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Mathematics Practice in Carpet Laying

Author(s): Joanna O. MasingilaSource: Anthropology & Education Quarterly, Vol. 25, No. 4 (Dec., 1994), pp. 430-462Published by: Wiley on behalf of the American Anthropological AssociationStable URL: http://www.jstor.org/stable/3195859 .

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Mathematics Practicein CarpetLayingJOANNA . MASINGILASyracuse UniversityBuilding upon previousresearchconcerningmathematicspractice n everydaysituations, this articlestudies the mathematicalconceptsand processesused incarpet aying. Mathematicspractice n carpet aying is characterizedthroughadiscussionof the mathematicsusedby carpet ayersin estimationand installa-tion activities, in an effort to describeand detail how people actively givemeaningto and use mathematics n themidstof ongoing activities in relevantsettings. ETHNOMATHEMATICS, MATHEMATICS PRACTICE, EV-ERYDAY MATHEMATICS, OUT-OF-SCHOOLMATHEMATICS, AC-TIVITYTHEORYPrior to the last 15 years, mathematics was generally considered asculture-free knowledge. However, recent anthropological studies havefound that mathematics does indeed have a cultural history (Bishop1988; D'Ambrosio 1985b; Wilder 1981). Furthermore, mathematicslearning "is not limited to acquisition of the formal algorithmic proce-dures passed down by mathematicians to individuals via school. Mathe-matics learning occurs as well during participation in cultural practicesas children and adults attempt to accomplish pragmatic goals" (Saxe1988:14-15).From this perspective, knowledge is not "a factual commodity orcompendium of facts" but rather takes on the character of a "process ofknowing" (Lave 1988:175). Learning and doing mathematics is an act ofsense making and comprises both cultural and cognitive phenomenathat cannot be separated (Schoenfeld 1989). As Cobb noted, "Cognitionis context-bounded.... [T]he elaboration and coordination of contextsis essential to the achievement of the most general of goals, the construc-tion of a world that makes sense" (Cobb 1986:5).Part of every culture are the everyday happenings of the peoplebelonging to that culture. As such, mathematical thinking and learningoccurs in this everyday practice. As Lave explained, everyday "is not atime of day, a social role, nor a set of activities, particular social occasions,or settings for activity. Instead the everyday is just that: what people doin daily, weekly, monthly, ordinary cycles of activity" (Lave 1988:15).For decades conventional wisdom has viewed schooling as responsi-ble for replacing "the (presumably) faulty and inefficient mathematicalknowledge acquired by people" in everyday life (Lave et al. 1989:67).The assumption has been that "young people learn what they need to

Anthropology & Education Quarterly 25(4):430-462. Copyright ? 1994, AmericanAnthropological Association.

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Mathematics racticen CarpetLayingknow in the classroomand thenapply hisknowledge in the workplaceand in othersettingsof modernlife"(Scribner nd Stevens1989:1).

However, this view was not always dominant. In the early 19thcentury,school mathematicscurriculawere basedupon happeningsinthe workplace(i.e.,mathematicsused in branchesof commerce).Sincethat time the roles have been reversed. The contemporaryview is thatschool mathematics should be the mathematicsthat will preparestu-dents forits application n all life situations,replacingthe intuitiveandnonintentionally nvented mathematicsof everyday(Cohen 1982).Thiscontemporaryview, oftencalledthefunctionaltheoryof school-ing, has been formulated from a theoretical model of cognition andlearningthatassumesthatintellectualfunctioning s uniform acrossallactivities,settings, and cultures.However, thereis a growing body ofresearchon mathematicspractice in everyday situations that "chal-lengestheview thatschool is the centralsourceofeverydaymathematicspractice" Laveet al. 1989:45).This researchdemonstratesthatmathe-maticalproblemsin out-of-school situations areoften approachedeffi-ciently and creativelywithout using school-taught proceduresin anyobservable manner (e.g., Carraheret al. 1985;de la Rocha1985;Mur-taugh1985b;Saxe1988;Scribner1985).These andotherstudiescallintoquestionthe common beliefand assumptionthatschoolingis "afontoftransferable bilities"(Lave1988:xiii).Thisresearchalso challenges the theoriesof cognitionthat view themindas a "systemof symbolic representationsand operationsthat canbeunderstood nand of itself,in isolation fromothersystemsofactivity"(Scribner 988:1). nthe UnitedStates,researchand theoryon cognitivedevelopmenthas been dominatedby Piaget'stheoryof developmentalcognitive psychology that considers individual development largelydevoid of cultural nfluences(Millroy1992).Thepreviouslymentionedbody of research on cognition in everyday practicepoints toward theneed forstudying cognitionwithin a culturalcontext.Research n the last 15 years has indicated a burgeoninginterest inexaminingthemathematicspractice n:(a)distinct ultures(Bishop1979,1983, 1988;Brenner1985a, 1985b;D'Ambrosio1985a,1985b;Ferreira1990;Gerdes1985,1986,1988;Lancy1983;Posner1982;Saxe1979,1981,1982, 1988, 1991); and (b) everyday situations within cultures (Carraher1986;Carraher t al. 1985,1986,1987;de Abreuand Carraher 989;de laRocha1985, 1986;Fahrmeier1984;Harris1987a,1987b,1988a,1988b,1988c;Lave 1977, 1985, 1988;Lave, Murtaughand de la Rocha 1984;Millroy1990,1992;Murtaugh1985a,1985b;Petitto1982;Reed and Lave1979,1981;Saxe 1988, 1991;Schliemann1986;Schliemannand Acioly1989;Scribner1984a,1984b,1984c,1984d,1984e,1985,1988).Whereasthefirstbody of researchhastended to lookat themathematicspracticeof a whole culture,researchersexaminingmathematicspractice n ev-erydaysituationswithincultureshavefocusedon one situationorworkcontext(e.g., groceryshopping,carpentry).

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Anthropology EducationQuarterly Volume 25,1994Mathematicspractice ndistinctcultureshas been termed ethnomathe-matics.UbiratanD'Ambrosio,a Brazilian mathematicseducator, has

been credited with introducingthe conceptof ethnomathematics, hatis, the influence that sociocultural factors have on the learning andteachingof mathematics Scott1985).Theprefixethno-refers to "identi-fiable culturalgroups, such as national-tribal ocieties, labor groups,childrenof acertainagebracket,professionalclasses,and so on"(D'Am-brosio 1985a:44).Scott (1985) ncorporated he meaning of ethno-withmathematicscontentin his definition of ethnomathematics:Ethnomathematicsies at theconfluenceof mathematicsand culturalanthro-pology.At one level, it is whatmightbe called "math n the environment" r"math n thecommunity."At another,related evel, ethnomathematicss theparticular andperhaps peculiar)way of classifying,ordering,countingandmeasuring. Scott1985:2]In studying differentcultures,Bishop has argued that there are sixfundamentalmathematicalactivities that "areboth universal, in thatthey appearto be carriedout by every culturalgroupever studied,andalso necessary and sufficient for the development of mathematicalknowledge"(Bishop1988:182).These six activitiesare counting, locat-ing, measuring, designing, playing, and explaining;mathematics,as

culturalknowledge, "derivesfrom humansengaging in these six uni-versal activities in a sustained, and conscious manner" (Bishop1988:183).The studies that have looked at mathematics practice in distinctcultureshelp to furtherourunderstandingthatmathematics s andcanbedevelopedbydifferentcultures,and informus about themathematicspracticeof thesecultures.Closelytiedto the ethnomathematics esearchis researchabout mathematicspracticein everyday situations withincultures.MathematicsPractice n EverydaySituations

