An Ethnographic Study of the Mathematical Ideas of a Group of Carpenters (Journal for Research in Mathematics Education. Monograph, N.? 5)

ISSN0883-9530

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National Council of Teachersof Mathematics

A MonographSeries of the National Council of Teachersof Mathematics The JRME monograph series is published by the Editorial Panel as a supplement to the journal. Each monographhas a single theme related to the learningor teachingof mathematics.To be considered for publication,a manuscript should be (a) a set of reportsof coordinatedstudies, (b) a set of articles synthesizing a large body of research, (c) a single treatisethatexamines a major research issue, or (d) a report of a single researchstudy thatis too lengthy to be publishedas a journal article.

Series Editor FRANK K. LESTER,JR., IndianaUniversity, Bloomington, IN 47405 Series Editor-Elect DOUGLAS A. GROUWS, University of Missouri, Columbia,MO 65211

Associate Editor DIANA LAMBDIN KROLL,IndianaUniversity, Bloomington, IN 47405 EditorialPanel CATHERINE BROWN, University of Pittsburgh;Chair MICHAELT. BATTISTA, Kent State University, Ohio Any person wishing to submita manu- PAUL COBB, PurdueUniversity scriptfor considerationas a monograph MARTIN L. JOHNSON, University of should send fourcopies of the complete Marylandat College Park manuscript to the monograph series CAROLNOVILLIS LARSON, University of Arizona editor.Manuscriptsshouldbe no longer than 200 double-spaced typewritten PATRICIA S. WILSON, University of Georgia pages. The name, affiliations, and ALAN OSBORNE, Ohio State University; of each qualifications contributing BoardLiaison

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Copyright? 1992 by THE NATIONAL COUNCILOF TEACHERSOF MATHEMATICS,INC. 1906 AssociationDrive, Reston, Virginia22091-1593 All rightsreserved

Data Libraryof CongressCataloging-in-Publication Millroy, Wendy L. An ethnographicstudy of the mathematicalideas of a groupof carpenters/ by Wendy Lesley Millroy. cm. -- (Journalfor researchin mathematicseducation. p. ISSN 0883-9530; no. 5) Monograph, Includesbibliographicalreferences(p. ). ISBN 0-87353-341-0 1. Geometry. 2. Carpentry--Mathematics.3. Mathematics--Study and teaching. I. Title. II. Series. QA464.M49 1991 370.15'651--dc20 91-39453 CIP

The publicationsof the NationalCouncil of Teachersof Mathematicspresenta variety of viewpoints. The views expressed or implied in this publication, unless otherwise noted, should not be interpretedas official positions of the Council.

Printedin the United States of America

iii Table Of Contents Dedication .......................................................................................................................

vii

Acknowledgements ........... ....................................................

vii

Abstract ..........................................................................................................................ix Chapter One: Introduction ........................................ ....................... 1 Overview ...................................................................................................................1 The Cultural,Social,andPoliticalComponentsof Mathematics........................ 4 Mathematicsin EverydaySettings:"EverydayCognition"................................. 4 6 Settingfor the PresentResearch ........................................ ....................... An Ethnographic MethodologicalFramework: Approach................................... 7 A Constructivist EpistemologicalFramework: Approach................................... 9 Mathematicsin EverydaySettings:Ethnomathematics ....................................... 10 The paradoxembeddedin ethnomathematics ................................................ 11 A Strategyfor ConductingPracticalResearchin Ethnomathematics .................. 11 16 Organizationof the Monograph ............................................................... Chapter Two: Conceptualizingthe ResearchProblem ............................................. 17 Introduction................................................................................................................ 17 CultureandCognition...............................................................................................17 .............................................................22 Knowing-in-ActionandReflection-in-Action The Epistemologyof Constructivism........................................................................23 Whatis CultureandWhatConstitutesa CulturalGroup?.......................................... 28 The Cultural,Social,andPoliticalAspectsof MathematicsandMathematics Education...................................................................................................................30 Researchon the CreationandUse of Mathematicsin EverydaySettings ................. 39 EverydayCognition ............................................................................. 39 Ethnomathematics ..................................................................... 45 Statementof the Problem ...............................................................50 ResearchObjectives................................................................... 51 Chapter Three: In the Workshop ....... ........................................................ 53 Introduction....................................................................... 53 Contextof the ResearchSetting ....... ........................................................ 53 Castof Characters........................................................................................... 55 The Ownerof the Business................................................................56 The Artisans......................................................................................................... 57 The Apprentices................................................................... 64 The Laborers........................................................................................................ 66 Historyof the Workshop...... ......................................................... 68 Chapter Four: Methodology ............ ................................................... 69 Introduction................................................................................................................ 69 EthnographyandEducationalResearch................................................................... 69 ObtainingEntree ..... ...................................7.......................1. InitiationandAcceptance............................................................... 75 Focus on Problematics..............................................................................................82 DataCollectionMethods ...........................................................................................83 ParticipantObservation................................................. .................... 83 Interviews ............................................................................................................86

iv Additional Methods ................................................ .....................................

88

The Collectionof Artifacts.............................................................................. 88 ResearcherIntrospection........................................ .................. 89 The EmotionalDemandsof Ethnographic Research:SomeCopingStrategies ....... 90 Exit ............................................................................................................................ 91 Chapter Five: Results and Analysis:Mathematizingby the Carpenters ................. 93 Introduction........ 93 .................................................................. PartOne:The Episodes .................................................... 93 .............. The Establishmentof Unitsof Analysis .......................................................... 93 Criteriafor the Selectionof Unitsof Analysis................................................. 94 Introduction to the Episodes ........................................ .................. 95 The MainCharactersin the Episodes.......................................................... 96 Episode 1: Drawinga stardesignandlocatingit on a rectangularboxlid .......... 96 Episode2: How to findthe centerof the box lid .................................................. 98 ................ 100 Episode3: The staris crookedandtwo squarerulesaredifferent Episode4: Jackexpressessome opinionsaboutmathematics............................ 101 Episode5: Whenis a squarerulesquare?.......................................................... 101 Episode6: Jackandthe "right"answer .......................................................... 110 Episode 7: "Do it by proportion"-the table with the inlaid pattern..................... 111 Episode 8: Clive, Mr. S, and the dovetail joints ................................................... 113 118 Episode 9: Mr. S's ruler .......................................................... 119 Episode 10: The rattanchair .......................................................... 120 Episode 11: The Tulbagh chair .......................................................... 121 Episode 12: Jack avoids a question .......................................................... Episode 13: Mr. S draws multipointed stars ....................................................... 122 125 Episode 14: Replacing a brass escutcheon .......................................................... 126 Episode 15: Table legs from a plank ................................................................... Introduction to Episodes 16, 17, and 18................................................................. 128 Episode 16: How much does this plank of mahogany cost? ................................ 129 Episode 17: Clive designs a strategy for calculating the cost of a plank ............. 134 Episode 18: Jack invents a convenient unit to help solve an unfamiliar 137 problem ......................................................................................................... Comments on Episodes 17 and 18 .......................................................... 140 ........................ 141 Episode 19: Why isn't "straightup" a full 100 degrees? Episode 20: Clive and Jack talk about straightness ............................................. 145 148 Part Two: Analysis of Episodes .......................................................... 149 Grouping of Episodes ......................................................................................... 149 Categorization Within Groups of Episodes .......................................................... Conventional Mathematical Concepts That Were Embedded in Carpentry Practices in the Workshop..................................................................................150 158 Summary ........................................................................................................ The Carpenters' Conceptions of Mathematics and Feelings About M athematics .......................................................................................................158 160 Summary ........................................................................................................ Significant Characteristics of the Carpenters' Mathematizing ............................. 160

Chapter Six: Results and Analysis: The Workshop-a Classroom Where Mathematics Is Taught as an Action .......................................................... Introduction ............................................................................................................... The Teaching Methods of the Carpenters .................................................................. The Carpenters' Tools .......................................................... Learning to Mathematize Like a Carpenter ............................................................... The Development of My Sense of Touch ............................................................

167 167 167 170 171 171

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LearningHow to "UseMy Eye" .........................................................................173 LearningWhento MeasureandWhento Compare.............................................175 RoutineActivitiesin theWorkshopThatDid Not EncourageMathematizing.......... 177 The Workshopas a Classroom ........................................ .................... 178 182 Summary.................................................................................................................... .................... 183 Chapter Seven: Discussion ........................................ Introduction................................................................................................................ 183 EpistemologicalImplications:Knowingin Action ...................................................183 EducationalImplications ............................................................ 187 The Meaningof Understanding a Concept ..........................................................188 as a ModelforInstruction .........................................................190 Apprenticeship MathematicsEducationas Praxis .......................................................................191 192 MethodologicalImplications..................................................................................... Limitationsof the StudyandSuggestionsfor FutureResearch ................................ 194 195 Epilogue ..................................................................................................................... References ........ ...........

........................................................................... 197

205 Appendices .......................................................................................................... Conversations........................................205 AppendixA: Scriptfor "Semi-structured" Appendix Appendix Appendix Appendix Appendix

B: Script for StructuredInterviews .......................................................... C: How to Draw Jack's "Square"Star ..................................................... D: Jack's Modification of the Steps for a Square Star ............................. E: Jack's Eight-pointed Star ............................................................ F: Mr. S's Eight-pointed Star ....................................................................

List of Figures Figure 1. Scheme of strategy guiding present study ................................................ Figure 2. The constructivist cycle ............................................................................ Figure 3. The hierarchy: People in the workshop .................................................... ............................................. Figure 4. The square star ............... Figure 5. Positioning the square star on the jewelry box lid ................................ Figure 6. Jack's modification of the square star ................................................... Figure 7. "Squaring"a box ......................................................... Figure 8. Square rule or carpenters' square ............................................................ Figure 9. First version of Mr. S's model to check whether his square rule w as true .................................................................................................... Figure 10. Jack's first attempt to check whether his square rule was true ................. Figure 11. This demonstrates the flaw in Jack's argument ........................................ Figure 12. Jack's second (and correct) attempt to check whether his square rule w as true .................................................................................................... Figure 13. Inverted "V" instead of a straight line when the angle is greater than 90 degrees ................................................................................................ Figure 14. "V" shape instead of a straight line when the angle is less than 90 degrees ................................................................................................ Figure 15. Second version of Mr. S's model to check whether his square rule w as true .................................................................................................... Figure 16. The inlay patternon the back apron of the table ...................................... Figure 17. Dovetail joints .......................................................................................... Figure 18. The sliding bevel is used for marking angles other than 90 and 45 degrees ................................................................................................ Figure 19. Mr. S's method of preparing to place dovetail joints ................................

206 207 208 209 210

12 26 56 96 97 97 99 102 103 105 106 107 108 108 109 111 114 114 115

vi Figure20. Step 1: Drawa line at the midpointof the plank ..................................... 115 Figure21. Step2: Marka pointat a chosendistancefromthe left edge ................... 115 Figure22. Step 3: Markpointsto theleft andrightof the centerline, the same .... 116 distancefromthecenteras the firstpointis fromthe left edge Figure23. Step4: Dividethe left half andthe righthalf of the plankin half again 116 andrepeatstep 3 ....................................................................................... 117 Figure24. Markinggauge .......................................................................................... Figure25. The chairseatframein whichholes areplacedfor threadingrattan ....... 119 Figure26. The backof a Tulbaghchair .....................................................................121 Figure27. Mr.S's firstattemptat drawinga five-pointedstar.................................. 122 122 Figure28. Lookingfor a pentagon ............................................................................ 123 Figure29. Mr.S drawsa pentagon............................................................................ Figure30. Mr.S's secondattemptat a five-pointedstar............................................123 Figure31. Mr.S suggestsusinga compassto ensurethateach of the five sides will be thesamelength ............................................................123 Figure32. Dividingthecircleintofive equalparts ...................................................124 Figure33. The brassescutcheonon thebuffetdoor ..................................................125 Figure34. My diagramof whatBrianvisualized.......................................................127 Calculating the area of a rectangular room .............................................. Brian's diagram representing area ............................................................ Brian's diagram representing volume ....................................................... Brian's sketch of the gates he built............................................................. My diagram of what Clive visualized ....................................................... Jack's diagram of the narrowerplank he visualized .................................. Jack's diagram representing volume ......................................................... My diagram of what Jack visualized ........................................................ Jack called this a "90-degree angle" ......................................................... Jack called this a "100-degree angle" ....................................................... Moving the vertical arm clockwise through 90 degrees ("bending it all the way down") ............................................................ Figure 46. My diagram of Jack's description of angular measure .............................. Figure 47. The dial on the tablesaw in the workshop ................................................ Figure 48. Try square: The wooden handle sits flush against the edge of a surface and the metal blade indicates a 90-degree angle ...................................... Figure 49. Using a try square ............................................................ Figure 50. Analyzing Mr. S's method for placing dovetails in terms of parallel lines and transversals ........................................................................................ .............. Figure 51. Mortise-and-tenon joint ............................................................

