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Volume 33

. January 1986

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ISSN 0004-136XThe ARITHMETIC TEACHER is an official journalof the National Council of Teachers of Mathematics. It is a forum for the exchange of ideas and asource of techniques for teaching mathematics ingrades kindergarten through eight. It presentsnew developments in curriculum, instruction,learning, and teacher education; interprets the resuits of research; and in general provides information on any aspect of the broad spectrum of math-ematics education appropriate for preservice andin-service teachers. The publications of the Na-tional Council of Teachers of Mathematics presenta variety of viewpoints. The views expressed orimplied in this publication, unless otherwise noted,should not be interpreted as official positions ofthe Council.EDITORIAL PANEL

John F. LeBlanc, Indiana University, Bloomington, IN47405, Chairman

Francis (Skip) Fennell, Western Maryland College, West-minster, MD 21157

Marsha W. Lilly, Aliet Independent School District, Alief,TX 77411

Edward Rathmell, University of Northern Iowa, CedarFalls, IA 50614

Charles S. Thompson, University of Louisville, Louisville,KY 40292

Joan E. Worth, University of Alberta, Edmonton, AB T6G2G5

Genevieve M. Knight, Coppin State College, Baltimore,MD 21216, Board Liaison

Harry B. Tunis, 1 906 Association Drive, Reston, VA22091, Managing Editor

STAFFJames D. Gates, Executive DirectorHarry B. Tunis, Managing Editor

Joan Armistead, Editorial CoordinatorAnn M. Butterfield, Editorial AssistantLynn Westenberg, Editorial Assistant

Charles R. Hucka, Director of Publication ServicesRowena G. Martelino, Advertising Manager

Robert Murphy, Computer Services ManagerJoseph R. Caravella, Director of Membership ServicesBetty C. Richardson, Director of Convention ServicesArt direction by William J. Kircher and Associates

Correspondence should be addressed to the Arithmetic Teacher,1906 Association Drive, Reston, VA 22091 . Manuscripts should betyped double-spaced throughout, with wide margins, on 81/2 X 11paper and with figures on separate sheets. No author identificationshould appear on the manuscript. Five copies are required.

Permission to photocopy material from the Arithmetic Teacher isgranted to classroom teachers for instructional use, to authors ofscholarly papers, and to librarians who wish to place a limitednumber of copies of articles on reserve. Permission must besought for commercial use of content from the journal when thematerial is quoted in advertising, articles are included in books ofreadings, or charges for copies are made or profit is intended. Useof material from the Arithmetic Teacher, other than those casesdescribed, should be brought to the attention of the National Coundl of Teachers of Mathematics. The Arithmetic Teacher will notparticipate in the unauthorized reproduction of any computerizedcourseware that bears an explicit or implicit copyright claim.

The Arithmetic Teacher (ISSN 0004—1 36X), an official journal ofthe National Council of Teachers of Mathematics, is publishedmonthly, September through May, at 1 906 Association Drive, Reston, VA 22901 . Dues for individual membership in the Council are$35.00, which includes one journal subscription. For an additional$13.00 an individual member can also receive the MathematicsTeacher, the other official journal of the Council. Rates for studentmembers are one-half the regular rates. For mailing outside theUnited States, add $5.00 for the first journal per membership and$2.50 for the second official journal ($1 .50 for Journal for Re-search in Mathematics Education). The institutional membershiprate for one journal is $40.00. Multicopy subscribers pay $13.00 foreach additional copy going to one address. For mailing outside theUnited States, add $5.00 for the first copy and $2.50 for each additional copy per membership. Airmail rates for institutional and mdi-vidual membership are available on request. Life and retired membership information is available from the NCTM HeadquartersOffice. Dues support the development, coordination, and deliveryof the Council’s services, including $13.00 for each ArithmeticTeacher and Mathematics Teacher subscription and $2.00 for anNCTM News Buiietin subscription.

