j. deaf stud. deaf educ.-2006-chen-262-6

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Endnote

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


2/5

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,

Math in Motion 263


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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

264 Journal of Deaf Studies and Deaf Education 11:2 Spring 2006


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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

Math in Motion 265


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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.),

Oxford handbook of deaf studies, language, and education

(pp. 464477). New York: Oxford University Press.

National Council of Teachers of Mathematics (1989). Curricu-

lum and evaluations standards for school mathematics. Reston,

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.

Boston: Allyn & Bacon.

Traxler, C. B. (2000). The Stanford Achievement Test, 9th

edition: National norming and performance standards for

deaf and hard-of-hearing students. Journal of Deaf Studies

and Deaf Education, 5, 337348.

Received September 1, 2005; revisions received October 27,

2005; accepted October 31, 2005.

266 Journal of Deaf Studies and Deaf Education 11:2 Spring 2006
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