Thereappear obe twocommonthreadsrunningthrough he researchliteratureon mathematicspracticein everyday situations within cul-tures.First, hefactthatproblemsare embeddedin realcontextsthataremeaningfulto the problemsolvermotivatesand sustainsproblemsolv-ing activity.Second,in solvingproblemsthat ariseor are formulated neveryday situations,problemsolvers often use "mathematicalproce-duresandthinkingprocessesthat arequitedifferentfrom thoselearnedin school.Furthermore, eople'severydaymathematicsoften reflectsahigher level of thinkingthanis typically expected or accomplished nschool"(Lester1989:33).Lave (1988)found evidence that mathematicspracticein everydaysettingsdiffersfromschoolmathematics n a varietyof ways. Inevery-day settings: a)people look efficaciousastheydeal with complextasks,(b)mathematicspractice s structuredn relationtoongoing activityand

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setting, (c) people have more than sufficient mathematical knowledgeto deal with problems, (d) mathematics practice is nearly always correct,(e) problems can be changed, transformed, abandoned, and/or solvedsince the problem has been generated by the problem solver, and (f)procedures are invented on the spot as needed.Whereas school learning emphasizes individual cognition, pure men-tation, symbol manipulation, and general learning, everyday practicerelies on shared cognition, tool manipulation, contextualized reasoning,and situation-specific competencies (Resnick 1987). Researchers whohave investigated how persons solve problems in school-like situationsand solve mathematically similar problems in everyday contexts foundthat in the former situation people "tended to produce, without ques-tion, algorithmic, place-holding, school-learned techniques for solvingproblems, even when they could not remember them well enough tosolve problems successfully" (Lave 1985:173). When the same peoplesolved problems in situations that appeared different from school, theyused a variety of techniques and invented units with which to compute(Lave 1985).The majority of research on mathematics practice in everyday situ-ations within cultures has investigated the use of arithmetic and geome-try concepts and processes. To extend this research to a situation usingmeasurement ideas, I investigated the mathematics practice of a groupof carpet layers, in an effort to detail how people "actively give meaningto, and fashion, processes of problem solving in the midst of ongoingactivities in relevant settings" (Lave 1988:63). I chose the everydaysituation of carpet laying because measurement is a central concept andmeasuring is a central process in carpet-laying work. Measurement as aconcept is the idea that characteristics of objects can be quantified (e.g.,the space inside a region can be quantified as the number of square units,or area) whereas measuring as a process is the action taken to quantifythose characteristics.Measurement is different in some fundamental ways from arithmeticand geometry. For example, measurement units are determined moreclearly by cultural convention than are aspects of arithmetic and geome-try practice. Gay and Cole (1967) found, for example, that measures ofvolume, in units that were socially established, were used by the Kpellein Liberia in situations where the amount of a given material wasimportant. Buying and selling rice, the staple food of the Kpelle, weretwo such situations.

The local tradeuses what is called a sdmo-ko, salmoncup,"fordealingrice.It is the large size tin can (U.S.#1) in which salmon is normallypacked....Thecup the traderuses to buy ricehas the bottomrounded out by long andcarefulpounding,but the cup he uses to sellrice does not have the roundedbottom.This is the source of his profit. [Gayand Cole 1967:64]

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Anthropology& EducationQuarterly Volume 25,1994Measurements also a key partof the schoolmathematicscurriculum.In fact, the Curriculumnd Evaluation tandardsfor SchoolMathematics

stated that "measurement s of central importanceto the curriculumbecause of its power to help children see that mathematics s useful ineveryday life and to help them develop many mathematicalconceptsand skills"(NationalCouncilof Teachersof Mathematics[or NCTM]1989:51).However,measurementknowledge in school is often limitedto memorizing formulas (e.g., P = 2(1+ w) and A = lw) and learningmeasurementskills such as how to use a ruler,whereas measurementknowledge in the workplace is primarilycomprisedof concepts andprocessescentralto measuring,such as areaexpressedas squareunits,estimation,spatialvisualization,minimizingerror,and efficiency.Structureand Aim of the Study

Various conceptual, theoretical, and methodological frameworksguided the conceptualization,design, and conduct of this study-acultural framework of ethnomathematics,an epistemological frame-work of constructivism,a cognitiveframeworkof activitytheory,andamethodological rameworkof ethnography.Iwill elaborateon the cog-nitive frameworkhere and discuss laterhow this frameworkwas usedin collectingand interpreting he data.Cognitive ramework

Tofocusthisstudy in exploringcognition nculture,Iused thetheoryof activityas a guiding framework.Thetheoryof activityhasits originsin theworkof the SovietpsychologistVygotskyand hasbeendevelopedover the yearsby his successors,especiallyLeont'ev.Activitytheoryisa "theoretical rameworkwhich affords the prospectof an integratedaccount of mind-in-action" Scribner1984a:2).Thistheory emphasizesthat humansactwithin socioculturalcontextsthatneedtobe consideredwhen studyingcognition.Insteadof studying psychological entities such as skills, concepts,information-processing nits, reflexes,or mentalfunctions,the theoryof activityfocuseson the unit of activity.One of the key characteristicsof anactivity s that t "isnot determinedor even stronglycircumscribedby thephysicalorperceptualcontext nwhichhumansfunction.Rather,it is a socioculturalinterpretationor creation that is imposed on thecontextby the participant(s)"Wertsch1985:203).AimoftheStudyThegeneralaim of this study was to develop a betterunderstandingof mathematicspracticein everyday situations.To this end, I focusedmy attentionon one particular ype of mathematicspractice-mathe-maticspractice n carpet aying.Morespecifically,my aim was to iden-tifythe mathematics onceptsandprocessesusedin the contextofcarpet

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laying and examine what this data can add to the study of everydaymathematics. My motivation for this study and future research is todevelop a model for closing the gap between doing mathematics inschool and out of school.Fieldwork

In order to achieve the two aims, I spent an average of four hours aday, five days a week, for seven weeks during June, July, and August1991, observing and informally questioning the employees of a carpet-laying business, Miller's Floor Coverings (a pseudonym), in the mid-western United States. I used four methods of data collection in myfieldwork: participant observation, ethnographic interviewing, artifactexamination, and researcher introspection.Data Collection

Although I did not participate in the actual laying of floor coverings,I participated in the discussion and decision making accompanying anyestimation or installation job. My participation was not, however, thatof contributing ideas or making suggestions. Rather, I asked questionsto clarify what I heard, and I participated in casual conversation. I alsoparticipated by going to the estimation and installation sites, holdingtape measures, helping to chalk lines, and carrying equipment andmaterials.I observed Miller's employees through the entire floor-covering pro-cess for a variety of situations: (a) residential and commercial settings, (b)one-room jobs up through entire buildings, and (c) carpet, tile, hardwood,and base installations. I chose to observe work tasks completed by bothestimators and installers. I observed the estimators taking field measure-ments, making sketches, and deciding on best estimates. I observed theinstallers interpreting the estimator's sketches and estimates, measuring,deciding on best installations, and installing floor coverings. Of the sixactivities identified by Bishop (1988),I focused on measurement and locat-ing in examining mathematics practicein carpet laying.Criteria for selecting work tasks to be observed and analyzed includedthat the tasks: (a) involve person-world transactions, (b) be essential to,if not constitutive of, job performance, and (c) involve observable modesof solution, as well as the actual achievement of solutions. These criteriahave been used successfully in other studies to select candidate tasks forcognitive analysis (e.g., Scribner 1984b). I examined (and made copiesof) all sketches and calculations made by estimators and installers, aswell as blueprints used for commercial floor covering jobs, for informa-tion that might aid me in my research. Along with creating an expandedaccount of my field notes, which included observations, interviews, andartifact notes, I also recorded my reflections, feelings, reactions, insights,and emerging interpretations daily in a journal.

