A Problem Solving Approach to Mathematics for Elementary School Teachers, 4th ed. by RickBillstein; Shlomo Libeskind; Johnny W. LottReview by: Patrick B. ScottThe Arithmetic Teacher, Vol. 38, No. 6, FOCUS ISSUE: Reaching ALL Students (FEBRUARY1991), p. 64Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194780 .

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nished by students' notes, hard copy of proce- dures and graphical output, and structured audiotaped interviews with the students.

In addition to exploring Logo's potential "to facilitate understanding of some specific math- ematical concepts," the researchers were par- ticularly interested in the "nature and extent of collaborative learning," students' problem- solving strategies, "the nature and conse- quences of teacher intervention in the learning process," and "gender differences in pupils' approach to programming and problem solving with Logo."

A particularly practical chapter, "The Teach- er's Role," outlines categories of intervention, teacher-directed tasks, and teaching episodes. Examples include strategies to move students beyond their self-constructed boundaries and to use mathematics from other contexts in the problem at hand. "Gender Issues" makes the familiar case for high value being assigned to well-defined, product-oriented goals. "Girls' apparent lack of emphasis on product can easily lead to their work being undervalued." The re- searchers believe that the girls' comfort with process and extensive experience with loosely defined tasks made them "more able later to break down well-defined tasks into modules." They point out that more research is needed in this area.

In the final chapter, Hoyles and Sutherland conclude that "our work highlights the crucial influence of intervention in the learning pro- cess" and that "the presence of Logo does not deprive teachers of responsibilities but rather presents new ways for thinking about the teach- ing/learning process."

This book should be a "must read" for math- ematics educators eager to meet the challenges of the NCTM's Standards. I look forward to the publishing of the projects, intervention strate- gies, and teachers' problem-posting strategies in the context of the middle school curricu- lum. - Elizabeth Janssen, Albuquerque Public Schools, Albuquerque, NM 87106.

Mathematics and Gender, Elizabeth Fennema and Gilah C. Leder, eds. 1990, vii + 214 pp., $17.95 paper. ISBN 0-8077-3001-7. Teachers College Press, 1234 Amsterdam Ave,, New York, NY 10027.

This book is a collection of articles by several leading authors in the field of gender and math- ematics. Gender differences in mathematics have been extensively researched for approxi- mately fifteen years, since these differences are known to cause economic problems for women.

In chapter 1, Elizabeth Fennema examines the issue of equity. She defines and examines equity from three viewpoints: equality of ( 1 ) op- portunity, (2) treatment, and (3) outcome. She also introduces the Autonomous Learning Be- havior Model (ALB), which suggests that dif- ferences are caused by the lower participation in ALBs by females. Related issues are more thoroughly examined in later chapters.

Chapter 2, by Gilah C. Leder, overviews gen- der differences in mathematics and where they exist. In chapter 3, Lindsay A. Tartre examines

the effects of internal influences not usually ad- dressed by the ALB model, in particular, spa- tial visualization and orientation skills.

Margaret A. Meyer and Mary Schatz Koehler collaborate on chapter 4, which discusses the ALB model as a possible explanation for devel- opmental gender differences in internal belief systems. The next chapter, by Peter Klooster- man, examines the relationship among attribu- tions, performance following failure, and moti- vation; and he submits an attributional theory of motivation, which is explained in terms of beliefs.

Chapter 6 by Koehler, chapter 7 by Leder, and chapter 8 by Fennema thoroughly discuss treatment, including classroom environment and organization, teacher-student interactions, and teachers' beliefs and expectations, respec- tively.

Finally, Leder and Fennema summarize past research and note areas deserving of further re- search. What types of mathematics are suited for certain populations? What anomalies occur when the issue is examined from purely a fe- male perspective? How do atypical classrooms affect gender differences? What are character- istics of classrooms where no differences exist?

This work should be examined by every teacher who desires to revamp the classroom so as to limit gender differences. Researchers will find it a valuable tool from both a content and extensive referencing viewpoint. - George Nat- trass.

