Exploring Elementary Mathematics by Julian WeissglassReview by: Tommie A. WestThe Arithmetic Teacher, Vol. 28, No. 1 (September 1980), pp. 55-56Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189356 .

Accessed: 12/06/2014 22:49

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 62.122.76.48 on Thu, 12 Jun 2014 22:49:20 PMAll use subject to JSTOR Terms and Conditions

RaviGGDing and '7iaoDing

New Books for Pupils Edited by Tommie A. West Montgomery County Public Schools Rockville, Maryland

Prices on books and materials are included with reviews for general information only. Prices change and actual prices may vary from those listed. Please check with the individual publishers and producers.

Analyze. Eleanor Villalpando. 1979, $6.95. Synthesize. Eleanor Villalpando. 1979, 64

pp., $6.95. Think Ink Publications, 3421 North 44th Street, Phoenix, AZ 85018.

Although Analyze is not a book of mathematics, this collection of reproducible exercises does present clever activities which call upon thinking skills characteristic of mathematical problem solving. The following exercises and activities are included in this group: a family tree exercise, tangram-like exercises, word puzzles, mazes, classification activities, detecting similarities and differences activities, logic puzzles, word plays, patterns, and codes.

The treatment is strictly nonstandard and quite creative. Gifted students of upper elemen- tary and beyond will find this collection enter- taining fare.

Synthesize is an extension of the kinds of exer- cises found in Analyze. There are more analo- gies, classification exercises, word plays, codes, and games. Many of the exercises in this collec- tion, however, are more open-ended and call upon the reader to create something, given the conditions. It is cleverly developed, and will pro- vide a few hours of fun for more able learners.

Brain PleaserS. Marie Masse. 1979, 56 pp., $3.50. Logic and Learning, 260 Ipswich Road, Topsfield, MA 01983.

This booklet joins the host of publications pres- enting problems in logical deduction. Its focus is on the younger student.

Fifty pages of logic problems are contained in the typescript copy; half of the hundred prob- lems are nonnumerical and half are numerical. The clues given are couched in both positive and negative terms - "It is . . ."; "It is not . . ." The problems, varying from quite simple (grades 2-3) to quite complex (high school), are usually framed in a rectangular box on the page. Some

of the problems are illustrated by primitive pen and ink sketches.

Although the format appears to be prepared for limited copy permission, there is no state- ment that such is granted. Whether the book's use is for teacher resource or student workbook is not clear.

Computer Alphabet Book. Elizabeth s. Wall. 1979, $8.95. Bayshore Books, P.O. Box 848, Nokomis, FL 33555.

Computer literacy is the catchword of the 1980s for mathematics curriculum makers. This picture dictionary is an effort at bringing some elemen- tary information about computers to young stu- dents. It is a lexicon of terms occasioned by the initial letters and presented in alphabetical or- der. The problem with this arrangement of terms, however, is that the terms are presented out of a logically sequenced presentation about computers. Whether this alphabetic hodge podge of terms will be comprehensible to the audience for whom it is appropriate, or appropriate for the audience for whom it will be comprehensible is a matter of question.

CryptarithmS. Josephine Andrée and Rich- ard Andrée. 1978, 186 pp., $3.60 (individual member price $2.88).

Instructor's Manual for Cryp- tarithms. Josephine Andrée and Richard Andrée. 1979, 82 pp., $2.50 (individual member price $2). Published by Mu Alpha Theta. Na- tional Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091.

Husband and wife team Richard and Josephine Andrée have coauthored a booklet that is certain to become a classic for aficionados of logic puz- zles. Cryptarithms are defined as "puzzles made by substituting letters for digits in a simple arith- metic problem. The puzzle is to discover the original digits."

The authors begin by describing the step-by- step solutions to very simple cryptarithms and proceed to more difficult ones. They describe the logic and trial-and-error solutions to the con- struction and use of the more systematic "possi- bilities table" for more complex puzzles. A most useful chapter tabulates properties of numbers and cites generalizations that are time-savers for the problem solver.

Chapter 5 serves up a smorgasbord of other puzzles and riddles, and develops what for many students could be their first introduction to in- direct proofs using the "let's suppose" strategy for cryparithm solution. In chapter 6, the authors apply the strategy of cryptarithm solution to the problems of everyday life, such as constructing schedules of activities and "goal" charts for deci- sion making. They also supply the solutions to all the puzzles posed in the preceding chapters.

Chapter 7 gives clues for creating cryptarithms with suggestions on how to tell if a puzzle solu- tion is unique. In chapter 8, as an extension of the basic cryptarithm format, the authors discuss hidden cryptarithms, cryptarithms in bases other than ten, and cryptarithms involving square roots. The final chapter discusses the application of computers to solving cryptarithms. A program

for solving the following cryptarithm is given:

one + two + three = seven

A bibliography of related materials is provided at the end of the book, together with a page of grids to use for possibility tables.

The authors' explanatory style is a study in elegance, as they seek to unfold the mysteries of the cryptarithm solution and construction. The book will be of interest to all who are interested in the challenge of logical thinking. It is a "must" for the teacher of the gifted.

