BUILDING PER - University of Toronto

68 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

J O A N M O S S A N D B E V E R L Y C A S W E L L

T HERE WAS AN AIR OF EXCITEMENTand anticipation in the grade 5/6 class as thestudents consulted with one another and putthe final touches on their percent measure-

ment dolls. The doll-making unit, a favorite with thestudents, was a culminating activity in an ongoing re-search project for learning rational number and pro-

portion. The students, who attend a laboratory schoolassociated with the Ontario Institute for Studies in Ed-ucation, and their teacher, Beverly Caswell, had justspent the last five mathematics classes working on themeasurement, design, and building of these dolls.They had been invited to present their creations to agroup of preservice teachers and to explain the math-ematics that had been involved.

Alice was the first to speak. “Hi everybody. This isPaige McMillan,” she said, holding up her colorfuldoll. “She is 60 cm tall and she is proportional. She isa scale model of what the average of our grade 5/6class’s proportions look like. She and the other per-cent dolls in our class are made of the average mea-surements of body percents of our entire class. Howwe did it was we measured our heights and then wemeasured lots of body parts and found out what per-cent, say, the height of our head (from the chin tothe top of our foreheads) was compared to ourheight. So it’s about an eighth of my total body,’cause I am 140 cm tall and my head height is 17.5cm so it’s 12 1/2%. So for our doll Paige, we had tomeasure 7 1/2 cm for her head height ’cause that is1/8 of 60 cm. Then we averaged the percents of ourentire class and we got to make our dolls.”

When Nick presented his Pinocchio doll, he ex-plained that “the whole doll is proportional, even

JOAN MOSS, [email protected], is anassistant professor in the Department ofHuman Development and Applied Psychologyat the Ontario Institute for Studies inEducation (OISE) at the University ofToronto. Moss’s research and teaching areasinclude the development of mathematicsunderstanding and the learning and teachingof rational number concepts and mathematicsteaching in the elementary grades. Beforemoving to OISE, Moss taught grades 2–4 atthe Laboratory School of the Institute of Child

Study. BEVERLY CASWELL, [email protected], teachesgrades 5/6 at the Institute of Child Study in Toronto and is pur-suing a master’s degree in Applied Cognitive Science at OISE.Aside from her special interest in mathematics research in theclassroom, she has also been part of the Institute for KnowledgeInnovation and Technology (IKIT) where her work has focusedon building knowledge in science and mathematics.

Connecting Linear Measurement

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Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

VOL. 10, NO. 2 . SEPTEMBER 2004 69

the nose,” but he acknowledged that there was aproblem with the fishing pole. “I realize now that itreally isn’t proportional. See, it is shorter than thedoll but my own fishing pole at home is about 150%of my height, so this one’s way off.”

The preservice teachers were eager to learnmore. How long had it taken the students to makethese dolls? What was the sequence of activities thatled to their completion? What kinds of activities hadpreceded the doll building? As they could hear fromthe students’ descriptions, although the doll-makingproject had its basis in linear measurement, themathematics that the students had used also in-volved performing rational number calculations,translating among fractions and percents, and work-ing out mathematical averages. How had these top-ics been approached, and why did the children usepercents to describe the mathematical relationshipsinstead of fractions as was usually done with bodyproportion activities of this kind? (e.g., Burns 2000).

In this article, we describe the percent-doll unitthrough details of the lessons and accounts of the stu-dents’ learning. To set the context for these lessons,we begin with a brief discussion of our ongoing re-search project and how linear measurement served asa starting point for our work on rational-number learn-ing with the students in our research classrooms.

Linear Measurement to Support LearningRational Numbers

AS THE NCTM (2000) POINTS OUT, THE LEARNINGof measurement not only gives students opportunitiesto apply appropriate tools, techniques, and formulaeto determine measurements but the study of mea-surement facilitates the learning of other topics andallows students to see the connections among mathe-matical ideas. In our current research program inmiddle school classrooms, we have been implement-ing and assessing an approach to teaching the diffi-cult topic of rational numbers (decimals, percents,and fractions) and the aligned topics of ratio and pro-portion using representations involving linear mea-surement. We know from substantial research thatlearning about rational numbers and proportion isoften difficult for students: the learner must movefrom the familiar additive reasoning that underlieswhole numbers and counting toward the more com-plex multiplicative reasoning that underpins rationalnumbers and proportions (Carpenter, Fennema, andRomberg 1993). Many students appear to confusethese forms of reasoning (Hart 1988).

