A Problem-solving Approach to Mathematics for Elementary School Teachers by RickBillstein; Shlomo Libeskind; Johnny W. LottReview by: Mary T. WhalenThe Arithmetic Teacher, Vol. 35, No. 3 (November 1987), pp. 57-58Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194270 .

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instance, the three errors F, G, and H have the same description:

" problems finding common

denominators." What is the difference between these errors? Are they the same? If they are the same, then why flag them with three letters? The error analysis printouts make distinctions between them, but these errors, and possible student processes leading to these errors, should be clearly defined in the manual. The diagnostic information provided is very limited.

On the positive side, the package does pro- vide a class summary (by topic area only), which could aid in class grouping. Also, the decimal tests offer both horizontal and vertical examples; however, no other tests in this pack- age are so thorough. These encouraging signs are not sufficient to induce me ever to use the package. - Robert N. Ronau, University of Louisville, Louisville, KY 40292.

New Books For Teachers From NCTM

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Learning and Teaching Geometry, K- 1 2, Mary Montgomery Lindquist and Al- bert P. Skulte. 1987, xi + 250 pp., $16 cloth. ISBN 0-87353-235-X. National Council of Teachers of Mathematics, 1906 Association Dr.,Reston, VA 22091.

Mathematics educators of grades K-12, as well as college and university faculty, will find this 1987 geometry yearbook a worthwhile addition to their mathematics library. The editor notes in the preface that geometry is both speculative and practical and that readers will find material on which to speculate as well as material that they can use in a practical way in the class- room.

Twenty articles are organized into five cate- gories: perspectives, problem solving and appli- cations, activities, geometry related to other mathematics, and teacher preparation.

Some questions we might ask after reading the articles include the following: On what level of the Van Hiele model are our students work- ing? Is our high school geometry course rele- vant? Should we insist on Euclidean rigor in our courses? How can we integrate the computer into our geometry lessons? How can we relate geometry to other activities of the elementary school student's day?

I highly recommend the 1987 Yearbook to all readers of the Arithmetic Teacher. - Mary T. Whalen, University of Northern Iowa, Cedar Falls, IA 50614.

Edited by Diane Thiessen University of Northern Iowa Cedar Falls, IA 50614

From Other Publishers Another 20 Mathematical Investiga- tions, Barry Bastow, Jill Hughes, Barry Kis- sane, and Rob Randall. 1986, 20 pp., $4 paper. ISBN 0-949278Ί1-4. Mathematical Association of Western Australia, Department of Mathe- matics, University of Western Australia, Nedlands, WA 6009.

Another 20 Mathematical Investigations is a sequel to 40 Mathematical Investigations, also published by the Mathematical Association of Western Australia. The problem-solving inves- tigations are appropriate for upper elementary and junior high school students. The theme of this book is definitely "investigate."

Each investigation is intended to provoke inquiry. In most cases a situation is given but a question is not asked. The situations are de- signed to encourage different interpretations. The students are to determine the questions as well as to investigate their questions.

Each of the twenty investigations is printed in a card format on a half page. The layout of each card is simple, and diagrams or examples are used to illustrate each situation. For example, in "Rectiles" a 4 x 3 rectangular grid and a 2 x 6 grid are shown, and the following statement is presented: "Here are two different ways of making rectangles from twelve square tiles. Investigate." Another example shows a first- quadrant grid with the point (3, 4) plotted. The accompanying statement is "The point shown has a Y-value that is one more than its X-value. Investigate." Other topics include divisors, fractions, geometry, and patterns.

Hints and extensions are included for each investigation, but the authors encourage teach- ers to use these in moderation because hints tend to lead the students and it is preferable to give the students more experience in problem posing and exploration. Examples of hints for the rectile investigation include such questions as "For which numbers of tiles can only one rectangle be made?" and "For which numbers

of tiles can an odd number of rectangles be made?"

The authors claim that exploration, problem posing, generalization, and extension are im- portant processes. Their activities reflect their philosophy: "Investigating is not just getting the right answers but asking the right ques- tions." They have done an excellent job of posing problem-solving activities that can be readily used by other teachers. - Diane Thies- sen.

A Problem-solving Approach to Mathematics for Elementary School Teachers, Rick Billstein, Shlomo Libeskind, and Johnny W. Lott. 1987, xviii + 782 pp., cloth. ISBN 0-8053-0865-2. BenjaminlCum- mings Publishing Co., 2727 Sand Hill Road, Menlo Park, CA 94025.

This mathematics textbook for preservice ele- mentary education teachers follows closely on the second edition published in 1984. Readers are referred to the review of the second edition that appeared on page 42 of the Arithmetic Teacher in November 1986.

Topics covered in chapters 1-7 include Polya's four-phase problem-solving process; sets, functions, and logic; standard operations and properties with whole numbers and inte- gers; number theory; operations and properties with rational numbers; and percents. Chapters 8-13 deal with probability; statistics; introduc- tory geometry; constructions and transforma- tions in geometry; measurement using the met- ric system; and coordinate geometry. Chapter 14 returns to problem solving.

The appendices consist of an introduction to the computer and the BASIC language, an introduction to Logo turtle graphics, and basic compass constructions. As in previous editions, answers to odd-numbered exercises are in- cluded in the text.