Figure 35. Figure 36. Figure 37. Figure 38. Figure 39. Figure 40. Figure 41. Figure 42. Figure 43. Figure 44. Figure 45.

130 131 131 133 135 138 138 139 142 142

143 144 145 146 146 152 177

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DEDICATION This workis dedicatedto all thosein SouthAfricawho have struggledandwho still strugglefor a worthwhileeducation,for freedomfromoppression,andfor a just systemof government.

ACKNOWLEDGEMENTS I wish to acknowledgethe guidanceof my SpecialCommitteeat CornellUniversity: ProfessorsJereConfrey,DavidHenderson,JayMillman,andJohnVolmink.I also acknowledgethe financialsupportof a partialFulbrightScholarship,whichenabledme to conductthe researchreportedhere.Manythanksarealso due to my husband,MarkConstas, for his emotionalandeditorialsupport.

AN ETHNOGRAPHICSTUDY OF THE MATHEMATICAL IDEAS OF A GROUP OF CARPENTERS

by Wendy Lesley Millroy University of Northern Colorado Greeley, Colorado

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

ix ABSTRACT The researcherconducteda six-monthethnographicstudyas an apprenticecarpenter in CapeTown, SouthAfrica,to documentthe validmathematicalideas thatareembedded in the everydaywoodworkingactivitiesof a groupof carpenters.A secondaryobjectivewas to examineandto give a firsthandaccountof the teachingandlearningof mathematical ideas in the contextof the researcher'sapprenticeship. Finally,the studyoffers methodologicaltechniquesfor identifyingmathematicsin thoughtandactionandfor differentiatingthe mathematicsfromroutineapplicationsof procedures. The resultsshowedthatmanyconventionalmathematical conceptsareembeddedin the practicesof the carpenters.Theymadeextensiveuse of suchconceptsas congruence, symmetry,proportion,andstraightandparallellines in theireverydaywork.Furthermore, the carpenters'problemsolvingwas enhancedby theirstrengthin spatialvisualization. Theirexplanations,in the formof convincingarguments,showedthe sequential,logical reasoningthatis relatedto the needin mathematicsfor proofandsubstantiation. The resultsalso showedthatthe carpenters'mathematicshas severalunique characteristics: therewas tacitmathematical knowledgein theiractions,andreflectionon actionsled themto articulatetheirtacitknowledge;decontextualized questionsposedwere revisedinto concrete,contextualizedproblematics; andtheirideas wereframedby the contextof the workshopandcarpentrytools.Comparison,usingthe senses of touchand sight, was preferredto measuringandusuallyresultedin optimalsolutions.To solve a problemsuch as "Howmanytablelegs can be cut fromthis plank?"spatialvisualization practiceswere used to constructfunctionalunits,producingan optimalresultthatcould not be obtainedwith formalprocedures. Theresultsarepresentedas a seriesof 20 narrativeepisodes,followed by an analysis.The epistemological,educational,andmethodologicalimplicationsof theseresults are discussed.

Chapter 1 INTRODUCTION Overview Mathematical knowledgehas commonlybeenportrayedas consistingof universal truthsthatexist independentlyof peopleandthatarediscoveredby mathematicians through a processof formalreasoning.Mathematical reasoning,unlikeany othercognitiveactivity, is believedto be a decontextualized activity,tied to a formalsystemthatrelies upona specificallydefinedset of symbols.Theseideashaveled to a view of mathematicsas divorcedfromordinaryhumanactivityanddevoidof social,cultural,andpolitical considerations. The social, cultural,andpoliticalaspectsof the developmentof mathematicalideas have consistentlybeen overlookedin historicalreconstructions of the growthof mathematics.Contributions fromdiversenon-Westerncultureshavebeen underrepresented in prejudicedhistoricalaccounts,andit has long beencommonlyacceptedthatmost of the ideasreferredto by the label "mathematics" emanatefromWesterncivilizations(Joseph, 1987). The extentto whichpeoplehaveboughtintothe dominantWesternview of what countsas mathematicsis broughtinto sharpfocus by reflectinguponthe remarkable uniformityof school mathematicscurriculathroughoutthe world.In the analysisof the mathematicscurriculaof 22 educationsystemsfromcountriesaroundthe world (e.g., USA, Associationfor Japan,Hungary,Thailand,Nigeria,Scotland,Swaziland),the International EducationalAchievementhas reportedthatthereis a substantial"commoncore"in most areasof mathematicalcontentacrosscurricula(Travers& Westbury,1989).Althoughthis standardsfor achievement may be deemeddesirableby thosewho wish to set international in school mathematicscourses,suchuniformityunfortunately emphasizesthe formaland decontextualizedandignorestherelevanceandenrichmentthatculturalandcommunity practicescould bringto individualcurricula.

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Fromthe pointof view of anyoneholdinga formalistconceptionof mathematics, the activitiesof learningandteachingmathematicsmustbe concernedwith comingto know, or helpingsomeoneelse come to know,therulesandconventionsof the universally acceptedformalsystemandthe strategiesthatcan be broughtto bearon problems,so as to discoveranswersthataresupposedlyuniversallycorrect.In followingthis model of teachingandlearningmathematics,the authorityandexpertiselie with the teacher,andthe emphasisis on the students'absorptionof alreadyexistingideas.Littleroomis allowedfor students'own informalconceptionsof mathematical ideas.Mathematical knowledge generatedin contextsoutsideof classroomsis usuallynot acknowledgedas relevantto the academicsetting,since suchknowledgemay notconformto the tenetsof whatcountsas of teachers.Steffe (1986) comments "genuine"mathematicsaccordingto theinterpretation that"theformalistview of mathematicsencouragesmathematicsteachersto become mathematicallyinactive,"so thattheirsearchfor mathematical meaning"canbeginandend of rulesandprocedures" with the formal,symbolicpresentations (p. 3). As a result,Steffe suggeststhatsuch teachersmay be shockedto learnthatthe rulesfor the multiplicationof signednumbersthatthey haveportrayedas unassailabletruthsareactuallyconventions. As a consequenceof unhappychildhoodexperiencesin the mathematicsclassroom, manypeopleexperiencea sense of alienation,coupledwithfearanddespair,at the mere TheNationalResearchCouncil's(NRC)recentreport mentionof the word"mathematics." to the nationon the futureof mathematicseducationpresentsa dismalpictureof the current situation: Mathematicsis not seen as somethingthatpeopleactuallyuse, butas a best of school.Formostmembersof the forgotten(andoftenpainful)requirement public,theirlastingmemoriesof schoolmathematicsareunpleasant-since so oftenthe last mathematicscoursethey tookconvincedthemto takeno more. (NRC, 1989, p. 10). Threeof everyfourAmericansstopstudyingmathematicsbeforecompleting careerorjob prerequisites.Moststudentsleave schoolwithoutsufficient in mathematicsto cope eitherwithon-the-jobdemandsfor problempreparation for mathematical literacy.(NRC, 1989, p. 1). solving or withcollege requirements

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My deepconcernaboutthe widespreadfailureof mathematicseducationgave impetusto the presentstudy.Over25%of all high schoolstudentsin the U.S. dropout beforegraduating;amongblack,Hispanic,andnativeAmericanstudents,the dropoutrate oftenexceeds 50%.As observedin the NRCstudy,"Mathematics is the worstcurricular villainin drivingstudentsto failurein school.Whenmathematicsacts as a filter,it not only filtersstudentsout of careers,butfrequentlyout of schoolitself' (NCR, 1989, p. 7). In SouthAfrica,mathematicspresentsa similarstumblingblockto manystudents,in particularto blackstudents. I believe thatthe failureof mathematicseducationis due in partto the sterile, narrowdescriptionof whatcountsas authenticmathematics.Manypeoplefeel themselves to be excludedfromparticipation in genuinemathematical activity.Whenan individualis faced with a mathematicalproblemto solve, it is not unusualto hearher1apologizefor not using "real"mathematicsas she proceedsto use an efficientandinnovative(butnot schooltaught)methodto reacha satisfactorysolution(Lave, 1985).Mathematicseducationis furtherhinderedby a belief thatonly a smallproportionof humankindis capableof fully mathematics.An elite coterieof professionalmathematicians is seen to own understanding the prizedbody of knowledgeandto controlthe creationof new mathematicalknowledge. have Throughouthistory,the individualsadmittedto the innercircleof mathematicians been predominantly whitemales.Today,whitemenearn74%of the doctoraldegreesin the mathematicalsciencesawardedto U.S. citizens,while blackmenearn5%of these degrees. Despitethe factthatwomenearn46%of all mathematicsbaccalaureates awarded,they earn only 20%of the doctoraldegreesawarded(NRC,1989, p. 22). Terselystated,whatcounts as mathematicalknowledgeis narrowlydefined,andthe access to this knowledgeis throughan even narrowerdoorway.

1Althoughfemininepronounswill be usedthroughoutfor the sake of uniformity,they are intendedto includethe masculinecase as well.

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The Cultural, Social, and Political Componentsof Mathematics I proposeto arguefor a broaderconceptionof whatcountsas mathematics,drawing on the literatureto supportmy claimsthatmathematicsis culturallybased,in thatall culturesgeneratemathematicalknowledgeto suit thegoals andpurposesof communities who wish to solve problemsandto imposeorderin theirlives (Bishop,1987, 1988a, 1988b; D'Ambrosio,1985, 1987;Fasheh,1982, 1988;Gerdes,1985, 1986, 1988a, 1988b;Stigler& Baranes,1988);thatmathematicsis sociallyconstructedin the contextof a community, wheremeaningis negotiatedandconventionsareagreedupon(Bishop,1985, 1988a;Cobb, 1986a;D'Ambrosio,1985, 1987;Gerdes,1985, 1986;Lakatos1976;Steffe, 1986; Tymoczko,1980,1986);andthatmathematicshas apolitical component(Fasheh,1982, 1988;Gerdes,1985, 1988c;Harris,1987, 1988a,1988b;Kallaway,1984;Mellin-Olsen, 1987). Access to mathematicsworldwideis anythingbutdemocratic.Race andgender differencesin mathematicsparticipation andachievementarenot consequencesof free choice or of some innatelack of capability.Rather,thesedifferencesarethe consequences of an imbalanceof powerandthe destructiveeffectsof oppression,authoritarianism, and social conditioning. Mathematicsin Everyday Settings:"EverydayCognition" Researcherswho haverecentlyinvestigatedpeople'suse of mathematicsoutsidethe confinesof the classroomhavebeguna paththatcouldlead to a less exclusiveview of mathematics.Therecentresearchintocognitionin everydayactivitieshas concentratedon people'suse of mathematicalknowledge,andit urgesrecognitionof the importanceof activityandenculturationto learning(Brown, Collins,& Duguid,1989).The practiceof mathematicshas beenexploredin the contextsof a varietyof everydaysettings.Two main groupsof researchershaveexploredthe use of mathematicsin settingsoutsideof schools. They arethose interestedin "everydaycognition"or "cognitionin practice,"whereLave where (1985, 1988)is a key figure,andthoseinterestedin "ethnomathematics," D'Ambrosio(1985, 1987)andGerdes(1985, 1986, 1988a,1988b, 1988c)arekey figures.