Theindex for each volume appears in the May issue. The Arithmetic Teacher is indexed in the Current Index to Journals in Education and Education Index. Second-class postage paid at Reston,Virginia, and at additional mailing offices. POSTMASTER: Sendaddress changes to the Arithmetic Teacher, 1906 AssociationDrive, Reston, VA 22091 . Teiephone (703) 620—9840. TheSource: STJ228; CompuServe: 75445,1161. Copyright © 1985,The National Council of Teachers of Mathematics, Inc. Printed inthe U.S.A.

Cover story: Students in Joan Eschner’s fifth-grade class atWinchester Elementary School, West Seneca, New York, constructpolyhedra from sticks and clay. For a creative use of thesemodels, see “Build a City” (September 1985).

Readers are encouraged to submit nonreturnable color slides ofchildren involved in elementary school mathematics forconsideration for use on the cover.

Member of the—

Educational PressDREDS$ Association of America

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FEATURES

One Point of View: Let the Learning Disabled Learn 2Nancy Bley

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Mathematics for the Learning Disabled Child in the Regular Classroom 5Carol I. Sears

The Value of Informal Approaches to Mathematics Instruction and Remediation 14Arthur I. Baroody

The Low Achiever in Mathematics: Readings from the Arithmetic Teacher 20James H. Vance

The Number Namer: An Aid to Understanding Place Value 24Ian D. Beattie

How Our Decimal Money Began 30Robert G. Clason

Children’s Conceptual Understanding of Situations Involving Multiplication 34Ana Helvia Quintero

More Patterns with Square Numbers 40David J. Whitin

Y Is for Yacht Race: A Game of Angles 44John W. Butzow

Mathematics Attitudes of Elementary Education Majors 50Joanne Rossi Becker

DEPARTMENTSa,”

Research Report: The Process of Counting 29Marilyn N. Suydam

Problem Solving: Tips for Teachers 38Oscar Schaaf

Computer Corner 52Margie Mason

Reviewing and ViewingComputer Materials, Barbara Signer; New Books for Pupils, A. Dean Hendrickson;New Books for Teachers, Randall I. Charles; Etcetera, Carol Novillis Larson

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Readers’ Dialogue 4

From the File 58Ruth A. Meyer and James E. Riley

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NCTM Board of Directors: F. Joe Crosswhite, Ohio State University, Columbus, OH 43210, President. John A. Dossey,Illinois State University, Normal, IL 61 761 , President-elect. James D. Gates, NCTM, Reston, VA 22091 , Executive Director.Joan L. Akers, San Diego County Office of Education, San Diego, CA 921 1 1 . Albina S. Cannavacioio, Hamden Hall CountryDay School, Hamden, CT 0651 7. Philip C. Cox, Walled Lake Central High School, Walled Lake, Ml 48088. Marilyn L. Hala,South Dakota Department of Education, Pierre, SD 57501 . Louis G. Henkei, West Ottawa Middle School, Holland, Ml 49423.Donald W. Hight, Pittsburg State University, Pittsburg, KS 66762. Margaret J. Kenney, Boston College MathematicsInstitute, Chestnut Hill, MA 021 67. Genevieve M. Knight, Coppin State College, Baltimore, MD 21216. Katherine P. Layton,Beverly Hills High School, Beverly Hills, CA 90212. Mary M. Lindquist, Columbus College, Columbus, GA 31993. Louise M.Smith, Isle of Palms, SC 29451 . Ronald Wittner, Re9ina Board of Education, Regina, 5K 54R 358.Honorary President: John R. Ciark, Professor Emeritus, Columbia University, New York, NY 10027.

One-step word problems are usual-ly included in the mathematics curriculum so that students can learn toapply a new arithmetic operation. Re-cent studies (Carpenter, Hiebert, andMoser 1979; Quintero 1980), however, point out that the main difficultychildren have with these problems isunderstanding the situation describedin the problem and not in determiningthe correct arithmetic operation toapply. So in teaching word problems,we should place more importance onhelping children understand the situation described in the problem. To dothis , we must be aware of the differentsituations that can be modeled by anoperation as well as the different con-cepts needed to understand each situ-ation.