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Anthropology& EducationQuarterly Volume25,1994Data Analysis

The data were analyzed using the theory of activity (Eckensberger andMeacham 1984;Leont'ev 1981;Wertsch 1985) as a framework. Leontev(1981) has specified several distinct but interrelated levels of analysispresent in the theory of activity, with a specific type of unit associatedwith each level. The three levels are activity, goal-directedaction, andoperation.Among the activities in which humans engage are several thathave been mentioned by Vygotsky's students (e.g., El'konin 1972):play,instruction, and work. I chose the activities that occur in the workplaceto focus on for reasons of both significance and strategy. Work is obvi-ously significant; it is basic to human life in all societies and all culturesand occupies a great part of an adult's time.When I considered examining mathematics practice in everyday situ-ations, the strategies involved in such research also pointed in thedirection of the workplace. Occupations such as carpet estimating andinstalling are "highly structured and involve tasks whose goals arepredetermined and explicit" (Scribner 1984a:3). Following Scribner'sexample, I used occupations,work tasks, and conditions to represent thethree levels of analysis-activities, goal-directedactions,and operations.The two occupationsfor which I collected data were estimator andinstaller. The worktasks n the estimator's job that interested me include:(a) taking field measurements, (b) making sketches, and (c) deciding onbest estimates. Those in the occupation of installer include: (a) interpret-ing the estimator's sketch and estimate, (b) measuring, (c) deciding onbest installations, and (d) installing.Like most situations in everyday life, the processes of estimating andinstalling floor coverings have many constraints that must be taken intoconsideration during these processes. These constraints are the condi-tions that I focused on for this study. Constraints that I observed includethat: (a) floor covering materials come in specified sizes (e.g., most carpetis 12' wide, base [vinyl piece glued around the perimeter of a room] is4' long, most tile is 1' by 1'); (b) carpet pieces are rectangular; (c) carpetin a room (and often throughout a building) must have the nap (thedense, fuzzy surface on carpet formed by fibers from the underlyingmaterial) running in the same direction; (d) consideration of seamplacement is very important because of traffic patterns and the type ofcarpet being installed; (e) some carpets have patterns that must matchat the seams; (f) tile and hardwood pieces must be laid to be lengthwiseand widthwise symmetrical about the center of the room; and (g) fillpieces for both tile and base must be six inches wide or more to stay inplace.I analyzed the field data through a process of inductive data analysisusing two subprocesses that Lincoln and Guba (1985) have called unit-izing and categorizing. The work tasks that I observed were chosenthrough purposive sampling and were changed as the study designemerged. Several times per week I reflected upon the estimation and

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Mathematics racticen CarpetLayinginstallation work tasks that I had observed thus far and made samplingdecisions to: (a) observe certain work tasks again, (b) ask specific ques-tions of certain respondents, (c) observe unfamiliar work tasks, and (d)discontinue the observation of work tasks for which I felt I had enoughdata.Mathematical Concepts

I observed four areas of mathematical concepts used by estimatorsand/or installers: measurement, computational algorithms, geometry,and ratio and proportion. In the following sections, each of these areasis discussed and data are presented to illustrate the use of some of theseconcepts.'Measurement

The concept of measurement was involved in most of the work doneby Miller's estimators and installers. I observed the presence of thisconcept in the following categories of use: finding the area and perimeterof a region, drawing and cutting 45? angles, and drawing and cutting90? angles.Finding the Area and Perimeter of a Region. Because carpet pieces arerectangular, every region to be carpeted must be partitioned into rectan-gular regions. The areas of these regions are then computed by multi-plying the length and width. Floor-covering businesses are generallyonly concerned with the perimeters of regions when base is to beinstalled; base might be installed around the perimeter of a room aftertile, vinyl floor covering, or commercial carpet has been laid. Calculatingthe perimeter of a region is essential when figuring a job estimate thatincludes base. This might be done either from a blueprint or from a fieldmeasurement.Perimeters are measured on blueprints by tracing the outline of therooms with a measurement trundle wheel. When completing field meas-urements, however, estimators seldom measured along the walls todetermine the perimeter. Instead, since most rooms are rectangular, thelength and width were used to calculate the perimeter. Dean,2 an esti-mator, explained how he estimated the amount of base needed for aroom. "Suppose you have a 12' by 8' room. Then 12 plus 12 plus 8 plus8 equals 40," explained Dean as he used an adding machine. "There'sprobably one door; so you subtract three feet. That's 37 feet. Divide thatby four because base comes in four-foot sections. That equals 9.2 pieces;so you have to make that 10 pieces of base, or 40 feet." When I askedDean what he would do if the room were not rectangular, he replied,"Well, if I can't figure it out from the length and width, then I measurethe lengths of all the walls and add 'em up."

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3'

3 `.-/'~ 1 t piecelaid

Figure1Installinghardwood in a 45?-anglepattern.Drawing and Cutting 45? Angles. An angle of 45? is a common anglemeasurement used in floor-covering installations. One reason for thefrequent use of 45?angles is that this angle is easier to estimate than otherangle measures less than 90?.Furthermore, a cut of 45? is more aestheti-cally pleasing, in some situations, than cuts of other angle measures sincefloor covering situations often involve cuts made at a corer with anangle measure of 90?.In this case, a 45?-angle cut is on a line of symmetry.The following example illustrates one situation that I observed involv-ing 45? angles.During the last two weeks of my fieldwork I stopped in from time totime to observe one of the installers, Jack, laying hardwood pieces (each2" by 12") in a large corridor of a building on the campus of the localuniversity. Since some of the wood was to be laid in a herringbonepattern (at a 45? angle), some measuring needed to be done prior to theinstallation. After Jack had found the intersection of the center lines ofeach hallway (see Figure 1) and checked to see if the walls were square,he began working to establish a 45? angle. (Consider the center lines ofeach hallway as the axes and use the orientation shown in Figure 1.)Jackmeasured three feet in both the positive x- and y-directions and mademarks at (3,0) and (0,3). Using his tape measure and pencil as a compass,Jack then measured three feet up from (3, 0) and three feet to the rightof (0, 3) and established a mark at (3, 3). Jack established the point (-3,

z4

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Mathematics racticen CarpetLaying-3) in a similar fashion. He then drew the line segment connecting (3, 3)and (-3, -3), which passed through (0, 0).

Jack: incethese aresquares,the angles arerightangles,and so the diagonalcuts therightangles in halfand makes45-degreeangles.Joanna:Whydid you maketwo three-footsquares?Jack: could have done it with one, but the othersquaremakes the diagonalmoreaccurate.Joanna:Whydid you make three-footsquares?Jack:No particular easonexcept thatit's easy to work with-not too big ortoosmall.

Although he mentioned nothing about coordinate geometry, Jackcertainly used these ideas in constructing the squares and drawing the45?angle. To start the installation of the hardwood, Jackplaced one pieceof wood on the diagonal that he had drawn such that one bottom cornerof the wood was at the origin and the bottom edge lay right along thediagonal. He then proceeded to lay other pieces, as shown in Figure 1.Drawingand Cutting 90?Angles. Along with the use of 45? angles, anglesof 90? are commonly used in floor-covering installations. The followingexample used the hardwood installation discussed above.

