Matching Instruction to Children's Thinking about DivisionAuthor(s): Linnea WeilandSource: The Arithmetic Teacher, Vol. 33, No. 4 (December 1985), pp. 34-35Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194111 .

Accessed: 10/06/2014 15:32

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 188.72.96.26 on Tue, 10 Jun 2014 15:32:56 PMAll use subject to JSTOR Terms and Conditions

Matching Instruction to Children's Thinking

about Division By Linnea Weiland

Jane was sharing 792 pennies among 8 children. Although base-ten blocks and paper and pencil were available for use, she chose to argue without the support of concrete mate- rials.

gram. Children are not taught algo- rithms in their first three years but rather are encouraged to use their own arguments to solve problems (Madell and Stahl 1977, ix). This post- ponement of formal teaching of algo-

tens, and ones and distribute these groups. In the following excerpts, Jane used this argument. She began by arguing without the support of ma- terials and then represented part of the dividend in blocks to complete her

Jane, seven years old, had not been "taught" a division algorithm, but she was part of a study that exposed her to many partitive division problems. Over a period of a year, she was interviewed (as were the eighteen oth- er subjects) about such problems of increasing dividends and asked to solve them in any way she wished.

The subjects for the study attended an independent school in New York City, the Village Community School, which has a unique mathematics pro-

Linnea Weiland is currently working to develop microcomputer software at Children' s Comput- ing Workshop, a division of Children' s Televi- sion Workshop in New York, NY 10023. She was a mathematics and early childhood consul- tant at the Educational Improvement Center/ Northeast in New Jersey, a teacher at Kean College of New Jersey, and a teacher in the Chicago Public Schools.

rithms allows teachers and research- ers to gain insight into the arguments children create to solve problems. Sharing these arguments with teach- ers can provide the opportunity for planning instruction that can be matched to the way children think.

The children used five distinct argu- ments to solve the partitive division situations. The arguments used by the children were probably related to the type of division problems posed - par- titive problems rather than measure- ment problems - and to the concrete materials made available for their use - base-ten blocks.

The most frequently used argument was an analogue of the standard dis- tributive algorithm. All the subjects in the study used this argument at some time during the interviewing. A child using this approach would represent the dividend as bundles of hundreds,

arguments.

Interviewer: If 9 people are to share 927 pennies, how many will they each get?

Jane: 900 to each ... 27. Well, they'll get 900 each [note that she mistak- enly reports the amount used up when she distributes 100 to each of 9 people], and now I have to divide up the 27. My father's going to the hospital. He has an ulcer - four of them. It's rather major, but he isn't going to die or anything. He can still kiss me and stuff. [As she spoke, she took 2 tens and 7 ones.] I'm gonna trade these guys [the tens] in. [She threw back the 2 tens and took 20 ones.] Now I'll divide everything up into nines. Nine people and each of them already has 100. [Note that she corrected her previous error on her own without even noticing. She

34 Arithmetic Teacher

LET'5 SŒ. SO F/RSTLfrs) l'r rs Λ ,]| I'M Ш/NG /Γ/ 7ZUřň / 32 LEFT. N0WGIV£4 ' TAKE 90 FROM ̂^ - <-- Ji*) 4- JI I DON'T ΤΗ/ Ν Κ /TW/U . > TO EACH. NONfifFT ) вАСНЮО. s^lWUYQOim**'

^^ 14 WORK. TAKf SOF У ( TOLD YOU IT WORKED / - - 1 л-^^ ч_2^^^^ ш ΤΗε 72 то тне*-У V out/ so /τ 's 99. у

JANE IS TOLD ТО SHARE 792 PENNIES AMONG d CHILDREN*! /шШк, '· J)1«£m^(^t /~«^K

This content downloaded from 188.72.96.26 on Tue, 10 Jun 2014 15:32:56 PMAll use subject to JSTOR Terms and Conditions

passed out the units 1 by 1 so they each had 2.] How many should I have left? 18? No. I used 18. I should have 9 left. [She gave 1 more to each, simultaneously reporting the answer.] 103.

She was able clearly to verbalize her procedures for the standard distribu- tive argument. She kept track of the part she did mentally as she distribut- ed blocks and chatted about her fa- ther's health. The children usually ex- pressed this argument with blocks and solved it most concretely. Base-ten blocks seem to suggest this argument.

A second argument used by the children also involved distributivity, but in this case, the child was not sharing bundles of hundreds, tens, and ones, as Jane did in the standard distributive argument. To solve 792 + 3, one child immediately reported the answer as 99 by considering 792 as 800 - 8, dividing the 800 by 8, and then subtracting 1 (8/8) from the 100. This solution, as well as the one de- scribed in the very first excerpt, can be quicker in some cases and less tedious than the standard distributive argument.

