ACADEMIA Letters Origami as a General Education Math course: Before and During a Pandemic Jeanine Meyer Origami, paper folding, associated with Japan, but with invention around the world, now stimulates considerable research activity in theoretical mathematics, computer science and engineering. I have been doing origami as a craft off and on since I found the instructions for the flapping bird in the Mathematical Games section of Scientific American while a teenager. I have used the task of producing origami instructions as teaching examples in my programming courses. Recently, I have developed an origami general education mathematics course. This led to a book project: Origami with Explanations and More Origami with Explanations has been published. The origin story of the course and the books starts with learning the Dollar Bill Rosette by Martin Kruskal at a session organized by Mark Kennedy to occupy the time waiting for tickets at Shakespeare in the Park in New York City. Over time, I experienced what I call an origami-inspired adventure in limits and number theory. The folding starts with estimating 1/11 along the edge of a dollar bill and then improves the estimate by reducing the error. This is similar to a common method for dividing an edge into thirds and I use it in making the Kissy Fish model by Junior Fritz Jacquet. This surprising result can be shown using algebra and also using a long rope. The procedure works for 11 in that it hits all 10 intermediate points along the edge. I wanted to know if there was a name for the numbers that work. I wrote a program to simulate the folding procedure and tried it out for numbers up to 1000. I put the results into the Google search box and got a hit! There was a match for a set of primes called the Reptend Primes base 2. I proved that the two sets do match. Nothing strange happens above 1000. More accurately, the two defining procedures are the same, though in reverse order. I am sure that Martin Kruskal, a highly regarded mathematician (though known in origami circles chiefly as Academia Letters, February 2021 ©2021 by Academia Inc. — Open Access — Distributed under CC BY 4.0 Corresponding Author: Jeanine Meyer, jeanine.meyer@purchase.edu Citation: Meyer, J. (2021). Origami as a General Education Math course: Before and During a Pandemic. Academia Letters, Article 245. https://doi.org/10.20935/AL245. 1 the son of Lillian Oppenheimer and the husband of Laura Kruskal) had proved what I figured out, but I am happy that I did it on my own. I gave a talk on my adventure to our Number Theory class at Purchase College. My chair suggested that I design and teach an origami class that would satisfy the State University of New York general education requirement for mathematics. My first reaction was that the math was too difficult. After all, the target audience would be students that are so weak and scared of math that Precalculus, Statistics, and even our courses such as Communicating Quantitative Information (math in the news) and Mathematics for Contemporary Life were too much. However, after further thought, I changed my mind. Yes, there is high level mathematics done for and with origami. Look at the work of Robert Lang or Rona Gurkewitz and others. However, doing origami forces (in a gentle way) attention to symmetries, spatial relations, and changing a piece of paper from being essentially two-dimensional to three-dimensional. Moreover, origami provides opportunity to explore basic algebra, geometry, and trigonometry in concrete ways that can introduce or re-introduce the concepts. One group of lessons is to calculate the lengths of parts of the final model. I do this in the very first class when we determine the size of a hem that holds the Magazine Cover Box together. The MCB is made from a rectangle that is not a square and the size of the hem is one-quarter the difference between the lengths of the sides. Next, I show the size of the Star Basket, a good model for holding candy. Its final dimensions can be calculated using geometry and trigonometry. Students appear to agree when I say that our fingers are doing the math. This is repeated for several other models. Our view of mathematics is not just calculations. In the classes and in the books, the students are asked to compare models and the folding of models. For example, when we have covered the major origami bases (the initial steps that are shared by many models), the students are asked to identify common elements in the crease patterns. I learned from reading Origami Angelfish to Zen by Peter Engel that the creases made by the kite, fish, bird, and frog, have common parts and, in fact, display a sequence of patterns. Students are given peeks at more complex math. I do not claim that all the general education students could repeat the proof that a certain folding procedure does produce improved estimates. My strong impression is that they go from being surprised by the result, which is appropriate, to accepting it by what they experience and by looking at the algebra. When I describe how I really wanted to know about the numbers “that work” as 11 does for the Dollar Bill Rosette and wrote a Python program and then came up with a formal proof, they are seeing someone doing mathematics and enjoying it. The students get a peek at the issue of what conditions are required to be met for a model to fold flat by doing some simple exercises and learning the Shen Basket, a beautiful model Academia Letters, February 2021 ©2021 by Academia Inc. — Open Access — Distributed under CC BY 4.0 Corresponding Author: Jeanine Meyer, jeanine.meyer@purchase.edu Citation: Meyer, J. (2021). Origami as a General Education Math course: Before and During a Pandemic. Academia Letters, Article 245. https://doi.org/10.20935/AL245. 