role of representation

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492 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL L Vol. 13, No. 8, April 2008

The student understood that 1/10was quite small. He recognized that5/12 is less than 1/2 and probably alsoknew that 3/5 was greater than 1/2.He compared two fractions close to 1

by accurately picturing the size of themissing piece. Based on our experi-ences, we concluded:

Before operating with fractions, studentsneed to understand what a fraction means.This involves understanding the part-whole model for fractions and the ability to

judge the relative size of a fraction.

The ability to order fractions playsan important role in students estima-

tion skills. The following questionwas posed to the sixth graders withinindividual interviews after two weeksof instruction:

Marty made two types of cookies.He used 1/5 cup of flour for onerecipe and 2/3 cup of flour for theother. How much flour did he use inall? Is it greater than 1/2 cup or lessthan 1/2 cup? Is the amount greaterthan 1 cup or less than 1 cup?

(Although 1/5 cup is not as realis-tic a measure as 2/3 cup of flour, weused that value to encourage studentsto apply ordering ideas related to

comparing unit fractions. As you seein the students answers, knowing that1/5 is less than 1/3 gave students theinformation needed to conclude thatthe total amount of flour must be less

than 1 cup.)Here are some student responses,with additional questions placed inbrackets:

He used more than 1/2 cup because

2/3 is more than 1/2. He used less

than 1 cup. Because 1/5 is less than

1/2. [Why would that be enough

to know that the answer is less than

1?] Because 1/5 is 1 out of 5 and so

its smaller pieces. [What would you

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 2 Sixth-grade students thinking on homework assignments about estimation


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Vol. 13, No. 8, April 2008 L MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 493

need to add to 2/3 to equal 1 cup?]

1/3; 1/3 is bigger than 1/5.

It is already more than 1/2 because 2/3

is greater than 1/2. Its going to be less

than 1 cup because 1/5 is not equal to

1/3 so it wont cover that space.

More than 1/2 cup because 2/3 is

already bigger than 1/2 so plus 1/5

would be over 1/2. Its less than 1

cup. Because 2/3, 1/5 doesnt fill up

the missing piece in 2/3. It just fills

up a little pieces of it so it wouldnt

equal a whole.

Students used 1/2 as a benchmark,

compared unit fractions, and deter-mined amounts to add to 2/3 to makea whole. Students were able to dothis because they constructed strongmental images for fractions. Based onstudent responses to estimation taskslike this one, we concluded:

Estimation and visualization are im-portant. These abilities will help studentsmonitor their work when finding exact

answers.

Figure 2 shows examples of sixthgraders thinking on estimation tasksdrawn from homework assignedbefore students were asked to findexact answers to fraction addition andsubtraction problems. Students wereasked if the solution given in eachproblem was a reasonable estimate.Notice that students coordinatedseveral ordering ideas to judge if theanswers to addition and subtractiontasks were reasonable. Their workwith fraction circles helped themjudge reasonableness, because theywere able to visualize the relative sizesof the fractions and combine them bymentally adding on or taking away.Students were able to visualize almostany fraction, regardless of whetherthey had actually seen that fractionwith the fraction circle model. Note

the students response in figure 2h.Although there was no fraction circledivided into hundredths, he knew thathundredths were very small pieces,and he only had two of them, so he

explained that 2/100 is pretty muchnothing.We were impressed with the depth

of understanding that this group ofsixth graders showed regarding frac-tion meaning, their ability to judge therelative sizes of fractions using mul-tiple strategies, and their use of orderideas to estimate fraction addition andsubtraction. We strongly believe thatthe fraction circle model gave studentsthe mental imagery for fractions that

supported their understanding. Thisunderstanding provided a strong foun-

dation on which to build the conceptof a common denominator for addingand subtracting fractions.

In planning the lessons for this lat-est teaching experiment, we consid-

ered which representation might bestdemonstrate the common denomina-tor procedure for adding and sub-tracting fractions. We looked at othermodels, such as pattern blocks, chips,and dot paper, which had been usedby other NSF-supported curricula.We looked closely at the positives andnegatives of these models using datawe collected during another fractionstudy. We found that students haddifficulty simply representing frac-

tions with the pattern blocks and wereunable to use them to add fractions.