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I shallreferbrieflyto the workof thoseresearchersinterestedin "everyday cognition,"broadlygroupingtogethertheworksof Lave (1977, 1982, 1985, 1988);Reed andLave (1979);Petitto(1979);Lave,Murtaugh,andde la Rocha(1984); Schliemann (1984); Scribner(1984, 1985);Murtaugh(1985);de la Rocha(1985);Carraher,Carraher, andSchliemann(1985);CarraherandCarraher(1987);Saxe (1988); andSchliemannand Acioly (1989). (Fora detailedreviewof this literaturesee Chapter2.) In general,the two mainaimsof this groupof researcherswere,first,to explorethe transferof school knowledgeto out-of-schoolsituations,andsecond,to critiqueconventionalcognitive theory,by using social andculturaltheory,andthe theoryof practice.In orderto achieve these goals, the arithmeticpracticesof, amongothers,tailors,cloth merchants,child market vendors,dietingcooks, groceryshoppers,bookies,carpenters,anddairyworkerswere studied. This genreof researchprovidesvaluableempiricalevidencethatpeoplefrequently constructandapplyinnovativeandcreativemethodsof theirown to reachsatisfactory solutionsto real-lifeproblems.Schools,therefore,obviouslydo not have the monopolyon teachingmathematics.The methodsusedto solve problemswerechosento suit the goals andpurposesof the problemsolvers;for instance,efficiencyandlabor-savingconcerns shapedthe strategiesemployedby thegroceryshoppers,dietingcooks, anddairyworkers. The workconductedin practicalsettingsalso providesevidencethatmathematical knowledgeis developedwithina contextandis framedby thatcontext.Forexample,useful unitswere inventedthatmadeuse of the social andphysicalenvironmentsto facilitate calculations:tailorsin Liberiacomputedin "trousers'worth"of cloth, anddairyworkers used case pricesas units(Lave,1985,p. 173). A furthervaluablecontributionfromthis workis the suggestionthatan modelis a viablemodelfor teachingandlearning,andnot only for the apprenticeship teachingof practicaltradeskills as is the case traditionally(Lave, 1982).In an model,instructionis conceivedof as a joint problem-solvingevent in which apprenticeship

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a moreexpertindividualprovidessupportor "scaffolding"to extendthe skills of a novice to higherlevels of competence(Rogoff& Gardner,1984).Learningis embeddedin activitieswithinsocial andphysicalcontexts. Thereareseveralways in whichthe presentstudyextendsthe workin "everyday the focus has been cognition."In all the studiesdiscussedin the precedingtwo paragraphs, on arithmeticpracticesthataregeneratedby peoplein theirworkor homeenvironments. No attemptshave beenmadeto studygeometry(exceptperhapsfor the workof Scribner, who describeshow dairyworkersavoidedusingarithmeticby relyingon spatial visualizationto solve problems).In fact,Laveet al. (1984, p. 68) statethatone of the severalreasonsfor focusingon arithmeticwas that"arithmetic activityhas formal propertieswhich makeit identifiablein the flow of experiencein manydifferentsituations," andLave (1988, p. 5) statesthat"it (arithmetic)has a highlystructuredandincorrigible lexicon,easily recognizablein the courseof ongoingactivity"(Lave,1988, p. 5). I decidedto makegeometrythe focus of my studyfor exactlythe oppositereasons. I believe thatthe richnessof geometryin ourphysicalenvironmentsprovidesfertile opportunitiesfor peopleto developcomplexgeometricknowledgethatdoes not look of geometric anythinglike the formalgeometrywe learnin schools.An understanding conceptscan be expressedin a varietyof differentways, throughan appreciationof the aestheticin designandthroughthecreationof a sense of balancein for example,art, architecture,or furnituredesign.D'ArcyThompson(1961, p. 269) remindsus of Galileo's aphorism,"TheBook of Natureis writtenin thecharactersof Geometry."Whereas arithmeticconsistsof proceduresfor carryingoutcalculationsandhas a limitedvocabulary andexpressionof geometryis not for describingthose procedures,the conceptualization similarlyconstrained. Setting for the Present Research In orderto exploremy intuitionsandtentativeformulationsabouthow geometryand othermathematicalknowledgemay be constructedandusedin everydayactivities,it was

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necessaryto decideupona suitablesite for investigationandan appropriate methodologyto guide the investigation.I chose to conductfieldworkin a carpentryworkshopin Cape Town, SouthAfrica,with a groupof carpenterswho representeda diversespectrumof SouthAfricanculturalbackgroundsandlanguagegroups.Oneparticularfurniture workshopwas chosenas the site for fieldworkfor severaldifferentreasons.The workmanshipdisplayedin the furnitureproducedwas of an extremelyhigh quality.The workshopwas not highlymechanized-muchof the constructionof the furniturewas done with old-fashionedhandtoolsaccordingto the old traditions,andhand-madearticleswere highly prizedby the group.The excellentqualityof the designs,the skillfulconstruction, and the fact thatthe carpenterscreatedbeautifulfurniturefromroughplankswith little assistancefrommechanizedsystemsconvincedme thatthey hadto be using mathematical, andin particular,geometricalideasto guidetheiractivities.Only one of the artisanshad receivedformaltrainingat a technicalschool.The otherartisanshadlearnedthe craftof handeddownfromfatherto son, or woodworkingin traditionalinformalapprenticeships, simply "onthejob"while workingwith moreexperiencedcarpenters.All of the carpenters' betweenthe ages of about12 to 16 years,so the formalschoolinghadbeeninterrupted grouphadhadlittle involvementwith schoolmathematics. Two questionswereespeciallyintriguingto me. Whatdid the mathematicsused to guide carpentryactivitieslook like, andwheredid theseideas come from?In orderto addressquestionsof this nature,I hadto evaluatethe methodologicaloptionscritically beforechoosingan appropriate methodologyto guidemy investigation. MethodologicalFramework:An EthnographicApproach My examinationof the "everydaycognition"literaturegave rise to threeotherissues thatservedas furtherpointsof departure for the presentstudy.The firstissue is thatthe methodologyused in the studiesmentionedabovecouldnot challengewhatshouldor shouldnot be countedas mathematics.The generalmethodologicalapproachwas to conductnaturalisticobservationof the subjectsat workandto note theirperformanceon the

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tests or verbaltests arithmetictasksencounteredduringthe workingday.Pencil-and-paper of arithmetictasksof a similar,but"academic," varietywereconstructedandadministered, andcomparisonsof performanceweremadebetweenschoolarithmeticandwork arithmetic.Schoolarithmeticwas thususedas a kindof benchmark.Second,no explicit attemptswere madeto examinetheepistemologicalrootsof the inventedarithmetic(that lookeddifferentfromschool-taughtarithmetic)witha view to presentingit as a viable alternativesystem.Lave (1985) commented,"Theremaybe a qualitativelydifferent organizationof arithmeticin differentsettings;the proceduresobserveddid not look, feel, or soundlike school arithmeticperformances" (p. 173).Theanalysisperformedwas not Third,most criticalenoughto provideevidenceof alternative,validconceptsof arithmetic.2 of the studiesdid not investigatethe ways in whichthis arithmeticknowledgeis taughtand learnedoutsideof schools. Thesethreepointshighlightedthe needfor thecarefulselectionof botha methodologicalframeworkandan epistemologicalframeworkto guidemy research.Since I wantto arguefor a broaderconceptionof whatcountsas validmathematics,it was importantthatI gatherevidencethatcouldbe seen as a challengeto theconventional, narrowdefinitionof mathematics.An ethnographic approach,as describedby Eisenhart methodologicalchoice thatwouldallow (1988), was selectedas beingthe mostappropriate me to explorethe kindsof issues I hadin mind.Manyof the tenetsof ethnographyderive fromthe interpretivistphilosophicalstance,whichhas at its heartthe ideathat"allhuman experience"(Eisenhart,1988, activityis fundamentallya social andmeaning-making p. 102) andthatobjectsdo not haveany intrinsicmeaning;rather,the meaningof objects

of the conceptof 2Forexample,I arguethatthe carpentersdisplaya validunderstanding volumedespitethe fact thatthey do notknowhow to calculatevolumenumerically of theconceptis of a visualnature (Episodes15 to 19, Chapter5). Theirunderstanding andis not linkedto numbers.Theyareableto use theirconceptof volumeto takeactions andto solve complexproblems.Theirconceptof volumeis very differentfromthat whichis conventionallytaughtin schools,andif judgedaccordingto the academic of volumewouldremaininvisible. yardstick,theirunderstanding

9

"lies in the actionsthathumanbeingstaketowardthem"(Denzin,1978, p. 7). and andethnographyprovidedme withthe frameworkfor understanding Interpretivism interpretingthe carpenters'actionsandpracticesin termsof mathematics,without constantlyandexplicitlyinvokingconventionalmathematicsas a standardof comparison. Eisenhart(1988) sumsup whatshe meansby interpretivism: Fromthis perspective,meaningsandactions,contextandsituationare inextricablylinkedandmakeno sense in isolationfromone another.The "facts"of humanactivityaresocial constructions; theyexist only by social agreementor in a contextandsituation.Whatcountsas marriage, consensusamongparticipants genderroles, teaching,calculation,therightanswer,or whateverdependson the ways (andwhether)thesethingsaredefinedandusedin humangroups.. .it makes no sense for the interpretivist to do thingslike catalogbeliefs aboutmathematics withoutalso consideringthe contextsin whichtheseideas areimportant.(p. 103) EpistemologicalFramework:A ConstructivistApproach The constructivisttheoryof knowledgecreationprovidedme with an epistemologicalframeworkaroundwhichI couldstartto makesense of coordinatingthe of mathematics(Cobb,1985, 1986b;Confrey,1985, existenceof differentrepresentations 1987, 1991;Kaput,1985;Steffe, 1986, 1988;von Glasersfeld,1982,1984).Constructivism ideasthroughrepeatedcycles of providesa modelfor theconstructionof mathematical actionstakento resolvea problematicandreflectionson thoseactions(Confrey,1985, to emphasizethatin constructivist 1987). Confrey(1991) uses the term"problematic" to theorythe problemdoes not haveindependentstatus;it is ratherseen as a "roadblock" wherean individualwantsto be. Thus"problematic" is usedto referto the individual's "roadblock." Constructivism encouragesandacknowledgesthe legitimacyof students'own inventionsandexplanations,fostersthe acceptanceof multiplerepresentations of ideas, and emphasizesthe importanceof actionandreflection(fora moredetaileddiscussionof constructivism,see Chapter2). Unfortunately,theEnglishlanguagedoes not have a wordfor the activityof mathematics.The only way in whichwe can referto this multifaceted "doing/making" humanphenomenonis via the noun"mathematics," whichseems to referto a staticbody of

10 knowledgethatalreadyexists out in the world.We needa wordto expressthe conceptof an activeprocessof creatingandusingmathematical ideasor tools. I shalluse the verb whichembodiesactionandrefersto the experienceof creatingandusing "mathematizing,"3 mathematicalideas. Mathematicsin EverydaySettings: Ethnomathematics The secondgroupof researcherswho haveexploredthe creationanduse of mathematicsin settingsoutsideof schoolsarethoseinterestedin ethnomathematics. I have groupedtogetherthe workof Fasheh(1982, 1988),Bishop(1985, 1987, 1988a, 1988b), D'Ambrosio(1985, 1987),Gerdes(1985, 1986, 1988a,1988b,1988c),AscherandAscher (1981, 1986),Harris(1987), andBorba(1987a, 1987b).In general,the two mainaims of this groupof researchershavebeento explorethe mathematicsthatis createdin different culturesandcommunitiesandto describethismathematics.In orderto achievethese goals, the mathematicalpracticesof nationalgroups,peopleliving in shantytowns,fishermen, basketweavers,rugweavers,knitters,andseamstresseshavebeenstudied.This genreof researchprovidesconvincingtheoreticalargumentsthatdifferentformsof mathematics resultfromdifferentways of thinkingthatexist acrossculturalgroups.Practical,empirical evidenceof this mathematicshas also beenprovided-for instance,the quipusof the Incas studiedby AscherandAscher(1981, 1986);the mathematicsusedby fishermenand peasantsin Mozambique,describedby Gerdes(1985, 1986, 1988b);andthe mathematicsof childrenin a shantytownin Brazil(Borba,1987a,1987b).This genreof researchhas often carriedhintsandwarningsthatthereis also mathematicsthatwe arenot able to perceive. Forinstance,AscherandAscher(1986, p. 139) warnthat"asWesterners,we areconfined

has croppedup infrequentlyoverthe years.The earliest 3Theterm"mathematizing" referenceI couldfind was attributedto HermannWeyl. Kline(1980, p. 6) andDavis and Hersch(1981, p. 60) quoteHermannWeyl,consideredto be one of the greatest mathematicians of this century,as sayingin 1944,"'Mathematizing' maywell be a creativeactivityof man,like languageor music,of primaryoriginality,whose historical decisionsdefy completeobjectiverationalization."

11 in whatwe can see andwhatwe can expressto ideasin some way analogousto our own." Fasheh(1988, p. 2), whose illiteratemotheris a seamstress,observed: My mother'smathwas so deeplyembeddedin the culturethatit was invisible througheyes trainedby formaleducation....The mathshe was using was beyond my comprehension... It was a shockto me to realizethe complexityandrichness of my mother'srelationshipto mathematics.Mathematicswas integratedinto her worldas it neverwas into mine. Theparadoxembeddedin ethnomathematics. My critiqueof this literaturelies in the fact thatit seems to embedan intriguingparadoxto whichlittle attentionhas been paid.The paradoxunfoldsas follows: 1. Ethnomathematics is concernedwiththe studyof the differentkindsof mathematicsthatemergefromdifferentculturalgroups. 2. It is impossibleto recognizeanddescribeanythingwithoutusing one's own frameworks. 3. The paradox:How can anyonewho is schooledin conventionalWestern mathematics"see"any formof mathematicsotherthanthatwhichresemblesthe conventionalmathematicswith whichshe is familiar? A Strategy for ConductingPracticalResearchin Ethnomathematics The paradoxdescribedabovegave rise to a dilemma,andit becameclearto me that in orderto proceedwith the explorationof the mathematicsof an unfamiliarculture,I wouldhave to navigatea passagebetweentwo dangerousareas(see Figure1). The founderingpointon the left representsthe overwhelmingnotionthat"everythingis mathematics"(like beingsweptawayby a tidalwave!),while the founderingpointon the rightrepresentsthe constrictingnotionthat"formalacademicmathematicsis the only valid of peoples'mathematical ideas"(like beingstrandedon a barrendesert representation island!).Partof the way in whichto ensurea safe passageseemedto me to openly

12 ETHNOMATHEMATICS conductedin a setting ** DANGER

** DANGER **

**

"ONLYFORMAL MATHEMATICS IS VALID"

"EVERYTHING IS MATHEMATICS"

[

e.g., carpentryworkshop

|

MATHEMATICS

RECOGNIZABLE |

I

I NOTRECOGNIZABLE Rationaleto describe this:

Strategy to describe this: Use frameworkssuggestedby: 1. Bishop(1988) 2. Constructivism 3. Owncriteria 4. Krutetskii(1976) to examinethe activitiesof the carpenters

Claimthat: 1. Mathematicsis sociallyconstructed. 2. Mathematicsis communicated. Ouestion: How is the carpenters' mathcommunicated? Answer: Throughthe medium of apprenticeship Therefore: Researchermust becomean apprentice anddescribeher learningexperience.

