Read This! The MAA Online book review column Arithmetic for Teachers: With Applications and Topics from Geometry by Gary R. Jensen Reviewed by William P. Berlinghoff

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.

• The unconventional agreement (early in Ch. 1) to regard multiple appearances of the same symbol as distinct elements of a set confuses objects with their names and invites confusion later, such as in the idea of non-disjoint unions and the definition of whole number addition by representative sets.

• The definition of whole number on page 13, an essential foundational piece, is circular. What does "number of elements" mean if you don't know what "number" means yet?

• Section 1.3, Measurement, contains far too much factual information packed in without sufficient motivation or mnemonic assistance. It almost reads like a handbook, rather than a textbook — length, weight, volume, angle, time. A glaring instance of the inadequacy of this approach is the introduction of metric measure with no explanation other than "The official relation between the English and metric systems is 1 inch = 2.4 centimeters (cm)". [exactly?]

• Chapter 3 deals with important topics — explaining why arithmetic algorithms work — but its style is unnecessarily formal and artificial. Students (future teachers) need flexibility (PUFM); the formality of the approach throughout this book actively discourages that. Labeling various algorithms as theorems gives them "sacred cow" status as THE right way to compute.

• "The base ten numeral of a number is built by repeated applications of long division." (p. 102) What?! Long division is a base-dependent algorithm; it is a process for dealing with numerals written in some base. Moreover, a student does not have to know long division to know how to group by tens.

• The base-ten divisibility tests of Theorem 4.22 are particularly useful for this audience. They should not be buried as a single multipart theorem in an extensive section on factors and primes, most of which focuses on number-theoretic properties that are independent of base representation.

• Much of the material in Chapter 4 (Number Theory) is critically important for future elementary teachers. However, the excessive formality of its presentation virtually guarantees that most of those future teachers will not appreciate its significance, even if they succeed in learning the theorems. This is a chapter of missed opportunities for relevance.

• The definition of rational number gets Chapter 5 off to very bad start. "The fraction p/q represents the point on the number line arrived at by dividing the unit interval into q equal parts and then going p of these parts to the right from 0. This point is called the value of the fraction. A rational number is the value of some fraction." This says that rational numbers are geometric objects — points on a line. This attempt to sidestep equivalence classes of fractions invites more misunderstandings than it avoids. Moreover, it's a definition that only results in positive numbers. This is not clearly flagged as an intermediate step in creating the entire rational number system.

• Finally, in Chapter 7, we get to negative numbers, but only to negative integers. The negative rationals are relegated to seven lines at the end of this chapter, tossed off with a do-it-again comment: "The development of the preceding five sections can be repeated, without change, for the rational numbers." Besides being too little too late, this is mathematically disingenuous. It conceals the important structural differences between the integral domain Z and the field Q.

• The emphasis on number-line diagrams throughout the book is good for some students, but not good at all for others. The attempt to use this as a one-motivation-fits-all device is not helpful. Alternative strategies are needed, but not provided.

Publication Data: Arithmetic for Teachers: With Applications and Topics from Geometry, by Gary R. Jensen American Mathematical Society 2003. Hardcover, 383 pages, $59.00 ($47.00 to AMS members.) ISBN 0-8218-3418-5.

William P. Berlinghoff (billberlin@aol.com) teaches at Colby College. His books include Math through the Ages and A Mathematics Sampler.

Posted October 9, 2004.

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