The people of the Navajo Nation understand that mathematics plays a key role in education. Their passion for educating their children about mathematics takes many avenues through application problems, pedagogy, arts, and the joy and beauty of mathematics. The effort of the community has produced an eye-opening series of Navajo Math Circles that are rich with interactive self-discovery projects for students used across the entire Navajo Reservation. The Forward is by Dr. Henry Fowler, a Navajo native, who discusses his journey when, in 1972, he was sent to a Bureau of Indian Education (BIA) boarding school in Kaibito, Arizona to learn basic academic classes to how he met Tatiana Shubin. Shubin introduced Fowler to Math Circles and Fowler wanted to pass along the knowledge he gained to his people by shedding new light on mathematics in grades K-12. As pointed out on page viii, “Math Circles helped open the opportunity for me to address the dismal outlook of the Navajo high school performance in mathematics. As a direct result of the Math Circles, I am more aware of my surroundings and how they impact teaching delivery and reception.” Fowler’s experience with Math Circles is one of many positive outcomes of not only the power of collaboration but also a renewed positive spirit and love of mathematics.

As pointed out in the Introduction, the Navajo Nation is known as the Diné Bikeyah (“Diné” means “people” and “Bikeyah” means “land of” and is home to more than 200,000 Diné and is bounded by four sacred mountains in Arizona, Utah, Colorado, and New Mexico. There exists a major gap between Indigenous and non-Indigenous people. The 10th Annual AP Report to the Nation (College Board, 2014) reports that though “American Indian/Alaskan Native” students represent 1% of the graduating class of 2013 in the US, only 0.6% of AP test takers and only 0.5% are successful AP test-takers. In addition, the gap in education grows from generation to generation and many students hear the stereotype that Indigenous people are either not good at mathematics or are not college material. Returning to Dr. Fowler’s story in the Forward, in grade school, he was labeled as a special education student. However, he overcame many obstacles and by the collaborative effort with Shubin, Math Circles seem to be the answer to for math education in the Navajo Nation. The Diné Bikeyah’s Math Circles and the Math Teachers’ Circles all involve an awareness to respect one’s heritage, identity, and culture to change the perspective of learning mathematics.

The book contains 18 scripts – problems and problem sets that were used in the Math Circle sessions on the Reservation. The book has a lot of eye appealing color images – some notable are Math Blocks Figures 65-66 on pages 118-119, Constructing the Icosahedron Figure 70 on page 127, A Rhombic Triacontahedron Figure 76 on page 136, and One SOMA Solution Figure 112 on page 209. Some of my favorites while reading the text are the Toilet Paper Math, The Cookie Monster Problem, and Pancake Problem. The Toilet Paper Math script starts with three ways people use toilet paper:

**The Halves Method:** Fold a strip in half, fold the result in half and repeat some number of times.
**The End Method:** Fold some paper from one end and fold that over repeatedly.
**The Chaotic Method:** Crumple it up.

The script lays out situations for middle and primary students with some very interesting results – one of them being that of Britney Gallivan’s solution to repeated paper folding based on semi-circular ends relating thickness \( t \), length \( L \), and the number of folds \( n \):

\( L = \frac{\pi}{6}t (2^{n}+4)(2^{n}-1) \)

This result is very impressive to see from a high school student. The *Cookie Monster Problem* can be targeted for students who just learned how to count or extended to graduate students. Here, the main question the “Cookie Monster” is faced with is, according to the given rules, how many steps are needed to empty all given jars or cookies? A geometric representation of the jars is given in Figure 47 on page 94 along with a proof to Proposition 1: If \( 2^{k} \leq n < 2^{k+1} \) , then \( M(n)=k+1 \), the “Greedy Monster Algorithm (GMA), and a Binary Algorithm. The *Pancake Problem* looks at open-ended problem called “prefix sorting.” The main question is: What is the maximum number of flips required to put a stack of n pancakes in the desired order? In Figure 88 on page 177 gives the reader a table on the number of flips needed to sort each possible stack of 3 pancakes.

This book is filled with great problems and shows the creativity not only in the making of the problems but the quality and different approaches of the solutions. They are challenging but they lead the students to exchange ideas with other students, promote collaboration, critical thinking skills, make connections between mathematical topics, and make friendships. There are various types of solutions including pictures, charts, proofs, and logic. The Navajo Math Circle is doing great things to motivate teaching mathematics in the Navajo Reservation and is an example of an effective and successful Math Circle. I highly recommend this book for all those interested in pairing up with a Math Circle or starting your own. These problem sets can also be used as outreach to recruitment events for colleges and universities to use in local and surrounding high school and middle schools for students who have an interest in mathematics.

Peter Olszewski, M.S., is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at

[email protected]. Webpage:

www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.