Figure1. Schemeof strategyguidingpresentstudy.

13 engagedin by the carpentersthere acknowledgethatwhenI examinedthemathematizing would be examplesof mathematical ideasandpracticesthatI wouldrecognizeandthatI would be able to describein termsof the vocabularyof conventionalWesternmathematics. However,it was likely thattherewouldalso be mathematicsthatI could not recognizeand for which I wouldhaveno familiardescriptivewords.A parallelsituationis describedby Becker(1989), who discussesthe experienceof learninga new language: a distantone, confrontsthe Comingto know anotherlanguage,particularly knowerwith silences,thingsin one languagewhichhaveno counterpart in another.These silencesareoftenobscurebecauseone fills themin as one actively hears(exuberances)or doesn'tnoticethemat all becausethe distinctionsdon'texist in one's own language(deficiencies).(p. 14) It was my aim to tryto noticethe silenceswith whichI was confrontedwhencomingto know the mathematizingof the carpentersandto find a way to give a voice to some of those silences, a voice in whichthe carpenters'ownvoices couldbe heardandvalued. I proposeto set out below the rationalethatI adoptedto select, describe,andanalyze each of these two categoriesof mathematizing, as a suggestionfor how practicalresearchin ethnomathematics Theconceptualization of describingtwo categories may be approached. of mathematizinghadseveralfortunateoutcomes.The choice of ethnographyas the most appropriate methodologywas reviewedandrefined,andfromthis choice emergeda with whichI wouldnot be familiar. strategyto explorethe formsof mathematizing Spradley'scomment(1980), "Ratherthanstudyingpeople, ethnographymeanslearning from people"(p. 3), took on greatersignificance.I realizedthatthe only way in whichI could hopeto find a way to understand the mathematicsthatwouldnot be recognizableto my eyes (shapedby manyyearsof studyingandteachingformalacademicmathematics) would be throughmy learningexperiencesas a carpentryapprentice. My assumptionis thatthe carpenters'mathematicsis sociallyconstructedandthatit is communicatedthroughthe mediumof apprenticeship. In orderto experiencethe communication,it was essentialthatI shouldbe a learnercarpenterin the workshop.One of my conjecturesis thattacitmathematical knowledgedrivesthe physicalactivitiesof

14 designingandconstructingfurniturein the workshop,so it was vital thatI come to know thosephysicalactivitiesmyself.The knowledgethatI woulddevelopthroughthe medium of actionwouldgive me an experienceon whichto reflect,bothduringthe activityand afterward.Frommy reflectionwouldemergea way to describemy experience.To this end, I workedas an apprenticeat the workshopon a dailybasisfor a periodof five months, receivinginstructionfromseniorartisanson an informalbasisduringthe courseof the workingday. The apprenticeship systemin the workshopprovidedme with an opportunityto learnsomethingnew with a consciousawarenessof Vygotsky'stheoryof learning (Vygotsky1962, 1978, 1981;Wertsch,1979, 1984;Wertsch,Minick,& Ars, 1984). Vygotsky'stheoryemphasizesthe socialnatureof people'slearningandarguesthat"the individualresponseemergesfromthe formof collectivelife" (Vygotsky,1981, p. 165). In the workshop,underthe tutelageof moreexperiencedcarpenters,I benefitedfromthe collectivewisdomof the groupandlearnedto mathematizelike a carpenter.Describingmy learningexperiencesas an apprenticeprovideda way to talkaboutthe mathematizingthat the carpenterstaughtme usingthe tools, symbols,andmetaphorsof theirculture.This rationaleprovidedme some protectionfromthe dangerof beingconfinedto describingonly those aspectsof the carpenters'mathematizing thatresembledthe academicmathematicsin whichI had beenschooled.Theresultsof thispartof the studyarereportedin the formof an essay in Chapter6. In orderto avoidthe "everythingis mathematics" danger,I used severalframeworks throughwhichto view the activitiesof thecarpenters.Bishop'swork(1988a, 1988b),in which he describesthe six "environmental activities"of counting,measuring,locating, designing,playing,andexplainingandfromwhichhe claimsthatall mathematicalideas can be generated,provideda startingpoint.However,findingactivitiesin the workshopthat fittedBishop'scategorieswas not enoughto claimthatsuch an activitywas an exampleof mathematizing.Therewerecertainlymanyactivitiesthatinvolved,for example,measuring

15 Some woodworkingoperations or countingbutthatI did not wantto call mathematizing. had been routinized,andmanywereperformedin a rotefashion,ratherlike a studentwho solves a calculusproblemcorrectlyby imitationandrotememorybutwithoutany is occurringin eitherof these I wouldnot like to claimthatmathematizing understanding. two cases.

AlthoughBishop'scategoriesprovideda frameworkfor the kindsof activitiesthat may be fruitfulto explore,it seemedclearthathow the activitieswere performedwas of importanceto my research.Forthis reason,I set up some rigorouscriteriato applyto my observationsin orderto test whetheror not I couldarguethatI hadfoundevidencethatthe ideasthatwouldyield to description. carpenterswere generatingandusingmathematical A basicrequirementfor the inclusionof an incidentin the workshopas an example of mathematizingwas thatthe carpenters,individuallyor in a group,wereengagingin a thatI lookedfor in cycle of actionandreflectionon a problematic.Othercharacteristics encounterswere thoseexamplesthatrequiredthecarpenterto explainsomethingto me verballyor nonverbally,to offersome argumentfor verification,to appraisesome work critically,or to act as a tutorproviding"scaffolding"to assistan apprentice,"inthe formof ideal modelingof a performance,hints,reminders,explanations,or even missingpieces of knowledgeto assistthe apprentice'staskexecution"(Gott,1988,p. 99). The above of mathematizing,but requirementsarenot meantas an exhaustivelist of the characteristics servedthe functionof allowingme a methodto sift throughmy experiencesand observationsandto concentrateon thosethatyieldedrichresults. As an analytictool, I referredto the workof Krutetskii(1976), who listed nine "basiccharacteristicsof mathematical thought"(1976, p. 87) basedon his studieswith Russianschool children(for a full listing,see Chapter5). An analysisof the carpenters' of mathematicalthinkingthatwere problem-solvingactivitiesrevealedcharacteristics similarto those describedby Krutetskii.Theresultsof this partof the studyarereportedas a series of episodes,followedby an analysisof thoseepisodes,in Chapter5.

16

Organizationof the Monograph The monographis organizedin the followingway:Chapter2 containsreviewsand critiquesof the relevantliterature.Chapter3 describesthe "castof characters"-thepeople who workin the workshop,as well as a briefhistoryof the workshop.In Chapter4, I set out the methodologicalfoundationsfor theresearchanddiscussthe datacollectionmethods. revealedby the carpentersas they Chapter5 relatesthe storyof themathematizing solved problemsthatarosespontaneouslywithintheirdailyroutineat the workshop.In Chapter6, I tell my own storyof how I learnedto mathematizelike a carpenter.In Chapter 7, some conclusionsandsuggestionsfor furtherinvestigationsarediscussed.

Chapter2 CONCEPTUALIZINGTHE RESEARCHPROBLEM Introduction This chapterreviewsin detailthe literatureintroducedin the overviewin Chapter1. of the present As the literatureis examinedandcritiqued,a broaderconceptualization researchproblemwill emerge.The chapterendswitha formalstatementof the problemand a descriptionof the goals of this researchproject. Cultureand Cognition Lave (1988) arguesthatourthinkingon cognitionhas been shapedby psychologists As a resultof a sortof andourthinkingon culturehas beenshapedby anthropologists. divisionof laborbetweenthe two disciplines,therehas beena lack of meaningfuldiscourse betweenthem.Followingthe theoryof developmentalcognitivepsychologydevelopedby Piaget,researchandtheoryon cognitivedevelopmentin Americanpsychologyhave been dominatedby an examinationof the individual'sdevelopment,withoutmuchconsideration of societalandculturalinfluences.Culturehas oftenbeentreatedas somethingto be factoredout in a laboratorysetting,ratherthanas an integralpartof cognition(for 1979). exceptionsto this approach,see Bronfenbrenner, In the cross-culturalworkdoneby cognitivepsychologists,particularlythatdone withinthe Piagetianframework,thelack of attentionto the complexitiesof culturehas led to certainproblems.Cole andScribner(1974, p. 173) list the followingproblemsthat settings: pervadecognitiveresearchin cross-cultural 1. Thereis a greatreadinessto assumethatparticularkindsof tests or experimental situationsarediagnosticof particular cognitivecapacitiesor processes. 2. Psychologicalprocessesaretreatedas "entities"thata person"has"or "doesnot have"as a propertyof thatperson,independentof the problemsituation.They arealso consideredto operateindependentlyof each other.

17

18 3. Closelyrelatedto (1) and(2) is a readinessto believe thatpoorperformanceon a particulartest is reflectiveof a deficiencyin, or lack of, "the"processthatthe test is said to measure. 4. Evidencefromotherdisciplines(especiallyanthropologyandlinguistics)is usuallynot takenintoaccountin makinginferencesaboutthe cognitive processesthata given culturalgrouphas or uses. 5. The complexityof the culturalgroupsandinstitutionsstudiedis very often grosslyoversimplified. One unfortunate resultof the pointsaboveis thatmuchof the ethnographicwork andcross-cultural doneby anthropologists psychologistshas beenbasedon the conceptof "culturaldeprivation"(Keddie,1973).Forexample,Bishop(1988a,p. 21) describesthe seminalwork,TheNew Mathematicsandan OldCulture,by AmericanresearchersGay andCole (1967), as "clearlya bookwrittenby foreignerscomingfroma culturethatsees itself to be superiorto the Kpelleculturein some way." Lave's (1988) statedaimis to developa moreadequatemodelof cognitionin a culturalcontext.To this end,Laverecommendsthe studyof cognitiveactivityin everyday of cognition"(1988, p. 1) as a theoryof settings.She conceivesof a "socialanthropology practicethatchallengestheconventionalassumptionsaboutthe impactof schoolingon everydaypractice.Herresearchin everydaysettings(forexample,in a grocerystore)led herto rejectthe theoryof learningtransfer,since mostof the groceryshoppersinterviewed in the studywere ableto performinnovativeandflexiblementalcalculationsin the store when decidingwhichproductto purchasebutdid not performwell on pencil-and-paper tests consistingof "school"versionsof similarproblems. Lave arguesthatcognitioncannotbe seen as beingseparatefromthe social world, the peopleacting,the settingsof theiractivity,andtheiractions.She attemptsto developa "theoryof everydayactivity,"wherethe term"everyday"does not have its customaryplace in the "unsungcategoryof humbledomesticactivities"(1988, p. 14). Rather,it refers

19 simplyto the ordinarydaily,weekly,or monthlycycles of humans,like the activitiesof a teacherandherpupilsin the classroom,the workof a scientistin herlaboratory,a person or a groupof carpentersworking.Lave (1988, shoppingfor groceriesin the supermarket, p. 1) arguesstronglyagainstseparatingtheindividualfromcollective,culturalaspectsof cognition,assertingthat"'cognition'observedin everydaypracticeis distributedstretchedover, not dividedamong-mind, body,activityandculturallyorganizedsettings (whichincludeotheractors)."Cultureandcognitionarethusinextricablyintertwined. The workof Lavehas stronglinkswiththe ideasof the Russianpsychologist Vygotsky(1962, 1978, 1981),whose theoryof cognitivedevelopmenthas raisedthe interestof educationalanddevelopmentalpsychologistsin the U.S. duringrecentyears. Vygotsky'stheoryemphasizesthatcognitivefunctioningoccursfirston the social level, betweenpeople,andthatthechildtheninternalizesthis in individualdevelopment. Vygotskydoes not claimsimplythatthe mentalprocessesof an individualdevelopwithina social milieu;rather,accordingto WertschandRogoff (1984), Vygotskyviews individuals'mentalprocessesas havingspecific organizational propertiesthatreflectthe social formfromwhichthey derive.The composition, structureandmeansof actionareinternalizedfromtheirsocial origins.This means variationin the organizationof individualpsychologicalfunctioning.Forexample,a child who has participated in jointproblemsolvingwill use the same task thatprovedeffectivein groupproblemsolving whensolving such a representation problemindependently.(p. 2) Vygotsky(1981) arguesthatthe mainfocus of psychologyshouldbe "toshow how the individualresponseemergesfromthe formof collectivelife" (p. 165). In contrastwith the psychologicalinvestigationsof Piaget,who measuredthe level of mentaldevelopmentof a child as beingindicatedby thestandardized problemsthatthe child was able to solve alone, Vygotsky(1962) studiedtheresultsof allowingchildrento workin collaborationwith an adulton problemsolutions.Usingthis approach,he discoveredthatone of two eight-yearold boys who workedon problems"incooperation" was able to solve problemsdesigned for twelve-year-olds,while the othercouldnot go beyondproblemsintendedfor nine-yearolds. He introducedthe term"thezone of proximaldevelopment"(1962, p. 103), whichhe