Fig. 3 Students errors with chip and dot paper models

dot spacing

1

4 2

3

3

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Students reaction to the chip and dotpaper model was interesting in thatthey used it to reinforce the incorrectstrategy of adding numerators anddenominators (see fig. 3). Notice that

the students did not use a commonunit to show each fraction. Insteadthey used a unit equal to the numberof parts in each denominator. Theyadded the number of parts shaded andthe total number of parts to calculate3/7 as the answer.

Students who tried to model ad-dition with chips or dot paper didnot realize the need to express bothfractions as equivalent using the samenumber of chips in the unit or the

same number of square units in thedot paper model. In other words, theydid not realize the need to expressboth fractions as equivalent with acommon denominator. When consid-ering what the students did with thechips and dot paper to try to modelfraction addition, we realized thatnothing is obvious in the model itselfto show the need for finding commondenominators. To add two fractions

accurately using chips and dot paper,students need to know already thatthey have to find a common denomi-nator and realize how the denomina-tor relates to the final answer.

We believe that students wouldunderstand common denominators ifthey constructed for themselves theneed to find common denominators.The fraction circle model made thisneed more obvious than other modelswe examined. Therefore, in our currentteaching experiment, we began fractionaddition and subtraction with frac-tion circles and estimation. We askedstudents to show 1/2 (a yellow piece)plus 1/6 (a pink piece) on their frac-tion circle unit (the black circle). Weasked, How much of the whole circlewas covered? The discussion led to anestimate that the answer was obviouslygreater than 1/2 but less than 1. Stu-dents had difficulty naming the exact

amount shown on the black fractioncircle. One student suggested that toget the exact answer, you need to haveall the pieces the color of the smallestpiece. We concretely exchanged the

yellow piece (1/2) for 3 pinks (3/6).Students previous work with equiva-lence ideas based on the fraction circlemodel supported this step of findingcommon denominators. Studentseasily saw that 4/6 of the whole circlewas covered, which became an A ha!moment for some. At this point, itwas tempting to jump right into theprocedure but we held off. Findinga common denominator is critical tounderstanding how to add and subtract

with symbols. To truly grasp this idea,a majority of students need extendedperiods of time with the fractioncircles, determining for themselves theneed to change the different pieces onthe circle to those that are of the samecolor. Students solved many problemsusing the fraction circles to add andsubtract before they made the connec-tion between exchanging circle piecesand finding common denominators

symbolically. We concluded:

Students need to experience acting outaddition and subtraction concretely withan appropriate model before operatingwith symbols.

Moving from a concrete rep-resentation to a symbolic one isnot as simple as it seems. Writtenlanguage and verbal language playimportant roles in helping studentstranslate from concrete to symbolicrepresentation. Students need timeto describe what they do with theirconcrete models before they areable to use symbols to meaningfullyrecord their work. Figure 4 illustrateshow sixth-grade students use writtenlanguage to describe what they didwith the fraction circles. The idea ofexchanging equivalent fractions tofind common denominators is shown

in these examples. For some students,recording with symbols meant usingnumbers. For others, it meant writingin words.

Students had many opportuni-

ties to connect their actions with thefraction circles to pictures of fractioncircles to symbols. Students writ-ten explanations were often messy,which sometimes led to errors. Aftermuch discussion about, and guid-ance from, Principles and Standards forSchool Mathematics(NCTM 2000), weconcluded that students own record-ing systems would be more meaning-ful than a teacher directed one: It isimportant to encourage students to

represent their ideas in ways that makesense to them, even if their first repre-sentations are not conventional ones(NCTM 2000, p. 67). For example,many sixth-grade students used arrowsto show the change from the givenfraction to an equivalent form. Thisseems to be a reasonable way to showwhat is being done with the procedure.Figure 5 shows a few examples of howstudents recorded their fraction addi-

tion and subtraction. We concluded:

Making connections between concreteactions and symbols is an important

part of understanding. Students shouldbe encouraged to find their own way ofrecording with symbols.

