Making division meaningful and logical

Making division meaningful and logicalAuthor(s): LELON R. CAPPSSource: The Arithmetic Teacher, Vol. 9, No. 4 (APRIL 1962), pp. 198-202Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186618 .Accessed: 17/06/2014 20:11

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 [email protected].

.

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 91.229.248.154 on Tue, 17 Jun 2014 20:11:01 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctmhttp://www.jstor.org/stable/41186618?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

Making division meaningful and logical

LELON R. CAPPS University, California Professor Capps is a member of the school of education, University of California at Santa Barbara, University, California.

A. considerable quantity of research is available to indicate that meaningful teaching results in greater insight and, consequently, superior achievement in the mastery of a process or skill. Unfortu- nately, present practices lag behind in im- plementing much of what is known about the psychology of learning.

In arithmetic, the trend is toward in- creased emphasis on the understanding of the arithmetical process, minimizing the teaching of rules as the only means of solving a particular algorism. Current methods divide themselves generally into two pro- cedures: (1) teaching the understandings and aiding the children to generalize and formulate the rules for solving the algorism, referred to as the meaningful approach; and (2) teaching the rules as a series of steps to follow in order to derive the correct answer, with little or no atten- tion devoted to developing understandings. The latter approach is commonly called the mechanical method. Regardless of which method is used, some of the arithmetical processes are more difficult to teach to children than are others.

It is generally agreed that the most complex of the four fundamental processes is division. The general trend in the teaching of division is to introduce it as a series of subtractions. This shows its inverse rela- tionship to multiplication as a series of ad- ditions.

Also, this can be easily illustrated by solving a number of simple problems using the subtractive principle and then pro- ceeding to more complex problems:

24/ 72 -24 1

48 -24 1

24 -24 J_

0 3 (Answer) This method develops the basic under-

standings, but problems still present themselves in arriving at the most efficient manner of solving the more complex division algorisms. Also, there is a greater chance of making errors since several mul- tiplications and subtractions may be done before the answer is found. The following example will serve to clarify this:

34/ 952 -340 10

612 -340 10

272 -170 5

102 -102 _3

0 28 (1) Another method would make the sub-

tractions in the following manner:

28

34/ 952 -68

272 -272

00 (2) Arguments in favor of method (1) over

method (2) generally center about prob-

198 The Arithmetic Teacher


http://www.jstor.org/page/info/about/policies/terms.jsp

lems of estimation, placing the quotient figure, and omitting digits from the quotient.

It is the purpose of the following illus- trative lesson to suggest a way of teaching division meaningfully in the most efficient way. A necessary prerequisite for the children is an understanding of division as a series of subtractions and also of the fundamental structure of our number system. The lesson is designed to capitalize on previous understandings and experiences of the children in developing the most efficient method of dividing one number by another.

The group used in testing the lessons consisted of 25 fifth-grade students en- rolled in the elementary school at the Uni- versity of Minnesota. The average IQ was approximately 120 with three children who fell below 110. AU IQ measures are based on the WISC or Binet.

Previous to introducing the lesson, some time was spent in various ways of express- ing numbers:

1987 one thousand, nine hundreds, eight

tens, seven ones nineteen hundreds, eight tens, seven

ones one hundred ninety-eight tens, seven

ones nineteen hundred eighty-seven ones The example 32/9792 was then placed

on the board with the question as to how this example might be solved.

The children, having been exposed to the subtractive idea, immediately sug- gested subtracting. The solution was ar- rived at in the following manner:

32/ 9792 -3200 100

6592 -6400 200

192 -192 _6

00 306 The question was then asked by the

teacher, "Where might we meet such a problem as this in life?" From the puzzled look on the faces of the children, it was apparent they did not understand the question, so the teacher gave the directive, "Someone make a verbal problem from the algorism."

One child's response was, "How many thirty-two's are there in nine thousand seven hundred ninety-two?" The lesson continued as follows:

T. Yes, but this is really just putting the symbols into words. Les make a problem such as John has three cents and Jim has five cents. How much do they have together?

C. A man died leaving an estate of $9792. He wanted to divide the money equally among 32 charities. How much would each charity get? (Written on the board)

T. What facts do we know in this problem?

C. The amount the man had to divide. T. What other facts do we know? C. The number of charities the man

wanted to donate to. T. What could we say instead of the

number of charities? C. The number of groups he wants. T. How do we solve this problem? C. Divide 9792 by 32. T. What is this number called in divi-

sion (pointing to 32)? It has a special name.

C. It is called the divisor. T. (Placing the problem on the board

as follows: 32/9792.) Now, what do we want to find when we know the number of groups we want to make and the amount which we want to divide?

