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Second-Grade Children^ Understanding of the Division Process

Sue Brown School of EducationUniversity of Houston-Clear LakeHouston, Texas 77058-1098

This study investigated children’s intuitive understanding ofdivision problems as reflected by their ability to solve verbaldivision problems. The hypothesis is that prior to formalinstruction many children can solve a variety of problemsinvolving both measurement and partitive division. This studydiffers from earlier studies in that it included problems inwhich: (a) the remainder is zero, (b) the remainder is non-zerobut it is not used in the solution, and (c) the remainder is non-zero and is used in the solution.

The impetus for this study was the following question askedof 13-year-old students on the Third National Assessment ofEducational Progress (1983).

An Army bus holds 36 soldiers. If 1128 soldiers arebeing bused to their training site, how many buses areneeded?

Only 25% of the 13-year-old students correctly solved theproblem. The most popular answer, 31.33 or 311/3,was givenby 29% of the students. Since division was so difficult for the13-year-old students, the question is posed, Is divisionappropriate for elementary school age children?

Research on children’s understanding of division is limited.Zweng (1964) studiedtheperformanceofsecond-gradechildrenon a 12-item division test. She considered both measurementdivision and partitive division. In measurement division, thedivisor indicates the size of a subunit that is measured againstan original unit to determine how many times it is contained inthe original unit. For example, "Susie has eight cookies. Sheis going to give two cookies to each friend. How many friendswill get cookies?" In partitive division, the divisor indicates thenumber of equal-sized subsets to be derived from a given set,while the quotient indicates the size of each subset. Forexample, "Billy has eight cookies. He is going to give his twofriends the same number of cookies. How many cookies willeach friend get?" Zweng found partitive division problems tobe more difficult for second graders. In analyzing the strategiesused by the students, Zweng found that twice as many partitiveproblems were solved by use ofa grouping strategy as opposedto a sharing strategy. That is. in solving the Billy problem, thechildren using a sharing strategy would give a cookie to childone. a cookie to child two, another cookie to child one, andwould continue this process until all cookies had been given

This research was supported in partby a grant from the MelroseThompson Foundation.

away. The children using a grouping strategy would count outa set oftwo cookies, a second set oftwo cookies, and a third setof two cookies.

Scott (1963) asked third-grade children to distinguishbetween word problems with measurement and partitiveinformation. Hefound that thesechildren wereable to distinguishbetween the two types of division. Scott also found that thechildren had "profound intuitive notions about division prior tothe initiation of formal instruction in division" (p. 747).A study conducted by Fischbein, Deri, Nillo, and Marino

(1985) with fifth-, seventh-, and ninth-grade students in Pisa,Italy, concluded that initially, in grade five, students were mostsuccessful in solving partitive division but by grade nine, "thequotative (measurement) model had become stable andinfluential" (p. 14).

Sato (1984) examined the performance of 5- to 8-year-oldchildren in Japan. He asked children to solve the followingmeasurement division problem, "If you have six marbles andgive two marbles to each person, how many people wouldreceive marbles?" (p. 52). The corresponding partitive divisionproblem was, "If you have six marbles and give the samenumber of marbles to two people, how many marbles wouldeach person receive?" Sato found the measurement divisionproblems to be easier for the children. Sato also noted that theyoungerthechildren, the greaterdiscrepancy occurringbetweenthe understanding of measurement and partitive division.

Vest (1986), working with third- and fourth-grade children,found the children could distinguish between measurementandpartitive division. In addition, when they wereasked to indicatetheir preference for a division type, the majority of the childrenindicated no preference for one above the other.

These studies are limited in that each focused on zero-remainder problems. This study examined the performance ofchildren on two additional types of division problems: (a) theremainder is not-zero but is not used in the solution and (b) theremainder is non-zero and is used in the solution.

Method

Subjects

The subjects for this study were 47 second-grade childrenrandomly selected from a middle-class suburban public school.Interviews with mathematics teachers prior to thestudyindicatedthe children had received no formal instruction in division.

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The Division Process93

Figure 1. Measurement and partitive division problems.

1. Measurement - 0 Remainder Paul wants to put 16 cups ofpunch on a table but he can only cany 2 cups at a time.How many trips does Paul have to make to carry all the cups ofpunch to the table?

Snoopy collected 13 bones. He wants to putthem in his doghousebutcan only carry3 bones at a time. How many trips does Snoopy have to make to get all the bonesinside his doghouse?

2. Measurement - Remainder Used

Susie wants to put 16 cokes back in their cartons. Each carton holds 6 cokes.How many cartons will Susie fill?

3. Measurement - Remainder Not Used

Casper the friendly ghost collected 15 treats on Halloween night. He wants to giveall his treats to his 5 ghost friends. Casper wants each friend to have the samenumber of treats. How many treats will each of his friends get?