In addition to the numerical solutions of the cryptarithms included in the text, the instructor's manual gives in detail the logical steps used in arriving at the solutions. It also identifies useful "chalkboard" cryptarithms for introducing the topic. Newspaper release forms for advertising this minicourse, bulletin board displays, repro- ducible possibility tables, and an extended bibli- ography are included in the manual, too.

Exploring Elementary Mathematics. Julian Weissglass. 1979, 279pp., $14.95. W. H. Freeman and Company, 660 Market Street, San Francisco, CA 94104.

This lab-text is designed as a text or text supple- ment for the college-level mathematics methods class for preservice teachers. It is a laboratory manual for the physical modeling of abstract mathematical concepts and procedures.

Attribute blocks are used to depict sets and set operations. In the chapter on numerations sys- tems, base blocks are used to depict numbers in positional systems. Addition and subtraction are developed through Cuisenaire rods and the Chi- nese abacus. The concept of multiplication is de- veloped through Cuisenaire rods and geoboards, while the multiplication algorithms include the lattice method and the partial products through Cuisenaire rods.

Division is restricted to the measurement con- cept, since the model is the Cuisenaire rod. The algorithm for division is rationalized by the base ten blocks, but the "array" model which the au- thor uses (see p. 69) will be lost on all but the most sophisticated. The two factors (missing and given) and the product are depicted as three-di- mensional base ten blocks. This is inconsistent with either the measurement or the partition in- terpretation of the algorithm.

Number theory (factorization) is depicted by using towers of Cuisenaire rods. These also are used to show odd and even numbers.

The chapter on topology presents the classic Königsburg bridge discussion. The modeling of fractions and operations on fractions is shown through arrays, strips, and Cuisenaire rods. Posi- tive and negative numbers are developed on a numberline and through a pedagogically pow- erful "counter" model. Multiplication is ratio- nalized through the "backward projector" de- vice.

Geometric concepts in chapter 12 are treated in the traditional nonmetric manner: shapes, lines, planes, and inside and outside. Lines, an- gles, and triangles are shown through Mira ma- terials with the usual construction tools used.

The chapter on area, developed by using the geoboard, is an excellent treatment of this topic,

September 1980 55

This content downloaded from 62.122.76.48 on Thu, 12 Jun 2014 22:49:20 PMAll use subject to JSTOR Terms and Conditions

with direct application to the classroom. The chapter on decimals, however, is disappointing. Instead of using the meter stick as a model for decimals, or even the base blocks, the author re- verts to an abstract, place value treatment. The chapter on measurement treats the metric system very briefly, but does begin with measuring ac- tivities.

The dessert of this text is supplied near the end, in a chapter titled "Tesselations: Mathe- matics in Art and Floor Tilings." Transforma- tions are treated algebraically and are depicted on the coordinate plane, while probability and statistics are treated experimentally. The final chapter called "Mathematical Explorations" contains some odds and ends not included in previous chapters - Moebius strips, for example.

On large, this text is an excellent laboratory manual for preservice teachers. The notation is somewhat formal in the current climate, and there are parts that are more appropriate for the teacher than for the elementary student. If the teacher will use the models that are powerful for explaining the concepts and procedures of ele- mentary arithmetic and avoid using those mod- els that are not, then this book can be a very use- ful resource for teachers and teachers-to-be.

Fantastic Voyage - Whole Number Division. Lawrence W. Swienciki. 1979, 48 pp., $4.95.

Fantastic Voyage - Decimal Divi- sion. Lawrence W. Swienciki. 1979, 48 pp., $4.95.

Safari Into Whole Numbers. Lawrence W. Swienciki. 1979, 45 pp., $3.95. A. R. Davis & Co., P.O. Box 24424, San Jose, CA 95154.

Fantastic Voyage - Whole Number Division is a collection of two-page exercises for drill in whole-number division. The first page of each exercise presents an interesting historical para- graph on a variety of topics - Egyptian, Mayan, and Viking tidbits, Irish folklore, and Ashanti legend. The exercises that follow are coded to answers that are used to resolve a self-check problem. The problem formats are varied - crossword puzzle, coded message, colored pat- tern, or dot-to-dot. The range of exercises per- mits this material to be used over a period of time to enliven dull practice.

Although the text is typewritten, with run-on headings, and the art work is amateurish, the treatment is interesting and will be welcomed by teachers who need to provide this kind of drill.

Decimal Division is a second volume of the Fantastic Voyage booklets. The theme of Ashanti legend turns to that of Indian folklore, picks up a Hawaiian theme, and concludes with Chinese topics. The skills addressed are decimal-division skills, and the exercises include both mixed and selected practice. Divisors are whole numbers through three-place decimal numbers.

Safari follows the format of its sister volumes. The range of skills is quite wide, from basic facts to the four operations (addition, subtraction, multiplication, and division) with four-digit numbers in each of the operations.

The topics in the introductory paragraphs range from Houdini to the pharaohs. The table

of contents lists the operations, but not the diffi- culty of the exercises.

