Unified Mathematics: Content, Methods, Materials for Elementary School Teachers by ArnoldL. Fass; Claire M. NewmanReview by: Edith RobinsonThe Arithmetic Teacher, Vol. 24, No. 1 (JANUARY 1977), p. 53Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189163 .

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symbols (curly brackets, commas, and so on); in con- trast, the conventional sets of geometry - curves and regions- are traditionally viewed as single entities with no attention to their membership. Thus the juxta- position of these topics gives a more complete charac- terization of the term set. Concomitantly, set oper- ations become something more than games with marks on paper.

The informality of the treatment makes the caution offered in the preface a necessity - the book must be read carefully. The significant parts of any discussion are singled out, but can be missed if the reading is superficial.

The necessity for careful reading may be considered a drawback by some people; the book is also quite lengthy. A few inconsistencies show up - there is a difference between an exercise used to motivate a suc- ceeding topic, and an exercise that assumes a fact not in evidence. A few instances of the latter are found in the problems at the ends of chapters. Traditional use of the word intersect (in geometry) to mean "non- empty intersection" is made without comment, and is especially disturbing in an otherwise careful dis- cussion.

Topics in Contemporary Mathematics. Jack R. Britton and Ignacio Bello. 1975, xv + 487 pp., $13.95. Har- per & Row, 49 E. 33 Street, New York, N. Y. 10016. The strong feature of this book is the exercises.

These are plentiful, interesting, and challenging; they also incorporate many applications from contempo- rary life.

Unfortunately, the expository portions are quite disappointing. There is, for example, a lengthy in- troductory chapter (chapter 2, 38 pages) on logic, yet in succeeding sections proofs and definitions are given without proper foundation. The proof that for in- tegers, ab = 0 => a = 0 or b = 0 (p. 139) requires division of both sides of an equation by b, but division of integers is not defined, so the significance of the assumption that b ф 0 is lost. For the student who never understood proofs in mathematics, this cannot be enlightening. Again, subtraction of integers on the number line is defined to be the opposite of addition (p. 141), but addition of integers on the number line is not discussed.

Distinctions such as those between number and nu- meral are often claimed to be antics with semantics; however, the importance of distinguishing between a physical situation, its mathematical representation, and the symbolism used to express it is exemplified in the following quotation (p. 144):

Thus, with the integers alone, a famous Betty Cracker recipe calling for 3 cups of flour could not be cut in half, because 3 -r 2 is not an integer. Likewise, you could not share a piece of candy equally with a friend, because there is no integer to represent 1 4- 2.

Real life supplies the necessary counterexamples to these assertions.

Unified Mathematics: Content, Methods, Materials for Elementary School Teachers. Arnold L. Fass and Claire M. Newman. 1975, xii + 465 pp., D. С Heath & Company, 125 Spring Street, Lexington, Mass. 02173. As indicated in the title, this is intended as a multi-

purpose text, but the content loading seems sufficient to warrant its inclusion here.

The major premise for the content is (p. 38) that mathematics is essentially a language. As a con- sequence, the learning of mathematics becomes a kind of information processing, for one learns another lan- guage by mimicry or by learning its rules and how to apply them.

There are several drawbacks to the dissemination of information approach to the learning of mathematics. One is that exemplification must often be contrived. To illustrate that 6 X (8 X 4) = (6 X 8) X 4 with a pile of blocks, for example (p. 102), the count must pro- ceed from front to back in one case, from right to left in the other. The fact that this model cannot be made to work without appeal to the commutative property for multiplication is obscured, sacrificed to the aim of making it work. It would seem more instructive for the student to perceive the inherent difficulty with the model, since that would contribute to his under- standing of spatial relationships. In contrast, the asso- ciative property for multiplication is not such a diffi- cult idea that its verification should be contrived.

Also, there is always the danger of giving mis- information. An instance is that of stating (p. 17) that for sets, "the braces enclose the elements that are separated by commas." For the ensuing example it is tempting to visualize the Great Lakes between braces, with a giant comma being dragged to Buffalo (Niagara Falls?) to separate Lake Erie from Lake Ontario.

Out of the main stream . . .

Consciousness and Creativity: Transcending Science, Humanities, and the Arts. Bill Romney. 1975, 278 pp., $5.00. Ash Lad Press, P.O. Box 396, Canton, N.Y. 13617.

An interesting book, for reader reaction is not likely to be moderate.

Writing with the zeal of a new convert, the author manages to involve the reader in examining his own position on a number of critical issues - from dealing with a daydreaming student to the assignment of marks and the role of the teacher.

The current press for accountability makes it almost mandatory that educators confront their value sys- tems. Some may prefer the confrontation to take place on a mental level, with the pros and cons neatly listed. But divested of all emotional loading, the resolution of an issue becomes a policy, not a value. The highly personal style in which this book is written may not appeal to all tastes; the book does, however, deal with educational values, not policies.

January 1977 53

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