A Survey of Paper Cutting, Folding and Tearing in Mathematics Textbooks for Prospective Elementary School Teachers

Origami or the art of paper folding receives substantial endorsement from current reform initiatives in mathematics education. Particularly, at the elementary school level, the National Council of Teachers of Mathematics in its Principles and Standards for School Mathematics recommends that students use paper folding to perform their initial investigations of symmetry and congruence in geometry (NCTM, 2000). Since prospective elementary school teachers need a solid foundation in the mathematical basis of these reform initiatives, textbooks for these future teachers should contain paper-folding applications. Although origami spans academic disciplines, paper folding has increasingly been applied to mathematical systems and structures (Hull, 1996; Auckly and Cleveland, 1995). Moreover, origami is a unique mathematical activity because one can take a sheet of paper and transform it into a three-dimensional object. This concrete experience in spatial reasoning can then be transformed into a lesson on symmetry or graph theory (Peterson, 1995). Educators have also applied paper folding to such diverse mathematical objects as logical structures (Cipra, 1998), axiomatic systems (Alperin, 2000), and tessellations with geometrical figures (Stewart, 1999). Clements and Battista (1992) defined school geometry as the study of spatial objects, relationships, and transformations that have been formalized. As students participate in the physical opportunities afforded by origami and other manipulations with paper, they gain a foundation from which to build the formalizations needed in school geometry. Particularly, students benefit from experimenting and exploring with physical materials and models, and learning opportunities that require students to visualize, draw, and compare figures in various configurations help to develop spatial sense. Since geometry at the elementary and intermediate school levels aims to help students describe, relate, and represent objects in the environment, students can benefit from activities involving paper cutting, folding and tearing. Put another way, transforming a flat piece of paper into a three-dimensional object is essentially a manifestation of spatial reasoning, a key component of geometrical learning. Thus, origami along with paper cutting and tearing can allow students to create, modify, and investigate specific geometric shapes such as polygons and polyhedra. The use of origami in school programs is also one of the notable differences between American and Asian educational systems. Given the better performance of Asian children in international comparisons in mathematics, it may be worthwhile to give attention to the potential impacts origami can have on the development of US students' mathematical thinking. Rooted in Asia, origami reflects the creativity and aesthetics of those cultures. By taking part in origami and other paper manipulation activities, American students can gain an appreciation of Oferent cultures and alternate ways of observing and studying geometric figures. Cooney (2001) points out, that in our internal, national zeal to improve mathematics education in the US, we should not overlook the attributes and approaches of other cultures that have demonstrated high student achievement in mathematics. Clearly, Asian societies and their contributions to origami and other manipulations with paper are a case in point. Silverman and Manzano (1996) further noted that what is accomplished by using origami is no less than the planting and nourishing of the seeds of geometric thinking. Among the geometric concepts that are embedded in origami are similarity, congruence, measurement, and construction. Understanding these and other basic concepts and the developmental ways that children learn allows the teacher to facilitate activities which are rich in exploration, application, representations, communications and mathematical reasoning. Silverman and Manzano conclude that paper folding provides a highly engaging and motivating environment within which children extend their geometric experiences and powers of spatial visualization. …