A third argument, which involved halving, is illustrated next. Of course, only problems with divisors that are powers of 2 (2, 4, and 8) can be solved correctly with this argument. As an example of this argument, the child took half of a half to divide by 4, and half of a half of a half to divide by 8.

Interviewer: If 8 trucks are to deliver 328 crates, how many crates should be loaded on each truck?

Jane: How many crates? Interviewer: 328. Jane: What's an eighth of 100? Well,

half of 100 is 50, and a quarter of 100 is 25, and you can't divide 25 equally. 24 equally is O.K. 12 to everybody, and 1 left over from everybody. No, with 4 left over from the entire 100. Now do the same. 24 with 8 left over takes care of 200, and then 36 with 12 left over takes care of 300. What is 28 and 12? ... Hum, 40. O.K. 40 divided by 8 is 5. What was the answer to the hundreds? 30 what?

Interviewer: 36. Jane: That's easy. 41.

In this excerpt, Jane combined a dis- tributive argument with a halving ar- gument. She was able to follow her own argument carefully, remembering that the remainder from the 25 halved needed to be multiplied by 4 to take care of an entire 100.

A fourth argument used by the chil- dren was a measurement argument. In this subtractive approach, the child subtracted bunches or groups the size of the divisor and counted the number of bunches. In the excerpt that fol- lows, Jane converted "24 shared among 8" into a measurement prob- lem that asked how many groups of 8 in 24.

Interviewer: You have 24 crayons that have to be divided equally into 8 piles. How many will be in each pile?

Jane: This is going to be easy, but I might as well (take the blocks). [She took 2 tens and 4 ones.] 3. [She marked off 8 on each ten as she said the answer.] There are 4 left over on both of these (tens), and 4 and 4 makes 8. So it's 3.

Often this argument led children to make errors, since the switch back to the partitive viewpoint caused confu- sion. Such childen might report the answer as 8 (the number of piles) rather than 3 (the number in each pile). These errors illustrate the diffi- culty children have in reconciling measurement and partitive division.

Solving the same combination at another time, Jane replaced the mea- surement argument described previ- ously with the fifth argument used by the children in the study. This last argument relies on recognizing multi- plication and division as inverse oper- ations. Without a mastery of multipli- cation facts, this argument requires an ability to keep track of a complex situation. In the following excerpt, the child guessed at a quotient, multiplied to check her guess, and then adjusted the guess appropriately. Interviewer: 24 children form 8 equal

teams. How many children on a team?

Jane: [She tried 2, counting how many 2s on her fingers.] 2, 4, 6, 8, 10, 12, 14, 16. [She then used her fingers to count silently from 16 to 24 to see

how many were left.] 3. 3 times 8 is 24.

Jane, as well as the other subjects in the study, was able to carry out argu- ments in an efficient and confident way. Although the children encoun- tered difficulties as they argued to solve the problems, the errors they made had some basis in meaningful strategy. By recognizing the argument that a child is making, the teacher can intervene in a way that helps the child without interrupting the flow of the argument.

The children's extensive experi- ence in creating their own arguments may have made the eventual teaching of the standard distributive argument a fairly routine matter. The frequent use by the children of the standard distributive argument to solve these partitive division problems indicates that in the circumstances given, the selected argument naturally suggested itself. The use of other arguments can help the child understand the benefits of the standard argument. After ex- pressing the standard distributive ar- gument with blocks, the transition to representing the argument in numer- als was accomplished in four short teaching sessions (Madell and Wei- land 1976). In this situation the dis- tributive algorithm is well matched to what the children do on their own. This practice suggests the importance of giving children opportunities to cre- ate their own arguments and of allow- ing them the opportunity to verbalize these arguments in clinical interviews. If time is spent interviewing children individually, instruction can be matched to the way they think.

Bibliography Madell, Robert, and Elizabeth Larkin Stahl.

Picturing Numeration: From Models to Sym- bols. Palo Alto, Calif.: Creative Publications, 1977.

Madell, Robert, and Linnea Weiland. "A De- scription of the Difficulties Selected Eight Year Old Children Have in Learning the Distributive Algorithm for Division." Re- search Reporting Session, 54th Annual Meet- ing of the National Council of Teachers of Mathematics, Atlanta, Georgia, 1976. ERIC Document ED 120001.

Weiland, Linnea. "A Description of How Se- lected Seven Year Old Children Learn to Reason to Solve Partitive Division Prob- lems." Ph.D. dissertation, New York Uni- versity, 1977. Dissertation Abstracts Interna- tional 38A (April 1978): 5976. W

December 1985 35

This content downloaded from 188.72.96.26 on Tue, 10 Jun 2014 15:32:56 PMAll use subject to JSTOR Terms and Conditions