C. How many in each group? T. What word do we use to express the

number in each group? C. Size of the group. T. Our problem looks like this -

(known) number of groups of group (unknown)

/whole (known)

April 1962 199



when we know the number of groups and the whole to be divided and want to find the size of the group we are using. What did you actually do when you subtracted 3200 as you worked the problem using the series of subtractions? (Children do not reply.) Did you give the 3200 dollars to one charity?

No, we divided it among the thirty- two charities.

T. How much did each one get? C. One hundred dollars. T. Could we have given each one

more than that to begin with? C. Yes. T. Could we have given each one a

thousand dollars? No. T. Why couldn't we? How many

thousand dollars does the man have? C. He only has nine. T. How many thousand dollars would

he need before he could give each charity a thousand dollars?

C. Thirty-two. T. How many hundred dollar bills

could he give each charity to begin with? (Children are again unable to answer.) How many hundred dollars did the man have?

C. Ninety-seven. T. How many hundred dollars would

he use each time he gave one hundred dollars to each charity?

C. Thirty-two. T. How many would he use if he gave

each charity two hundred dollars? C. Two times thirty-two is sixty-four. T. How many hundreds would he use

if he gave three hundred dollars to each charity?

C. Ninety-six. T. How did you get ninety-six? C. Multiplied three times thirty-two. T. How can we show in our answer

that each charity has received three hundred dollars? Remember, we only want to use the number three to show this.

Put the three above the seven because that is the hundred's place.

T. How much has been used if he gives > each charity three hundred dollars?

C. Ninety-six hundred dollars. T. What must we do to show that we

have used this amount? Subtract it. T. How will we write the ninety-six

hundred dollars? (A pupil comes to the board and writes 9600 under 9792.) Do we really need the two zeros?

C. No, because if we just write 96 in the hundreds' place it will mean 96 hundreds. We can just erase the two zeros.

T. Then our problem looks like this : 3

32/ 9792 -96

Now, how many hundred dollars did he have left?

C. One. T. How did you get that? Subtracted ninety-six from ninety-

seven. T. Could you give each charity

another hundred dollars? C. No, you would need 32 and you

only have one left. T. What would you do if you had

more than thirty-two hundred dollars left? C. Put another one in each group and

put a four where the three is. T. Where do I record the one hundred

dollars he has left? C. Under the six in the hundreds'

place. T. Our problem looks like this now:

3 32/ 9792

-96 1

After you have divided the hundred dollars, what is the next logical step?

. divide the ten dollars. T. How many ten dollars has he? . has nine. T. Yes, in the tens' place. What about

the hundred dollars in the hundreds' place? Can we just forget about this?




. Oh, I know! Make it into tens and you will have nineteen tens.

T. What is the simple way to do this in our problem?

C. Write the nine after the one. T. Yes. Sometimes people say, "Bring

down the next number." What does this 19 mean?

C. Nineteen tens. T. Can you give each of the thirty-

two charities ten dollars? C. No. He has only nineteen ten dol-

lars, but he could change those to ones. T. We are a little ahead of ourselves.

How are we going to show that we couldn't give each charity ten dollars?

C. Put a zero above the 9 in the tens' place.

T. Fine. Now our problem looks like this:

30 32/ 9792

-96_ 19

So far, how much has each charity received?

Three hundred dollars. T. How do we have it written in our

answer? C. Thirty tens. T. How else could we say it using

hundreds? C. Three hundreds and no tens. T. Let's go on with the suggestion of

changing the nineteen tens to ones now. How many ones do we have?

Oh! Nineteen tens is one hundred ninety ones and you have two ones in the ones' place so you have a hundred ninety- two ones.

T. Now we have our problem written this way:

30 32/ 9792

-96 192

All we had to do was write the two after the nineteen or "bring down the two."

Now, how many one dollar bills can we give each charity?

Six. T. How many will he use if he gives

six to each charity? Take six times thirty-two. T. How many is this? C. Six times thirty is one hundred

eighty and six times two is twelve. That makes one hundred ninety-two.

T. We write the six in the ones' place and subtract the one hundred ninety-two ones. Does he have any ones left to divide?

C. No. We're finished. Before the assignment was given, a set

of steps was developed which the children could use as a guide. They were: 1 Look at the number of groups you want

to make. (Divisor) 2 Look at the whole you are going to di-

vide and see which is the largest amount you can place in each group to begin with. (Thousands, hundreds, tens, etc.)

3 About how many of these can you put in each group?

4 Find how many you have used by multi- plying your estimate times the number of groups.

5 Subtract the number you have used from the number in the whole.

6 "Bring down" the next number and re- peat the process.