4. Partitive - 0 Remainder

Eighteen children aregoing on a field trip to the zoo. Four mothers are going todrivethe children. How many children will the teacher put in each car so that all thechildren get to the zoo?

5. Partitive - Remainder Used

A clown has 15 balloons. He wants to give the same number of balloons to 6children. How many balloons can each child get if the clown gives each child thesame number of balloons?

6. Partitive - Remainder Not Used

Instruments

A 6-item division test was designed to analyze students’understanding of division. Three test items were measurementdivision and three were partitive. Experts were consulted toidentify those questions representative ofthe different situationsthat exist for division. The items in the test are given in FigureJ.

Procedures

Children were tested individually in a quiet area apart fromthe classroom. The order of the problems was randomized foreach child. The problems were read initially to each child andwere re-read in their entirety as often as necessary. The childalso had a written copy oftheproblem for reference. A set of20wooden cubes was available for use in working the problems.Children were encouraged to solve the problem any way theychose. Theresearcherobserved whether thechild was successfulin solving the problem and also noted the strategy used to solvethe problem. If the strategy was not obvious, the researcherasked the child to describe how the answer was obtained. Eachchild required approximately 15 minutes to complete all sixitems. Table 1 shows the percent ofcorrect solutions for all sixproblems.

Thenumberofproblems ofeach division typesolved correctlyand the strategy used in obtaining the solution are given inTables 2 and 3. Students were most successful in solving themeasurement problem with a remainder of 0. This problem was

Table 1

Percentage of Correct Solutions (N = 47)

RemainderType

0 RemainderRemainder Not UsedRemainder Used In Solution

Partitive

704743

Measurement

874555

Table 2

Number of Correct Measurement Responses and StrategyUsed

Problem TypeGroupingby MultiplesMental

0Remainder41Remainder NotUsed18Remainder Used in Solution25

correctly solved by 87% of the students. All students used theblocks and a grouping by multiple strategy. The studentsplaced their blocks in groups of two and counted groups toobtain the answer 8. Students were least successful in solvingthe measurement problems where the remainder was not used.Only43% of the students correctly solved this problem. Threeofthe students solved theproblem mentally while theremainderofthe students used blocks and agrouping ofmultiples strategy.

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The Division Process94

Table 3

Number of Correct Partitive Responses and Strategy Used

Problem TypeGrouping by MultiplesSharingMentalUndetermined

0Remainder20Remainder NotUsed9Remainder Used in Solution16

1384

Children were not as successful with the partitive problemsas they were with measurement problems but over one-third ofthe students solved the partitive problems correctly. Childrenused both the grouping by multiple strategy and the sharingstrategy in solving theproblems. Forexample, in the Casper theGhost problem, 20 children counted out one set of 5 blocks, asecond set of 5, and a third set of 5 to obtain the answer 3. Theother 13 children made five sets of one and added one by oneuntil all blocks were distributed.

The partitive problem in which the remainder is used in thesolution was themost difficult problem for the children to solve.Fourchildren obtained an answer of5.5.5,3. Using a groupingstrategy, these children counted out one set of 5, a second set of5, a third set of 5, and had 3 left. The other 14 children obtainedan answer of4,4,5,5. Three different techniques were used toobtain this answer. Six children counted one set of4, a secondset of4, a third set of 4, and a fourth set of4. The two left wereplaced in the third and fourth group. Four children counted outa set of 4, a second set of 4, a set of 5, and a second set of 5 toobtain the answer of 4,4,5, 5. Four children used the sharingstrategy. These children made 4 sets of 1 and continued todistribute the blocks one by one until all were placed in one of4 sets.

Discussion

The results of this study indicate that second graders showan understanding of division prior to instruction in division.Students are most successful in solving a measurementproblemin which the remainder is 0 and least successful in solvingpartitiveproblems in which the remainder is used in the solutionbut even this difficult problem type was solved correctly by43% of the second graders.

These results suggest that prior to any formal introduction todivision, children can solve some division problems. Childrenusually solved the division problems using either a grouping orsharing strategy. The strategies used by children in solving thepartitive problems are consistent with thosereported byZweng.The grouping strategy was used more frequently than thesharing strategy. Yet the grouping technique does not correctlymodel the action of the problem. In the Casper the Ghostproblem, the child that counted out one set of 5, a second set of5, and a third set of 5 obtained the correct answer usinginappropriate modeling.

The use of a single strategy, the grouping by multiplestrategy, in solving measurementdivision problemsand multiplestrategies^ grouping by multiples and sharing when solvingpartitive divisionproblems is consistentwith theresults reportedby Zweng. In measurement situations, the model ofrepeatedlyremoving equal-sized groups is the only approach appropriatefor solving this problem type. In partitive division, however,the children can use either the sharing strategy or the techniqueofremoving equal-sized groups to obtain a correct answer to theproblem.