20 definedas "thedistancebetweenthe actualdevelopmentallevel as determinedby independentproblemsolvingandthelevel of potentialdevelopmentas determinedthrough problemsolving underadultguidanceor in collaborationwithmorecapablepeers" (Vygotsky,1978,p. 86). In the exampledescribed,the firstchildhada zone of fourandthe secondhada zone of one. Vygotsky(1962) claimedthatthe childwith the largerzone of proximaldevelopmentwouldfarebetterin school.Throughthe introductionto the child of the use of society's tools andtechniquesandtheirpracticein the child's zone of proximal developmentwith moreexperiencedmembersof society,individualdevelopmentis guided by the social world(Wertsch& Rogoff, 1984). The Vygotskianschoolconsiders"activity"as the basic unitof studyin psychology. The notionof activityfocuseson the socioculturallydefinedcontextin whichhuman functioningoccursandrefersto an actual,identifiableactivityas opposedto a generic notionof humanactivity(Wertschet al., 1984).Some of the activitiesmentionedby Vygotskianpsychologistsareplay,instructionor formaleducation,andwork.Wertschet al. (1984, p. 155) discussthe differencebetweenthe activitysettingof formalschooling,where "learningfor learning'ssake"is valuedanderrorsareexpectedandsometimeseven encouraged,andthe activitysettingof a "workor laboractivity,"whereerror-freetask performanceis of overallimportance.In the workcontext,the activitysettingis structured wherethe novice's so thatlearningmay occurin a formsimilarto apprenticeship, performanceis carefullymonitoreduntilit is faultless.Vygotsky'stheoryis of particular interestfor the presentstudyin the carpentryworkshop,wherelearningoccursin a system of apprenticeship. The idea of "scaffolding,"a processwherebya child learningto mastera problemis given supportby an adult,was developedby Wood,Bruner,andRoss (1976) andis closely relatedto the conceptof the zone of proximaldevelopment.Rogoff andGardner(1984) use the notionsof "scaffolding"andthe "zoneof proximaldevelopment"to explorecognitive developmentin a studyof mothersassistingtheirchildrento preparefor a memorytest on

21 the categorizationof objects.Theydescribethe assistanceas integratingbothexplanation (whenthe adulttalksandthe childlistens)anddemonstration(whenthe adultcarriesout the taskandthe childwatches)withtheparticipationof the childin the task.Rogoff and Gardner(1984) characterizeinstructionthatuses scaffoldingas in the task... (whichfunctions) carefulguidanceandgraduatedparticipation constructin the courseof as a deliberatebuttacitprocesswhichthe participants in the transferof informationand communication.Theyproceedopportunistically skills in a purposeful,flexibleway, makinguse of pragmaticaspectsin the context to developmeansof instruction.Instructioncan be conceivedof as a complex, tacitprocessdevelopedin the particularproblemsituation,ratherthanas an explicitrecipefor problemsolutionthatis availableout of the contextof action. (pp. 102-103) of the teachingandlearningthatoccursin the modelof This thoroughcharacterization and is extremelyvaluable.The integrationof explanation,demonstration, apprenticeship participationin the modelis an importantpointthatis not madeexplicitin otherdiscussions modelof teachingandlearning,such as the work of the wideruse of the "apprenticeship" by Brown,Collins,andDuguid(1989). Brownet al. (1989), drawingon theresearchintocognitionin everydayactivity, as an proposethe ideas of "situatedcognition"4andof "cognitiveapprenticeship" alternativeto conventionalschoolpractices,arguingthat"knowledgeis situated,beingin parta productof the activity,contextandculturein whichit is developedandused"(p. 32). into "authentic" Brownet al. suggestthatstudentsbe "enculturated" practicesthrough butwhat activityandsocial interactionin a way thatis similarto craftapprenticeship, is not clearlystated.The classroom preciselyshouldbe taughtto achieveenculturation practicesof two exemplarymathematicsteachersaredescribedandareused to derivethree of cognitiveapprenticeship" (Brownet al., 1989, p. 38): "proceduresthatarecharacteristic namely,beginningwith a taskembeddedin familiaractivity,stressingthatheuristicsarenot absolute,andallowingstudentsto generatetheirown solutionpaths.AlthoughI agreethat the apprenticeship modelhas characteristics thatmay be usefulto tryto incorporateinto 4Theidea of "situatedcognition"generatedbothexcitementandskepticismin education circles. Forcritiquesof the article,see Palincsar(1989) andWineburg(1989).

22 classroominstruction,I do not thinkthatthe threecharacteristics describedby Brownet al. can be said to be derivedfromthe traditionalapprenticeship model.Social interactionand collaborativelearningareaddedas two moreimportantfeaturesto be fosteredin classrooms.Thesefeaturesare characteristic of apprenticeship learningandhavebeen shownto be fruitfulin theclassroom.However,it is notenoughto simplydescribesome of the seeminglymoredesirableattributesin learningsituationsandto give this collectionthe new label "cognitiveapprenticeship." Forthe apprenticeship modelto be seriously consideredas an alternativeto conventionalclassroompractices,therewill have to be muchdeeperanalysesof whatit meansto be an apprentice,whatit meansto be a andof the dynamicrelationshipthatdevelopsbetween "master/mistress" practitioner, novice andexpertin the learningandteachingprocess,suchas in the workon the "reflectivepractitioner" by Schon(1983, 1987).I intendin the presentstudyto drawupon my experiencesas a carpentryapprenticeto commenton how I believe some of the characteristicsof sucha teachingandlearningsituationcouldbe used in the mathematics classroom. Knowing-in-Actionand Reflection-in-Action Schon(1983, 1987),in his proposalto redesignprofessionaleducationfor "reflectivepractice,"has writtenextensivelyabouthis researchinto "theexperienceof learningby doingandthe artistryof good coaching"(1987, p. 17). Schon(1987) proposes thatthe university-based professionalschoolsshouldlearnfromthe manytraditionsof educationfor practice,suchas studiosof artanddesign,conservatoriesof music anddance, in the crafts,all of whichemphasizecoachingand athleticscoaching,andapprenticeships learningby doing.He drawson the workof Dewey to supporthis argumentthatstudents can be coachedratherthantaughtwhattheyneedto know.Dewey (1974) stated: He has to see on his own behalfandin his own way the relationsbetweenmeans andmethodsemployedandresultsachieved.Nobodyelse can see for him andhe cannotsee just by being"told,"althoughthe rightkindof tellingmay guidehis seeing andthushelphim to see whathe needsto see. (p. 151)

23 Manyof ouractionsareintuitiveandspontaneousandreveala tacitknowledgethat is difficultto describeexplicitly.Schon(1983, p. 49), in describingthe behaviorof successfulpracticingprofessionals,claimsthat"ourknowingis in ouraction ... The Some examplesof workadaylife of the professionaldependson tacitknowing-in-action." this knowledgethatis implicitin actionincludejudgementsof distancein golf or ballthrowing,the use of tools, or therecognitionandcorrectionof a "poorfit"in design (Schon, 1983).We oftencannotdescribetherulesthatgovernthese actions,althoughof coursewe andafterthe fact. can reflecton ouractionsbothwhile we areacting(reflection-in-action), It is sometimespossible,by observingandby reflectinguponouractions,to describethe tacitknowledgethatis impliedby ouractions.Schon(1987) statesthatwhich,when we Knowingsuggeststhe dynamicqualityof knowing-in-action, describeit, we convertto knowledge-in-action. (p. 26) Whateverlanguagewe mayemploy,however,ourdescriptionsof knowing-inactionarealwaysconstructions.Theyarealwaysattemptsto putinto explicit, symbolicform,a kindof intelligencethatbeginsby beingtacitandspontaneous. (p. 25) I hypothesizethatthe intuitiveactionsof the carpentersreveala tacitknowledgeof mathematicsthatis difficultfor them,or for an observer,to describeexplicitly.I shall use to suggestthe dynamicqualityof this knowing-in-action.By the word"mathematizing" observingtheiractionsandperformingthe actionsmyself as an apprentice,by encouraging the carpentersto reflectupontheiractions,by reflectinguponmy own actions,andby reflectingupontheirreflections,I intendto describethe mathematizingimplicitin the actionsrelatedto carpentry. The Epistemologyof Constructivism Piaget'sworkteachesthatthe essentialway of knowingthe worldis not directly throughoursenses, butprimarilythroughouractions.Actionis understoodas being all behaviorby which we cause a changein the worldaroundus or by which we changeour own situationin relationto the world(Sinclair,1987). Suchbehaviortransformsthe relationshipbetweentheknowerandthe knownandsometimesresultsin the construction

24 of new knowledge,awareness,andmeaning.Newly constructedactionschemesremain viableuntilwiderexperienceraisesnovelquestionsandcreatessituationsin which the actionschemesareinadequateandrequirefurthermodification. As pointedout by Confrey(1991),constructivistsbase theirdefinitionof the constructionof knowledgeon Piaget'sprincipleof intellectualadaptation.Intellectual adaptationis an exchangebetweena personandhis or herenvironmentandinvolves assimilationandaccommodation. Assimilationoccurswhenthe cognitivesystemdealswith environmentaleventsin termsof its existingstructures,andaccommodationoccurswhen the cognitivesystemmodifiesitself as a resultof environmental demands(Ginsburg& Opper,1969).Childrenarealwaysactive,attemptingto understandthings,to structuretheir experience,andto bringcoherenceandstabilityto theirworlds.Throughassimilationand accommodation,they endeavorto reachan activeharmonywith the environment,which Piagetcalls "equilibrium" (Ginsburg& Opper,1969,p. 172).Von Glasersfeld(1982) claims thatPiaget'sconceptsof actionscheme,assimilation,andaccommodationhave been so as to conformto the traditional widely misunderstoodandhavebeenmisinterpreted beliefs of realists.Von Glasersfeld(1982, 1984)putsforwarda view of "radical constructivism" (whichexplicitlyrejectsthe notionthatknowledgemustmatchan absolute reality)as the epistemologicalview he considersto be mostin harmonywith Piaget'swork. thatprovidesa frameworkfor allowingmultipleand It is "radicalconstructivism" of knowledgeto co-exist.Von Glaserfeld(1984) describesa competingrepresentations theoryof knowledge"inwhichknowledgedoes not reflectan 'objective'ontological reality,butexclusivelyan orderingandorganizationof the worldconstitutedby our experience"(p. 24). Whereasrealistsexpectthatscientificandeverydayknowledgeprovide a truepictureof the realworld,independentof anyknower'sexperience,radical constructivistsbelievethatthe humanactivityof knowingcan lead only to a conjectural of the world;thuswe constructourown realities(von Foerster,1985;von interpretation Glaserfeld,1984).Insteadof the traditionalconceptionof a "match"betweenknowledge

25 andreality,von Glasersfeldconceivesof a "fit"betweenknowledgeandreality.He uses the metaphorof a key, or morethanone key, ableto "fit"the samelock, to illustratethe human endeavorof constructingourownkeys to fit the locks of ourexperientialworld,the testing groundfor ourcognitivestructures(vonGlasersfeld,1984).Ourknowledgeremainsuseful andviableso long as it fits the demandsandexpectationsof ourexperiences.Knowledge thatcollapsesunderthe demandsof ourexperientialworld"becomesquestionable, unreliable,useless, andis eventuallydevaluatedas superstition" (von Glasersfeld,1984, p. 24). is frequentlycriticizedas beingsusceptibleto relativism;skeptics Constructivism claim thatsince thereis no "certainknowledge"to appealto, theneverybody'sknowledge mustbe equallyvalid (Confrey,1985,p. 5). However,suchcriticismignoresthe fact that the "roleof others"in the constructionprocessis a fundamentalaspectof constructivism withothersandby interpretingthe acts of (Confrey,1985,p. 5). Throughcommunication others,we receivefeedbackaboutourconstructions. Througha processof argumentation andnegotiationwithincommunities,peopletryto reachsome consensusof meaningabout theirpersonalconstructs(Cobb,1986a,1986b;Confrey,1985) Cobb(1985), Confrey(1984a, 1984b,1985, 1987, 1991), KamiiandDeClark (1984), Labinowicz(1985), Steffe (1986, 1988),andSteffe,Cobb,andvon Glasersfeld (1988) have appliedthe epistemologyof constructivismto mathematicseducation.The constructionprocessof mathematical knowledgedescribedby Confrey(1987, 1991) occurs in partas the resultof repeatedcycles of actionandreflectionon a problematic(see Figure 2). Confrey(1991) gives a detailedexplanationof hermodelof the constructivistcycle. I shall paraphrase herexplanationin the paragraph thatfollows. Forthe constructivist,a problemdoes nothave an independentexistence;it is definedsolely in relationto its solver.The problemis seen as a "roadblock" to wherean individualwantsto be. The term"problematic" is usedto referto the individual's "roadblock." Whenan individualacknowledgeshis or herproblematic,equilibriumhas