In our research teaching, we promote the idea ofmultiplicative relations by designing activitiesgrounded in linear measurement contexts in which

to Learning Ratio and ProportionCENT DOLLS:


students consider ideas of relative amount and full-ness. Moreover, in our instruction, rather than in-troduce rational number concepts through the tra-ditional approach of using decimals and fractions,we use students’ everyday knowledge of percentsas the starting point. For example, students com-pare heights in relative terms using language thatinvolves percent (“She is about 75% as tall as he is”),or they compare the fullness of containers relativeto the whole (“That cup is approximately 25% full”).Further, they go on to use their informal knowledgeof percents and a strategy for halving and doubling

numbers to performcalculations usingpercents : “Yourheight is 160 cm,then 50% of yourheight is 80 cm, and25% of your heightis 40 cm.” Our re-s e a r c h f i n d i n g shave revealed thatby using these par-

ticular measurement contexts and percents, stu-dents naturally consider the ratio relation of one di-mension to another and also consider proportionalrelations (Kalchman, Moss, and Case 2001; Mossand Case 1999, 2002). Thus, they think about themultiplicative and not the additive or absolute rela-tions involved. (For a fuller discussion of additiveand absolute thinking, see Lamon 1999 and Moss,in press.)

In the sections that follow, we describe the se-quence of activities that the students engaged in overtwo weeks to make their percent measurement dolls.Since the scope of this article does not allow for a de-scription of the full curriculum (see Moss 2000 and2001 for details of the curriculum), we begin our ac-count of the unit with the description of a series ofstring measurement activities in which the studentstook part. We believe that these activities can serve asa good starting place for teachers and students inter-ested in making percent dolls in their own class-rooms. We would like to point out that the halvingand doubling strategies the students use for estimat-ing and calculating were not taught to them at anypoint but were methods that the students had learnedbefore beginning the activities discussed here.

String Challenges: Guessing Mystery Objects

THE DOLL-MAKING UNIT STARTED WITH WHAT we called “string challenges.” First, we presented stu-dents with various lengths of string cut to representdifferent percentages of the heights of mystery objectsin the classroom. The activity was introduced this way:

TEACHER: [Holding up a small piece of string.] Ihave here a length of string that I have measuredand cut so that it is 25% of the length of a mysteryobject in the classroom, and it is within your sight.Any ideas as to what the mystery object might be?

STUDENT: I think that it is the length of desktop ormaybe the length of the Bristol board on the wall.

TEACHER: How did you figure that out?STUDENT: Well, I just imagined moving the string

along the desk four times, and I think it works. [Thestudent then carefully moved the string along the deskand was able to confirm her assertion.]

Next, the students worked in pairs and chal-lenged their classmates to find the lengths ofstrings that corresponded to percents of the lengthof their own mystery objects. Two students, Beckyand Scott, for example, chose the classroom pencilbin as their secret object. First they measured theheight, 46 cm, then calculated that 25% of the lengthwould be 11 1/2 cm. They cut a piece of string tocorrespond to that value and challenged their class-mates to guess the mystery object. The students en-joyed this activity a great deal, so we expanded it,and the students produced what they called “per-cent families.” They used their secret object mea-surement as a base, then cut a series of strings thatrepresented the benchmarks of the halving per-cents. These strings were then labeled and tapedonto large sheets. As they had done with the mys-tery object activity, the students presented theirpercent families to one another, reviewed the calcu-lations they had done, and challenged their class-mates to guess their mystery object. Figure 1shows an example of a completed percent familiesdisplay. The students who produced this item firstmeasured their object, then cut strings that repre-sented the halving benchmarks of 75%, 50%, 37.5%,25%, 12.5%, and 6.25%.

These measurement exercises and the visual dis-plays that the children created—mounted on thewalls around the classroom—naturally evoked dis-cussion of proportions. Students remarked, “Ourstring lengths are different even though all of ourpercents are the same.”

Body Percents

THESE STRING CHALLENGES, WITH THEIR FOCUSon estimation, measurement, and percents, servedas a good introduction to discovering body propor-tions—an idea adapted from a lesson designed byMarilyn Burns (2000)—and then to designing andbuilding the percent dolls.

