Origami is the art of paper folding. By a sequence of folds, a flat piece of paper is turned into a stylized animal, flower, box, or other recognizable object, generally 3-dimensional and often with moving parts or serving a utilitarian purpose. The final object is called a 'model'. Origami is associated with Japan, but it is practiced all over the world. The classical models include the water bomb, crane, and flapping bird. In recent times unit or modular origami, in which geometric constructions are built up from so-called modules, has become popular. Origami is both a craft and an art.
Origami as practiced in the United States and elsewhere, has developed a certain culture, largely influenced by Lillian Oppenheimer, Alice Gray, and Michael Shall who over time established Origami USA. In this culture, everyone is potentially a teacher as well as a student; a high value is placed on sharing. Similarly, care is taken to giving credit to creators, people who add variations to models, teachers, collectors, and people who write down directions and diagrams. It is to be noted that the recommended practice in origami circles goes beyond the letter of the law concerning intellectual property. Certain attributes of the nature of origami and this culture provide the potential for its use in teaching (and doing) mathematics.
In order to take full advantage of the potential of origami for supporting teaching and learning of mathematics, it is critical to have a broad, modern view of the goals of mathematics instruction. We are not just talking about teaching students arithmetic or the terminology of geometry. Starting around 1989, reform activity in K-12 education began that culminated in the development of a new set of standards for mathematics. In these standards, the focus is on:
The mathematics reform movement is consistent with the general educational theory called constructivism. This theory views learning as not being a simple transmission of information from expert to student, but as the construction of individual frameworks by the student through the assimilation of new information with old. Most of these ideas are based on the lofty goal that teaching and learning is successful only if students retain concepts and skills and can apply what they have studied to new situations. To put this another way, teaching and learning must shift from concentration on the surface features of memorizing terms and rote-practice of procedures to invoking deep enough understanding of concepts to support students' applying, comparing, and adapting what they have. This change is most likely to happen if the learning process itself involves confronting problems and applying concepts.
This view of learning opens the way and even creates a demand for changes in curriculum. Origami is a prime candidate for satisfying some of this demand. It even creates artifacts to take home. The multi-cultural and multi-generational nature of origami--it is practiced all over the world and by people of all ages--is an additional attribute that makes it appealing for the classroom.
A significant goal of current reform efforts is not simply an improved acquisition of mathematics knowledge, but an improved epistemological view. Students would view mathematics as concepts and procedures that are continually evolving in a mathematics community. Origami is reflective of this, with new models being formed based on old ones, modelers learning from one another, and communication as a key part of the process. In fact, origami can be considered a subset of the overall mathematics community and discipline.
In this paper, we will describe a set of general strategies that can be used for enhancing K-12 mathematics education through the making of origami models and then apply them to a specific set of models, ranging from simple to intermediate in complexity.