Strategies for any model

Here are general techniques that can be applied to any model and fit very well in the culture of origami, in which every student is a potential teacher.

• The most basic use of origami is to have students teach models to other students. Various strategies exist for this. You can divide the class into groups and teach each group one model in a set of models of similar complexity. Their anchoring task is to teach the model to the people in the other group. They can work together, trying different approaches, and then agree on one presentation, or they can each find a partner in the other group. Notice: even if there is one 'designated teacher', he or she may be using ideas from many students in the final lesson. Of course, you must manage the process over time so that the same students are not doing all the talking. You can listen and give feedback to rehearsals or you can just let them do it. During rehearsals or after the 'real' lesson, discuss explicitly the uses of language. Point out that specialized language - 'jargon' - serves a definite purpose. Origami and mathematics each have their own jargon and are highly useful in describing folding. Your students may develop their own jargon. Having students develop their own terms and reconciling those with the traditional terms illustrates both the arbitrariness and usefulness of conventions. Discuss also the viewing angle of the audience and the use of gesture.
• Assign students the task learning new models by consulting the many books available (see the bibliography) and also a growing number of Web pages.
• As a natural follow-up to (oral) teaching, give students the task of preparing directions using their own writing and diagrams. Preparing directions at the level found in books is a challenge. However, it is possible to make acceptable diagrams using a variety of methods. Computer based systems can be used and this can serve as an opportunity to encourage students to refine and polish their work. Many people use specialized tools to produce drawings, but even the basic draw tools can be beneficial. Other options are to scan in hand drawn diagrams, scan actual models in development, or use a digital camera to produce images. These images can be marked up using a computer drawing program. Finally, students can produce hypermedia: text, images, animation, and sound linked together, with the navigation under the partial control of the user. Several different techniques for producing diagrams are included in this paper.
• Ask students to consider beforehand what will be the results of making a fold. Ask them to visualize it 'in their minds'. Encourage them to pose generalizations on the effects of folds. For example, folding an edge to a parallel edge divides an area in half.
• Encourage students to compare models to models and folds to folds. Certain models (or partial models) are called bases in origami. Similar folds can be applied to different models in a way analogous to functions or transformations applied to different objects in mathematics.
• Ask students to describe and keep track of symmetries in models as the folding proceeds.
• Encourage students to ask themselves why a particular model works. For example, in the models described below that begin with a rectangle that is not a square, the model would not work if you began with a square. Similarly, you can ask students how there can be movement in action models.
• Ask students to compute a specific measurement of the final model in terms of the original dimensions. Moving from linear to two and three dimensions, ask students to determine what regions of the paper end up as specific regions of the final model. When in the folding procedure does a model become (truly) three-dimensional?
• The following strategy or lesson for origami is not especially mathematical, but it is important: neatness truly does count. In contrast to many traditional academic topics, there is less risk of over emphasizing neatness to the detriment of critical thinking or of failing to distinguish between mistakes of accuracy and mistakes in fundamental concepts. The distinction between failings in surface features and failings in deep concepts is clear, even explicit, in origami. Furthermore, the importance of neatness is pragmatic enough that most students will learn this lesson on their own. A mathematical connection is to note when folds can be less than exact.

Origami activity motivates the explicit use of geometric terms. However, it is important to keep in mind that mathematical learning can be taking place even in the absence of mathematics terminology.

Conclusion

Origami does not require fancy equipment. Ordinary paper, even used paper, works well, especially for the early stages of learning. However, when students become teachers, it can be advantageous to make use of computer technology to make diagrams and other teaching aids. Several methods were demonstrated in this paper. Such computer use often serves to make the activities even more appealing to students and to raise the standard for student work.

Origami can be an answer to the demand from educators and from others for activities and entrée points to mathematical discourse and applications called for by the National Council of Teachers of Mathematics and other reform efforts. One of the most frequently stated goals of such educational reform efforts is to change the role of teacher from lecturer to guide, from the teacher being the only source of knowledge in the classroom to the condition of the classroom being a community in which everyone is an enthusiastic, responsible, and contributing teacher and learner. In the origami community, we have long appreciated the pleasure and value of being teachers and learners together and we can bring these experiences to the classroom. Origami as an activity, with its open-ended nature, communication, and interconnectiveness, is reflective of the epistemology of mathematics that we want our students to experience.