Improvements and additions can be noted throughout the textbook. Major improvements in this edition include the following:

REFEREES! i~lk An Invitation from the V^iflnitt

Editorial Panel ^^MKvi The Editorial Panel invites members interested in serv- vJil I 1 If ing as a referee of manuscripts to request guidelines ijVi I I If/ and a personal data sheet from NCTM. Ill |Д I Ш

Each of the articles that appears in the Arithmetic III 1 1 ΤΐΖ/ίΐΛ Teacher is judged for content and style by at least three '''ll Ί/ IW referees. The success of the Arithmetic Teacher is very M4t --Z.f^4L much dependent on these volunteer efforts. I '

November 1987 57

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1. Reorganization of chapter 1 (on problem solving). Puzzle-type problems that cannot be solved by the usual Polya strategies are not part of the problem set but are presented alone in a "Time Out" section. A section on using the calculator as a problem-solving tool has been moved from the appendix to chapter 1 .

2. Chapters 10 and 11 include Logo instruc- tion, and problems involving the writing of Logo procedures are added to the problem sets in the geometry sections.

3. Probability and statistics are separated into two chapters, with stem-and-leaf plots and mis- uses of statistics included in the statistics chap- ter.

I have used the second edition of this text with preservice elementary school teachers. Sufficient material is included to cover two three-hour semester courses in mathematics content, supplementing with more work on problem solving, estimation and mental- computation skills, and laboratory activities. -

Mary T. Whalen.

Learning Math with Logo, Rudy v. Neufeld. 1986, 1 student text (232 pp.) and 1 teacher resource (136 pp.), $29.95; 5-14 stu- dents texts, $24.95 ea., 1 teacher resource; 15 or more student texts, $19.95 ea., 1 teacher resource; + $2.00 shipping, paper. Logo Pub- lications, 7 Conifer Crescent, London, ON N6K2V3.

The student text is a series of Logo activities and investigations designed to be an introduc- tion to computer literacy and an aid in imple- menting the mathematics curriculum for grades 4-7. The texts can be used with Apple Logo, Terrapin Logo (Apple), and Commodore 64 Logo; prior computer or Logo knowledge is not necessary. The activities are designed to help students learn mathematics using a guided- discovery approach. Concepts treated in the activities include number lines, estimation, op- erations, integers, parallel lines, polygons, de- grees, pi, circles, variables, recursion, slides, flips, turns, tessellations, and area. The stu- dents also learn how to format disks, save procedures, and read from disks.

The student text can be used by individuals or for group instruction. Each lesson includes an introduction, a precomputer lesson, a guid- ed-discovery activity, and an extended-learning activity. Students are encouraged to solve prob- lems by thinking with the turtle and by walking through the commands before using screen commands. The teacher resource book states clear objectives for chapters and sections. Each section includes grade-level identification, mathematics concepts covered, the Logo com- mands required, the Logo commands learned, materials needed, and necessary editing in- structions. A precomputer lesson, a guided- discovery activity, and an extended-learning activity are then detailed. Often follow-up sug- gestions are included. Suggested answers to

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questions in the student text are given. Blackline masters that correspond to certain activities in the student text are also found throughout the teacher resource book.

The activities emphasize the user being a problem solver and the controller of the turtle. The teacher resource book states that many of the activities are not precisely described and are meant to be open-ended to encourage the students to "play turtle" in investigating ideas. The activities also include good ideas for using many concrete materials. Not having access to many computers would not prohibit teachers from using these materials.

In summary, these books offer upper elemen- tary school teachers ideas for introducing math- ematical concepts using Logo while not teach- ing many Logo commands. They offer students the chance to learn about learning. - Cheryl A. Ross, Hudson Community Schools, Hudson, I A 50643.

Logo Works: Lessons in Logo, 5. Cory andM. Walker. 1985, 262 pp. + 141 pp. in teacher's manual + reproducible worksheets + disk; student book (paper) + disk, $29.95; teacher's manual (paper), $12.95. ISBN 0-927510-04-9 and 0-927510-05-7. Terrapin, 222 Third St., Cambridge, MA 02142.

The student book is a series of forty-six lessons for developing the Logo language and extending students' knowledge of geometry, arithmetic, computer programming, computer literacy, and problem solving. It is designed for use with Terrapin, Apple, or Commodore Logo. The lessons provide a balance of classroom struc- ture with Piagetian learning for those teachers who wish a quality Logo program for their students. The student text seems to be written for upper elementary and junior high school students new to Logo, although it could be used in senior high school. The activity disk that accompanies the student book contains sample files to which the students are directed in the lessons.

The activities include Logo primitives, writ- ing procedures, editing, managing workspace and files, variables, recursion, and problem solving. Concepts of polygons and of circles are treated in separate chapters. Each student les- son contains a narrative presenting new ideas, activities to practice those ideas, and explora- tions to challenge students further. The book is easy to read with numerous Logo diagrams appearing as they would on a screen and with important words and sections highlighted or in boldface type. The questions in the lessons encourage students to use higher-level thinking skills.

The separate teacher's manual is very useful for teachers who may not themselves be totally at ease with teaching Logo. Each lesson corre- sponds to one in the student text and includes lesson objectives, vocabulary, additional infor- mation, ideas for expanding students' thinking, points to emphasize, ways to guide computer activities, and potential problem areas. The manual also includes some reproducible pages for record keeping, chapter covers for a student

58 Arithmetic Teacher

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