Elementary-School Mathematics: A Modern Approach for Teachers by J. Houston BanksReview by: WILLIAM C. LOWRYThe Arithmetic Teacher, Vol. 13, No. 7 (NOVEMBER 1966), pp. 597-598Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187173 .

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Reviews Clarence Ethel Hardgrove

Books and materials

Elementary-School Mathematics: A Mod- ern Approach for Teachers, J. Houston Banks, Boston, Mass.: Allyn & Bacon, 1966. Cloth, xiii + 398 pp. The author states in the preface that he

used the recommendations of the Panel on Teacher Training of CUPM, Mathematical Association of America, as guidelines in the selection and organization of the con- tent for this text, which is designed for the mathematical training of elementary school teachers. A comparison of the topics in the text with those outlined by the panel re- veals that the basic recommendations for the course on the structure of the real-num- ber system and the course on the basic concepts of algebra are met by the text. Other writers may present this content differently, but this text is guided closely by the report of CUPM.

The first seven chapters, essentially that part of the text that deals with the develop- ment of the number system, is rather in- formal and intuitive in approach. The eighth chapter ushers in the algebra, which is presented a bit more formally. The author does a good job of discussing and giving many examples of the ideas he de- velops, and his sets of exercises are de- signed to strengthen the grasp of these ideas. Each chapter ends with a good, con- cise summary. There are several places in the text where the author points out how the ideas being developed are useful to chil- dren in learning mathematics.

At places the author's detailed explana- tion could have given way to a develop-

ment in which the student works out the material for himself. Examples of this are a rather detailed explanation of a method for changing a numeral from base ten to base eight (pages 117-18), the discussion of the structure of modular seven arith- metic (Chapter 2), and the treatment of the function defined by y = - (Chapter Ì4). At such places the material takes a rather tedious turn when it could have been presented in a more interesting fashion. In other expositions, however, the author does take the reader through some fascinat- ing developments. His presentation of se- quences of triangular, rectangular, and cubic numbers, starting on page 144, is a good example.

Chapter 6, on rational numbers, is one of the better treatments of this topic to be found in texts of this kind. The author starts with three basic interpretations of fractions and proceeds to develop the prop- erties of rational numbers consistent with these three interpretations. His discussions on measurement and on how to locate a point corresponding to a given rational number on a number line are well done. The latter is often neglected or slighted in such texts. A section on linear program- ming and a somewhat stronger treatment than usual of fractions in bases other than ten add to the value of the book.

There are places in the text where the brevity makes the presentation of little value except to those who know the ma- terial. The treatment of negative expo- nents (pages 110-11) is an example; and

November 1966 597

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the brief references to the terms "axis," "parabola," "focus," and "directrix" in Chapter 10 may better have been omitted. The brief treatment of the algorithms for operating with whole numbers in Chapter 4 is also disappointing.

It is virtually impossible for a first print- ing of a text to be free from errors. While it is not a reviewer's purpose to proofread, reference to the following errors may help for the second printing. A list of field properties is given on pages 222-23. Sev- eral of the references to this list in the proofs of theorems on pages 226-30 are to the wrong properties. On pages 278-79, the example given for finding the solution set of a quadratic inequality is x2 - Ix + 12 < 0, but the solution presented is for jt2- 7* +12 >0.

This text for elementary teachers has many points to be commended. Teachers of courses in which the material is appro- priate will want to consider the book.

WILLIAM C. LOWRY

University of Virginia Charlottesville, Virginia

Building Mathematical Competence in the Elementary School, Peter Spencer and Marguerite Brydegaard. New York: Holt, Rinehart & Winston, 1966. Cloth, 402 pp. This book, primarily a methods text-

book, is one of the few books of its type that college students will find interesting to read, since it is not encumbered with un- necessary trivia. The book is written in a manner that would allow a methods teacher to supplement the material with his own ideas. In fact, it will be necessary to intro- duce more methods for teaching specific concepts than are suggested in the book. Most of the methodology is relegated to the sample lessons that are included at the end of each chapter. In most instances, only one technique is suggested for teaching a given concept.

The book begins with a general rationale and justification of the inclusion of numer- ous illustrative lessons.

Chapter 2 contains a brief discussion of quantification of measures, variability, and functions. Functions are treated as "rela- tionships of dependency," or as "if ... then . . ." situations.

Chapter 3 is one of the better chapters in the book. It gives some clear and con- cise illustrations of set operations, employ- ing Venn diagrams and other illustrative symbolism. It also includes a chart which defines useful set terms and symbols. The chapter also has a clear discussion of sub- sets as they relate to telephone area codes. Some of the properties of numbers sys- tems, i.e., closure, commutativity, associa- tivity, and distributivity are treated and illustrated with Venn diagrams. This is well, done and should be clear to the stu- dents.

In Chapter 4, the authors take a very sensible attitude toward the use of the terms "number" and "numeral" without "nit picking" on the proper use of these terms.

Their explanation of changing numerals from one base to another is not very clear. The inclusion of some dot patterns to help the students visualize the division method advocated by the authors would improve the discussion.

Spencer and Brydegaard are to be com- mended for their use of proper notation and the way they handle the inconsistencies that usually occur in the use of symbolism related to equivalency and congruency.

In Chapter 5, the authors deal with a development of the properties of numbers systems as they relate to addition and multiplication. They have treated all of the major properties except for inverse ele- ments. Their treatment of number prop- erties throughout the book is a bit con- fusing. It would be clearer if they would designate the number system with which they are dealing, i.e., fields, groups, etc.

Chapter 6 involves an interesting dis- cussion of the relation between language

598 The Arithmetic Teacher

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