4

5'

4

If

Figure2Using a righttriangle.

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Anthropology& EducationQuarterly Volume 25,1994As mentioned previously, tile or hardwood must be installed so as tobe lengthwise and widthwise symmetrical about the center of the room.

The center of the room is established by the intersection of the lines thatbisect the length and width of the room (see Figure 1). Before Jackbeganlaying the hardwood in the herringbone pattern, he used these lines tosee if the hallway walls were perpendicular to each other. Jack deter-mined the center of the room by measuring and chalking lines. He thenmeasured three feet in the positive y-direction and four feet in thepositive x-direction and made marks at the points (0, 3) and (4, 0) (seeFigure 2).Jack asked me to hold the end of the tape measure at one mark, andhe extended the tape measure to cross the other mark. He seemedpleased when this measurement was five feet. I asked Jack what thatmeant-that the length from one mark to the other was five feet. Hereplied, "Well, that's the way you figure it out: 3,4, 5 [tracing a trianglein the air]." I asked why that meant that it was square. Jack laughed,"That's something from geometry, right? a + b = c-no! It's somethinglike that. What I know is that a 3-4-5 triangle makes a 90? angle, and sothe walls that I measured from are square." Jack based this procedureon the Pythagorean theorem, which provides a relationship between thelengths of the sides of a right triangle, even though he could not remem-ber the relationship or the name of the theorem that he probably learnedin school. I then asked what happens if the distance from one mark tothe other is not five feet. Jackreplied, "Well, then you gotta be aware ofthat and make some adjustments. The main thing is to know if they'resquare or not so you won't be surprised."ComputationalAlgorithms

I observed a number of computational algorithms that estimatorsused in measurement situations for determining the quantity of materi-als needed for an installation job: estimating the amount of carpet,estimating the amount of tile, estimating the amount of hardwood,estimating the amount of base, and converting square feet to squareyards. For example, to estimate the amount of tile needed for one room,the maximum length and width of the room must be found. Steve, aninstaller, explained the rest of the process to me this way: "Well, say yourroom is 11'7"by 19'5".Then you'd make it 12' by 20' because you alwaysround up to the nearest foot. Even if it's 11'1", you have to make it 12'.Then you figure the area (length times width) and divide by 45 'causethere's 45 tiles in a box." Since each of the tiles are 1' by 1', a box of 45tiles contains 45 square feet of tile. Thus, taking the square footage (withlength and width rounded up to the nearest foot) of the room dividedby the number of square feet of tile per box gives the number of boxesneeded.All the estimators converted square feet to square yards by dividingsquare feet by nine. This algorithm is essential in the carpet-laying

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Figure x

Dean'siagrame3

Dean's diagram.business since measurements are taken in feet but carpet must be or-dered from a supplier in square yards. In a conversation with Dean Iasked him why this conversion worked.

Joanna:fyoujustknow thelengthand widthof aroom,how do you findhowmanysquareyardsof carpetyou need?Dean:Well, f the roomis 12'by 8', thenyou take 12 times 8 divided by 9.Joanna:Whyo you divide by 9?Dean:That'stheway you convertsquarefootage to squareyardage.Joanna: kay,but wheredoes the9 come from?Dean: don't know. [Pauses.]MaybeI don't understand he question.Wheredoes the9 come from?Joanna: eah,why don't you divide by 8 or 6?Dean:Well,when you have squarefootage [draws iagram ith3-by-3grid; eeFigure ], each of these squares s a squarefoot and thereare 3 feet in a yard[putsx's inside he three quaresn therightcolumnof thegrid]and then 3 feetacross[putsx's inside hree quaresn thetoprow].So that makes9.By using a diagram Dean was able to illustrate, although not fullyarticulate, that in one square yard there are nine square feet and toconvert from square feet to square yards involves dividing by nine.Geometry

In addition to the 3-4-5 right triangle discussed earlier, I also observedthe use of another geometry concept: constructing a point of tangencyon a line and drawing an arc tangent to the line. This particular installa-tion involved completing a tiling job by installing base. At one pointduring the installation, Jackcame upon a pipe sticking out of the bottomof the wall. He placed the base piece to be installed around the pipe atthe top of the pipe but in line with the adjacent piece of glued base (seeFigure 4). Jackmade two vertical cuts in the base piece: one on each sideof the pipe. Then he placed the base piece to the left of the pipe so thatthe piece was in the correct position vertically but not horizontally.

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Baseplaced ntopofpipeand utvertically.

Baseplaced djacentopipeandmarkedorizontally.

Baseturnedver,horizontal ark xtended,rccut nbase.

I)FigureCuttingbasetofit aroundpipe.

Jackmade a small mark(notacut)on theedge of the basepiece in linewith the top of the pipe. He then turnedthepiece over and marked n astraightline (makinga faintline) to meet the verticalcuts made earlier.Fromthose verticalcuts,Jackcutin an arc so thatthearc was tangenttothe horizontal ine.He thenglued the basepieceandput it in place; t fitvery well aroundthe pipe. I questionedJackaboutwhat he had done.

Joanna:How did you know how to cut that arc so it would fit so well?Jack:Oh,afteryou've done it forawhile,you justget a feel for it. I knew thatif I cut justup to the markit would make a circle[sic],and then it would fitaroundthepipe.Whereas n most geometryclassesstudents constructa line tangenttoa given circle, n this instanceJackconstructeda point of tangencyon agiven line and drew an arc tangent to the line. Note that it was not

necessaryin this case to constructa circle because the pipe was at thebottom of the wall, touching the floor. Thus the piece of base did notneed to go aroundthe bottomside of the pipe.Although he did not articulate his, it appearedobvious to me thatJackknew fromexperiencethat,if he cut from the two verticalcuts inan arc up to a certainpoint on the horizontalline, a symmetricalarc

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Mathematicsracticen CarpetLayingwould be formed. Jackchose the point of tangency such that a segmentdrawn from the point of tangency would be a perpendicular bisector ofthe segment connecting the endpoints of the two vertical cuts. He chosethis point of tangency by estimating; there was no measuring involved.However, it was clear to me that he knew what point of tangency on thehorizontal line would give him the best-fitting cut.Ratioand Proportion

I observed the use of ratios and proportions during estimating situ-ations. In one instance, Gene (an estimator) was measuring a blueprintin the process of preparing an estimate for a commercial job. Theblueprints were drawn in a scale of 1/4 inch to 1 foot, and Gene wasusing a drafting ruler to measure the maximum length and width of eachroom. When he noticed that I was observing him, Gene explained whathe was doing.Gene:The scale here is one-fourthof an inch to one foot, and this here rulerhas that scale on this side. So when I measurethe length of a roomon here[touchingheblueprint],he rulershows what the actual measurementsof theroomare.Joanna:How aboutwhen you find the perimeter?What do you do since thetrundlewheel is in the scaleof one-eighthof an inch to one foot?Gene: ometimeswe get printsdrawn in that scale;so then you justread it.Butforthisone, sincewhenyou measurean inchon here the wheel considersiteightfeetbut it'sreallyonlyfourfeet,you have to takehalfthe measurementthat the wheel shows.Gene was mentally working with the following proportion: (trundlewheel measurement in feet) - (8 feet per inch) = (unknown measure-ment in feet) - (4 feet per inch). He had simplified the expression so thathe only had to divide the trundle wheel measurement by two to find the

unknown measurement in the desired scale: unknown measurement infeet = (4 feet per inch) x (trundle wheel measurement in feet) + (8 feetper inch) = (trundle wheel measurement in feet) - 2. Although Gene didnot speak of or write a proportion, he used a proportion that he hadsimplified to be a one-step division calculation. However, Gene indi-cated through his explanation that he understood why the calculationprocedure that he used produced the desired information.Analysis

