Teaching Elementary School Mathematics: Methods and Content for Grades K-8 by FrederickH. Bell; Mathematics for Elementary Teachers: A Content Approach by Ruth E. HeintzReview by: Margaret S. MatchettThe Arithmetic Teacher, Vol. 28, No. 8 (April 1981), p. 47Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191894 .

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lenge in mathematical thinking. Prerequisite arithmetical skills are minimal. This is material made for students who like puzzles.

New Books for Teachers Edited by Max S. Bell University of Chicago Chicago, Illinois

Applications in School Mathe- matics. Sidney Sharron, ed. 1979 Yearbook of the National Council of Teachers of Mathematics, viii + 243 pp., $12.* The Council, 1906 Associa- tion Drive, Reston, VA 22091. (^Individual NCTM members are entitled to a 20 percent dis- count on all NCTM publications.)

The editors of the 1979 yearbook on applications have collected twenty essays and assigned them to four themes: 1. What are applications? 2. Why include applications in school mathematics? 3. How can applications be brought to the class- room? 4. What issues are related to applications? Each essay may be read independently of the others.

Some essays are general in orientation and present a philosophical and psychological basis for the study of applications. For example, the opening essay in the yearbook delineates diffi- culties in presenting applications in the class- room. The main problem is said to lie not with the mathematics to be applied but with finding interesting settings for those applications that are at the right level for the students. Another article makes a case for a program that admits a broad range of objectives and modes of instruction with an early emphasis on exposure rather than mastery. The yearbook's final essay, an anno- tated bibliography, provides one way to expedite the classroom teacher's search for applications. It is designed for mathematics teachers who are looking for applications in other fields that make use of the mathematics they teach.

Essays on specific applications are included for all levels of instruction. The following are ex- amples: describing, classifying, or measuring cer- tain human characteristics; using a story to pro- vide the real or physical situation for applying mathematics; integrating mathematics with the study of other cultures; establishing a general procedure for the development of mathematics applications based on ever-changing socioeco- nomic problems; the mathematics of finance re- visited through the hand calculator; the study of the circle as a special case of a curve of constant width; constructing an elementary mathematical model of an underground cellar; bringing the outside world into the classroom in simulated situations. This yearbook could be useful to anyone who

wants applications to play a larger role in school instruction. - John Dalida, Woodward School, Dearborn, Michigan.

Problem Solving in School Mathe- matics. Stephen Krulik, ed. 1980 Yearbook of

April 1981

the National Council of Teachers of Mathematics, xiv + 241 pp., $12.* The Council, 1906 Associa- tion Drive, Reston, VA 22091. (*Individual NCTM members are entitled to a 20 percent dis- count on all NCTM publications.)

This yearbook on problem solving is similar in format to the previous yearbook on applications. It is addressed to what the editor, Stephen Kru- lik, calls one of the basic skills that students must take from school for use throughout their lives. The twenty-two essays span issues of interest to teachers at all school levels.

The more general articles focus on placing problem solving in a curricular context. The opening essay by George Polya, master teacher of problem solving, is titled "On Solving Mathe- matical Problems in High School," but can be read with profit by all teachers of mathematics. Another article explores the interpretation of "problem solving" as a goal, as a process, and as a basic skill. Other essays discuss the following topics: teacher and lay opinions about teaching problem solving, problem posing, pictorial lan- guages in problem solving, using the calculator in problem solving, supplementing and under- standing textbook problems, problem solving through recreational mathematics, a new ap- proach to the measurement of problem solving skills, symmetry in problem solving, and re- search reports on problem solving. The volume closes with an annotated bibliography divided into six sections: bibliographies, general works, suggestions for teachers, puzzles and recreations, mathematical discussions, and collections of problems. - John Dalida, Woodward School, Dearborn, Michigan.

Teaching Elementary School Mathematics: Methods and Content for Grades K-8. Frederick H. Bell. 1980, xvi + 582 pp., $16.15. Wm С Brown Company Publishers, 2460 Kerper Boulevard, Dubuque, I A 52001. Mathematics for Elementary Teach- ers: A Content Approach. Ruth e. Heintz. 1980, 492 pp., $16.95. Addison- Wesley Publishing Company, 5851 Guion Road, In- dianapolis, IN 46254.

Of the two books at hand, one is explicitly de- signed for a combined content and methods course, the other for a content course, for ele- mentary teachers. The question of how much emphasis in teacher training should be given to content and how much to method is very inter- esting, but it is probably not relevant in this re- view.

Both books recognize that the mathematics of the elementary school goes - or at least should go - beyond computational skills and measure- ment formulas. Both incorporate material on probability and statistics, on the integers, and on the use of calculators. Heintz includes a section on transformations. Bell does not, but he incor- porates considerably more work on coordinate geometry.

Given the wide range of mathematics that can be incorporated in the elementary grades, it would probably be unreasonable to demand that a text cover all possible topics in detail. Neither

author makes clear a principle on which the choice of material is based. The Pythagorean Theorem appears in neither text.

The Heintz text is systematic and has an at- tractive format. Of particular value are well-se- lected, nonroutine problems. A five-step strategy for solving problems is stressed. Though the rec- ognition of the importance of problem solving is commendable, the strategy appears somewhat simplistic. Step 3, for example, is "collect ideas." How best to teach problem solving is an open question, but it is likely that no single strategy is adequate. The Heintz text would be useful pri- marily where a careful and methodical approach is needed. Though it contains only occasional references to the classroom, its examples and presentations would be useful to many classroom teachers.

The Bell text is considerably richer not only in the additional methods material but also in the mathematical content. The latter is not always carefully presented, however, and it suffers from a format that is not very inviting or illuminating. A few inaccuracies were noted. For example, the term incommensurable is misused (p. 31). Bell's inclusion of compound interest is desirable, but the explanation of it is not very clear. Good bib- liographies follow each chapter. The methods sections contain a wealth of ideas and are help- fully specific. The teacher in search of sugges- tions for classroom activities should find this book a valuable resource.

Neither book would be an entirely satisfactory text. The Bell book would require considerable amplification and exposition. The Heintz book does not offer enough material for the well-pre- pared college student. - Margaret S. Matchett, University of Chicago Laboratory School, Chi- cago, Illinois.

Teaching the Gifted and Talented in the Mathematics Classroom. Kevin a Bartkovich and William С George. 1980, 48 pp., $4.50. National Education Association, Washing- ton, D.C.

Essentially a "how to" manual, this booklet pres- ents the program ideas developed by Julian С Stanley's Johns Hopkins Project, the Study of Mathematically Precocious Youth. Whether your concern is a single gifted student or small groups of gifted students, this booklet describes procedures that have already proved successful in a number of schools.

This monograph, which favors acceleration of gifted students, advocates that to qualify for en- trance into a special program the "gifted youth be in the upper one percent or one-half of one percent in mathematical reasoning in his or her age group." If a small population base necessi- tates less selectivity, the authors do not recom- mend dipping "below the upper five percent for entrance into special programs in mathematics." Most school systems identify their gifted at the end of sixth or during the seventh and eighth grades.

The text contains some intriguing ideas and is well worth reading. References are cited throughout for the reader who wants more infor- mation. - John Dalida, Woodward School, Dear- born, Michigan. W

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