Carpenter, Hiebert, and Moser(1979), Greeno (1978), and Nesher(1979) have analyzed the situationsthat can be modeled by addition andsubtraction. They also have studiedwhich concepts and relationships in-

volved in these situations are difficultfor children to comprehend. This article presents a similar analysis for multiplication. It also suggests some activities that can help childrenunderstand these concepts and relationships.

MultiplicationWord Problems

Consider the following problems inwhich multiplication might be used:

1 . Juan has five pieces of candy.Mary has three times as many asJuan. How many pieces of candydoes Mary have?

2. A store sells five apples for a dollar. Joan bought three dollarsworth of apples. How many applesdid she buy?

4. Jane has five blouses and threeskirts of complimentary colors.How many outfits can she make’?

All these problems can be solved bymultiplying 5 X 3 . Yet each problem isdifficult for different reasons.

The situations described in the preceding problems exemplify four typesof structures found in multiplicationword problems (Schwartz 1976).

1 . Extensive quantity X Quantifier(ExQ)

Example:

5 pieces of candy x 3 times as many(candies)

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2. Ratio x Extensive quantity(RxE)

Example:

5 pieces of candy/bag X 3 bags

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Children’s ConceptualUnderstanding of

Situations Involving MultiplicationBy Ana Helvia Quintero

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At a birthday party there were fivepieces of candy in each surprisebag. Three surprise bags were given to each child. How many piecesof candy did each child receive?

Ana Helvia Quintero is an associate professorof mathematics at the University of PuertoRico, Rio Piedras Campus, PR 00931 . Besidesteaching she has been active doing research onmathematics learning.

34 Arithmetic Teacher

To help children understand the sit-uations modeled by multiplication, wefirst need to identify in each of thesesituations the concepts and relation-ships that cause children difficulty. Aprevious study (Quintero 1980) identifled these difficulties through children’s drawings. Children were askedto represent the situation described inproblems of the same structure as thefollowing:

A store sells boxes of candy con-taming twelve pieces each. Marybought four boxes of candy. Howmany pieces of candy did she get?

In this study, almost all the children’s drawings fell into the threecategories that follow:

2. Representations where they drewone or both of the elements described in the problem but not therelationship between them (43.6percent)

For example:

LILILE000000000000 LI El

3 . Representations of only the con-text of the problem (10 percent)

For example:

Mary in the store.

The types of drawings that the children produced gave us insight into thelevels of understanding that they hadof the problem. The children’s draw-ings showed that there were—

. those who understood the problemand visualized the operations required for its solution;

. those who had difficulty with thenotion ‘ ‘pieces of candy in eachbox,” and

. those who did not have a clear ideaof what task they had been asked todo or of which of the facts wereneeded to do it.

Once we have identified the typesof difficulties that children experiencein solving a problem, we can developactivities and materials designed fordifferent levels of understanding thatwill help children to surmount thesedifficulties. For example, the majorityof the children who had difficultieswith the previous problem did notunderstand the notion of ‘ ‘pieces ofcandy in each box. ‘ ‘ This notion in-volves some understanding of ratio—that of a sample in a homogeneousgroup ; that is , for understanding thenotion of pieces of candy in each box,you have to know that if a store sellstwelve pieces of candy in each box,then you will find twelve pieces ofcandy in any of the boxes you buy.The following set of activities can helpto develop this notion and children’sunderstanding of the situation described in problems of the type “6children per teacher x 5 teachers”(RxE).

Activity 1 . Understanding theconcept ofa sample in ahomogeneous group

Pairs of cards are prepared. In eachpair, one card presents a probleminvolving the notion of a sample in ahomogeneous group, whereas the other card presents a similar situation notinvolving the notion of a sample. Forexample:

Each card is presented individuallyto the child, and the questions on thecard are discussed. After discussingthe cards separately, the pair of cardsis presented to the child, and the similarities and differences are discussed.

Activity 2. Developing the ability torepresent correctly the situationdescribed in the problem

The following materials are requiredfor this activity:

(a) A set of index cards with a wordproblem written on each one

(5) Sets of four index cards for eachword problem with a differentdrawing on each one. One draw-ing should be the correct representation of the problem, whereasthe others should be incorrect rep-resentations. These can be basedon the correct and incorrect draw-ings the children produced whentheir difficulties were being diagnosed.