The body proportion activities began with theteacher asking this question, “Take a look at my

The measurementexercises and

visual displays evoked discussions

of proportions


foot. Can you please tell me what percent of mybody height you think my foot is?” (The answersprovided by the students ranged from 10% to 20%.)“I have already measured my height—I am approxi-mately 168 cm tall. Let’s cut a string that is as longas my foot. Here it is. Let’s measure it. It is 23 cm.So now what is that as a percent?” Most of the stu-dents relied on a halving strategy to compare thefoot with the length as a percent, then they rea-soned as follows: “50% of our teacher’s height is 84cm, 25% of her height is 42 cm, and 12 1/2% is 21cm.” This strategy would give them an approxima-tion of the percent of the foot compared with theteacher’s height. As one student, Maya, expressed:“I did the calculations for 12 1/2% and that is 21 cm,so 23 cm is pretty close. So, I’d say that your foot isnearly 12.5%.” Silas, a grade 6 student, continuedwith these comments: “But your foot is 23 [cm] so itis a bit more. So if you add 1 more percent that’s1.68 [1% of 168]; you get even closer.”

Before the students worked on their own mea-surements, they practiced by estimating other bodyproportions of the teacher. For example, after mea-suring, they discovered that her arm span wasnearly 100% of her body height, that the length ofher waist to foot was about 60% of her height, andthat the circumference of her head was 37.5%.

Measuring Body Parts and CalculatingPercentages

AFTER MEASURING AND ESTIMATING SOME OFthe percentages of the teacher’s body parts, the stu-dents then went on to find the measurements andpercentages of their own body parts. With blankdata sheets in hand, the students worked in pairs tohelp each other measure and record the informa-tion on the data sheets. To help with their calcula-tions, the teacher suggested that the students cre-ate a list similar to the percent families list they hadcreated based on the measurements of their mys-tery object. This time, however, they would usetheir own heights as 100% bases. Again, theirmethod was to use a halving operation to find these“benchmark” quantities. As the list in figure 2 illus-trates, students started with 100 and used a seriesof halving operations to calculate 50%, 25%, and12.5% of their height. They also used halving basedon the 10% quantity to determine what 5% and 2.5%was of their height. However, further calculations

Fig. 1 Mystery object percent families

Fig. 2 Body percents

Body Percents

Name: Bev Your Height: 168 cm

What is the measurement of your arm span? 168 cmWhat percent is that of your height? 100 %

What is the measurement of your foot? 24 cmWhat percent is that of your height? 14 %

What is the measurement of your hand? 17 cmWhat percent is that of your height? 10 %

What is the measurement of your baby finger? 6 cmWhat percent is that of your height? 4 %

What is the measurement of your head circumference? 60 cmWhat percent is that of your height? 35.7 %

What is the measurement of your head length? 21 cmWhat percent is that of your height? 12.5 %

What is the measurement of your shoulder width? 40 cmWhat percent is that of your height? 25 %

What is the measurement of your waist to floor? 98 cmWhat percent is that of your height? 60 %

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were needed to compute the intermediate percentsof 75, 37.5, and so on.

How did the students find these percents? Acloser look at the addition calculations that Kenziejotted down in the right margin of his worksheet(see fig. 3) shows how he calculated these percent-ages. To find 75% of his height, Kenzie added 72 cm(50% of his height) to 36 cm (25% of his height) tofind the correct answer that 75% of 144 is 108. Simi-larly, he found the answer to 37.5% of his height bysumming 36 cm (25%) and 18 cm (12.5%) to deter-mine that 37.5% of his height is 54 cm.

Students Discuss Proportions

AS THE STUDENTS WORKED IN PAIRS TO MEASUREand calculate their percentages, we noticed thattheir conversations contained references to propor-tional reasoning:

PAULO: Hey, my head height is the same percent-age as Darnel’s even though we’re not the sameheight and we had different answers for the cen-timeters tall. What did you get for your foot length?