FOCUS ОП Calculator Math. Chuck Lund and Margaret A. Smart. 1979, 64 pp., $4.95. Activity Resources Company, Inc., P.O. Box 4875, Hayward, CA 94540.

This booklet, designed for use with the hand- held calculator, contains a variety of single-page exercises (with limited copy permission granted to the teacher).

Most of the exercises are concerned with rounding numbers - rounding to a given place or to estimate sums, differences, products, and quo- tients. There is no instruction for the rounding skills, but the calculator is used to verify the cor- rectness of the rounded numbers in a self-check- ing format. Exercises in identifying place values in whole and decimal numbers along with order- ing decimals are also given in a self-checking form.

Some of the exercises are designed to develop generalizations from patterns generated by easy calculations with the calculator. Thus, the con- cept of adding zeros (or moving the decimal point) in multiplication by powers of ten is gen- erated by a series of calculator multiplication ex- ercises; and divisibility rules for 2, 5, 10, 3, and 9 are generated by tables completed with the use of the calculator.

Some exercises use the calculator to do the arithmetic for common calculations - changing fractions to decimals, balancing a checkbook statement, calculating averages over a series of events, and calculating tax. A most useful set of exercises develops skill in locating the decimal point in decimal calculations.

FOCUS ОП Percent. Margaret A. Smart. 1978, 64 pp., $4.95. Activity Resources Com- pany, Inc., P.O. Box 4875, Hayward, С А 94540.

This collection of single-page sets of exercises will provide an excellent supplement to any text- book treatment of the difficult concept of percent and its calculations. Beginning with exercises in constructing ratios and proportions and pro- ceding through ratios with denominators of 100, the booklet treats almost every computational procedure and problem situation connected with percent.

One of the best features of this publication is its ample use of pictorial models for the concepts developed. Problem solving is not slighted and the text presents abundant practice exercises. It is attractively and interestingly published.

Despite its many good features, this text has its deficiencies. It is unfortunate that percents less than 1 and more than 99 are not treated here. This is where many students fail.

Funny Flashers - Addition and Sub- traction Facts. 1979, $5.95.

Funny Flashers - Multiplication Facts. 1979, $4.95. Skills W Stuff, 18011 Starmont Lane, Huntington Beach, CA 92649.

Shades of flash cards! These publications, with limited copy permission, are a variation on a fa-

miliar theme, the flash card. They might well be subtitled "The Complete Book of Basic Facts."

Each of the texts provides a pretest and post- test for basic combinations, a set of flash cards (by factor or addend family), worksheets for each of the tables, a record keeping system for memorization, a bulletin board idea, a letter to parents, a matrix (table), a bingo grid, certifi- cates of achievement (one for a single table, and one for the complete set of tables), and a game- board.

The theme, which is carried throughout the multiplication volume, is baseball and the flash cards are baseball replicas. The theme of the ad- dition/subtraction booklet is not clearly defined, for each of the flash cards, which are to be cut out of the heavy stock, is a "funny,*' anthropo- morphic character. Trees, flowers, ghosts, nuts, clouds, strawberries, boats, and peanuts are among the characters depicted.

Informal Geometry Activities Spirit Masters. / D. Gawronski, G. R. Prigge, and K. E. Vos. 1979, $16.75. J. Weston Walch, Publisher, Box 658, Portland, ME 04104.

Although this set of spirit master worksheets bears a title similar to the title of a set of visual masters, has a similar but not congruent glos- sary, has a similar but not congruent table of contents, and is written by the same authors, it is not correlated to the pages of the visual master set.

The set of spirit masters is misnamed. Rather than a set of activities, it is a collection of work- sheet exercises related to the concepts named in the worksheet titles. Each of the worksheets con- tain a collection of questions and directions re- lated to the concepts of the worksheet. Answers to the questions are included in the stapled teacher pages, which also contain a table of con- tents and a glossary.

The graphics of this set are well done, and a teacher might well find it a useful supplement to a standard text treatment.

Making Math Matter. Ida Mae Heard. 1979, x + 66pp., $8.95. Mafex Associates, Inc., 90 Cherry Street, Box 519, Johnstown, PA 15907.

What mathematics for the kindergarten? The au- thor found that the planned activities described in this photo-text resulted in greater achievement (on the Comprehensive Mathematical Inventory Test for a group of culturally deprived students of Dentón, Texas) than activities provided through an incidental and "unplanned" pro- gram.

Although this text is basically a description of a program that was the treatment in an experi- mental study, and the organization and content parallels that of a doctoral thesis, it can be a very useful tool for prospective teachers of young children.

The activities are grouped into sets - numbers, numeration; measurement; and geometry. The brief passages (activities) describe what children did, how they did it, and what they learned by doing it. The important insight to be gained from this book is the view of the wide range of

56 Arithmetic Teacher

This content downloaded from 62.122.76.48 on Thu, 12 Jun 2014 22:49:20 PMAll use subject to JSTOR Terms and Conditions