As the first line of its Preface says, Arithmetic for Teachers is intended as a textbook for a college mathematics course for prospective elementary school teachers. When this reviewer saw the AMS logo on its cover, he hoped that this book might present the power and elegance of fundamental mathematical ideas in an inviting, intuitively thought-provoking way that would give future teachers a firm, flexible grasp of the mathematical ideas they will be teaching. Unfortunately, such is not the case.
There is a fundamental tension in presenting mathematics to prospective teachers of grades K–6. On the one hand, it is important to emphasize the ways in which the basic ideas of numbers, arithmetic, and elementary geometry form a logically consistent fabric. On the other hand, it is equally important to recognize that a disproportionately large percentage of prospective elementary teachers are math-averse in one way or another, and the presentation should seek to lessen the aversion so that it is not passed along to another generation of students. The structure of this book leans heavily toward the former, while the explanations sometimes try to address the latter. The result is a text with the trappings of rigor, but not the substance.
In its accurate description of the book's approach, the Preface implicitly forewarns of the difficulties to come. An early clue is its necessary but insufficient characterization of Liping Ma's now-famous PUFM [Profound Understanding of Fundamental Mathematics] as "know how and know why," which trivializes Ma's much deeper analysis of the flexibility and multiple approaches that teachers need in knowing how and why. Then on page viii the author says, "The presentation of material in this book is in the old-fashioned style of definition, theorem, proof used in Euclid's Elements." The obvious, unanswered question is: Why is that the best approach for people who need this material to teach young children in far less formal, more intuitive ways? Presumably as a concession to that audience, the author goes on to say, "[A] proof is nothing but an explanation for why a theorem is true, and everybody wants to know that... I have presented most proofs in the form of a specal case that exhibits the general argument." This oversimplification lays the groundwork for one of the most insidious, pervasive flaws in this book, the trust-me-this-example-is-typical "proofs" of theorems. Here is a prominent instance of the danger of trying to serve the two masters of mathematical rigor and elementary intuitive comfort simultaneously. If typical examples suffice, why bother with the artifice of calling them theorems? That certainly is not faithful to Euclid's Elements, nor does it serve to illustrate the rigorous nature of mathematical proof. p> Many theorems incorporate a process into the statement of the theorem, making it difficult for the reader to focus on what is algorithmic procedure and what requires logical justification. These theorems are generally more forbidding than helpful. Their "proofs" are really just common-sense justifications (good pedagogy, but not rigorous mathematics), but the presentation structure is counterproductively formal. Theorem 1.45 on page 19 typifies this. On pages 16–19 the author describes base-ten numeration in a context and from a perspective compatible with his intended audience. Unfortunately, the attempt to capture these ideas as a theorem undoes most of the good work generated by his informal discussion. Some of the proofs trivialize the very notion of proof in other ways. For eample, the proof of Theorem 1.65 on page 33 depends on believing that the angle bisector shown is a line of symmetry, thereby begging the question.
Looking past the style of presentation, a number of specific content items are worrisome. Here are some instances of concern.
It is painful to write such a negative review. This reviewer knows from personal experience that it is a difficult, often thankless job to write good text materials for future elementary teachers. Gary Jensen deserves credit for even trying to tackle the task. And there are some bright spots in his book. In particular, Chapter 8 on clock arithmetic is good, within the parameters of the book's overall style, and Chapter 9 on RSA Encryption is novel and interesting. Unfortunately, these chapters are late in the game and optional, even for the author himself. The essential concepts for this audience are in Chapters 1–7, and this is where the book's many weaknesses lie.