ED'S rEGYFORSOLVINGDIVISIONPROBLEMS

Guershon Harel and Meriyn Behr

P rior to formal instruction, chil-dren possess considerable un-derstanding of arithmetic and the

strategies for solving basic arithmeticproblems. Carpenterand Moser (1982),for example,have foundthatmany chil-dren can successfully solvewordproblems that involve additionand subtractionbeforetheyre-ceive any instruction in suchproblems.](ouba(1989)identi-fied fIfty-sixappropriatestrate-gies and several inappropriate.strategies used by first-, second-, andthird-gradepupils to solvemultiplicationand division

problems.](nowledgecan be con-

structed by pupils in various waysand at various times. Probablymore often than we realize it,pupils have some conceptions jabout many topics on which'they have not had school in-struction. For example, before apupil receives any formal instruc-tion on division, he or she has had expe-rience with sharing. A sharing problemmight be solved by a one-by-one distribu-tion of the objects to be shared or bymultiple di~tribution (see, e.g., ](ouba[1989]). Pupils also develop sophisti-cated strategies that go beyond modelingan intuitive or informal interpretation ofan operation (Resnick 1986). This article

Guershon Harel teadl£s at Purcbu! University, West Lafay-

ette, IN 47907. His research interest is in advanced

mathematical thinldng and elementary mathematics, in-

cluding children's learning of the concepts of multiplica-

tion, division, rational number, and proportion. Merlyn

Behr teaches at Northern Illinois University, DeKalb, IL

60115. His interests include research on middle school

children's learning of mathematics---<?specially multipli-

cation, division, rational-number concepts, and opera-

tors-and their development of proportional reasoning.

38

discusses one such strategy for solvingdivision problems.

Ed is a second grader who was reportedby his teacher to be slightlyabove averagein mathematics and average in othersubjects. By the time we interviewed him,

Ed's formal instructionhad includedaddition and subtraction without

regrouping. ill addition, Ed wasinformally exposed to the basicnotion of multiplication as re-peated addition and to the mean-

ing of division as sharing equally,but he was never taught any division

strategy. Edwas orallygiven theproblem''How much is

forty-two di-vided by

seven?" His answer was, "Fortydivided by ten is four; three andthree and three and three are

twelve; twelve plus two is four-teen; fourteen divided by two isseven; two plus four is six." Ed

gave this same solution and an-swer each time he was asked to

explain his solution. When asked wherethe "3" came from, he responded, "Tomake the ten a seven [apparentlymeaning'subtract three from ten to yield seven']."

To make sure that Ed had a generalstrategy and didn't get the answer "byaccident," we orally gave him anotherquestion similar to the first one, "Howmuch is seventy-two divided by eight?"Interestingly, the structure of his solutionwas exactly the same as for the first one:"Seventy divided by ten is seven; seventimes two is fourteen; fourteen plus two issixteen; sixteen divided by two is eight;two plus seven is nine. The answer isnine."

It was very difficult to get more infor-mation from Ed. Again. when asked about

where the "2" came from, he responded,"The two from the ten."

Ed was orally given a third problem:"How much is fifty-six divided by eight?"He was urged to go slowly and explainwhat he was doing. His solution had thesame structure as the other two, "Fiftydivided by ten is five; five times two isten; ten plus six is sixteen; sixteen dividedby two is eight; two plus five is seven.Theanswer is seven." This time we were able

to capture more of his solution process:''The twos come from the ten. They haveto give back two, so I have two and twoand two and two and two, which is ten.And the six that is left, that makes six-teen." When asked who has to give backtwo, he answered, "Those who got ten,they need to get only eight."

Amazed at what Ed was doing, wewere still not sure that the solution strat-

egy was generally valid. We examinedthe validity of the strategy by lookingcarefully into Ed's responses,.from whichwe were able to identify the conceptualbase for this solution process. Let us lookat the solution of 42 divided by 7 in termsof drawings to represent what might bethe pupil's mental model and show howthe solution might be demonstrated withmanipulative objects.

Ed's solutionuses a measurement rather

than partitive interpretation of division;that is, forty-two objects are put intogroups of seven, and the answer to thedivision is then the number of groups. Hissolution starts with forty-two objects. illsome manner he mentally distributed fortyof these objects into four sets of ten,leaving two objects not distributed, asshown in figure 10. Remembering that hewas to make groups of only seven, hetakes back three objects from each groupof ten (''three and three and three and threeare twelve"). These twelve elements andthe two left from the forty-two ("twelveplus two is fourteen'') must be distributed

ARITHMETICTEACHER

(a) Forty-two objects form sets often, leaving two objects undis-tributed.

(b) Three objects are taken fromeach set of ten and put withthe two left over.

(c) The fourteen elements on theright are distributed into twogroups of seven.

NOVEMBER1991

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(see fig. 1b). These fourteen elementsweredistributedintotwogroupsof sevenobjects (''fomteen dividedby two is seven,"which apparently means that because1472 = 7, then1477 = 2). Combiningthe four groups of seven with the twogroups of seven gives the six groups ofseven ("two plus four is six. The answeris six"). Seefigur~ 1c.

