Geometry: An Investigative Approach by Phares G. O'Daffer; Stanley R. Clemens; LaboratoryInvestigations in Geometry by Phares G. O'Dafler; Stanley R. ClemensReview by: Edith RobinsonThe Arithmetic Teacher, Vol. 24, No. 5 (MAY 1977), pp. 398-399Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189313 .

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Each part of the picture to be colored is labeled by number sentences. The children are then provided a code that tells them what color to make that part of the picture. Ten pages of the book use letters corre- sponding to the colors of Cuisenaire rods.

Teachers who are looking for a different way to provide addition and subtraction practice should find this book useful. Although the colored rods are needed for a few pages, the other pages can easily be used with primary children.

New books for teachers

Edited by Edith Robinson, Asheville, North Carolina

For the professional mathematics educator

The Psychology of Mathematical Abilities in School- children. V. A. Krutetskii. Translated by Joan Teller. Edited by Jeremy Kilpatrick and Izaak Wirszup. 1976, xvi + 417 pp., $22.00. The Univer- sity of Chicago Press, 5801 Ellis Avenue, Chicago, III. 60637. Most readers who would be interested in this book

are doubtlessly aware of its availability. It is a valuable reference on several counts. There is, for example, a bank of problems (Chap. 8). These are the problems used in the experiments and they are categorized not only by subject (arithmetic, algebra, and so on), but also by correspondence to the steps in the problem- solving process. Krutetskii stresses, however, that these are not diagnostic tests, but research tests. There is also a discussion of the history of research in mathe- matical abilities, a review of foreign studies, and an extensive reference list. The latter contains both Soviet and non-Soviet entries.

The book is written in an anecdotal style reminis- cent of Piaget and, especially with respect to the mathematically capable, these reports help illustrate the variety of styles of mathematical thinking. With respect to the very capable, the descriptions contain some implications for pedagogy, although these are not singled out. Drawing a picture, for example, does not help all students; some prefer not to try to illus- trate the problem. And one cannot help wondering if the very capable student, with a penchant for thor- oughly investigating a problem, might not be thwarted by a twenty-minute class.

The editors have given an excellent critique in their introduction; this should not be overlooked.

Research on Mathematical Thinking of Young Chil- dren. Leslie P. Steffe, ed. 1975, v + 202 pp., $3.90. National Council of Teachers of Mathematics, 1906 Association Drive, Reston, Va. 22091. Most readers are probably already familiar with this

book. The studies reported, for the most part, are based on doctoral dissertations and largely on Piage- tian theory.

The studies are concerned with order relations, classification and seriation, equivalence relations, and the like, with children from four to eight years old.

There is also a final chapter by Lovell that discusses and summarizes the studies. Lovell further outlines some areas of needed research.

A most useful reference for the researcher.

For the content course

Two new geometry books are reviewed here. Both represent a departure from the type of üfext that has been familiar for many years. These books have sev- eral features in common: in neither is there much proof, neither is primarily concerned with Euclidean geometry in the traditional sense, both contain inter- esting exercises that are experimental in nature, and both contain suggestions for geometry activities for children. Furthermore, neither represents a mis- cellaneous collection of topics. Although the organiza- tions of the two books differ, both reflect nontrivial mathematics. Either should be interesting to prospec- tive teachers. The capsule reviews that follow are in- tended to give readers some idea of the differences in emphasis, organization, and so on.

Geometry: An Investigative Approach. Phares G. O'Daffer and Stanley R. Clemens. 1976, xvi + 445 pp., $12.00. Addison-Wesley Publishing Co., 2725 Sand Hill Rd., Menlo Park, Calif. 94025.

Laboratory Investigations in Geometry. Phares G. O'Daffer and Stanley R. Clemens. 1976, viii + 128 pp., $4.00. Addison-Wesley Publishing Co., 2725 Sand Hill Rd., Menlo Park, Calif. 94Ó25. An attractive and well-illustrated book. There is considerable attention to geometry in the

environment, which accounts for many of the illustra- tions. What used to be called informal geometry (in- cidence, parallelism, and so on) is included, but oc- cupies only a small part of the book.

The major portion of the book deals with tessella- tions of the plane, properties of polygons, motion geometry, and topology. There is some work with measurement, mostly using metric units.

Problems for students stress the search for patterns, making and testing conjectures, and making drawings of arrangements. The laboratory manual would seem to be a most useful adjunct; it contains needed pat- terns, dot paper, and geoboards.

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The text is very readable, and the exercises are unambiguous.