3. Ratio X Ratio (R x R)

Example:5 pieces of candy/bag x 3 bags/child

or

4. Extensive quantity X Extensivequantity (E x F)

Example:5 blouses x 3 skirts

Rosalva went to a store. The clerktold her, “The bags of candy costfifty cents, and twelve pieces ofcandy are in each bag. “ If she buysone bag of candy, how manypieces of candy will she get?

Later, Mary goes to the store. Shealso buys a bag of candy. Howmany pieces of candy will Maryget? —JJoe is preparing surprise bags forhis birthday party. He puts one ortwo toys and a bunch of candy ineach bag.

Louise got a set of jacks and sevenpieces of candy. Can you tell howmany pieces of candy Janice got?

1 . Correct representations (36 per-cent)

For example:

January 1986 35

Students are to select the picture that corresponds to the problem.

Word Problem Representations

Fig. 2 Students are to select the word problem that corresponds to the picture.

Joan in the store

Representation Word Problems

Procedure. This activity is present-ed as a game. The teacher (or an olderstudent) presents the word problemtogether with the drawings to thechild. The child chooses the drawingthat he or she thinks is the correctrepresentation and explains why it isthe correct one. A correct first answerearns ten points. For an incorrectresponse, the teacher explains whythe drawing is incorrect and gives thechild a second opportunity to choose.A correct second answer earns fivepoints.

This procedure is repeated, withthree points earned for a correct response on the third selection. If thechild makes the wrong choice this

time, the teacher again explains whyhe or she is wrong and then why theremaining drawing is the correct rep-resentation of the word problem.

Activity 3

This activity is the reverse of activity2. A drawing is given to a student whomust choose which one of four wordproblems describes the situation itrepresents. See figure 2.

Activity 4

This activity uses cards made to looklike dominoes. Word problems, pictorial , or numerical expressions are on the separate halvesof the cards. Students are to find theircounterparts and match them up. For

example, a correct sequence is shownin figure 3 . The word problems in thisgame can involve a variety of simpleaddition and multiplication problems.

Conclusion

These activities are suggested as away to help children understand thenotion of a sample in a homogeneousgroup, a concept essential to under-standing R x E problems. This con-cept is not common to all multiplication problems. The E x Q and E x Eproblems, for example, do not involvethis notion. To solve E x E problems,students must understand the conceptof combination (as in the problemwith five blouses and three skirts).

Fig. 1

A store sells 6 apples for one dollar.Joan bought 4 dollars worth of apples.How many apples did she buy?

‘ 6

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Six students and three teachers arein an art class. How many people arethere?

There are 6 students for eachteacher in an art class. There are 3teachers. How many students arethere?

Eighteen people are in an art class.Three of them are teachers. Howmany students are there?

See figure 1 for an example.

There are 18 students for eachteacher in an art class. There are 3teachers. How many students arethere?

36 Arithmetic Teacher

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Fig. 3 Mathematical dominoes

A store sells 5pieces of candy ineach box. Mary

3 + 4 bought 4 boxes ofcandy. How manypieces of candydoes she have?

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This concept is also difficult for children, but the difficulty that they havewith combinations is different fromthe one that they have with a samplein a homogeneous group.

Children’ s drawings illustrate thedifficulties that they have with theconcept of combinations. A set ofactivities can be developed to helpchildren with this concept. Activities2, 3 , and 4 can be adapted to teach thisconcept by using E X E problems andtheir representations.

This article underscores the importance of children’s understanding thesituation in a problem so that they cansolve it. They need to be aware of thedifferent types of situations modçledby an operation and the conceptsneeded to understand each situation.

3x3 2+6

Bibliography

Carpenter, T. P. , J. Hiebert, and J. M. Moser.The Effect of Problem Structure on First-Graders’ Initial Solution Processes for Simipie Addition and Subtraction Problems.Technical report no. 516. Madison, Wis.:Research and Development Center for mdi-vidualized Schooling, 1979.