By (a) analyzing the occupations of estimator and installer, (b) exam-ining the work tasks of taking field measurements, making sketches,deciding on best estimates, interpreting the estimator's sketch and esti-mate, measuring, deciding on best installations, and installing, and (c)noting the constraints that must be considered during these processes, Iwas able to classify the mathematics concepts that I saw the respondents

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Anthropology&EducationQuarterly Volume 25,1994in this study use. Mathematics concepts are an important component ofthe mathematics practices of people, and the concepts discussed aboveare essential to the mathematics practice of carpet layers.Although the carpet layers were not always able to articulate themathematics concepts that they were using in their mathematics prac-tice, by studying their interaction with their environment I was able toexamine their knowledge in action and make my interpretation of theconcepts that they were using. Through observing numerous constraint-filled actions of the estimators and installers involving finding area interms of square feet, estimating the amount of base needed, and estab-lishing 45? and 90? angles, I concluded that measurement was a perva-sive mathematics concept in this context. I also gained insight into thebreadth of the concept of measurement in the floor-covering context.I categorized the computational algorithms as an important mathe-matics concept in this context after observing that the estimators andinstallers use them continuously to determine quantities needed andnoting their measurement foundation. Although Jack did not indicatethat he knew the concept of constructing a point of tangency on a lineand drawing an arc tangent to the line, his actions led me to concludethat he was using this concept. He knew that this procedure workedfrom his experience, and he knew what point of tangency would workthe best. By observing him during this work task as he confronted aconstraint that he needed to deal with, I was able to see that he knew themathematics concept. Likewise, by analyzing Gene measuring a blue-print, I was able to understand how he used ratios and proportions tochange from one scale to another.Mathematical Processes

Besides the use of mathematical concepts, the estimators and installersmade use of several mathematical processes: measuring and problemsolving. These two processes are discussed in the following sections, andsome of the ideas are illustrated through examples from the field data.Measuring

As would be expected, the process of measuring is widespread in thework of floor covering estimators and installers. Although being able toread a tape measure is vital, other aspects are equally as important inthe measuring process: estimating, visualizing spatial arrangements,knowing what to measure, and using nonstandard methods of measur-ing.Estimating. Although all measurements can be considered estimates(e.g., a measurement is only as accurate as the instrument used), floor-covering estimators and installers make use of estimates in a much moreconscious manner. For example, while preparing a bid for a commercialjob from a set of blueprints, Gene told me that all these measurements

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Mathematics racticen CarpetLayingare estimates: "That's why we're called estimators. All these measure-ments I get from here are rounded up to the nearest six inches. Since wehave to stand by our bid price (for a commercial job), we have to allowfor a difference between these prints and the actual building. Of course,you don't want to overestimate or you might not get the job."Field measurements are also rounded up-to the nearest inch. Todd,the installation manager, told me during a field measurement, "To thenearest inch is good enough for most of the jobs we do. A half-inch ofcarpet isn't going to cost the customer much money, and it gives us thelittle bit of extra we need. The only time we use fractions of inches iswhen we have to get an exact amount-like if the carpet is supposed tobe exactly 12 inches from the wall because they have a border, or if tileis being laid and there is supposed to be an equal amount of carpet oneach side of the tile."Estimates are made during blueprint and field measurements forreasons of time and cost efficiency. More precise measurements wouldrequire more time and effort in measuring and would slow down theprocess of preparing an estimate for a job. In installation situations,installers estimate angle measurements when needing to cut carpet, tile,vinyl, or base at a certain angle. Through experience they become quiteskilled at estimating 45? and 90? angles, as well as certain lengths suchas two inches or four inches. When installing carpet, a carpet piece mustbe laid on the floor so that it extends up the wall two inches. These twoextra inches are needed for trimming along the wall. On a wall with adoorway, the carpet piece is laid so that it extends up the wall fourinches. This extra carpet is needed for trimming around the door jamband for meeting other carpet or floor covering in the center of thedoorway.Visualizing SpatialArrangements.Spatial visualization plays a key role inthe measuring process. Visualization is used when deciding how cutsmight be made in a piece of carpet or when developing a mental pictureof how an installation job should look upon completion. Any estimationjob that will use fill pieces requires the estimator to visualize how thosefill pieces will be cut from a larger piece of carpet. Complicating thisprocess are the constraints of seam placement and matching the fill-piecenap with the nap of the adjacent carpet pieces. Figure 5 shows how Deanfigured a residential carpet estimate for a bedroom and bathroom. WhenDean started to estimate the carpet needed, the sketch he used was likethe top diagram in Figure 5 without the seam lines. He first figured thathe needed one piece of carpet 12' by 14'9"for the region labeled A. Deandecided the bathroom (region B) would require a piece of carpet 5' by4'8". However, since a customer must buy a piece of carpet 12 feet wide,the piece for B would be 12' by 4'8", making the total amount of carpet,so far, 12' by 19'5" (one piece 12' by 14'9" and one piece 12' by 4'8").Dean hoped to use the carpet that remained (after cutting a piece 5'by 4'8" from the 12' by 4'8" piece) to fill in the remaining area in the

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closets notcarpeted)

E 1 - _--i- seam

12' -

19' 5"44

~A~~~B

4' 8"19' 8"

A _B

4' 11"Figure5Spatial visuali7ation in estimating.

bedroom.Deanappearedto have a mentalpictureof how piecescouldbe cut from a largerpieceof carpetto fit into the threeregions(A,B,andC) as he sketchedarectangle orepresent hepieceof carpet12'by 19'5"(see the middle diagram in Figure 5). He drew a line to divide the

446

4' 6"

12'

12'

7'