Problems were then assigned and individual help was given.

The following day another problem was worked in a similar manner and more examples were solved. One of the most fre- quent difficulties was making the correct estimate; therefore, some time was spent in reviewing and explaining estimation.

Once children showed an adequate understanding of the process, the measurement situations were introduced in a similar manner, whereby the number of groups was to be determined.

The evaluation of this approach is sub- jective, but several points seem worthy of further consideration.

It was apparent in correcting the papers

April 1962 201



that the children had little difficulty in understanding the zero difficulty. Though in some cases they did not place the zero in the quotient, they left the appropriate place vacant. It was necessary to develop the habit of placing the zero in the quotient figure to make the number read cor- rectly. There were no other difficulties in placing the quotient figure or partial products in their respective place value.

About one-third of the group showed an understanding of the "partition" and "measurement" ideas at the completion of the time spent on using this approach to division.

The children were very capable of stat- ing verbal problems from the algorism once they had an opportunity and some experience in doing so. In most instances, the children stated the problem as the partition situation at first. They then re- worded the problem for a measurement situation.

In the estimation of the writer, the most valuable contribution of the approach was the opportunity provided the children to express a variety of thinking patterns in solving the problem. Though many ideas for solving the problem were erroneous, the children seemed to benefit from apply- ing them and seeing why they would not work. These kinds of opportunities seem to be plentiful in this approach to teaching division.

In summary, we must recognize that the preceding method in approaching the more complex division problems was used with an atypical group. However, there is no reason why it would not be applicable to a more normal sample of the popula- tion, providing some basic considerations are made concerning advantages and limitations.

The method assumes a program of meaningful arithmetic existed in the preceding grades. There could be only a limited suc- cess if this approach were attempted with a group which possessed only a limited understanding of our number system and the basic processes of arithmetic.

The following would appear to be desir- able outcomes of the approach described : 1 It develops knowledge and understand-

ing of our number system. 2 It provides a review of other arithmetic

processes. 3 It provides opportunities for explaining

the process of estimation. 4 It should aid in eliminating mistakes in

placing the quotient figure, particularly in problems involving zero in the quotient.

5 It provides the children with opportunities to construct problems from the algorism which may be of some help in fostering problem solving ability.

6 It develops the "partition" and "measurement."

7 It capitalizes on previous experiences and background of the children.

8 It allows the children some freedom in testing ideas they have for solving the problem.

9 It should aid in decreasing the errors in setting the partial products. Some of the limitations are:

1 There may be some initial difficulty in getting children to verbalize the problem from the algorism.

2 The method may not show similar outcomes when used in a normal classroom situation.

3 The children must show understanding of the subtractive idea and its variations which are basic to division.

4 The method assumes previous meaningful teaching and understanding of our number system.

5 The teacher must possess adequate understandings and be particularly skill- ful in directing questions to secure the desired outcome. Though the approach may not be ap-

plicable to each teacher's individual situation, it does appear that in many class- rooms the approach could be utilized to provide deeper understanding of the division process as well as a more efficient manner of solving division problems.




Article Contentsp. 198p. 199p. 200p. 201p. 202

Issue Table of ContentsThe Arithmetic Teacher, Vol. 9, No. 4 (APRIL 1962), pp. 177-233Front MatterEditorial commentsAs we read [pp. 177-179]From the Editor's Desk [pp. 179-179]

The Miquon mathematics program [pp. 180-187]Manipulative materials and arithmetic achievement in grade 1 [pp. 188-192]Shall we change our arithmetic program? [pp. 193-197]Making division meaningful and logical [pp. 198-202]The mathematics consultant: A suggestion for improving the elementary mathematics program [pp. 203-205]Three lessons in Soviet arithmetic, grade 5 [pp. 206-209]Two aspects of algebra [pp. 210-211]Upper-elementary-school children use statistics [pp. 212-214]In the classroomMathematical vignettesideas from here and there [pp. 215-220]

Experimental projects and researchA cooperative in-service teacher education program in the new mathematics for elementary schools [pp. 221-223]Newsletter: a means of mathematics communication [pp. 224-226]

ReviewsBooks and materialsReview: untitled [pp. 227-227]Review: untitled [pp. 227-228]Review: untitled [pp. 228-228]Review: untitled [pp. 228-229]Review: untitled [pp. 229-229]Review: untitled [pp. 229-231]

Professional dates [pp. 232-233]Teaching arithmetic with the overhead projector [pp. 233-233]Back Matter

Documents

Making division meaningful and logical