It was interesting for us to observe the level ofcomfort that the students had in this mixed classwhen they revealed their body measurements. Asmany teachers will agree, classroom measurementactivities involving body comparisons based on ab-solute differences can inadvertently make somestudents feel uncomfortable. For example, the task

of comparing heights of the students in the roommay disadvantage shorter children, although it ismathematically useful. In our project, students arelooking for percents and proportions comparedwith their own height. Thus, any comparisonsamong classmates are within a similar range, andself-esteem is left intact. For example, “My hand is10% of my height. What’s yours?” (Answers rangedfrom 7.5% to 12%.)

When all the students had finished calculatingtheir individual proportions, they pooled these per-centages and worked out the class average on alarge data sheet (see fig. 4). Once the average pro-portions were established, the idea emerged to usethese data to make a series of classroom dolls.These dolls would represent scale models of theclass average proportions.

Constructing the Dolls

THE CLASS THEN BRAINSTORMED IDEAS ABOUTwhat materials could be used to build these dolls.The teacher thought of pipe cleaners; students’ideas ranged from wooden dowels and cardboardtubes to swatches of material, yarn, Plasticine, andglitter. Next, the students worked in pairs to negoti-ate what the dolls would look like. First, they estab-lished a general design and listed materials theywould like to use. Most important, they selected theheight for their dolls. (In the end, the dolls rangedin height from 35 cm to 70 cm.)

With these decisions made, the doll buildingcould begin. However, a new set of calculations wasrequired. Students needed to convert the percents

Fig. 3 Benchmark percents based on a student’s height

Fig. 4 Class average percentages

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from the class average into a new measurement sothat the dolls’ body parts would be proportional inrelation to their heights. The students were eager tomake their dolls as representative of the class aver-age as possible, and they used a variety of strategiesto do so. Although some students filled in new datasheets in which they performed these conversions(see fig. 5), others worked out the measurementsas they built their dolls.

Alice and Paulo, for example, were consideringhow long to cut the wooden dowel that would serveas the arms and hands of their doll.

ALICE: Now let’s see. Our doll is going to be 40cm tall. I know the [average percentage] shoulderto fingertip is 44%, so hmmm, 10% of 40 cm is 4 cm,so 4 times 4 is 16. That is 40%, so it’s a bit more than16…. It is about 18 cm.

PAULO: Yeah, that makes sense because 50% of 40is 20 cm. So it must be pretty close. So let’s cut thatwooden stick 18 cm.

What about making a head for a doll? Joseph andhis partner, Krista, had decided to use a balloon forthe head of their doll. How would they calculate thedistance of the circumference, and how would theyexpand a balloon so that it had the exact circumfer-ence required? The following vignette shows howthey reasoned and accomplished this task.

JOSEPH: [Holding a balloon in his mouth, he be-gins to blow, then pauses to talk with his partner.]The class average head circumference is 37 1/2%,so that’s the 25% measurement plus the 12 1/2%measurement. It’s 3/8. Okay, our doll is 60 cm tall.So for our doll that’s 22 and a half cm. So I have to blow up the balloon so that the circumference is22 1/2. [Joseph then continued to blow up the bal-loon while his partner held the measuring tapearound the balloon.]

KRISTA: Okay, Joseph, slow down. You’ve alreadyblown the balloon to 20 cm. Just a little more. Stop.That’s too much, let some out. OK, we’re at 22.5 cm.Stop blowing!

Conclusion

THE DOLL-MAKING UNIT WAS A FINAL PROJECT INa series of twenty-five experimental lessons inwhich students built their understanding of ratio-nal number concepts through working with per-cents and linear measurement. Our goal for this re-search project was to present students with a seriesof activities that would foster a multiplicative un-derstanding of rational number concepts. A furtherquestion for this research was whether the mea-

surement activities that we designed would helpstudents to develop an understanding of propor-tional reasoning. In this project, as in our previousexperimental studies on rational number under-standing, students partici-pated in extensive pretestand posttest interviews inwhich their knowledge ofrational number conceptswas assessed. Quantita-tive results revealed thatstudents not only madesignificant gains in theirability to work with deci-mals, fractions, and per-cents, but that they alsomade substantial gains intheir abilities to performstandard proportional rea-soning tasks. Moreover,as can be seen in the pro-tocol of students’ reason-ing throughout this article, students gained a flexi-bility in moving among the representations ofrational number.