Eachpieceof C is4' 11"x 2' 2".waste

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rectangle into two sections: one 12' by 14'9" and one 12' by 4'8". Deanthen drew a line to divide the 12'-by-4'8" section into two parts: one 5'by 4'8" and one 7' by 4'8".Since each fill piece had to be 2'2" wide (to reach from carpet piece Ato the wall and have enough for trimming purposes), only three fillpieces could be cut from the 7' by 4'8" piece. Dean then realized that thiswas not enough carpet to fill the area: "Threepieces 4'8" long is only 14';that's not enough. We're going to have to make this second piece longerthan 4' 8"."Dean figured that since he needed 9 more inches, each piecewould have to be 4'11" long: "So, this whole piece [pointing to therectangle on the sketch] will be 19'8" long." Dean then drew anotherrectangle to represent a carpet piece 12'by 19'11"and made the divisionsas before (see the bottom diagram in Figure 5): "Eachof these fill piecesis goin' to be 2'2" by 4'11"-and that will do it."This diagram helped me visualize how the carpet could be placed sothat the nap would be running in the same direction, but Dean seemedto be able to picture this mentally. I remarked that I really had to thinkabout how the pieces go in (turning them in my mind as necessary) tomake sure that the naps were running the same way. Dean replied,"Figuring the fill was the hardest thing for me to learn how to do whenI first started estimating. You have to be able to put the carpet piecestogether in your head and have them all going the same way."Dean used a sketch to check his mental picture. Since the carpet laidin the bathroom would meet the bedroom carpet, the nap had to berunning the same way on both pieces. Similarly, the fill pieces had tohave the nap running the same way as the large piece in the bedroom.By sketching the carpet piece in this manner and drawing the cuttinglines, Dean was able to check this representation against the mentalpicture that he had formed earlier and see that the naps of all the pieceswere running in the same direction.Like the estimators, the installers made use of spatial visualization inmentally picturing how the carpet should be cut and installed beforebeginning their work. Even though the estimator had made a sketch andindicated the seam placement and direction of the nap, the installeralways checked the same information before starting the installation.Knowing What to Measure. Essential to the mathematical process ofmeasuring is the knowledge of what to measure. In order for measure-ments of an object to be of any use, they must quantify the desiredcharacteristics. For example, when an estimator needs to find the lengthof a room, he or she must be certain to measure the length at itsmaximum to avoid having too little carpet at the time of installation. Iobserved several situations in which estimators and installers used theirknowledge of what to measure to figure floor-covering jobs. Some ofthese situations involved measuring nonrectangular rooms. Knowingwhat were the necessary and sufficient measurements to take for a setof steps with one side exposed was another situation that I observed.

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t 6 ^ 12'2"I Hallway

Hallway2 Hallway1

5'1" 6'

Figure6Measuringintersectinghallways for a tile installation.Preparing for tile and hardwood installations requires knowledge ofwhat to measure since both of these floor coverings must be installed tobe lengthwise and widthwise symmetrical about the center of the room.One tile installation that I observed involved two parallel hallwaysintersecting a third hallway perpendicular to both. For purposes ofidentification, I will call the two parallel hallways 1 and 2 and the otherhallway 3 (see Figure 6). Steve was the installer for this job, and he tookseveral measurements before chalking lines, spreading glue, and begin-ning the installation. He measured the maximum width of each of thethree hallways and divided each of these measurements by two to findthe distance that each bisector would be from the respective wall.Steve realized that in this case measuring the length of each hallwayis unnecessary because of the arrangement of the hallways. Finding thewidth bisector and amount of fill for hallways 1 and 2 determines theamount of fill for the length of hallway 3. Likewise, finding the widthbisector and amount of fill for hallway 3 determines the amount of fillfor the length of hallways 1 and 2.Because the hallways intersected, it was also necessary to measure thedistance between the width bisectors of the two parallel hallways. If thismeasurement was not a whole number of feet, some adjustment wouldneed to be made so that the tile laid in each of the parallel hallwayswould meet and match the tile laid in the third hallway. It turned outthat the width bisectors of these two hallways were 12'2" apart, and soan adjustment had to be made before installing the tile.Steve knew this measurement was necessary; a less-experienced in-staller might only have measured the lengths and widths of each hall-

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way and started laying tile. If, as in this case, the distance between thewidth bisectors of hallways 1 and 2 was not a whole number of feet, theinstaller would face difficulty during the installation. He or she wouldchoose one hallway to begin (e.g., hallway 1). After laying all thewhole-tile pieces in hallway 1, work would begin on hallway 3. Theinstaller would encounter a problem when he or she would try to matchthe tile in hallway 3 with the chalk lines set up in hallway 2. This in turnwould negate the plan established forequal fill on both sides of the widthof hallway 2.Using Nonstandard Methods of Measuring. I observed a number of situ-ations in which installers measured in nonstandard ways-not involv-ing tape measures or other marked measuring devices. The majority ofthese situations involved object-to-object measuring, that is, measuringone object against another. For example, at one installation site Phil (aninstaller's helper) measured tack strips against the perimeter of theroom. The job involved installing carpet in a basement utility room. Thisroom had not had carpet in it previously, and so tack strips had to benailed down before the installation of the pad and carpet could begin.Tack strips come from the factory four feet in length and must be cut tofit. The perimeter of the room could have been figured fairly easily (theroom was a rectangle: 11' by 11'4"),and tack strips could have been cutto fit the room. However, Phil simply started at the door and laid andnailed pieces of tack strip along one wall until he came to a place wherea strip needed to be less than four feet long (approaching a corer). Hedid not measure the length needed for the strip but rather laid the stripfrom the corner back to the last piece nailed down and cut off the partthat overlapped. He continued around the room, measuring the stripsagainst the perimeter and cutting them accordingly.ProblemSolving

The mathematical process of problem solving is used by floor-cover-ing workers every day as they make decisions about estimations andinstallations. However, the problem solving that occurred in this contextis slightly different from how problem solving is typically defined.Problem solving is commonly thought of as the process of coordinatingprevious experiences, knowledge, and intuition in an effort to determinean outcome of a situation for which a procedure for determining theoutcome is not known (Charles et al. 1987). Problem solving in thefloor-covering context deviated from this definition in that proceduresfor determining outcomes were usually known. However, unfamiliarconstraints (e.g., a post in the middle of the room) and irregular shapesof rooms forced floor-covering workers to coordinate their previousexperiences, knowledge, and intuition to determine outcomes of situ-ations that they faced.



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Anthropology EducationQuarterly Volume 25,1994The problems that estimatorsand installers encountered requiredvariousdegreesof problem-solvingexpertise.As the shapeof the space

being measured (or in which floor-coveringmaterial was being in-stalled)moved away from a basicrectangular hape,the level of exper-tise requiredincreased.To solve problemsoccurringon the job, I ob-served Miller's Floor Coverings employees use four categories ofproblem-solvingstrategies:using a tool, using a picture,checkingthepossibilities,and using analgorithm.Usinga Tool.I observed both estimators and installers use tools thataided them in the problem-solvingprocessof decidingan estimationorinstallationjob. In particular,both estimators and installers used tapemeasures.Estimatorsused atapemeasure otakethemeasurements hattheyneed to makeadecision about how thecarpetshouldbe laid,wherethe seams shouldbe,andhow muchcarpet s needed. Installers ikewisemeasuredto checktheestimator'smeasurementsandmadeadjustmentsif necessary.A measurement trundlewheel and a draftingruler were tools thatwere used in the preparationof commercialbids. Gene used a trundlewheel with blueprintsto measurethe perimeterof roomsin whichbasewould be installed.The trundlewheel had the scaleof 1/8 inch to 1foot,and even though measurements akenon a blueprintdrawn in a scaleotherthan1/8 inch to 1 foothad to be converted,the trundlewheel wasusefulbecause it couldmeasurethe perimeterof anyshapequicklyandwithacceptableaccuracy.Becauseof the wheel feature hetrundlewheelcould be maneuveredeasily around corners and jutson the blueprintsto accuratelyassess the perimeterof the region.Genealso used a draftingrulerto measurethe maximumlengthandwidth of eachroombeing carpeted.A draftingruler has six measuringsides, each with two scales. Gene told me thatmost of the blueprintsheworkswitharedrawninoneof two scales:"Mostof theblueprintscomein 1/4 inch [to]1 foot or 1/8 inch [to]1 foot scales;so Imainlyuse thosetwo scales on theruler,butif Ineed the othersIhavethem."Thedraftingruleris a useful tool in measuringand convertingthe blueprintmeas-urements to feet. These measurementscan then be used with otherfactors(e.g., cost efficiency,seam placement, patternof carpet, type ofcarpet) o determinehow thecarpetshouldbe laidand how muchcarpetis needed for the job.Using a Picture.In all the floor-coveringsituations that I observed,pictures were used to help visualize the situation and function as aproblem-solvingtool. These pictures were either blueprintsor hand-drawn sketches.I observedPhil use a drawing to solve the problemofmatchingthepattern nacarpet n acommercial arpet nstallation.Thisinstallationwas in a universitydormitoryand involved several inter-secting hallways on a numberof floors and carpetwith a patternthatrepeatedevery threeinches.Thehallwayswere very long (e.g.,one wasapproximately125 feet long), and since the carpetwas sent from the

