overview of the use of origami
in teaching mathematics to deaf and hard-of-hearing stu-
dents. The author posits that in both the general and specialeducation settings, origami can be very useful for students
who are deaf and hard of hearing as many of them need to
see and feel to learn and are likely to be concrete learners.
Suggestions about how to conduct an origami math lesson
for deaf and hard-of-hearing students are included in this
article.
Many deaf and hard-of-hearing students show a signif-
icant delay in learning math (Stewart & Kluwin, 2001;
Traxler, 2000), and as a result, these students may not
be able to take advantage of many regular, unmodified
activities in math, they are often not given the oppor-tunity to develop skills within their ability, and they do
not learn to play math games or activities in the gen-
eral education classroom. Too often, unless a conscious
effort is made to accommodate, lessons on math con-
cepts may go unnoticed.
With some creative thought and imagination, how-
ever, curriculum adaptations can be made to meet the
needs of these learners. Through adapted activities
such math origami, which is about teaching mathemat-
ics using origami, it is possible to improve the stu-
dents math skills and enable them to learn and use
new and specific math concepts. The purpose of this
paper is to provide an overview of the use of origami in
teaching mathematics to students who are deaf and
hard of hearing. The proposal is that in both the gen-
eral and special education settings, origami can be very
useful for deaf and hard-of-hearing students, as many
of these students need to see and feel to learn and
therefore are more likely to be concrete learners. Sug-
gestions about how to conduct an origami math lesson
for students who are deaf and hard of hearing are also
included.
A successful inclusive classroom provides interac-
tive tasks based on learning styles and individual
needs. Froebel, Montessori, and Steiner as well as
many other founders of age-appropriate practice all
created rich, hands-on materials for children to ex-
plore and conceptualize. Other educators who estab-
lished early childhood as a legitimate time for guided
learning all emphasized the importance of manipula-
tive experiences of seeing, touching, and handling
things and of experiencing new sensations for infants
and young children and the dangers of introducing
them to the world of symbols too early in life.
Why Math Origami for Deaf and Hard-of-
Hearing Students?
Although learning can occur through both active and
passive involvement, much of students learning comes
from activities and projects in which the students are
active participants, rather than passive recipients of
knowledge given by the teacher. In math teaching, it
is important to make math concepts more visual for
all students, especially for deaf and hard-of-hearing
Correspondence should be sent to Kaili Chen, 1 Nanyang Walk, ECSE,
National Institute of Education, Singapore 637616 (e-mail: klchen@
nie.edu.sg).
The Author 2005. Published by Oxford University Press. All rights reserved.
For Permissions, please email: [email protected]
doi:10.1093/deafed/enj019
Advance Access publication on December 6, 2005
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students who have communication difficulties. There-
fore, the use of manipulatives, games, and activities is
essential.
In both the general and special education settings,
origami can be very useful for deaf and hard-of-
hearing students, as many of these students will benefit
from hands-on activities. My experience of teachingmath to deaf and hard-of-hearing students shows that
math origami activities do help these students under-
stand mathematical concepts better and, most of all,
motivate them and increase their math skills.
Origami is an art of paper folding. In Japanese, it
literally translates as ori (folding) gami (paper). In
traditional origami, constructions aredone using a piece
of square paper. Another form of origami is called
modular origami, it is a compound structure com-
posed of a number of individual units that are folded
from a single sheet of paper. Math origami activities
incorporate both mental and physical involvement in
the learning process; students have to use their hands
and brains at the same time. As students fold and crease
paper into creative shapes and models, they build their
fine motor skills and social skills such as paying atten-
tion to the speaker and following instructions. Origami
allows deaf and hard-of-hearing students to become
more involved, leads to lively class discussions, and
arouses their interest in the curriculum.
Mathematical concepts and thinking can also be
illustrated through this rich art form. Many teachers
have developed hands-on lessons that use origami to
make math come to life for their students. Hands-on
origami activities enable students to build their own
experiential base relating to the development of certain
mathematical concepts and to explore many geometric
forms while problem solving and constructing. Math
skills and concepts that are inherent in origami include
spatial visualization, intersecting planes, area and vol-
ume, mirror images, and many more. Origami can also
be used to teach symmetry. For instance, for many ofthe folds, whatever is done to one side is likewise done
to the other. Hands-on activities such as this make
math concepts visual and allow deaf and hard-of-
hearing students to create and manipulate basic geo-
metric shapes such as squares, rectangles, and triangles
that might otherwise be taught through lectures and
without personal and concrete experience.
Further, origami can be designed for users of all
ages. Young students can explore mathematical con-
cepts and develop a mathematical foundation even be-
fore they are introduced to the more formal concepts.
For older students, paper folding can help them to
master mathematical concepts that are more difficult
to grasp by other means. Regardless of the age,however, turning a flat piece of paper into a three-
dimensional figure can be an empowering experience
in spatial reasoning for all students.
Despite the many educational benefits and the
wealth of information on the subject of math and ori-
gami, math origami is a fairly new field; there are very
few references for the use of origami for students with
hearing impairment at the time of writing this article.
