Arithmetic

Paul Lockhart, the elementary schoolteacher turned research mathematician turned K–12 mathematics teacher, is known to many MAA members for his 2002 essay, “A Mathematician’s Lament,” which he expanded into a book in 2009. Keith Devlin shared the essay and Lockhart’s responses to reader feedback in March and May 2008. David Bressoud revisited its themes in 2014, and then-undergraduate Daniel W. Farlow looked back at the book in 2015.

Lockhart has since turned to writing books that show how to think about mathematical subjects with the spirit of active play and creativity that he recommends; Katherine Safford-Ramus reviewed his 2012 Measurement. The book under review here takes on arithmetic. In fifteen short chapters, he ranges from the role and language of counting, through historical arithmetic in a roughly chronological series of geographical locations, to arithmetical operations and objects. Numerical expressions and illustrations appear throughout the book; as someone interested in the history of mathematical instruments, I was delighted to find drawings of Curta and desktop calculators on p. 146. Each chapter contains questions asking readers to work out problems related to the concepts.

While the chapters encompass a variety of contexts for arithmetical activities, the theme running throughout the book is how and why humans group things. Lockhart gives some attention to the psychological and intellectual processes that drive our urge for groupings, but much of the material is the sort of tour through number systems that one might see in any history of mathematics or mathematics for the liberal arts textbook. Similarly, despite his opening description of algebra as “a fun and enjoyable activity of the mind and a relaxing and amusing pastime” (p. vii), he notes numerous ways that arithmetic appears in our daily lives and so goes beyond isolated manipulations of symbols done solely for recreation.

Although I expected that the questions for readers would tend toward the philosophical, given Lockhart’s emphasis on beauty and de-emphasis of utility, almost all of the questions are concrete and computational. For instance: “Make a chart of all sums and differences of Hindu-Arabic digit symbols. Do you notice any patterns?” (p. 55) “Which is larger, \(381 \times 44\) or \(598 \times 28\)?” (p. 107) “What if seventeen jars of milk need to be divided among six goatherds?” (p. 154) None of these issues are problematic in themselves, but they perhaps provide reminders that resolving the lament is a complex undertaking.

Lockhart seems to address any reader who would care to pick up the book, but it seemed to me that the ideal audience is probably an individual who is 12 to 14 years old, already fairly familiar with the concept of regrouping numbers but able to handle the volume of text and its vocabulary level. (My 9-year-old Diary of a Wimpy Kid and I Survived! fanatic is a couple of years from having the patience to appreciate receiving this book on gift-giving occasions.) The tone sometimes struck me as too condescending for adults, but undergraduate instructors might be able to use the book to enrich classes, as they might employ William P. Berlinghoff’s and Fernando Q. Gouvêa’s Pathways from the Past, or in situations when they find Amy Shell-Gellasch’s and J. B. Thoo’s Algebra in Context too extensive or technical for their students. Since Arithmetic is considerably less expensive than these volumes, it might suit a class for non-majors or education students that has several books on the reading list.

Amy Ackerberg-Hastings is an independent scholar who researches the histories of American and Scottish mathematics education, among other things. With Peggy Aldrich Kidwell and David L. Roberts, she published Tools of American Mathematics Teaching, 1800–2000, in 2008.