An analysis ofthe data indicated error patterns for incorrectresponses. In the Snoopy problem, the most popular incorrectanswer was 4. The children who obtained this answer madefour piles of three bones and had one bone left. One studentjustified 4 as the answer by stating that on the last trip. Snoopycould squeeze in that extra bone. In the carton problem, theincorrect answer most frequently reported was 3. Studentsobtained 3 by counting out a set of six, a second set of six, anda set of four. These students did not verify that they hadanswered the question, "How many cartons will Susie fill?"The incorrect answer, 5, was reported for the Casper problem.The students obtaining 5 counted five groups of three andreported the answer as five. Themost frequent incorrect answerto the balloon problem was 3. Students typically counted sixpiles, then added one at a time until all the balloons were gone.These students had three groups of three and three groups oftwo. Thestudents did not look back at the question, "How manyballoons can each child get if the clown gives each child thesame number of balloons?"

The performance ofthe 7-year-old children in this study canbe compared with the performance exhibited by the 13-year-oldchildren in the Third National Assessment of EducationalProgress. The Army bus and the Snoopy problem are the sametype, measurement-remainder used. Only 25% ofthe 13-year-old students correctly solved the problem, while 55% of the 7-year-old students obtained the correct answer.

Why weresecond-gradechildren whohad notbeen formallyintroduced to division more successful than the NationalAssessment of Educational Progress 13-year-old students? Avariety of reasons seem possible. First the problems are notcomparable motivationally. The Snoopy problem is moremotivationally interesting to a second-grade student than theArmybus problem is to a 13-year-old. Second,one suspects theattitude with which children approached the problems varied

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The Division Process95

between the two groups. When the second-grade students wereasked at the conclusion of the six problems if they had likedworking the problems, they replied, "Yes, they’re fun." It isdoubtful the 13-year-old students would respond in a likemanner. Did the 13-year-old students’ knowledge of therealities of life influence their answer? In real life, would a buswith a capacity of 36 carry only 12 soldiers or would the 12soldiers be placed in the other 31 buses? It appears that the 13-year-old students’ natural intuitive skills may have beenhampered by the classroom instruction they received in divisionproblem solving. Since the most popular answer to the soldierproblem was 31.33 or 31 1/3, it is evident that many studentsread the problem, formally translated it to a division problem,divided to get an answer, and ignored the implications of theremainder for the problem.

Suggestions for Teachers

This study raises many questions about children, the divisionprocess, and classroom instruction in division problem solving.The current classroom instruction in division presents divisionas a computational skill and division wordproblem solving as anapplication of the computational skill. The results of thisresearch suggest that since children possess an intuitiveunderstanding of division prior to formal instruction in division,this intuitive understanding should be used as a basis forintroducing division to be followed by instruction incomputational division.

When presenting initial lessons on division, classroomteachers may want to survey their students. By asking forinformation such as favorite toys, favorite cartoons, favoritefoods, and favorite hobbies, the teacher can develop relevantdata to use in designing questions of interest to the students.Specific division questions using students’ names and theirinterests can be written. For example, an initial division lessonwould involve measurement, 0 remainder problem. Considerthe question, "If Susie brings eight Girl Scout cookies to schooland she gives two cookies to each student, how many studentswill get cookies?" This question is relevant since Susie is astudent in the teacher’s class, she is a Girl Scout, and earlier inthe year, she was selling Girl Scout cookies.

When working problems, pay particular attention to stepfour of problem solving, Looking Back. Many students whoobtained incorrectanswers to theproblems in this study modeledthe problem correctly but failed to look back at the originalquestion. For example, the balloon question asked, "How manyballoons can each child get iftheclown gives each child thesamenumber of balloons?" Some students modeled the problemcorrectly, obtaining three groups of three and three groups oftwo but they reported the answer 3. An important fact in theproblem states that each child will get the same number ofballoons. When answering the question, this fact was notaccounted for. If students have the opportunity to work a seriesofrelevant division problems, they will then have the foundation

on which to build an understanding ofcomputational division.

References

Fischbein. E., Dcri. M.. Nello. M., & Marino, M., (1985). Therole of implicit models in solving verbal problems inmultiplication and division. Journal for Research inMathematics Education, 76(1), 3-17.

National Assessment of Educational Progress. (1983). Thethird mathematics assessment of the national assessment ofeducational progress: Results, trends, and issues. Denver,CO: Education Commission of the States.

Sato, S. (1984, August). Measurement division is easier thanpartitive division. Paperpresented at theFifth InternationalCongress on Mathematical Education, Adelaide, Australia.

Scott, L. (1963). A study of teaching division through the useof two algorisms. School Science and Mathematics, 63,739-752.

Vest, F. (1986). A study of teaching the measurement andpartition concepts ofdivision. Focus on Learning Problemsin Mathematics, 5(2), 61-69.

Zweng,M. (1964). Division problems and the concept ofrate.Arithmetic Teacher, 77(8). 43-47.

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Volume 92(2), February 1992