A Problem Solving Approach to Mathematics for Elementary School Teachers, 4th ed., Rick Billstein, Shlomo Libeskind, and Johnny W. Lott. 1990, xx + 945 pp., $37.95 cloth. ISBN 0-8053-0390-1. Ben- jamin! Cummings Publishing Co., 390 Bridge Pkwy., Redwood City, С А 94065.

A Problem Solving Approach to Mathematics for Elementary School Teachers, now in its fourth edition, is a widely used text for prospec- tive elementary school teachers. Continuing goals from previous editions are (1) "to present the appropriate mathematics in an intellectually honest and mathematically correct manner," (2) "to use the heuristics of problem solving as an integral part of mathematics," and (3) "to approach mathematics in a sequence which ini- tially instills confidence, then challenges stu- dents as they complete the exercises and prob- lem sets." New goals added to the fourth edition are (1) "to present the topics in the con- text of the Standards" and (2) "to significantly revise and expand the geometry coverage." The following chapters are included: "Tools of Problem Solving," "Sets, Functions, and Log- ic," "Numeration Systems and Whole Num- bers," "The Integers," "Number Theory," "Rational Numbers as Fractions," "Deci- mals," "Probability," "Statistics: An Intro- duction," "Introductory Geometry," "Con- struction, Congruence, and Similarity," "Motion Geometry and Tessellations," "Con- cepts of Measurements," "Coordinate Geome- try," and "Problem Solving Revisited."

In addition to appendixes that give "An In-

troduction to BASIC" and "An Introduction to Logo Turtle Graphics," some computer and calculator activities and commentaries are found throughout the book. The initial and final chapters on problemsolving are supported by classic and original problems in each chapter, and a "preliminary problem" motivates the stu- dent to study each chapter. In each chapter are well-placed historical notes that, although to a certain extent Eurocentric, do give some credit to women and non- Europeans for their contri- butions to mathematics. Model pages from ele- mentary and middle school texts broaden the perspective on mathematical topics and relate them nicely to the school curriculum.

The K-4 Standards are displayed on the in- side front cover and the 5-8 Standards on the inside back cover. The Standards is also quoted thirteen times in support of such topics as prob- lem solving, calculator use, mental arithmetic, estimation, probability and statistics, and vari- ous kinds of geometry. In most respects, par- ticularly with the emphasis on problem solving, the text does much to prepare future teachers to be able to teach according to the Standards. In addition to numerous problem sets, one finds twenty-six laboratory activities that give some experience, particularly in geometry, with learning mathematics concepts using manipula- tives. Some references are also made to using manipulatives for teaching certain concepts. However, it may be that a text for future ele- mentary school teachers should introduce more mathematical concepts with manipulative activ- ities. - Patriek B. Scott, University of New Mexico, Albuquerque, NM 87131.

Reshaping School Mathematics: A Philosophy and Framework for Cur- riculum, Nationúl Research Council, Mathe- matical Sciences Education Board. 1990, xiii + 59 pp., $7.95 paper. ISBN 0-309-04187-2. Na- tional Academy Press, 2101 Constitution Ave., N.W., Washington, DC 20418.

The stated goal of this project, developed by the Mathematical Sciences Education Board (MSEB), is to develop a framework for curric- ular reform, thus giving educators a common base from which to start. More specifically, two issues are emphasized: (1) changing perspec- tives on the need for mathematics, the nature of mathematics, and the learning of mathematics and (2) the changing roles of technology - cal- culators and computers - in instruction of math- ematics. These two issues are emphasized in light of the NCTM's Standards and the MSEB and National Research Council's (NRC) Every- body Counts.

The first chapter devotes itself to developing a rationale for change, citing (a) changes in the need, use, and role of technology and (b) soci- etal demands on, metacognition in, and eco- nomic considerations for, mathematics. Next, a ten-page philosophical perspective is given of why we teach mathematics, what is mathemat- ics, and what is mathematics education. An- other ten pages are devoted to formulating a rationale for the use of technology in the class- room and considerations to be made. Chapter 4

64 ARITHMETIC TEACHER

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