26 been disrupted,andthe tendencyis to worktowardscoherenceandstability.In this way, the problematicacts as a "callto action."The "callto action"servesas an invitationbothto the whilethe problemsolveris invitedto engage problemsolverandto the researcher/teacher: of the in actionandreflection,the researcher/teacher is invitedto seek an understanding problematicbeingsolved,the actionschosen,andthe natureof the reflections.The actions takento resolvea problematicmay involvesensory-motoractivityand/orcognitive operations.Reflectionservesto providefeedbackon theresultsof the action.Has the problematicbeenresolved?Does theproblematicneedto be modified?Whatis an choice for the nextaction(if necessary)to begina new cycle? Whenan action appropriate is seen to be repeatedlysuccessfulin movingtowardresolutionof the problematic,that actionmay be objectified,named,or turnedintoa tool andusedfor futureactions. problematic

action

reflection Figure2. Theconstructivistcycle. as a ready-madecommodityfrom Thusmathematical knowledgeis not transferred one personto another;rather,peoplebuildtheirknowledgeon the basis of theirexperience (Steffe, 1986). Accordingto Labinowicz(1985, p. 5), arithmeticalknowledgeis Cobb "constructedactivelyby the childin the processof adaptingto his environment." (1986b)statesthatin the constructivistmodelof instruction,classroomsareviewed as placeswhereteachersandstudentsnegotiateas theyattemptto constructsharedmeanings. in whichthe teacherlistens to the children, Learningis thusa processof communication assumesthatthe child's behavioris rational,andtriesto identifyandunderstandthe child's goals. Confrey(1991) describestwo basicguidingcommitmentsfor the constructivist

27 researcher/teacher; first,thatthe meaningof themathematicalinsightsconstructedby individuals"lies withinthe frameworkof the individual'sexperience,"andsecond,that the individual'sinventionsandexplanations"havelegitimateepistemologicalcontent." The firstassumptionof theconstructivistview of mathematicsis describedas follows (Confrey,1991): Constructivistsview mathematicsas a humancreation,evolving withincultural contexts.They seek out themultiplicityof meanings,acrossdisciplines,cultures, historicaltreatments,andapplications.They assumethatthroughthe activitiesof reflectionandof communicationandnegotiationof meaning,humanbeings constructmathematical conceptswhichallow themto structureexperienceandto solve problems.Thus,mathematicsis assumedto containmorethanits definitions, theoremsandproofsandits logicalrelationships... includedin it areits formsof its evolutionof problemsandits methodsof proofandstandardsof representation, evidence. As pointedout earlier,Piaget'stheoryof cognitivedevelopmentwas builtas an individualisticmodel.Socialandculturalaspectsof developmentwere exploredonly so far as they affectedthe individual.Sinceconstructivismdevelopedfromthe ideas of Piaget,it is not surprisingto find the collective,societalelementsunderrepresented in the epistemologicalframeworkof constructivism, especiallyin relationto peers.However, recentworkin constructivismis movingtowardaccordingthe social andculturalaspectsof knowledgeconstructiona muchhigherprioritythanpreviously(Cobb,1988;Newman, Griffin,& Cole, 1989;Yackel,Cobb,& Wood, 1990). Constructivismis seen as a vitalpartof the conceptualization of the present research.The simplemodelof actionandreflectionon a problematicin the constructivist cycle has practicaluses duringfieldworkdesignedto recognizeanddescribethe mathematicalpracticesof a communityof people,since problemsolving is one of the occasionsduringwhichtheirmathematizing is demonstrated. The identificationof, and engagementwith, a problematicservesas a signalto theresearcherto takecarefulnote of all the interactionstakingplace.Since,as Schon(1983, 1987)claims,tacitknowledge residesin actions,it is importantto noticeall the detailsof these actionsandto reflecton

28 themlater.It is in reflectingon the actionsandthereflectionsobservedthatthe possibility of describingthe tacitknowledgedevelops. The commitmentof constructivistresearchersto the belief thatthe inventionsand sometimesseeminglyobscureandclumsyattemptsof all peoplerepresentslegitimate mathematicalknowledge,howevertentativeandpartial,lessensthe tendencyof the researcherto "switchoff' whenthe argumentsseem irrelevantandimplausible.Thereis a theexplanationandvery often to discover greatertendencyto tryto follow andunderstand thatit is one's own way of hearingthatneedsmodification.As one listens,one learnsto of single concepts.Traditional appreciatethe powerof constructingmultiplerepresentations school mathematicscan moreeasily be seen as one possibleconceptionof mathematicsand benchmarkaccordingto whichall the mathematicalendeavorsof not an international humankindmustbe judgedrightor wrong. What Is Culture and What Constitutesa Cultural Group? Most anthropologists todayagreewiththe semioticconceptof cultureespousedby Geertz(1973, p. 5), who believesthat"manis an animalsuspendedin webs of significance he himselfhas spun."Geertzregardscultureas thosewebs. It is throughculturalpatterns andorderedgroupsof significantsymbolsthatpeoplemakesense of the eventsin their lives. These symbolsystemsareconstructedhistoricallyandaremaintainedthroughsocial interactionsamongindividuals.The systemof symbolscomprisinga culturecan be November1988)to some (R. Ascher,personalcommunication, interpretedor "translated" extentby an observer.This is the mainendeavorof anthropology,whichregardscultureas a contextin whichsocial events,behaviors,processes,andinstitutionscan be describedby an ethnography.In the studyof culture,symbolicactsare"diagnosed"to "ferretout the importof things"in the processof analysisof social discourse,muchlike the unapparent way a physicianwoulddiagnosesymptomsin a patient(Geertz,1973). Giroux(1981) criticizesthe workof Geertzfor contributinglittle to the of how politicalpowerstructuresthe socioeconomicclasses, institutions,and understanding

29 social practiceswithina society.He describesthe conceptof cultureas being "strippedof or sociologicalobjectof study," its politicaldimensions,""reducedto an anthropological and"anapologyfor the statusquo"(Giroux,1981,p. 26). He calls for an acknowledgement thatany distinctionbetweencultureandpoweris false. Culturecouldthusbe redefinedin termsof its relationshipto the dominantsocialformationsandpowerrelationsin society. This "politicized"conceptof culturewouldincludethe relationshipbetweenideology and the socioeconomicsystem.Giroux(1981) describesa "politicized"notionof culturemore fully thus: Culture,in this sense,wouldbe definednot simplyas lived experiencesfunctioning withinthe contextof historicallylocatedstructuresandsocial formations,but as "livedantagonisticrelations"situatedwithina complexof socio-political institutionsandsocial formsthatlimitas well as enablehumanactions.(p. 26) In countrieswherethe politicalandeconomicdominanceof one groupover all othershas beenentrenchedfor manyyears,theconceptsof powerandconflictarecentralto a discussionof culture.In SouthAfrica,one cannotdiscusseducationalissues andlearning opportunitieswithoutrecognizingandacknowledgingthe existenceof a myriadof unwrittenconstraintsandsocialtaboosrootedin racismin additionto the overtformsof discriminationlegislatedby apartheid.Althoughthe presentstudydoes not follow the ideas of Girouxfully, it is recognizedthatcultureis morethanthe expressionof thatwhichgrows out of the experiencesof peoplein theirdailysocial andworkingroles.Working communitiessharea culturedefinednot only by the systemof symbolsdeveloped historicallythroughthe practiceof theircraftor trade;politicalandsocioeconomicfactors also play a significantpartin shapingthecultureof a group.Thusit is claimedthata group of carpenterssuch as the one selectedfor this studyconstitutesa culturalgroup,although theremay be a diversityof languagesandreligionsamongmembers.The notionof culture used in the presentstudysupportsD'Ambrosio's(1985) broadconceptualizationof what constitutesa culturalgroup. Forthe purposesof the presentresearch,the culturein the workshopis seen to includethe "antagonisticrelations"thatexist at some level betweenthe artisanswho wish

30 to protecttheirturf(as well as theirsocioeconomicstatus)andthe workerswho strugglefor opportunitiesto learnandto improvetheirsocialandeconomicstatus.The abilityof all the employeesto shapetheirlives withinsocial,political,andeconomicconstraintsis also recognizedas partof the cultureof theworkshop. The Cultural,Social, and PoliticalAspects of Mathematicsand MathematicsEducation is a pan-cultural phenomenon"(1988a,p. 56) Bishop'sstatementthat"mathematics is supportedby the existenceof Chinesemathematics,Greekmathematics,Roman mathematics,Africanmathematics,Islamicmathematics,Indianmathematics,andNeolithic mathematics.By referringto this long list, Bishopconcludesthatdifferentculturesproduce differentmathematics.Furtherexamplesof mathematicsthatdiffersfromthatusuallyfound in Westerntextbookscan be foundin theworkby Zaslavzky(1973), Brenner(1985a, 1985b),andGerdes(1985, 1986, 1988a,1988b)in Africa;Bishop(1979), Lancy(1983), andLean(1986) in PapuaNew Guinea;andPinxten,van Dooren,andHarvey(1983) with literaturesuggeststhatall culturesdeveloptheirown forms the Navajo.Thiscross-cultural of mathematicsdependingon the demandsof theirparticularenvironmentsandthe goals andpurposesof the people. history,culturecontactandcultureconflict Bishop(1988a)explainsthatthroughout in the formof trade,religion,education,andwarfarehave donemuchto eliminatecultural diversityin the areaof mathematicsandhavealso contributedto the growthof as an internationalized "Mathematics" discipline.Bishop(1988a)emphasizesthat Mathematicalideas growas a resultof developmentstakingplacebothwithinculturesand betweencultures.The label,"Mathematics," (whichusuallymeansWesternacademic mathematics)is generallyusedas if all the ideasreferredto by the label were developedby of historicalrecords Westerncivilizations.This is notthe case, as an honestinterpretation will reveal(Bishop,1988a;Gerdes,1985;Joseph,1987).It is importantthatstudentsall over the worldbecomeawareof the factthat,with few exceptions,a stronglyEurocentric

31 view of mathematicshas permeatedtheirschooling.The historiesof societies outsideof the to mathematicshavebeen misrepresented, and Europeantraditionandtheircontributions the "imperialist/racist ideologyof dominance"thatunderpinsthe widespreadEurocentric view prevailsdespiteconvincingevidenceto the contrary(Joseph,1987, p. 16). Whatwe call "Mathematics" is madeup of contributionsfrommanycultures,manyof themnonWestern(Bishop, 1988a;D'Ambrosio,1987;Gerdes,1985). Bishop(1988a)refersto the manydifferentconceptsanddefinitionsof culturethat have been putforwardby culturalanthropologists. He also discussesthe workof the mathematicianKline(1980), whichfocuseson thecultureof mathematics,andthe work of mathematician/historian Wilder(1981), whichfocuseson the culturalgroupof mathematicians. Bishoprejectsbothof theseapproachesas modelsbecauseof their exclusionof nonmathematicians. Whilehis workrefersmainlyto culturalgroupsthatare nationalgroupsor languagegroups,he does not give a clearstatementas to whatconcepts of cultureandculturalgrouphe, as an educator,choosesto use. D'Ambrosio(1985) suggeststhatit maybe fruitfulto explorethe mathematical activitiesof smallercommunitieswithcommonpurposesandinterests.His conceptualizationof the kindof culturalgroupthatgeneratesits own formsof mathematics does not remainrestrictedto nationalgroupsbutis muchmoreencompassing.D'Ambrosio (1987) states: ourconceptionof "Ethno"encompassesall the ingredientsthatmakeup the cultural identityof a group:language,codes, values,jargon,beliefs,food anddresshabits, physicaltraits.... Culturalgroups,childrenof a certainage rangein a neighborhood,farmerscultivatingwheat,engineersin carfactories,andso on, all have theirown patternsof behavior,codes, symbols,modesof reasoning,ways of measuring,of classifying,of mathematizing. (pp.4-5) While bothBishopandD'Ambrosioclaimthatdifferentculturalgroupsgenerate differentformsof mathematics,theyemphasizethatit is througha very similarrangeof humanactivitiesthatthesedifferentformsresult(Bishop,1988a;D'Ambrosio,1985, 1987). The identityof the mathematicsdepends,for example,on the environment,the context,the focus of interest,the motivations,the formsof communication,the goals, andthe purposes