2 that becomes three-dimensional when three folds are reinforced to be mountain folds at each corner of its square base. The chair of the Mathematics/Computer Science department helped make the case locally and to the SUNY administration that our course met the bar to satisfy the general education requirement. For the first offering Spring 2019, I had three teaching assistants. One of them was Takashi Mukoda, who had served as a technical reviewer for 3 of my programming books. Between the first and second offering, Fall 2019, I began making notes and realized that this could be a book. I gave two talks at the Mathematics of Various Entertaining Subjects (MOVES) conference sponsored by the Museum of Mathematics in New York City and I was approached by the editor-in-chief of World Scientific Publishing, who was encouraging. The proposal was sent out for review and I was offered a contract. I recruited Takashi, who had graduated and returned to Japan, to be a co-author. The editor said it needed to be 2 books. The approach and the instructions are not just for college students. K-12 age young people have responded positively to the folding. The first and second offerings of the course, done in the traditional, face-to-face format, were successful. One assignment, for teams of two, was to learn a model and teach it to the class. Another team project was to research and give a report to the class on a cultural, historical or scientific topic. Origami used in space was a favorite. Students had to learn to follow origami diagrams. Students loved doing the origami. They reported being surprised that they liked the math. They also reported that they liked the time away from screens. Then COVID hit. I could not imagine doing the class using Zoom. We did not offer the class in 2020. However, here again, with encouragement from my chair and others, I changed my mind. The class is scheduled for Spring, 2021. Here are my plans. I decided that I needed to have faith in our books. They are required for the class. In a sense, the class will be flipped: in many cases, students will be assigned to learn or at least get started on learning the model before class. I also have recruited three (3) teaching assistants, all international students, two from Japan and 1 from Taiwan. I am grateful for them and they are very grateful for the opportunity. They will be satisfying their math gen ed requirement and spending time doing something “in their culture”. My plan is to make use of Zoom Breakout sessions to divide the class into small groups for helping with the folding. Zoom office hours also will be available. Learning from the many online origami classes (the annual convention of OrigamiUSA was named the Unconvention and was done using Zoom), I have a setup with two cameras. My laptop camera will cover my face and my iPad with its camera will be mounted to show my hands and the folding. Students will be assigned the task of teaching someone in their household (their pod) Academia Letters, February 2021 ©2021 by Academia Inc. — Open Access — Distributed under CC BY 4.0 Corresponding Author: Jeanine Meyer, jeanine.meyer@purchase.edu Citation: Meyer, J. (2021). Origami as a General Education Math course: Before and During a Pandemic. Academia Letters, Article 245. https://doi.org/10.20935/AL245. 3 a model and to document it using video. A second video assignment will be documenting doing what is called a modular: combining individual models to build a large model. They will indicate their choices of cultural/historical/scientific topic and I will pair them up. There will be periodic assignments to post their observations to Forums. Our objectives for the class remain the same. Students will learn origami and will see abstract mathematics become visual and practical. Folding can be done while under quarantine. It can be calming. Making something in the physical world to use, show, and share is delightful. Questions, comments and suggestions welcome. Sources OrigamiUSA, https://origamiusa.org/ Bennett Arnstein and Rona Gurkewitz, Beginner’s Book of Modular Origami Polyhedra: The Platonic Solids, Dover, 2008. Peter Engel, Origami from Angelfish to Zen, Dover Origami Papercraft, 1994. Robert J. Lang, Origami Design Secrets, A. K. Peters, Ltd., 2003. Jeanine Meyer and Takashi Mukoda, Origami with Explanations, World Scientific Publishing, Fall 2020. Jeanine Meyer and Takashi Mukoda, More Origami with Explanations, World Scientific Publishing, Fall 2020. Academia Letters, February 2021 ©2021 by Academia Inc. — Open Access — Distributed under CC BY 4.0 Corresponding Author: Jeanine Meyer, jeanine.meyer@purchase.edu Citation: Meyer, J. (2021). Origami as a General Education Math course: Before and During a Pandemic. Academia Letters, Article 245. https://doi.org/10.20935/AL245. 4