Greeno, J. G. ‘ ‘Some Examples of CognitiveTask Analysis with Instructional Implications. ‘ ‘ Paper presented at ONR/NPRDCConference, March 1978, California.

Nesher, P. “Levels ofDescription in the Analysis of Addition and Subtraction. ‘ ‘ Mimeographed. Haifa University, 1979.

Quintero, A. H. “The Role of Semantic Under-standing in Solving Multiplication WordProblems. “ Unpublished doctoral dissertation, M.I.T. , 1980.

Schwartz, I. L. “Semantic Aspects of Quantity. ‘ ‘ Mimeographed. Cambridge, Mass.:M.I.T. , Division for Study and Research inEducation, 1976.

. to enforce their mathematical con-cepts and

. to expose them to the vitality ofethnic and cultural diversity

Lead them to discover. The Hopi Rain Cloud. Korean Ko-no. Japanese Origami. Polynesian Lu-lu. Norwegian Julekurv. The Aztec Calendar, and many more.

Materials are conveniently reproducible.8’/2 X 11 inches, $0 pp. , #327, $5

NATIONAL COUNCIL OFcic TEACHERS OF MATHEMATICS

‘I 906 Association DriveReston, Virginia 22091

See the NCTM Materials Order form in theback of this issue.

MULTICULTURALMATHEMATICS MATERIALS

By Marina C. KrauseA wonderful collection of games andactivities from different parts of the world.Embark on multicultural trips with yourelementary school students

Mary had 2 piecesof candy. Janetgave her 6 more.How many piecesof candy does shehave?

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Statement of ownership, management and circulation (Required by 39 U.S.C. 3685(. 1. Title of publication, Arithmetic Teacher. A. Publication Number 0004136X. 2. Dateof filing, 1 October 1985. 3. frequency ofissue, Monthly—September through May. A. No. of issues published an-nually, nine. B. Annual subscription price, $40.00. 4. Location of known office of publication, 1906 AssociationDrive, Reston, Virginia 22091—1593. 5. Location of theheadquarters or general business offices of the publishers,same as #4. 6. Names and complete addresses of publisher,editor, and managing editor. Publisher, National Councilof Teachers of Mathematics, 1906 Association Drive, Reston, Virginia 22091. Editor, none. Managing Editor,Harry B. Tunis, 1906 Association Drive, Reston, Virginia22091. 7. Owner, National Council of Teachers of Math-ematics, 1906 Association Drive, Reston, VA 22091. 8.Known bondholders, mortgagees, and other security hold-ers owning or holding 1 percent or more of total amountof bonds, mortgages or other securities, none. 9. The purpose, function, and nonprofit status of this organizationand exempt status for federal income tax purposes havenot changed during preceding 12 months. 10. Extent andnature of circulation. Average no. copies each issue duringpreceding 12 months. A. Total no. copies printed, 36 333.Bi. Paid circulation, sales through dealers and carriers,street vendors and counter sales, none. B2. Paid circulation, mail subscriptions, 30 506. C. Total paid circulation,30 506. D. free distribution by mail, carrier or othermeans, samples, complimentary, and other free copies, 65.E. Total distribution, 30 571. fI. Copies not distributed,office use, left over, unaccounted, spoiled after printing,S 762. f2. Copies not distributed, returns from newsagents, none. G. Total, 36 333. Actual no. copies of sin8leissue published nearest to filing date. A. Total no. copiesprinted, 38 360. Bi. Paid circulation, sales through dealersand carriers, street vendors and counter sales, none. B2.Paid circulation, mail subscriptions, 30 603. C. Total paidcirculation, 30 603. D. free distribution by mail, carrieror other means, samples, complimentary, and other freecopies, 75. E. Total distribution, 30 678. Fl. Copies notdistributed, office use, left over, unaccounted, spoiled afterprinting, 7 682. f2. Copies not distributed, returns fromnews agents, none. G. Total, 38 360. 11. I certify that thestatements made by me above are correct and complete.James D. Gates, Business Manager.

January 198637