Review of Mathematical Expeditions: Exploring Word Problems across the Ages

Mathematical Expeditions: Exploring Word Problems across the Ages, Frank J. Swetz, 2012, ix + 192 pp, 59 halftones, 32 line drawings, paperback $30, ISBN-13: 978-1-4214-0438-7, Johns Hopkins University Press, 2715 North Charles Street, Baltimore, Maryland 21218-4363.

History testifies that all civilizations were concerned with solving mathematical problems. The problems pondered by our ancestors frequently involved their survival, ensured societal harmony, and prepared the way to the future…. Mathematical problem solving has a long and fruitful tradition. We are not alone in our efforts. (Swetz, 2012, p. 159)

Frank Swetz shares this observation at the close of his book, Mathematical Expeditions: Exploring Word Problems across the Ages, which is a lovely contribution that contains hundreds of historical problems, as well as case studies on interesting solution methods and cultural and contextual background for collections of problems. Although many may argue that the concerns of past civilizations for posing and solving problems may not resonate with students of today, I believe the book has the power to convince mathematics teachers to revisit the role of word problems in their teaching and their students’ learning.

The book contains 17 chapters, with the first two chapters serving as an overview of the development of historical problems over time and reasons for using historical problems as a resource for teaching mathematics. In the first chapter Swetz surveys problem development based upon different reasons and categories. For example, collections of problems may evolve based upon “societal concerns and a chain of situations demanding mathematical consideration” (p. 3) that include food, construction, labor, and trade and commerce. In Chapter 2, Swetz describes four features that make historical problems an excellent resource for teaching a variety of mathematical topics, including (1) problems are non-threatening and do not represent “something extra” in the school curriculum; (2) the mathematical context of problems is relevant to instructional needs; (3) the settings, historical milieu, and situational encounters provide intrigue and added motivation for students; and (4) historical problems promote appreciation of diversity (mathematical and cultural) and provide interdisciplinary contexts for exploration (p. 27).

Although there are 43 problems provided in the first two chapters alone, the remainder of the book is a mathematical treasure trove of historical problems. Chapters 3–11 include 233 total problems taken from a variety of civilizations from around the world (and from different times), from Ancient Babylonia, Egypt, Greece, and China, and from India, Islam, Medieval and Renaissance Europe, and Japan. Swetz provides another 201 problems in Chapters 12–16 from more modern contexts, including 18th and 19th century mathematical texts and the Farmer’s Almanac.  More importantly, however, for each of the chapters 3–16, Swetz ends with a “What are they doing?” section. Here, he selects a particular solution method or mathematical algorithm taken from the cultural or situational context of the chapter and presents an example to explain the mathematical idea. Furthermore, Swetz presents seven case studies in Chapter 17, in which he presents an historical problem, the historical solution, and the modern solution. Finally, for those readers who are a bit rusty in their historical problem solving practices, Swetz provides answers for the problems in Chapters 3–16.

In summary, every person interested in the history of mathematics should own a copy of Mathematical Expeditions. However, if you teach grades 6–16 mathematics or prepare future mathematics teachers, this book is an essential and valuable resource. Given the current landscape of the Common Core State Standards for Mathematics, particularly with attention to the Standards for Mathematical Practice, I believe the problems and case studies of Mathematical Expeditions could promote most if not all of the eight Standards for Mathematical Practice in middle and high school classrooms. Moreover, if I were to teach a “History of Mathematics” course for prospective teachers again, I would definitely assign Mathematical Expeditions as a required text!

See also the MAA Review by Charles Ashbacher.