The history of numeration is brought to the forefront in *How Math Works, A Guide to Grade School Arithmetic for Parents and Teachers.* G. Arnell Williams is a huge fan of using diagrams and pictures along with step-by-step breakdowns of each concept he discusses. Williams sets the stage by beginning with the “coin system,” which is basically the method of tally marks, the abacus, and Arabic numerals. The reader will also obtain an understanding of how the Romans had a difficult time coming to grips with the number zero, negative numbers, and how the Egyptians used doubling charts to multiply numbers. The examples in the book are simple to understand yet the history is more complex. When reading the book, the reader is sent on a journey through history that is compelling and downright fun. I found myself wanting more.

One of the central ideas of the book is that “One method can be spectacularly more advantageous to use than another method.” This is evident from the first page of tally marking examples to the last page dealing with the calculation of symbolic events using coins. William’s goal is for the reader to use symbols to simplify calculations. This, in turn, will help the reader master the art of solving problems in mathematics. Throughout history, mathematicians have always wanted the most precise and effective paths to solve problems. Doing endless calculations is not necessary if one is equipped with the knowledge of these symbols presented in the book. It gives one the upper hand and the more one knows the better.

In American schools, we are increasingly taking for granted how our number system came to be. Williams guides us through a historical journey and encourages his readers to take a step back and consider how certain ideas came about in mathematics. Some may say, “Why do that?” or “This will be too boring, I have enough trouble understanding the math I’m learning now.” The reader will be pleased to learn that Williams breaks down the history into manageable ideas for the brain to absorb. The climaxes of the book are Chapters 7 and 8, “Dance of Digits” and “The Highest Mathematical Faculties.” Here, Williams addresses the intriguing ideas of Carving up the Numerals, Training Zeros, The Lattice Method, The Standard U.S. Algorithm, Repeated Subtraction, A Three-Sided Coin, the ever important idea of Division by Zero, and Remainders. In particular, the examples of finding \(62 \times 37 = 2294\) on pages 124–126 and the discussion of 6/0 and 0/0 on pages 157–159 are pure examples of how, by using more sophisticated methods and having a deeper knowledge of mathematics, we can learn and properly teach concepts.

From a teaching standpoint, teaching fundamental concepts such as these can prove to be a challenge for students who need further explanation of concepts in order for them to make sense. As teachers, we sometimes take for granted that students will say they do not understand, when sometimes they may not in order to keep the lesson going. They do not want to hold up the lesson for fear of being made fun of by others. The methods presented in the book, if taught properly, may give students the confidence and drive to learn more.

I see this book being used in many different grade levels. Parents may need more motivation to read this text, as they may have had a bad experience from their schooling years. Teachers can certainly gain a lot from the book and can use all aspects to teach students how mathematics works. I also see the book being used for students entering math contests or for school projects on the use of symbols. For a college level class, a number theory class can certainly benefit from the book with all the material on remainders presented in Chapter 8.

Two things I would have liked to see accompany the book are lesson plans or ideas for classroom actives covering a wide range of grade levels. Suggestions could be made for teachers to motivate thinking and discovery for the students. Breaking away from how we have always done mathematics to consider why we do certain things in mathematics can only make our students stronger.

I believe we are going further and further along a path in education where we as teachers must focus on a set curriculum. Motivating deeper thought and discovery must occur, however, in order for the next generations to be ready for life after schooling, in order to be ready for global competitiveness.

One last thing I would like to see are comments from teachers after they have tried some of the numeration techniques used in the book. If Williams can get some feedback from teachers and how they used these ideas, this would strengthen the book’s validity. There is nothing wrong with diverging from a set curriculum in order to teach students different ways of solving mathematical problems or giving a brief history of topics. It provides an avenue of inspiration and wanting to learn more and a disservice if we don’t explain the “why.” Making students want to hear the “why” is another venture.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362^{nd} Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at pto2@psu.edu. Webpage: www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.