At some point, sixth-grade stu-dents need experiences with fractions(denominators less than 20) at thesymbolic level. Students need to beable to imaginefraction circles to helpsolve problems, but they must actu-ally calculate with symbols (numbers).We found that even though almostall students knew what to do to addand subtract fractions symbolically,their ability to do so easily withoutthe concrete model depended on howquickly they were able to recall theirfacts. Students who still counted ontheir fingers were at a disadvantage


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Vol. 13, No. 8, April 2008 L MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 495

rely on students being able to under-stand the need for a common denomi-nator to be able to use these models.For example, toadd 3/4 + 1/5 withchips, the student needs to under-stand that he or she has to representeach fraction using the same numberof chips. Determining that numberis equivalent to finding the commondenominator. Modeling each fractionwith that number of chips correspondsto finding equivalent fractions with thecommon denominator.

In our current work, we used thenumber line model for addition andsubtraction after students learnedto find a common denominator. We

Fig. 4 Connecting actions with circles and pictures to symbols

when working only with symbols. Wefound the following:

Students need easy recall of their multi-plication and division facts.

Our past experience observingstudents using chips and dot paperto add and subtract fractions and ournew work employing a number lineto add and subtract fractions led us toconsider the power of connecting theirnewly learned procedure for addingand subtracting fractions to a differentmodel as a way to reinforce the pro-cedure. As mentioned earlier, some ofthe models found in current textbooks

Fig. 5 Students recording systems

Fig. 6 Students description of using a

number line to add two fractions


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found that although it is an effectivemodel for reinforcing the procedurefor adding and subtracting fractions,it is less so for the introduction ofthe concept of common denomina-

tors. Once the idea of a commondenominator was developed withfraction circles, the number line wasintroduced to reinforce it. Studentsused their understanding of commondenominators to make sense of thenumber line as a model for addingand subtracting fractions. Considerthe students work in figure 6. Stu-dents were given a page of eightnumber lines partitioned differently:halves, thirds, fourths, sixths, eighths,

ninths, tenths, and twelfths. Theywere asked to show how to add 2/3and 2/9 on a number line. The deci-sion as to which number line to useto add the two fractions correspondsto finding the common denomina-tor. Identifying each fraction as an

equivalent amount on the numberline is supported by a students abilityto find equivalent fractions symboli-cally. Finding the total amount on thenumber line is the final step involving

adding fractions now that each areshown using common denominators.We believe the following:

Connecting the procedure to a new repre-sentation may be an effective strategy toreinforce the procedure.

SUMMARY

The concrete models we choose tohelp students build meaning for frac-tions and operations are important.

In our work, we have determinedthat the fraction circle model sup-ports students understanding of thepart-whole model for fractions andprovides students with mental repre-sentations that enable them to judgethe relative size of fractions. Students

are able to estimate answers to frac-tion addition and subtraction tasksand judge reasonableness of answersto fraction operation tasks becausethey have strong mental representa-

tions for these numbers and are ableto manipulate them mentally. In ourlatest teaching experiment, we devel-oped lessons to help students buildmeaning for finding exact answersto fraction addition and subtractionproblems using common denomi-nators. We found that the fractioncircles vividly demonstrate the needfor finding common denominatorswhen adding and subtracting fractionsand that the fraction circles show the

steps to exchanging given fractionswith equivalent ones with commondenominators. We will continue tostudy how fraction circles and othermodels contribute to students mas-tery of this difficult procedure.

BIBLIOGRAPHY

Cramer, Kathleen, and Apryl Henry.

Using Manipulative Models to Build

Number Sense for Addition and Frac-

tions. InMaking Sense of Fractions,Ratios, and Proportions, 2002 Yearbook

of the National Council of Teachers

of Mathematics (NCTM), edited by

Bonnie Litwiller and George Bright,

pp. 4148. Reston, VA: NCTM, 2002.

Cramer, Kathleen, Thomas R. Post, and

Robert C. delMas. Initial Fraction

Learning by Fourth- and Fifth-Grade

Students: A Comparison of the Ef-

fects of Using Commercial Curricula

with the Effects of Using the Rational

Number Project Curriculum.Journal

for Research in Mathematics Education

33 (March 2002): 11144.

National Council of Teachers of Math-

ematics (NCTM). Principles and Stan-

dards for School Mathematics. Reston,

VA: 2000.

Post, Thomas R., and Kathleen Cramer,

Childrens Strategies in Ordering

Rational Numbers.Arithmetic Teacher

35 (October 1987): 3335.L

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