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Mathematicsracticen CarpetLayingside seams/11

Lside seam/ side seam4

CB

E

side seamsideeamide seam

D

--- Dutt eam1 /

F

Figure7Carpet nstallationinvolving intersecting hallways.

supplier in rolls of varying lengths, all less than 100 feet, some butt seams(two carpet pieces seamed end-to-end) were necessary (see Figure 7).Phil, who earlier in the summer had been Jack's regular helper, wasnow two weeks into heading an installation crew with Matt as his helper.Because of his newly acquired status and his desire to prove himself asan installer, Phil was very thorough about sizing up a situation beforehe started the installation. In this case he saw that it was important toconsider carefully where to begin the installation and in what order toinstall the remaining carpet pieces given that the pattern must match atall the hallway intersections. The sketch had carpet pieces labeled Athrough F,but these letters were simply to label the pieces, not to suggestthe order of installation.All of the hallways were 4'4" wide and, while pieces B, D, and F were12' wide, and pieces A, C, and E were 4'8" wide. The carpet nap wouldbe running in the direction of the length of piece B.Phil talked aloud and

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Anthropology&EducationQuarterly Volume 25,1994tried different ways of numbering the six pieces of carpet on the sketch,indicating the order of installation and discussing the situation withMatt. Occasionally he would make comments directed toward me,explaining why a certain choice was or was not feasible. Finally, Phildecided that they would install the carpet piece labeled F first and thenD, matching the pattern at the butt seam. The third piece to be installedwould be E, and this piece must match the pattern in F at the side seam.This would require some careful installation since, with the hallway 4'4"wide and the carpet 4'8" and the pattern repeating every 3", there wouldbe very little room for maneuvering.Phil told me that he would then install piece B, matching the patternin E at the side seam. Piece C would then be installed. This would be themost difficult carpet piece to install since this piece must match thepattern in B and D. Phil remarked that he might have to work to shiftone of the pieces (B or D) to match at the side seam. Piece A would beinstalled last, cut into two pieces matching B at the side seams.Phil used the drawing of the hallways to visualize the installation andseam placements. By ordering the pieces and thinking about how install-ing the pieces in this order would affect the ease of matching the pattern,he was able to choose an appropriate order for installation.

Checkingthe Possibilities. I observed a number of situations in whichestimators or installers checked possible solutions when solving a prob-lem. These situations can be grouped into four categories: weighing costefficiency against seam placement, checking the amount of carpetagainst the area to be carpeted, deciding on roll cuts, and deciding ontile and hardwood work.The following example illustrates the problem-solving process ofweighing cost efficiency against seam placement. One carpet-estimatesituation that I observed involved a pentagonal-shaped room in a base-ment. I accompanied Dean as he took field measurements and figuredthe estimate. The maximum length of the room was 26'2", and themaximum width was 18'9" (see Figure 8). Dean decided that this roomwould have to be treated as a rectangle and figured how much carpetwould be needed by checking two possibilities: running the carpet napin the direction of the maximum length, and turning the carpet 90? sothat the carpet nap ran in the direction of the maximum width.In the first case, two pieces of carpet, each 12' by 26'4", would need tobe ordered. After a 12'-by-26'4" piece of carpet was installed, a piece of6'11"-by-26'4" carpet would be needed for the remaining area. Sinceonly one piece 6'11" wide could be cut from 12'-wide carpet, multiplefill pieces could not be used in this situation. Thus, a second 12'-by-26'4"piece of carpet would be needed for a total of 70.22 square yards. Theseam for this case is shown by a thin line in Figure 8.Turning the carpet 90? would require two 12'-by-18'11" pieces and a12'-by-4'9" piece for fill. The 12'-by-4'9" piece would be cut into fourpieces, each 2'4" by 4'9". The seams for this case are shown by thick lines

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22'4"

12'

I

.1 A1 +\ / /^ 18' 9"

seamsfor 26' 2"second \case X

seam for -first case

12'

FigurePentagonalasement oom.in Figure8. The total amount of carpetneeded for this case would be56.78square yards.This second case has more seams than the first,butthefillpieceseams areagainstthebackwall,out of theway of thenormaltrafficpattern.Thus,these seams do notpresenta large problem.Inbothcases there would be a seam in themiddleof the room.Thecarpet n thefirst case would cost at least 200 dollars more than the carpet in thesecond case. Dean weighed the cost efficiency against the seam place-ment anddecided that thecarpetshould be installed as described n thesecond case.UsinganAlgorithm. reviouslyI discussed the algorithmsthat are usedby estimators in measuringsituations to find the quantityof materialsneeded for an installationjob. These algorithmswere used, not in aproblem-solvingmanner,but ratherto obtain a numberrepresentingaquantityof material. Put anotherway, the estimators did not use thealgorithmsto help them makedecisions; hedecisions hadalreadybeenmade and the task was simply to find how much of each materialwasneeded.Unlike the use of algorithms in the foregoing sorts of estimatingsituations,Iobservedinstallersusinganalgorithm n situations nwhichit helpedtheinstallermakeadecisionconcerninganinstallationjob.The

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algorithm that I observed installers use in the process of solving theseproblems involved determining how tile should be laid in order that itbe lengthwise and widthwise symmetrical about the center of the room.The algorithm involved several steps. The first step was to measurethe maximum length and width of the room. These measurements werethen divided by two and expressed in feet and inches. The center of theroom was found by using these half-measures to measure out from therespective walls and chalk lines representing the length- and width-bi-sectors. The intersection of these bisectors was the center of the room.

However, if only these lines were used as a guide for laying the tile, aproblem might arise: Besides being lengthwise and widthwise symmet-rical about the center of the room, tile must also be laid such that the fillpieces are greater than or equal to six inches (in order to stay in place).Thus, lining up full tile pieces against the bisectors that establish thecenter of the room and extending the tile to the wall may result in fillpieces with width less than six inches filling the gap between the fulltiles and the wall.What was done to avoid this problem was to consider the maximumlength and width measurements one at a time, after they had beendivided by two. Expressing these measurements in feet and inchesprovided the installer with necessary information: The number of feetrepresented the number of full tiles from the center of the room to thewall, and the number of inches represented the width of the fill piece oftile to bridge the gap between the full tiles and the wall (tiles are 1' by1'). If the number of inches was less than six, the tiles were shifted sixinches (half of a tile). In other words, instead of a full tile edge touchingthe length-bisector (or width-bisector), the tile was placed so that thebisector ran through the center of the tile (see Figure 9). This shifted thetiles so that the fill tile on each of the two sides split by the length-bisector

14' 6"

6'9" 7'3"

?VFigure9Shifting tiles six inches.

