*The Mathematics that Every Secondary School Math Teacher Needs to Know* is a well-intentioned book. Its title seems to promise a book filled with essential insights that connect the mathematical coursework of future teachers with the content of secondary school curricula. Regrettably, the promise of those intentions is largely unrealized in this massive book.

Weighing in at 3½ pounds, this paperbound book of 732 letter-size pages is a sprawling, verbose compendium of secondary school math topics described in great detail. Its size and presentation as a classroom text make it inconvenient as a handy teacher reference. Its scope makes it unwieldy for use as a text in any single course, as the authors themselves admit, despite saying, “It was specifically written to accompany a culminating mathematics course for prospective secondary mathematics teachers.” The writing is prolix and sometimes condescending, diminishing its appeal as a reference book for current or future education professionals. The layout and typography also tend to be uninviting. There are large portions of unrelieved text in a wide single-column format with relatively narrow margins. Careless copy editing failed to catch the many errors in punctuation and grammar that interrupt the flow of reader comprehension.

The Preface says, “This book has multiple uses, ranging from a very helpful resource, to a text that accompanies any methods or mathematics course for pre- or in-service secondary mathematics teachers.” By trying to serve so many diverse purposes, this book serves none of them well. Its ambivalence of purpose is apparent in the uneven expectations of the reader’s background knowledge. The authors use elementary calculus (including the Intermediate Value Theorem and the Fundamental Theorem) without apology, presumably because this is a capstone course for a math major or minor preparing to be a teacher. However, they cover *de novo* many topics that should have been standard fare in the student’s earlier courses, such as induction, elementary number-theoretic ideas, vectors, rational and irrational numbers, the basic structural ideas of abstract algebra (e.g., associativity, commutativity, inverses), and the like. This might be appropriate if they were presented with a fresh perspective uniquely suited to high school teaching, but that is generally not the case. Despite some good, novel examples here and there, the treatment of these topics is not very different from that of many standard college texts. The “Launch” questions at the beginning of each section, intended to capture interest and motivate student involvement, reflect this uncertainty about reader background. Some of them are indeed thought-provoking, but far too many are either frustratingly vague or too shallow for senior-level students or teaching professionals. They seem unlikely to provoke the kind of deeper thinking and involvement that the authors seek.

To justify the need for proof, Chapter 1 begins with a fairly heavy-handed appeal to mistrust intuition, which seems oddly at cross-purposes with the emphasis on “reasonableness” in recent documents of the National Council of Teachers of Mathematics (NCTM). Detailed proofs of specific theorems occupy a significant part of the rest of the book. In many cases, applications of those theorems are relegated to problem sets, which the authors call “Student Learning Opportunities.” (What does this imply about the *rest* of the book?) Relatively little attention is given to helping students see unifying relationships that would bring some coherence and perspective to the many, many details that confront them. In fact, the authors apparently tried to keep the chapters quite independent of each other, as a favor to professors who want to choose what to include in their courses. That, however, is a disservice to the students, who need to see important connections across topical divides, a point of emphasis in recent NCTM policy documents.

A strength of this book is its many nice choices of illustrative examples, especially in the early parts of sections. The introductory sections in the various chapters often contain valuable insights, as well. A particularly good example of this is the “Issues with the Approaches to Probability” subsection of Chapter 12, which provides students with a big-picture perspective that relates the mathematical ideas to their common sense. It is too bad that there is not a lot more of this kind of thing elsewhere in other chapters.

Finally, there is the issue of “need to know,” which is quite different from “nice to know.” It would be nice if every secondary school math teacher knew all the things in this book (and more, besides), but they don’t *need* to know all of this. For instance, they don’t need to know the chapter on the Three Problems of Antiquity; knowing *about* them would surely be sufficient. They don’t *need* to know many of the topics that pop up late in chapters (e.g., Lissajous curves, fractal dimension, etc.) or many of the formal proofs of famous theorems. What they *do* need in a capstone course or reference book is a way to sift, organize and unify all the specific things they learned in college and all the specific things in high school texts. Unfortunately, that is not something this book provides.

William Berlinghoff is a retired professor of mathematics who most recently taught at Colby College. He is the author or co-author of several college texts, including *A Mathematics Sampler* and *Math through the Ages*, and was a Senior Writer of *MATH** Connections*, a Standards-based secondary core curriculum series. He is currently the managing editor of Oxton House Publishers in Farmington, Maine.