In-service Course for Primary School Teachers: Studies in Mathematics XIIIReview by: Forrest L. ColtharpThe Arithmetic Teacher, Vol. 15, No. 8 (DECEMBER 1968), pp. 742-743Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185894 .

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disservice to the education of their chil- dren."

The reader is told that the operations of multiplication and addition are indeed conceptually different. But it is not clear how one is to convince children of this when the product is calculated by count- ing set members in exactly the same way as in finding the number of members in the union of several equivalent sets. The symbolization may not be that of repeated addition, but the recommended concrete illustration with sets of objects is no differ- ent.

The two books are intended to be used together, with activities interwoven and re- lated, not sequentially. Certainly a teacher would be well advised to read both books before introducing the activities and thus take advantage of the many opportunities to relate the logical games to the activities concerned with sets. The format of the books does differ somewhat, however. Learning Logic, Logical Games is pri- marily a description of games and activi- ties that provide a basis for many impor- tant logical concepts. It is either assumed that the reader already is familiar with these concepts or will learn about them in reading about the games. On the other hand, Sets, Numbers and Powers has a good deal of expository material that in- structs the reader, and this is interspersed with suggestions for classroom activities. The appendix contains most of the actual descriptions of particular games and other activities.

Apart from this appendix, Sets, Num- bers and Powers contains much standard material which a teacher encounters in many other books. It seems to me that Learning Logic, Logical Games provides many more unique and original ideas. Both books will be invaluable to teachers as sources of good ideas for classroom activi- ties, but Learning Logic, Logical Games provides an excellent basis for develop- ment of some important concepts not likely to be provided by the standard mathe- matical textbooks. The teacher will find

both books easy to read, and the children will find the games fascinating.

Shirley Hill University of Missouri Kansas City, Missouri

In-service Course for Primary School Teachers: Studies in Mathematics XIII,

School Mathematics Study Group. Pasa- dena, Calif.: A. C. Vroman, 1966. Paper, 369 pp., $2.50.

The introductory chapter of this book gives attention to the problems of the "cul- turally deprived" or disadvantaged child. Based on psychological findings, a descrip- tion of the physical, social, and psychologi- cal environment from which these children enter school helps establish methods for teaching the essential concepts of mathe- matics. The language characteristics of the disadvantaged child and the implications of these characteristics for teaching these chil- dren are discussed in Appendix B. Appen- dix C contains the results of a comparative study of children who used the School Mathematics Study Group texts, Mathe- matics for the Elementary School, Books K and 1, during the school year 1964/65. The classroom teacher should certainly find this discussion of the culturally disadvan- taged child helpful in identifying and meet- ing the needs in this area of growing con- cern.

The mathematical content presented in this text is principally of relevance to the primary grades. Topics included in Books K and 1 of the SMSG texts are treated along with other topics deemed essential in the development of necessary concepts. Since these essential topics thread their way throughout the elementary program in the spiral approach, upper-level elementary school teachers who need to update their backgrounds in mathematics should find this development beneficial.

The concept of sets and set notation is developed and used to introduce the num- ber property. Sets of elements are com-

742 The Arithmetic Teacher

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pared and order relation established be- tween numbers. The set operations of union, intersection, Cartesian product, and complement are used to discuss the com- mutative and associative properties and are later used to develop the understanding of operations on numbers. The union of disjoint sets is used to introduce addition of numbers, and the cross product of sets used to develop multiplication of numbers. The inverse operations of subtraction and di- vision are developed through the principal operations of addition and multiplication. A section devoted to "Applications to Teaching" follows each development, and comments made on pedagogy or other sug- gestions appear to be helpful.

The sections on geometry cover both metric and nonmetric geometry and ex- tensively develop the concepts and termi- nology typical of contemporary programs. Numeration systems are discussed, with emphasis on place-value notation developed through use of nondecimal bases. The operations are introduced through ordered pairs and concrete interpretations, includ- ing sets, and the use of the number line helps to develop the desired concepts. The basic properties for the set of whole num- bers are developed extensively, while the set of rational numbers is simply intro- duced through concrete representations.

In my opinion this text presents a suffi- cient mathematical content to adequately prepare primary grade teachers to teach any contemporary elementary series.

Forrest L. Coltharp Kansas State College

Pittsburg, Kansas

Guiding Discovery in Elementary School Mathematics, C. Alan Riedesel. New York: Appleton-Century-Crofts, 1967. Cloth, 491 pp., $7.50.

In the preface, the author states the purpose of the book: ". . . to provide prospective and in-service elementary

teachers with illustrative situations that make use of modern mathematical con- tent and ideas to develop a guided dis- covery approach to teaching mathematics in the elementary school." From the state- ment of purpose and from examination of the book, it is clear that the book is in- tended for a methods course in elementary school mathematics.

The author has achieved well part of his stated purpose. The book begins with an illustrative lesson on modular arithmetic contrasting the use of an explanatory pat- tern and a guided discovery pattern. Some guidelines are stated for guided discovery. Then repeatedly throughout the book one finds illustrative lessons using guided dis- covery that seem to be taken directly from a classroom. Most of these lessons are excellent.

All the illustrative lessons indicate the author's strong commitment to adequate developmental work for pupils and to their active involvement in learning. It is clear that the teacher must listen to the re- sponses of pupils and reply to the pupils' own ideas and verbalizations. Responses of teachers in the illustrations reveal clearly the importance of a supportive and understanding teacher to build the child's confidence in his own thoughts and his own methods. The use of oral work is stressed.

Each lesson begins with a practical prob- lem, considered by the author to be im- portant for motivation of the learner. While there is renewed emphasis on ap- plications of mathematics and on the use of mathematical models, few leaders would go to the extreme of making every lesson have a practical orientation. Indeed, many mathematical topics have sufficient intrin- sic interest to provide needed motivation. Furthermore, many topics such as "be- tweenness" and repeating decimals are made more difficult by making them so closely related to the "real" world.

The illustrative lessons stress multiple approaches to computation algorithms. But there is little indication of which

December 1968 143

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