# Book Review: Learning Mathematics with Origami

by Charlene Morrow
edited by Thomas Hull
 Title Learning Mathematics with Origami Tung Ken Lam and Sue Pope Association of Teachers of Mathematics (ATM), UK, 2016 $20.25 Book and PDF from The Source$8.10 PDF only from The Source Polygons, Polyhedra/Three-Dimensionality, Modular Origami, Volume, Proofs US Grades 2-10 37; PDF (included in purchase price) includes 22 master slides

## Introduction

Rarely do you see the cover of a mathematics-related book showing a student front and center with his/her hands on a project, especially without the teacher hovering over the student, ready to point out mistakes. The cover of Learning Origami With Mathematics (LMWO) shows just that. The teacher is standing back, giving the student plenty of space, and in a reflective, observational stance. This image drew me in immediately and communicated volumes about the philosophy of the authors/teachers and the materials on offer.

The authors, who are deeply experienced in both origami and mathematics education, are enthusiastic and inviting, and yet demanding of the reader/teacher to engage with the activities described through active problem solving. As you begin to peruse the activities, you will notice that the authors do not structure your experience for you in detail. They pose problems, suggest a multitude of questions, and offer activities that are flexible, multi-topic, and non-hierarchical. The explorations challenge the teacher to think deeply, thus there is a great learning potential. Providing little in the way of pre-digested information gives teachers the opportunity to have the same experience that their students will have when folding and thinking about the mathematics that arises.

## How LMWO is organized

The format of LMWO is simple: Each page of the book contains one (or more) vignettes explaining an elementary math concept that can be explored by folding paper. Examples include folding in half (broadly defined), the possible geometry of a single fold, and folding fractions, angles, and polygons.

One of the main strengths of this book is that it fills a niche between origami and mathematics books that focus on either the elementary level (e.g., Math in Motion by Barbara Pearl), the high school level (e.g., Unfolding Mathematics with Unit Origami by Betsy Franco and Unfolding Mathematics with Unit Origami Boxes by Arnold Tubis and Crystal Mills), or the undergraduate level and beyond (e.g., Project Origami by Thomas Hull or Geometric Folding Algorithms by Eric Demaine and Joseph O’Rourke). LMWO contains materials for elementary through high school students.

Another strength of LMWO is that it covers a wider range of tasks and topics than other books, particularly those at the high school level mentioned above. Especially within the topics of fractions and polygons (see detailed comments in the next section below), there is a nice variety of activities that have offerings at multiple grade levels. On the other hand, most of the other books mentioned above (note especially Franco, Tubis & Mills, and Hull) give much more detailed guidance for in-depth exploration of a range of topics.

##### If you are a mathematics teacher and at least somewhat experienced in origami…
you will find LMWO to be a good compendium of classic topics. Since the book covers a wide range of grade levels, it is likely that not all activities and/or topics will be of equal interest to each teacher. Mathematics teachers can almost always find a "lesson" in any origami model, and the authors provide many excellent activities and also references for further reading and resources. The person who is both a mathematics teacher (or at least a teacher who is comfortable teaching mathematical ideas) and an experienced folder will get the best use from LMWO.

##### If you are a mathematics teacher with little or no experience in origami…
you will have to put quite a lot of effort into using the activities because LMWO does not teach in any depth about how to fold the models included. There is some guidance, but a lot of independent effort will be required to achieve a comfort level in the classroom. You will get the best and most immediate use of LMWO if you can partner with an experienced origami person.

A small caveat for novice folders: Sometimes reference points in diagrams in LMWO are not specified as to whether it is a halfway point, etc., and, in general, diagrams are not extremely detailed.

For novice folders who want to use LMWO independently, my suggestion is that you can learn most quickly by watching origami videos. One can find many useful and excellent videos on YouTube, but don’t become discouraged by trying to follow ineffective instructions. In addition to YouTube, an excellent source for videos is the Happy Folding site hosted by Sara Adams.

There are also several national organizations that offer resources, diagrams (sometimes free), conventions, and other learning opportunities. Two organizations that have great resources are OrigamiUSA and British Origami Society.

##### If you are a paper folder with no experience in teaching mathematics…
but have been asked by a mathematics teacher–or by anyone for that matter—about ideas for using origami in the classroom, LMWO contains a very nice collection of activities, each of which could be used to supplement a topic. An experienced folder could nicely partner with a non-folding teacher to offer an engaging experience in seeing mathematics via origami. If you have been asked this question and have not known how to respond, LMWO is a great resource.

## Comments on two topics covered in LMWO, especially for mathematics teachers

1. Fractions, starting on p. 7
These exercises provide excellent opportunities for students to gain traction in an area that is poorly understood by so many students. The folding here is simple, and the diagrams easily understood. Instead of being asked to memorize rules, which is rarely effective, students can see and touch fractions and fraction arithmetic through paper folding. Less space is devoted to connections to the more formal manner in which this information is usually taught, but enough for the teacher to see how to construct activities for the student to make these connections. There are many opportunities for all students—struggling or well-performing—and teachers to gain new insights. It’s fun and rewarding to see fractions in action! In my opinion, the price of the book is worth it for this topic alone.
2. Modular Polyhedra, starting on p. 22
While this topic also provides a wealth of possibilities for understanding our three-dimensional world, a much larger investment of time/practice is required for the teacher who is an origami novice. Most people can grasp how to fold the units for modular models until that moment where the unit needs to pop up into the third dimension. Even the seemingly simple Paul Jackson unit discussed on p. 22 can be challenging to construct, and its seeming simple-ness can make it frustrating. This struggle with working in the third dimension is not limited to students! If this is an issue for you, there is at least be a modicum of comfort in knowing that many people find modular origami difficult. The teacher who is experienced with this type of origami will be able to build immediately on the examples presented.

## Why use origami in teaching mathematics? An argument from the reviewer in support of books like LMWO

Folding, by itself, builds intuitive spatial understanding. Here is a short story about a former student of mine that might help make this point: Partway through a mathematics and origami course that I was teaching, a student told me a story: She was studying for the dental school entrance exam and struggling mightily with the spatial rotations portion of the practice test. This is a task that one has to do just in one's mind. About halfway through the course, she realized that her performance on this part of the exam had improved so much that it was now her best section!

## Bottom line

For those with considerable origami experience, it should be interesting to see some mathematical aspects of familiar origami models and exciting to use these models to enhance mathematical understanding, or even to introduce a new topic. The origami enthusiast should also be able to think in new mathematical ways about models not discussed in this book.

LMWO is a resource that can help facilitate students becoming excited about origami and wanting to learn to fold many more models. This is an additional benefit to the more explicit goal of helping students gain understanding of mathematical topics through origami. I highly recommend this book, a slim volume with extensive possibilities, as a resource for any motivated and creative teacher. At $22.50 for the print version that includes the PDF (OrigamiUSA pricing, but it's$20.25 for members) it is worth the price, but at $9.00 ($8.10 for members) for the PDF, you really can’t go wrong.