32 of each group,andtherewill be manydifferencesin theseareas.However,thereare strong similaritiesbetweenculturesas well. activities,which,he argues,formthe foundations Bishopdescribessix "universal" for the developmentof mathematicsandwhichareextremelywidespread,if not universally practicedin everyculture.Countingandmeasuringarebothconcernedwith ideas relating to number.The significantfeatureof countingis its discretenessin the associationof objectswith numbers,in contrastto the continuousnatureof measuring.Measuring thatareimportant.The involvescomparing,ordering,andquantifyingcharacteristics activitiesof locatinganddesigningfall intothe categoryof spatialstructuring,whichhas ideas.In locatingactivities, been of greatsignificancein the developmentof mathematical featuresof the environment-how andcartographical the emphasisis on the topographical spaceis conceptualizedandhow peopleandobjectsarepositionedin the spatial of objectsandartifactsthatare environment.Designingrefersto the conceptualization for use in ourhomesandleadsto the idea of "shape."Two activitiesthatlink manufactured us to our social environmentareplayingandexplaining.Playingis concernedwith social proceduresandrulesof performanceandalso stimulatesthe "asif" featureof imaginedand hypotheticalbehavior.Explainingrefersto the variouscognitiveaspectsof enquiringinto andconceptualizingthe environmentandof sharingthoseconceptualizations. Explanations establishmeaningfulconnectionsbetweendifferentphenomenain responseto the question "Why?" Thesecategoriesareusefulfor directingthe researcher'sattentionto thosehuman activitiesthatarelikely to be fruitfulareasof furtherstudyif one is interestedin withina communityof people.Membersof the documentinginstancesof mathematizing communitywho identifyproblematicsthatinvolvethesecategoriesof activity,andwho engagein the constructivistcycle, arelikely to constructor use some formof mathematical knowledgeto reachsolutions.

33 Fasheh(1982) pointsout thata prevalentmisconceptionin the teachingof mathematicshas been, andstill is, the beliefthatmathematicscan successfullybe taught withoutrelatingit to cultureor to the individualstudent.StiglerandBaranes(1988, p. 259) identifythreeprimarypointsat whichcultureinfluencesmathematicslearning.These points are (a) culturaltools, (b) culturalpractices,and(c) culturalinstitutions. Culturesprovidetheirmemberswith a varietyof tools that,throughprocessesof These internalization,becomea partof the individual'srangeof mentalrepresentations. can be usedfor thinkingaboutmathematicsandfor solving mathematical representations problems.Mathematicalthinkingis shapedby tools like languageas well as concrete culturalartifactslike the abacus,widelyusedin Taiwan,Japan,andotherAsiancountries, andlike the compassrose usedby medievalnavigators.Boththe abacusandthe compass rose provideda "representational structurethatenabledpotentiallycomplexcalculationsto be easily performedmentally"(Stigler& Baranes,1988). Culturalpractices,suchas the navigationaltraditionsof illiterateMicronesian islanders,wereshownto requiresuchtasksas estimatingandcalculatingdistancesand angularmeasure,while the fish marketingstrategiesof womenin Ghanamadeuse of the conceptof probabilities(Stigler& Baranes,1988).Otherexamplesof culturalpracticesare those given by Gerdes,suchas the Mozambicanpracticeof fish drying,which depends of the circleconcept(1985, p. 17);of housebuilding,which is based uponan understanding on an understanding of the propertiesof a rectangle(1985, p. 17);andof basketweaving, which uses the resultsof the Pythagorean theorem(1986, p. 12). StiglerandBaranes(1988) suggestthatculturalpracticesarestructuredin a way so as to eliminatethe need for repeatedcomplexcalculations.Tools andrulesaredevelopedso as to optimizeefficiency andease of performance. The institutionof schoolingis surrounded by a culturalcontextthattransmitsboth intendedandunintendedbeliefs,values,andpracticesto its participants.Some of the unintendedbeliefs arethatmathematicsproblemscan alwaysbe solved in less than 10

34 minutesandthattherewill alwaysbe an answerto a problemthatcontainsnumbers, regardlessof the meaningof theproblem;forexample,threeout of fourelementaryschool studentsgave a numericalanswerto the problem,'Thereare26 sheepand 10 goats on a ship. How old is the captain?"(Stigler& Baranes,1988,p. 288). Bishop(1985, p. 26) remindsus thatalthougha classroomis partof theinstitutionof the school,each classroom groupis a uniquecombinationof people,with"itsown identity,its own atmosphere,its own significantevents,its own pleasuresandits own crises.... It has its own history createdby, sharedbetweenandremembered by the peoplein the group." Referencesto the social aspectsof mathematicscan also be foundin the workof some mathematicians andphilosophers,who offera challengeto the traditionalconceptions of mathematics.The worksof Lakatos(1976) andTymoczko(1980, 1986) areseen as particularlyrelevantfor the presentstudy.Lakatos(1976) regardsmathematicsas intrinsicallyspeculativeandfallible,growing,like the Popperianview of science, as a refinementof theories,refutation,and patternof conjectures,criticism,counterexamples, furtherrefinement.He does not rejectformalmathematicsbutrejectsthe idea that mathematicsgrowsin the deductivepatternof formalization.It is claimedthatthe logic of mathematicaldiscoveryis naturallydeductiveonly if one restrictsmathematicsto formal axiomaticsystemswhereprogresscan occuronly in measuredlogical steps.This conceptionof mathematicsis falselyimpliedby formaldeductiveproofs,which are presentedin a way thatconcealsthe primitiveconjectureanddisguisesthe criticismand refutations.In the wordsof Lakatos(1976): In deductiviststyle, all propositionsaretrueandall inferencesvalid.Mathematics is presentedas an ever-increasing set of eternal,immutabletruths. air refutations,criticismcannotpossiblyenter.An authoritarian Counterexamples, andproofis securedfor the subjectby beginningwithdisguisingmonster-barring generateddefinitionsandwiththe fully-fledgedtheorem,andby suppressingthe primitiveconjecture,therefutations,andthecriticismof the proof.Deductivist style hidesthe struggle,hidesthe adventure.The whole storyvanishes,the successivetentativeformulationsof the theoremin the courseof the proofprocedurearedoomedto oblivionwhile theend resultis exaltedto sacred infallibility.(p. 142)

35 The book ProofsandRefutations(Lakatos,1976),most of which takesthe form of the of a discussionbetweena teacherandhis students,is a "rationalreconstruction" conjectureaboutthe relationship developmentof the prooffor the Descartes-Euler betweenthe numberof vertices(V), edges (E), andfaces (F) of polyhedra,namelythat V- E + F = 2. Lakatosgives a vivid illustrationof how mathematicalknowledgeis The studentsrepresent constructedfromthe interactionsof a groupof mathematicians. who triedto proveEuler'stheoremfromthe mid-18thcentury variousmathematicians throughoutthe 19thcentury.The socialexchanges,arguments,disagreements,attemptsto convince,attemptsto find a hole in another'sargument,criticisms,andelaborationsof another'sworkcannottakeplacein the absenceof a community,althoughthis pointis not explicitlymade.It seemsclearthatLakatossupportedthe idea thatmathematicalknowledge is socially constructedandis not individuallydiscovered,butit is doubtfulthathis concept of communitiesincludedpeopleotherthanprofessionalmathematicians. While Lakatosmadeno specificreferencesto the role of communityin the creation of mathematicalknowledge,Tymoczko(1986) centershis argumenton the social aspects of knowledgeconstruction.Tymoczkostatesthatthe primeconcernof the philosophyof mathematicsshouldbe to arguefor therole of thecommunityof mathematicians insteadof He arguesthatmathematicsis a human focusingon one individual,isolatedmathematician. activity,andhe urgesfor recognitionthatmathematicsis publicknowledgeandthatit is in theircommunitythatgroundsmathematical communicationamongmathematicians knowledge(Tymoczko,1980).However,Tymoczkoneglectsthe pointof view of the student,who is portrayedas the passiverecipientof the accumulatedwisdomof experiencedmembersof the mathematicscommunity.The studentis not consideredto be a participantin the creationof knowledgeandin the solutionof problemsbut as an initiate,to be moldedin the imageof the currentmembersof the elite communityof professional mathematicians. As with Lakatos,Tymoczko'sview of mathematicsdoes not seem to encompassthe creatorsandusersin theworldoutsideof the ivory tower:the children,

36 students,and"ordinary" peoplewho also createanduse mathematicsto explain,convince, andcommunicate. The mathematicseducatorSteffe (1986, 1988)andhis colleagues(Steffeet al., 1988) have developeda modelof how youngchildrenconstructthe numbersequenceusing theirearlycountingschemes.Thereis also evidencethatchildrencreatedifferenttypesof unitsandbase theirconceptualization of multiplicationon thoseunits.Steffe describesthis as partof the "mathematics of children"andvoices his explicitintentionthat"school mathematicswill at leastincludethe mathematicsof children"(1986, p. 138). Apartfromthe childrenandstudentswho havebeenleft out as genuinemakersand usersof mathematics,othergroupshavealso consistentlybeendeniedthe statusof fully Harris(1987)pointsoutthatin mosthistoriesof mathematics,women fledgedparticipants. mathematicians arescarcelymentioned.In fact,throughoutWesternhistorythe worksof womenmathematicians havebeen"ignored,robbedof creditandforgotten"(Alic, 1986). Harris(1988b)contraststhelow achievementof girlsin school mathematicswith theircapacityto do complicatedneedlework.In herresearchwith womenwho makeand use textilesat homeandin factories,Harris(1987) pointsout the similaritiesbetweenthe problemof shapingthe heel of a knittedsock andthe problemof enclosinga right-angled material.In essence, the mathematical cylindricalpipe in a factorywith heat-insulating problemsto be solved arethe same;however,theindustrialproblemwouldtypicallybe consideredto requiremathematical thinking,andthe sock problemwouldnot. Similarly, the complicatedsymmetricaldesignsthatilliteratewomenweaversconstructin theirrugs are not consideredas evidenceof anyformof intellectualthought.However,Harris(1988a) pointsout thatin culturessuchas in theGambia,wheremen do the weaving,the educationalaspectof weavingis claimed"undertheequationweaving= knowledge= power."Harris(1988b)concludesthatthe problemlies in who defineswhatmathematics is andin who defineswhatthe standardsare.In a male-dominated society,women's

37 mathematicsis not acknowledged.Fasheh(1988) madesimilarobservationsabouthis mother'smathematics: My illiteratemotherroutinelytookrectanglesof fabricand,with few measurementsandno patterns,cut themandturnedtheminto beautiful,perfectly fittedclothingfor people.... The mathshe was usingwas beyondmy comprehension.... Seeingmy mother'smathin contexthelpedme see my math in context,the contextof power.Whatkepthercraftfrombeingfully a praxisand limitedherempowermentwas a socialcontextwhichdiscreditedheras a woman anduneducatedandpaidherextremelypoorlyfor herwork.(p. 2) Gerdes(1988b)explainshow racistandcolonialideologynegatedthe capacityof Mozambicansto do mathematics.Theindigenouswisdomandmathematicalknowledgeof the colonizedwereignoredor ridiculedandhaveleft a legacy of self-doubt.The aftermath of yearsof repressiveeducationsystemsfurtherexacerbatethe problemsof mathematics education.Gerdes(1985) states: Duringthe yearsof Portuguesedomination,mathematicswas taught,in the interestsof colonialcapitalism,only to a smallminorityof Africanchildren.And those Mozambicansweretaughtmathematicsto be ableto calculatebetterthe hut tax to be paidandthe compulsoryquotaof cottoneveryfamilyhadto produce. They were taughtmathematicsto be morelucrative"boss-boys"in SouthAfrican mines.(p. 15) In SouthAfrica,the educationalpoliciesof the colonialera becamemoredeeply entrenchedwith the rise of theracistindustrialcapitalistsociety.As statedby Kallaway (1984): Above all, schoolsnegatedthe commonsense knowledgeof the colonizedby reinforcingthe self-imageof incompetenceandignorancefor those who did not go to school or thosewho failedat school.This oftenalso led to those who were schooledto despisetheirown cultureandtraditionsin favourof those of the colonizeror at very best to becomeambivalentabouttheirlinkswith the "traditional" pastor social milieu.Thisprocesshas been describedas a "culture of silence"-where the colonialelementin schoolingis its attemptto silence, to negatethe historyof the indigene,to rationalizethe irrationaland(to) gain acceptancefor structureswhichareoppressive.(p. 9) In the segregatedSouthAfricanschools,blackstudentsareoften taught manneras a set of formulasandrulesto be mathematicsin a particularlyauthoritarian memorizedanda set of problems(whichhaveno connectionto the students'lives) to be solved (Dostal& Vergnani,1984).Manystudentsbelievethatthereis only one "correct" answerandthattheirtaskis to findthis answerandto ratifyits truthwith the authority