As for the doll-making unit, the studentsseemed to enjoy the process and to value thelearning that they did. In their communicationswith the preservice teachers invited to view theproducts of their doll making, they showed a pridein their work, both mathematical and artistic. Per-haps one lesson gained from this teaching experi-ence is that working with measurement activitiesdoes indeed offer many advantages. Not only do

Fig. 5 Scaled percentages for building dolls

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In our project, students look for percents and proportions compared with their own height, so comparisons amongclassmates are largely avoided


measurement activities ground math-ematics in authentic tasks and hands-on experience, but they also promotethe understanding of many mathe-matics topics as well as highlight theinterrelations among them.

Throughout the many stages in-volved in building these dolls, thesegrade 5/6 students gained practiceperforming calculations and findingaverages in meaningful ways as wellas translating among percents andfractions to perform operations. Butperhaps most significant was the waythat our general approach to teachingrational numbers along with the doll-making unit appeared to help the stu-dents formulate their ideas about themeaning of ratio and proportion—top-ics known for the difficulties they pre-sent to students (Lamon 1999).

Sophia, a strong mathematics stu-dent, was able to articulate how mak-ing the percent dolls had led her to un-derstand proportion. She explained: “Aproportion is a part of something. It’s

basically what one part is of another,basically a fraction. So the head heightis 12 1/2%, so it’s 1.25 out of 10. And aproportion is something that staysconsistent through all different sizesand all different things. It’s consistentbecause it has the same relation, big-ger and smaller—it’s always related.Our dolls are a lot smaller than theimaginary big one [the class average],but their proportions, every part of thebody, is the same. So although they’redifferent sizes, they are the same sizeequivalent to the body.”

References

Burns, Marilyn. About Teaching Mathe-matics. Sausalito, Calif.: Math Solu-tions, 2000.

Carpenter, Thomas P., Elizabeth Fen-nema, and Thomas E. Romberg. Ratio-nal Numbers: An Integration of Re-search . Hillsdale, N.J.: LawrenceErlbaum Associates, 1993.

Hart, K. “Ratio and Proportion.” In Num-ber Concepts and Operations in the Mid-dle Grades, edited by J. Hiebert and M.Behr, pp. 198–219. Hillsdale, N.J.:Lawrence Erlbaum Associates, 1988.

Kalchman, Mindy, Joan Moss, and RobbieCase. “Psychological Models for theDevelopment of Mathematical Under-standing: Rational Numbers and Func-tions.” In Cognition and Instruction:Twenty-five Years of Progress, edited byS. Carver and D. Klahr, pp. 1–38. Mah-wah, N.J.: Lawrence Erlbaum Associ-ates, 2001.

Lamon, Susan J. Teaching Fractions andRatios for Understanding: Essential Con-tent Knowledge and Instructional Strate-gies for Teachers. Mahwah, N.J.:Lawrence Erlbaum Associates, 1999.

Moss, Joan. “Deepening Children’s Un-derstanding of Rational Numbers.” Dis-sertation Abstracts International, 2000.

———. “Percents and Proportion at theCenter: Altering the Teaching Se-quence for Rational Number.” In Mak-ing Sense of Fractions, Ratios, and Pro-portions, 2001 Yearbook of the NationalCouncil of Teachers of Mathematics(NCTM), edited by Bonnie Litwiller,pp. 109–20. Reston, Va.: NCTM, 2001.

———. “Introducing Percents in LinearMeasurement to Foster an Understand-ing of Rational-Number Operations.”Teaching Children Mathematics 9 (Feb-ruary 2003): 335–39.

———. “Pipes, Tubes, and Beakers:Teaching Rational Number.” In HowPeople Learn: A Targeted Report forTeachers, edited by J. Bransford and S.Donovan. In press.

Moss, Joan, and Robbie Case. “Develop-ing Children’s Understanding of theRational Numbers: A New Model andan Experimental Curriculum.” Journalfor Research in Mathematics Education30 (March 1999): 122–47.

———. “Developing Children’s Under-standing of the Rational Numbers.” InLessons Learned from Research, edited byJudith Sowder and Bonnie Schappelle,pp. 143–50. Reston, Va.: National Councilof Teachers of Mathematics, 2002.

National Council of Teachers of Mathe-matics (NCTM). Principles and Stan-dards for School Mathematics. Reston,Va.: NCTM, 2000. �

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