Geometry for Teachers. С Patrick Collier. 1976, xvi + 331 pp., $12.50. Houghton Mifflin Company, One Beacon Street, Boston, Mass. 02107. The book is organized into three sections: Informal

Geometry and Topology, Motion Geometry, and Metric Geometry. There is some development of top- ics, as, for example, the packing problems that recur. Experience with these problems motivates later topics in measurement.

Throughout the book there are sections called "Search." These are problems for students to in- vestigate and require experimentation, making con- jectures, constructing aids, and so forth. There are also review sections, quizzes, and tests.

There is quite a bit of emphasis on vocabulary, but little use of the symbols for ray, segment, and so forth, that have been characteristic of geometry texts of the last dozen years. The diagrams, however, show clearly which sets are under consideration.

The section on angle measure is of particular inter- est. It is different from the usual treatment and should help teachers faced with the difficulties of teaching the protractor.

On the international scene

Exploring Mathematics with Youngsters. Susan Nich- olas. 1968, 89 pp. World Education Markets, P.O. Box 30037, 7315 Wisconsin Ave., N.W., Washing- ton, D.C. 20014. There are some good ideas in this very short book,

but the accompanying text is trying in places. It is a book about teaching mathematics in kinder-

garten by an experienced kindergarten teacher. The pupil/teacher dialogues demonstrate how children can be made more observant and also how mathematical words can be introduced into casual conversation. There are also activities suggested for teaching the important relational words: high and low, first and last, over and under, and so forth.

Of particular interest to this reviewer was the dis- tinction made by the author between the effects of partitioning and joining sets. The author's children learn to recognize certain "standard" sets along with their names and position in an ordered arrangement. The children then experiment with, say, "the" five set, separating it in various ways. As a result, five becomes associated with the pairs 1 and 4, 2 and 3, 3 and 2, and 4 and 1. The author's point is that this is quite differ- ent from having children begin with 2 blocks and 3 blocks, put them together, and then count.

Unfortunately, the language used in the book is not always clear. For example, we find the following (p. 5): "When 1 is subtracted from one side and 1 is added to the other side, the answer remains the same." Here side apparently means addend. There is also some misuse of the term set as, for example, (p. 14), "All the sets containing 1 element are put into one bag: . . . ."

6 Thèmes pour 6 Semaines. André Мух. 1975, 341 pp., 35,00 francs. CEDIC, 93, Avenue d'Italie, 75013 Paris, France. The six themes are familiar ones - logic and sets,

relations, order and equivalence, measurement, nu- meration, geometry and topology - but the treatment is of interest. The book is intended for teachers, not children, yet the presentation is such that the math- ematics can be seen as suitable for children.

Applications are an integral part of the presentation of a topic. Several seemingly dissimilar situations are described, then adroitly analyzed so that the common- alities become evident. The analysis, of course, is a description of the underlying mathematics; the exam- ples are obviously suitable for children. The impor- tance of the mathematics is self-evident from the ex- amples, as is the applicability. There is no dichotomy of mathematics on the one hand, applications on the other - there is no need for this.

The use of diagrams is also of interest. These are plentiful and occur as flow charts, tree diagrams, ta- bles, and graphs, and are sometimes used in unusual ways. With the current interest in hands-on material, we sometimes forget that not all children can go di- rectly from the physical representation to the sym- bolic. Diagrams are especially useful in making the transition. They can also replace hands-on material for some children or some topic.

The problems in the text are interesting- there are some on cryptography, for example - and there is an appendix with supplementary problems and puzzles.

For everyone

Personal Liberty and Education. Monroe D. Cohen, ed. 1976, 272 pp., cloth, $8.95; paper, $4.95. Cita- tion Press/Scholastic, 50 West 44th St., New York, N.Y. 10036. This is a collection of articles, only four of which

were written specifically for this book, and only one addressed to mathematics. Nevertheless, it is a timely book.

Americans naturally believe in freedom. Increas- ingly, however, we are being forced to be more explicit about just what we believe, to take a stand on specific issues as they arise. And issues have arisen. Educators have perhaps felt this more than any other group - should students be free to speak as they choose, dress as they choose, study what they choose, attend school when they choose? We cannot dismiss these questions, nor can we pretend they are not issues.

The articles in this book speak to various aspects of freedom in education; textbook censorship, for ex- ample. And the article on mathematics education re- minds us that ability to do grocery store arithmetic is but one of the aims of instruction- intellectual free- dom should not be shrugged off as too idealistic. Free- dom to make choices, freedom to think for oneself, freedom to be creative, and freedom to learn more are as practical as calculating unit prices.

May 1977 399

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