Math in Motion: Origami Math for Students Who Are Deaf and Hard of Hearing

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1 Endnote Math in Motion: Origami Math for Students Who Are Deaf and Hard of Hearing Kaili Chen National 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 students. The author posits that in both the general and special education 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 significant 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 opportunity to develop skills within their ability, and they do not learn to play math games or activities in the general education classroom. Too often, unless a conscious effort is made to accommodate, lessons on math concepts may go unnoticed. With some creative thought and imagination, however, curriculum adaptations can be made to meet the needs of these learners. Through adapted activities such math origami, which is about teaching mathematics using origami, it is possible to improve the students 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 Correspondence should be sent to Kaili Chen, 1 Nanyang Walk, ECSE, National Institute of Education, Singapore ( klchen@ nie.edu.sg). hard of hearing. The proposal is that 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 need to see and feel to learn and therefore are more likely to be concrete learners. Suggestions 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 interactive 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 explore and conceptualize. Other educators who established early childhood as a legitimate time for guided learning all emphasized the importance of manipulative 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 Ó The Author Published by Oxford University Press. All rights reserved. For Permissions, please journals.permissions@oxfordjournals.org doi: /deafed/enj019 Advance Access publication on December 6, 2005

2 students who have communication difficulties. Therefore, 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-ofhearing students, as many of these students will benefit from hands-on activities. My experience of teaching math to deaf and hard-of-hearing students shows that math origami activities do help these students understand 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 are done using a piece of square paper. Another form of origami is called modular origami, it is a compound structure composed 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 attention 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 volume, mirror images, and many more. Origami can also be used to teach symmetry. For instance, for many of the 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-ofhearing students to create and manipulate basic geometric shapes such as squares, rectangles, and triangles that might otherwise be taught through lectures and without personal and concrete experience. Math in Motion 263 Further, origami can be designed for users of all ages. Young students can explore mathematical concepts and develop a mathematical foundation even before 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 threedimensional 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 origami, 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 investigating a wide range of questions relating to paper folding. Levenson (n.d.) has found that origami has shown that paper folding, particularly in the elementary school years, is a unique and valuable addition to the math curriculum. After all, the process of transforming 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 applied 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 telescopes 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,

3 264 Journal of Deaf Studies and Deaf Education 11:2 Spring 2006 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 communication 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 demonstrate and teach them some of the geometric properties of the square paper that are often exploited for folding first. For instance, the teacher can explain that a square is regular it 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 mathematical term will be used in the origami activity. The next step will be distributing the key directions (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 mountain 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 as the 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 learning. 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 arrangement will help the teacher address more of the students 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 origami). An origami base is a partially folded model from which several figures can be created and is usually named after a typical figure that can be created from 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

4 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 the series 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 become three dimensional and if there is a simpler procedure for folding a certain three-dimensional figure. However, challenges such as these will not be meaningful 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 manipulations (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. The ability 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 consider 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 illustrate the following origami axioms formulated by Huzita (1992): (a) given two points P 1 and P 2, we can fold a line connecting them. (b) Given two points P 1 and P 2, we can fold P 1 onto P 2. (c) Given two lines L 1 and L 2, we can fold line L 1 onto L 2. (d) Given a point P 1 and a line L 1, we can make a fold perpendicular to L 1 passing through the point P 1. (e) Given two points P 1 and P 2 and a line L 1, we can make a fold that places P 1 onto L 1 and passes through the point P 2, and (f) given two points P 1 and P 2 and two lines L 1 and L 2, we can make a fold that places P 1 onto line L 1 and places P 2 onto line L 2. These axioms then can be extended to their more general application in geometry (e.g., any two points can be connected to form a line. Furthermore, as the students become more proficient 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 Math in Motion 265 In summary, implementing origami in the classroom can mean rich, hands-on math experiences for students who are deaf and hard of hearing. Math origami can be used as an entry point to mathematical discourse 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

5 266 Journal of Deaf Studies and Deaf Education 11:2 Spring 2006 can help the public education fulfills its responsibilities of maximizing each student s opportunity to learn and succeed in the least restrictive environment. References Andrews, J. F., Leigh, I. W., & Weiner, M. T. (2004). Deaf people: 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 International Conference on Origami in Education and Therapy (pp ). London: British Origami Society. Levenson, G. (n.d.). The educational benefits of origami. Retrieved August 29, 2005, from / 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 ). New York: Oxford University Press. National Council of Teachers of Mathematics (1989). Curriculum and evaluations standards for school mathematics. Reston, VA: Author. Origami. (n.d.). Retrieved August 29, 2005, from serenevy.net/?page=origami-wheremath Stewart, 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, Received September 1, 2005; revisions received October 27, 2005; accepted October 31, 2005.