However, mathematicians have already been investi-
gating a wide range of questions relating to paper
folding. Levenson (n.d.) has found that origami has
shown that paper folding, particularly in the elemen-
tary school years, is a unique and valuable addition to
the math curriculum. After all, the process of trans-
forming a piece of scrap paper into a flying swan not
only makes children feel empowered, it also can be
used to link math and origami skills and help children
understand spatial relationships of three-dimensional
objects, investigate the symmetry, congruence, and
angles of geometry, and develop their analytical and
critical thinking skills.
In recent years, math origami theory has been ap-
plied to produce a wide range of practical applications
(Origami, n.d.), and new materials, new methods, and
new ideas have transformed the traditional world of
math origami. Many origamists have intentionally
ventured into the abstract world of mathematics,
assembling spectacular interlocking polyhedra or tile
mosaics. Other new technologies being developed
include paper product designs involving no adhesives,
better ways of folding maps, unfolding space tele-
scopes and solar sails, and many more (Origami, n.d.).As mentioned above, deaf and hard-of-hearing
students often have a delay in their math performance.
Therefore, teachers should choose the appropriate
adaptations of activity to meet the needs of the child
(Stewart & Kluwin, 2001). By using origami, teacher
can make lessons relevant, functional, and interesting.
In addition, math origami is appropriate for all ages,
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inexpensive to do, and will challenge students of all
ability levels. In summary, origami math as a hands-on
learning art is an innovative method for teaching math
to deaf and hard-of-hearing students at various levels.
Teaching Strategies and Accommodations for
Deaf and Hard-of-Hearing Students
Because math origami has so many educational
benefits, how can classroom teachers use it to teach
math through motions? The following section will
delineate general strategies that can be applied while
teaching math through origami.
First of all, the teacher will need to think through
the math concepts that are to be highlighted in the
activity. It will also be important for the teacher to try
the activity beforehand to anticipate any problems that
the students may encounter due to their communica-
tion difficulties. A completed model will be good
because it sets an example for them.
Most of the time, a single sheet of origami paper
(typically colored and square) is all that is needed for
the activity. Regular copy paper, gift wrap, magazines,
and other used paper can be excellent substitutes.
However, it is important to keep in mind that the
thinner the paper is, the easier it is to fold. In addition,
it is best to use paper where the two sides are easily
distinguished. If the students are young and are at the
early stage of learning, the teacher may want to dem-
onstrate and teach them some of the geometric prop-
erties of the square paper that are often exploited for
folding first. For instance, the teacher can explain that
a square is regularit has four 90 angles and four
sides that are of the same length. In addition, the
corner of the square takes up 90 of paper, the edge
180, and the middle 360. If the students sign, then
the teacher will need to make ample use of finger
spelling as it helps to indicate clearly which mathemat-
ical term will be used in the origami activity.The next step will be distributing the key direc-
tions (typically in diagrams) to the students. If the
students are beginners, it may be necessary to teach
them how to read origami diagrams. For example, it
may be helpful for them to understand that the two
basic and simplest origami folds (i.e., valley and moun-
tain folds) are generally marked by dotted lines. The
lines will either be a mountain fold (dot, dash, dash,
dot, dash, dash, etc.) or a valley fold (dash, dash, dash,
etc.). When folding a mountain fold, they should fold
behind along the line to simulate a mountain, and
when folding a valley fold they should fold it over itself
to simulate a valley. If more verbal instruction is
needed, the teacher will need to speak distinctly asthe diagrams may appear to be complicated and
confusing to beginners. Further explanation may be
necessary to help them create a mental picture of
math concepts involve in the origami activity.
Next, the teacher needs to demonstrate the folds
(with a bigger sheet of paper if necessary) and make
sure that the paper faces the way the students paper is
facing them. The teacher may need to go around the
classroom and provide more support for students who
need more help with following directions or have
difficulties understanding spatial relationships.
Deaf students are often socially isolated in regular
schools settings (Andrews, Leigh, & Weiner, 2004) and
therefore will need the support and encouragement of
teachers to develop positive relationships with peers,
which as a result will help with their academic learn-
ing. With this in mind, the teacher may want to divide
the class into small groups and let students who have
completed one fold assist others. Students can also
share their findings with one another. This arrange-
ment will help the teacher address more of the stu-
dents questions while at the same time foster peer
mediation. Successful cooperative learning activities
such as this gives students opportunities to use math
vocabulary in context and makes math an interactive
problem-solving experience.
In a typical origami activity, the next step will be
for the teacher to help students fold the bases (origami
bases are sometimes called fundamentals in ori-
gami). An origami base is a partially folded model
from which several figures can be created and is usu-
ally named after a typical figure that can be createdfrom it (e.g., bird base, kite base). Students can
understand origami better by means of the bases used
(as space is limited, diagrams of these bases are not
included in this article, but readers can find plenty of
information on the internet and origami books). It is
also essential that students fold on a smooth, hard
surface. Beginners will need to be reminded that
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neatness does count. They will have more fun and are
more likely to experience success if they apply these
origami principles.