Conclusions and

ImplicationsTeachers put forth a great deal of effort tounderstand and evaluate pupils' knowl-edge, especially after instruction. Butpupils' knowledge prior to instruction isalso important (Mack 1988). Good teach-ing should build on the pupil's existingknowledge, which forms the basis forbuilding new knowledge (Romberg andCarpenter 1986).

It is important for the teacher to recog-nize a pupil's attempts at a creative solu-tion of a problem and to praise suchinventiveness. Even when the inventiondoes not lead to correct answers, often itoriginates in some intelligent reasoningprocess. A thorough discussion of pupils'inventivenessin severaldomains of mathe-matics is given by Resnick (1987).

Whether the invention is correct or

incorrect, the teacher should capita1izeonthe knowledge constructed by the pupilswhen leading them to learn new know l-edge. The existing knowledge should berecognized,notignored.Resnick(1987)observes that pupils have the basic abilityto decompose and recompose quantitiesto represent such numerical operations assubtraction with regrouping in division.The ability to compose and decomposequantities is evident in the strategy that Edused to solve division problems. Anotherobservation to be made about pupils'manipulation of representations of quan-tity is the flexibility with which theycompose, decompose, and recomposethese quantities.

In addition to such flexibility, the divi-sion strategy described in the discussionhas three additional important features.First, the strategy can be represented withmanipulative aids. Second, as was dem-onstrated earlier, the strategy is appli-cable to a general class of division prob-lems. Finally, the solution of some prob-lems by this strategy might involve recur-

39

sive thinking. For example, the solutionoftheproblem136 + 8,showninthenextsection, includes the subproblem 32+8, which can in turn be solved by thesame strategy.

Examining Ed's Strategyin Algebraic TennsEd's strategy can be applied to solvedivision problems for which the divisorand the dividend are whole numbers, withthe restriction that the divisor must be less

than or equal to the dividend. Unfortu-nately, this strategy begins to exceedone's mental-arithmetic capabilitieswhentwo- and three-digit divisors are used, butthen a calculator can be used if practice isdesired or if a curious pupil just wants totry it out with big numbers.

We encoded Ed's solution in algebraicterms. Given any two-digit number 10x+y (e.g., 42, 4 x 10+ 2) and a one-digitnumber z (e.g., 7), it can be seen from Ed'sresponses that his solution process fordividing 10x + y by z is as follows:

(lOx + 10) + {[(l0 - z)x + y] + z},

that is,

(40 + 10) + {[(to - 7)4 + 2] + 7}.

A simplification of this expression showsthat it equals (10x + y) +'z, which provesthat the pupil's strategy is mathematicallyvalid for the whole class of division prob-lems in which the dividend is a two-digitnumber and the divisor is a one-digitnumber. However, we don't know if Edwould use this strategy to solve problemsin which the divisor is small (e.g.,42+2or 42 + 4). We give some examples thatuse bigger numbers to illustrate step bystep the generalizability of Ed's process.

Example 1: 136+ 8

130 + 10 = 13

10-8=2

13 x 2 = 26

136 - 130 = 6

26 + 6 = 32

32 + 8 = 4

13 + 4 = 17

The answer is 17.

Example 2: 192 + 24

180 + 30 = 630 - 24 = 6

40

6 x 6 = 36192- 180 = 1236 + 12 = 4848 + 24 = 26+2=8

The answer is 8.

Example 3: 930 + 186

760 + 190 = 4190 - 186 = 4

4 x 4 = 16930 - 760 = 170170 + 16 = 186186 + 186 = I

4+1=5The answeris 5.

To extendthis strategyfromone-digitdivisors to multidigitdivisors,the pupilmust first find the least multiple of 10thatis greater than the divisor then decomposethe dividend into two parts of which oneis a multiple of the least multiple of 10greater than the divisor. From there on,the process is exactly the same as that

used'by Ed to solve divisions with single-digit divisors.

References

Carpenter, Thomas P., and James M. Moser. "The Devel-

opment of Addition and Sublraction Problem-solving

Skills." In Addition and Subtraction: A Cognitive Per-

spective, edited by Thomas P. Carpenter, James M.

Moser, and Thomas A. Romberg, 9-24. New York:

Academic Press, 1982.

Kouba, Vicky L. "Children's Solution Strategies for

Equivalent Set Multiplication and Division Word

Problems. " Journal for Research in MathematicsEducation 20 (March'1989):147-58.

'Mack, Nancy K. ''Learning Fractions with Understanding:

Building upon Infonnal Knowledge." Paper presented

at the meeting of the American Educational ResearchAssociation, New Orleans, 1988.

Resnick, Lauren B. ''ConstructiIig Knowledge in School."

In Development and Learning: Conflict or Congru-

ence? edited by Lynn S. Liben, 19-50. Hillsdale, NJ.:Lawrence Erlbaum Associates, 1987.

-. "The Development of Mathematical Intuition."

In Perspectives on Intellectual Development: The

Minnesota Symposium on Child Psychology, voL 19,

edited by Marion Perlmutter, 154-94. Hillsdale, N.J.:Lawrence Erlbaum Associates, 1986.

Romberg, Thomas A., and Thomas P. Carpenter. "Re:

search on Teaching and Learning Mathematics: Two

Disciplines of Scientific Inquiry." I n Handbook of

Research on Teaching, 3d ed., edited by Merlin C.

Wittrock, 850-73. New York: Macmillan Publishing

Co., 1986. .

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