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Masingila Mathematics racticen CarpetLaying 455

18' 8"

7'781/2"

i- p

1' 7" 1'7"

X; o

4*

I) aa: "o CD

Figure10Measuringfor a tile installation.(or width-bisector) was greater than or equal to six inches, therebyhelping the fill tile to stay glued down.Sometimes both the length and width tile placements needed to beshifted, sometimes only one, and sometimes the measurements workedout so that no tile shift was needed. Along with considering the pre-viously mentioned constraints, other complicating factors often arose. Adiscussion of a situation in which this algorithm was used will help toillustrate its use.The first tile installation that I observed was at a commercial job sitewhere tile was being installed in a kitchen. The kitchen had cabinetsalong a large portion of the walls but also had spaces between cabinetsthat had to be tiled (see Figure 10). Steve was the installer figuring thisjob, and he first measured the maximum length of the room and foundit to be 18'8".Taking half of this gave 9'4". Since four inches is too shortfor the fill pieces, the tile had to be shifted six inches-making the fill oneach end 10". Thus, a measurement of 9'4" from one of the end walls



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Anthropology& EducationQuarterly Volume 25,1994toward the center of the room would reach the center of a full tile (afterthe tile was installed) instead of the edge of one.

Steve next measured from cabinet front to cabinet front across thewidth of the room and found that measurement to be 7'81/2".Half of thisis 3'101/4", nd so no shifting was necessary. However, the tile across thewidth of the room would extend, at some parts, all the way to the wallsince cabinets did not cover all the wall space. In considering the fill tilefor these cabinet-free spaces, Steve measured from the front of thecabinets back to the wall and found this measurement to be 1'7". Thus,from the width-bisector to a cabinet-free wall the measurement was5'51/". Since the fill tile for these spaces was less than six inches, Steveconsidered whether he should shift the tiles six inches across the widthof the room. After some calculations and muttering to himself, Stevedecided not to shift the tile because that would throw off the fill piecesfor the region from cabinet front to cabinet front. Besides, since most ofthe cabinet-free space along the walls would be later filled by appliances,Steve noted that it was more important to have the necessary fill for thespace along the cabinets.This example illustrates how installers used this algorithm to aid themin solving the problem of installing tile. The algorithm did not producea number that solves the problem, but rather it was a tool to be used formaking decisions about how the problem should be solved.

AnalysisJust as analyzing the occupations, work tasks, and conditions in-volved in floor-covering work allowed me to interpret the mathematicsconcepts used by the workers, I was also able to interpret the mathemat-ics processes that they used. I categorized these as the processes ofmeasuring and of problem solving. Not only did I recognize that meas-

uring is an essential process that is used continuously, but I gainedinsight into how the process of measuring is varied and context-bounded.By analyzing the different work tasks and conditions faced by theestimators and installers, I observed that measuring is multifaceted; itinvolves estimating, visualization, knowing what to measure, and theuse of nonstandard methods of measuring. Moreover, the measuringprocess is dependent upon the situation; different situations requiretaking different measurements, visualizing different arrangements, andusing different methods of measurement. By using the three levels of

analysis, I was able to understand what the carpet layers knew aboutmeasuring and how flexible they were in their thinking. I also was ableto examine the problem-solving skills and strategies that the estimatorsand installers used and classify them into several categories. Activitytheory provided the framework that allowed me to observe the interac-tions of the workers with their environment.

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Mathematics ractice n CarpetLayingDiscussion

As mentioned previously, Bishop (1988) has laid out what he called"six fundamental activities" that are engaged in by every cultural groupand through which mathematical knowledge is developed. This re-search focused on two of those activities: locating and measuring. Locat-ing involves "exploring one's spatial environment and conceptualisingand symbolising that environment, with models, diagrams, drawings,words or other means" while measuring is "quantifying qualities for thepurpose of comparison and ordering, using objects or tokens as meas-uring devices with associated units or 'measure-words'" (Bishop1988:182-183).My reason for using activity theory as a conceptual framework for thisresearch is that it allows one to address the relationship between know-ing and doing by proposing that the "starting point and primary unit ofanalysis should be culturally organized human activities" (Scribner1985:199). Focusing on the activities of floor-covering estimators andinstallers and examining the tasks involved in their work and the con-ditions that they encounter and with which they must deal in each taskprovided me the opportunity to see these persons using their knowledgein order to interact with their environment through activities of locatingand measuring; this systems approach is at the core of activity theory.To understand mathematics practice in the everyday context of carpetlaying, I needed to analyze the activities and the actions in which theyare embedded; activity theory provided the framework for this analysis.The research discussed here adds to other research on the mathemat-ics concepts and processes used in everyday situations by extending thatresearch to a measurement context. Other research has examined the useof (a) arithmetic (e.g., de la Rocha 1985;Murtaugh 1985b;Scribner 1984c)(b) geometry (e.g., Millroy 1992), and (c) rational number concepts (e.g.,Carraher 1986). This research on mathematics practice in the floor-cov-

ering context provides some insight into what is involved in measure-ment practice in everyday situations.Furthermore, this research combines some aspects of different typesof research in this area. Some researchers have focused on situations inwhich people develop strategies that become fairly routinized (e.g., Lave1988; Scribner 1984b, 1984c, 1984d), while others have examined situ-ations in which problem solving is adaptive (e.g., de la Rocha 1985;Saxe1988). The mathematics practice of the floor-covering workers involvesfeatures of both of these aspects. Many of the procedures and algorithmsused in this context have become somewhat routine for the carpet layers.For example, the installers know how to follow the algorithm describedpreviously for deciding how to lay tile. However, the numerous andvaried constraints that occur at different job sites force the carpet layersto be adaptive in their problem solving. At one location Steve was facedwith carpeting a room with a rectangular-prism post from floor to ceilingin approximately the center of the room. He needed to install the carpet

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Anthropology& EducationQuarterly Volume25,1994around the post and minimize the seams. Thus, this research builds onaspects from two strains of research on mathematics practice in every-day situations.Future research in this vein should examine how the gap betweenmathematics practice in school and out of school can be closed. Someresearch on mathematics practice in everyday situations has contrastedmathematics practice in school with mathematics practice in everydaysituations and noted the gap between these two (e.g., Carraher et al.1985; Masingila 1993). Knowledge gained in out-of-school situationsoften develops out of activities that: (a) occur in a familiar setting, (b) aredilemma driven, (c) are goal directed, (d) use the learner's own naturallanguage, and (e) often occur in an apprenticeship situation allowing forobservation of the skill and thinking involved in expert performance(Lester 1989). Knowledge acquired in school situations all too oftengrows out of a transmission paradigm of instruction and is largelydevoid of meaning because of lack of (a) context, (b) relevance, and (c)specific goal.It is my contention that the gap between in-school and out-of-schoolmathematics practice can only be narrowed after ways in which mathe-matics is meaningful in the context of everyday life have been deter-mined. As researchers work to understand mathematics practice ineveryday situations, we should also explore how teachers can buildupon students' out-of-school knowledge and engage them in solvingproblems containing real-life constraints so that their learning of mathe-matics is more meaningful.

JoannaMasingilaisanassistantprofessorof mathematics ducationatSyracuseUniversity.

NotesAcknowledgments.hisarticle s basedon theauthor'sdissertation,Mathemat-ics PracticendApprenticeshipnCarpet aying: uggestionsorMathematicsduca-tion, completed in May 1992 at Indiana University-Bloomington,under thedirectionof FrankK.Lester,Jr.1. Note that,although algorithmsare processesratherthanconcepts,I amincluding computationalalgorithmsas a conceptrather hana processbecauseI am interestedin the mathematical onceptof measurementunderlying hesealgorithms.2. Pseudonymsareused for all respondents.

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