Finally, to reinforce the math concepts that have
been taught, the teacher may want to ask the students
to unfold their origami projects to look at the patterns
and geometric figures they have created through theseries of creases and to recognize the sequence of steps
involved. According to Stewart and Kluwin (2001),
the recognition of patterns in mathematics along with
sequencing is a major concern for many deaf children,
so this activity will be useful.
Math origami abounds with open problems that
can be used to give our students opportunities to
contribute original ideas. For example, the teacher
may want to encourage students to compare models
to models and folds to folds. If the students are at the
upper level, the teacher may also want to ask them to
describe and keep track of symmetries in models as the
folding proceeds. Questions such as these will help to
build awareness and understanding of math concepts.
In addition, to challenge more capable students, the
teacher may want encourage them to create their own
variations and make their own diagrams and share with
the class how they did it.
To help the students understand linear and two and
three dimensions better, the teacher may want to ask
them when in the folding procedure does a model be-
come three dimensional and if there is a simpler pro-
cedure for folding a certain three-dimensional figure.
However, challenges such as these will not be meaning-
ful unless they are developmentally appropriate.
Deaf and hard-hearing individuals (who sign)
have an advantage over hearing individuals in making
math visual in the head and doing mental manipula-
tions (see Marschark, 2003). This strength makes
origami particularly appropriate for students who
are deaf and hard of hearing and might be especially
motivating and socially useful and reinforcing. Theability to visualize in the head is a talent that can be
strengthened with simple practice. With this in mind,
the teacher may want to challenge students to con-
sider beforehand what will be the results of making
a fold and ask them to visualize it in their minds and
encourage them to pose generalizations on the effects
of folds (e.g., folding an edge to a parallel edge divides
an area in half). To further help students develop
their math origami skills, teachers may want to illus-
trate the following origami axioms formulated by
Huzita (1992): (a) given two points P1 and P2, we can
fold a line connecting them. (b) Given two points P1
and P2, we can fold P1 onto P2. (c) Given two lines L1
and L2, we can fold line L1 onto L2. (d) Given a pointP1 and a line L1, we can make a fold perpendicular to
L1 passing through the point P1. (e) Given two points
P1 and P2 and a line L1, we can make a fold that places
P1 onto L1 and passes through the point P2, and
(f) given two points P1 and P2 and two lines L1 and
L2, we can make a fold that places P1 onto line L1
and places P2 onto line L2. These axioms then can
be extended to their more general application in ge-
ometry (e.g., any two points can be connected to
form a line.
Furthermore, as the students become more pro-
ficient in math origami, it also can be advantageous
to make use of computer technology to make folding
directions (e.g., diagrams) or other teaching aids. The
use of the computer often serves to make the activities
more appealing to students and improves their math
skills while at the same time raises the standard for
student work.
Though the foregoing suggestions are aimed at
helping deaf and hard-of-earing students to learn math
through origami, hearing students can benefit from
them as well because some hearing students may learn
math better when information is presented visually.
Therefore, math origami can be useful for both
hearing and deaf and hard-of-hearing students when
taught in the inclusive classroom settings.
Conclusion
In summary, implementing origami in the classroom
can mean rich, hands-on math experiences for stu-
dents who are deaf and hard of hearing. Math origamican be used as an entry point to mathematical dis-
course and applications called for by the U.S. National
Council of Teachers of Mathematics (1989) and other
reform efforts. Implemented as a math-based activity,
origami makes mathematics more visual and hands-on,
which we want our deaf and hard-of-hearing students
to experience. Hopefully, in this way, math teachers
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can help the public education fulfills its responsibilities
of maximizing each students opportunity to learn and
succeed in the least restrictive environment.
References
Andrews, J. F., Leigh, I. W., & Weiner, M. T. (2004). Deafpeople: Evolving perspectives from psychology, education, and
sociology. Boston: Allyn & Bacon.
Huzita, H. (1992). Understanding geometry through origami
axioms. In J. Smith (Ed.), Proceedings of the First Interna-
tional Conference on Origami in Education and Therapy
(pp. 3770). London: British Origami Society.
Levenson, G. (n.d.). The educational benefits of origami. Retrieved
August 29, 2005, from http://web.archive.org/web/
20011218014951/www.fascinating-folds.com/learningcenter/
educational.htm
Marschark, M. (2003). Cognitive functioning in deaf adults
and children. In M. Marschark & P. E. Spencer (Eds.),
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(pp. 464477). New York: Oxford University Press.
National Council of Teachers of Mathematics (1989). Curricu-
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VA: Author.
Origami. (n.d.). Retrieved August 29, 2005, from http://math.
serenevy.net/?page=Origami-WhereMathStewart, D. A., & Kluwin, T. N. (2001). Teaching deaf and hard
of hearing students: Content, strategies, and curriculum.
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Traxler, C. B. (2000). The Stanford Achievement Test, 9th
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Received September 1, 2005; revisions received October 27,
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