38 vestedin the teacheror the authorityvestedin theanswersprintedin a list at the backof the textbook.Approachesto textbooks,curricula,andinstructionin SouthAfricahave been importedfromEuropeandthe U.S.; thereareno discussionsof any Africancontributionsto mathematicsin texts.This all addsto thealienationof the studentsfromthe subjectmatter. Even whencarehas beentakento translatethe text into the mothertongueof the students, problemsremain(Berry,1985). Duringrecentyears,students,parents,andteachersfromblackcommunities educationandhavedemandedschoolingthatprovides have rejectedstate-controlled opportunitiesfor peopleto pursuetheirown goals andto realizetheirsocial, economic,and is a popularsloganamongblackSouth politicalpotential."Educationfor Liberation" Africans,andthereis community-based supportfor "People'sEducation."The workon includesthe basicmathematicsandstatisticsrequired,for example, "People'sMathematics" to analyzeandcriticizeimportantissues in communitiessuch as laborandrentdisputes, newspaperreports,anddifferentialgovernmentfundingof educationfor differentrace groups.Thus,this workformsa valuablecontributionto people'sgrowingawarenessof theirown exploitation(Breen,1986).The mostvitalissue is obviouslynot simply to point out injusticesbutto empowerpeopleto believein theirown abilitiesto mathematize.The destructiveeffects on blackstudentsof alwaysseeingschool mathematicsas comingfrom somewhereoutsideof theirown culture,as somethinga few hard-workingandsuccessful amongthemmay "borrow"anduse, butneverreallytakeownershipof, shouldnot be overlooked. Mellin-Olsen(1987) describeshow the mathematicsclassroomcan convertthe of some Norwegianstudents,who have passiveness,indifference,anddestructiveness rejectedmathematics,intoconstructiveactivities,leadingto inquiry,discovery, constructionof new perspectives,andenhancedsocialresponsibility.Forexample,when studentsin a largesecondaryschoolwantedto burntheirtextbooksbecausetheycontained "alot of rubbish,"Mellin-Olsen(1987) suggested,"Youshouldcollect some statistics.See

39 whatit is in this book thatmakessense to you. You shouldcollect some dataaboutit and reportbackto the publisher.Ask themwhy theypublishsuchrubbish"(p. 200). In this way, he hopedto turntheirnegativeenergytowardsactivitiesthatcouldgenerateself-directed learning. In thinkingaboutthe issues of authorityandconflictin mathematicseducation,I supportthe followingstatementmadeby Fasheh(1982): In short,I cameto believethatthe teachingof mathematics,like the teachingof any othersubjectin school,is a "political"activity.It eitherhelps to create attitudesandintellectualmodelsthatwill in theirturnhelp studentsgrow, develop,be critical,moreawareandmoreinvolved,andthusmoreconfidentand able to go beyondthe existingstructures; or it producesstudentswho arepassive, rigid,timidandalienated.Thereseems to be no neutralpointin between.(p. 7) Research on the Creation and Use of Mathematicsin Everyday Settings As discussedin Chapter1, the use of mathematicsin everydaysettingshas been studiedby two maingroupsof researchers.In the nexttwo sectionsI shallexamineand andelaborateon critiquethe workdonein "everydaycognition"andin "ethnomathematics" how questionsraisedby these areasof researchwill be addressedby the presentstudy. Everyday Cognition Lave's theoreticalargumentshavebeendiscussedin a previoussection.Lave and hercolleagueshave also produceda wealthof empiricalevidencethathighlightsthe relevanceto mathematicseducationof studiesaboutthe mathematicsused in everyday practice. tailorsandcloth merchantswho Earlyresearchwith "schooled"and"unschooled" hadlearnedtheirprofessionsvia apprenticeship systemsshoweddifferentresultsregarding the effects of schoolingon the abilityto do school-typearithmetictasksthatwere similarto the arithmetictasksperformedwithease on thejob. Lave (1977) andReed andLave (1979) studiedthe problem-solvingskills of Liberiantailors.Tailoringwas describedas a skill thatentailsmuchquantitativecomputation,suchas measuring,fitting,andcalculating payments.Both schooledandunschooledmastertailorswereequallyexpertat these tasks. Lave gave "tailoring"andnontailoring,"school-type"tasksto the mastertailors.All the

40 tailorsperformedequallywell on the tailoringtasks,butthe schooledtailorsperformed muchbetteron the school-typetasksthandidthe nonschooledtailors.Lave interpretsthe abilityof the nonschooledtailorsto solve some of the school-typetasksas evidencethatthe apprenticeship trainingallows"generalproblem-solving principles"to develop(1977, p. 179). However,the evidencesuggestedthatthe schooledtailorshaddevelopedmuch morepowerfulgeneralizedstrategiesfor solvingnovel problems. Contraryto thesefindingsthattraditionalpeoplefoundabstractmanipulationof numbersdifficultoutsideof familiarconcretecontexts,Petitto'sresearch(1979) with Dioulatailorsandclothmerchantsfromthe IvoryCoastshowedthatschoolinghadlittle or no effect on theirabilityto performnovel arithmeticproblemsthatwereanalogousto the or in practice.Petitto(1979) arithmeticcomputationskills developedduringapprenticeships investigatedthe arithmeticpracticesof the clothmerchantsandof the tailorsin theirusual context,in unfamiliarcontexts,andin abstractschool-typeproblems.Petittofoundher illiteratesubjectshadlittledifficultyin solvingarithmeticproblemspresentedin a purely skills wereorganizedinto specific numericalform,withoutcontext.Generalcomputational strategiesthatinvolvedthe numericalrelationshipsthemselvesanddid not dependon concretecontexts.Strategieslearnedin thecontextof practicalactivitywere foundto be transferableto directlyanalogousproblemsin novel settings.Petitto(1979, p. 197), however,pointedoutthat"thefamiliarandunfamiliarpracticalarithmeticproblemsmay be moreanalogousto eachotherin ourstudythanin Lave's"andalso thatthe schooledtailors in Lave's studyhadconsiderablymoreyearsof schoolingthandid the schooledtailorsin Petitto'sstudy. Schliemann(1984) studiedproblemsolvingamonga groupof professional carpentersanda groupof carpentryapprenticeswith a rangeof educationalbackgrounds. The carpentershadfrom0 to 5 yearsof formalschoolingandhadlearnedtheirprofessions while workingas assistantsto theownerof a workshop,usuallytheirown father.The apprenticeswereattendinga 3-yearcourseof instructionin carpentry,as well as attending

41 formalschool, andhadat least4 yearsof schoolmathematics.In the carpentryclasses, great andon how to calculateareaandvolume. emphasishadbeen laid on measurement Schliemannreportsthatin the study,each of thecarpentersandapprenticeswas given a clear,well-labelleddiagramof a bed andaskedto find out how muchwood wouldbe requiredto buildfive suchbeds. Althoughmost of the professionalcarpentersusedsuitablestrategiesto find a solutionanddisplayedan understanding of theconceptof volume,the apprentices'attempts were unsuccessfulandthe resultsobtainedwereabsurd.Althoughthey hadreceivedformal instructionon how to calculatevolume,mostapprenticeseithersimplyaddedtogetherall the dimensionsandgave a numericalanswer,or addedtogetherall the lengths,all the widths,andall the thicknessesto give the dimensionsof a huge blockof wood. Their formalknowledgeof the algorithmfor calculatingvolumedid not help themto solve the practicalproblem.5 SchliemannandAcioly's work(1989) with schooledandunschooledbookieswho runa lotterygamein Brazilshowedthatat work,regardlessof the level of schooling,the bookieswere extremelyefficient.Whenaskedto solve modifiedproblems,schoolingdid not influencethe use of specificalgorithmsfor performingarithmeticoperations,although the schooledbookieswerebetterableto analyzeandexplainverballythe relationshipsand mathematicalmodelsinvolvedin thelotterygames. Saxe (1988) investigatedthemathematicsof Brazilianchildcandysellers.His findingsshow thatsellerswith littleor no schoolingdevelop,in theirtrade,arithmetic practicesthatdifferfromthe arithmetictaughtin schools.Peerswho did not sell candy showedno evidenceof similarknowledge.Furthermore, it was foundthatyoung schoolgoing candysellersusedstrategiesanalogousto thoseusedin "currencyarithmetic"to solve additionproblemsat school.Forexample,28 + 26 = ? was reorganizedas practice-related

of volumein 5Comparethis situationto the discussionof the carpenters'understanding the presentstudy.See episodes 15-19, Chapter5.

42 (20 + 20) + (8 + 6) = ? This tendencyfadedwithtime,andoldercandysellersused standard school-taughtalgorithmsto solve schoolproblems.Overtime,however,candysellers developedincreasinglycomplexandpowerfulmathematicsrelatedto theirtrading activities,constructingconceptsof ratioandcontinuingto constructproblem-solving strategiesthatreliedon thestructureof the currencysystem. et al. (1985) conductedresearchwithchild street-vendorsin Recife, Brazil. Carraher Researchersposedas customersandaskedthevendorsto performa varietyof transactions on possiblepurchases,andlaterthe childrenwereaskedto takepartin a formalpencil-andpaperarithmetictest. As with Lave'sgroceryshoppers,resultsshowedthatproblemsolving in the marketplacewas significantlymoresuccessfulthanproblemsolving on the formal test.Therewas evidencethatthestrategiesbroughtto bearon daily tradingproblemswere not the routinesthatthe childrenhadlearnedin school.Childrenreliedon mental calculations,andthe preferredstrategyfor multiplyingseemedto be chainingsuccessive additions.In otherexamples,as the additionbecamemoredifficult,the child decomposeda quantityinto tens andunits,to add35 to 105,firstadding30, andthenadding5 later.In the formaltest, the childrentried,withoutsuccess,to follow the problem-solvingroutines taughtin school. Theseresearchfindingsraisetwo importantquestions:first,how is mathematical knowledgedevelopedfromeverydayexperience,andsecond,how can this out-of-school knowledgebe harnessedas a strengthandusedas a foundationfromwhichto develop mathematicsin the classroom? CarraherandCarraher(1987)attemptto addressthe secondquestion,arguingthat the strategiesusedin the arithmeticthatis learnedoutsideof schools ("naivearithmetic") are basedon the sameprinciplesas the arithmetictaughtin school.Forinstance,the decompositionandrepeatedgroupingmethodsusedby the youngstreetvendorsarebased on associativity,commutativity,andthe distributivepropertyof multiplication.Thusnaive arithmeticcan be extendedintoschoolarithmetic.However,as a resultof the differencesin

43 the situationsin which"naive"andschoolarithmeticarepracticed,problemsolvers experiencemoresuccess with "naive"arithmetic.In the marketplace,answersare evaluatedfor appropriateness andrejectedif theydo not makesense;at school, appealsto commonsense arefruitless,since conceptsareusually"strippedof situationalcomponents & Carraher,1987,p. 11). Schoolsmusttherefore duringthe learningprocess"(Carraher find contextsthatmakeconceptsmeaningfulandusefulin daily situations. Brenner's(1985a, 1985b)workwithVai elementarystudentsin mathematics classroomsgives an excellentexampleof how the Vai childrencombinetwo distinct arithmeticsystemsto masterthe schoolcurriculum.Vai childrenfromLiberialearnto count and performa varietyof arithmeticcalculationsin Vai as a naturalpartof learningtheir in homeandschool activities.Simultaneously,they are languageandthroughparticipation engagedin learningarithmeticin Englishat school andareusingalgorithmssimilarto those taughtin U.S. schools. Vai arithmeticis characterized by combiningstrategiessuch as extensivecounting, breakingproblemsinto smallerunits,convertingmultiplicationinto a seriesof addition steps, mentalcalculations,andmemorizedfacts.Brennerobservedthatthe childrenuseda combinationof Vai, school,andinventedarithmeticalgorithmsduringindividualtesting andclassroomlessons.Thosewho usedthe widestrangeof methodswere the most successful.Teacherswereall membersof the Vai groupandweresupportiveof the children'sstyles. The patternof teacher/student interactionsin classroomswas culturally andlessons werestructuredso thatchildrencould displaytheircompetence appropriate, ratherthantheirmistakes. The researchreviewedso fargives evidencethatin mostcases arithmeticthatis developedin everydaysettingsis usefulfor solvingnovel problems;in some cases, even noncontextual,school-likeproblemsweresolved.Therehas also beenconsiderablework done to explorethe oppositequestion.Of whatuse is school mathematicsin the solutionof real-lifepracticalproblemsencounteredby educatedadultsin everydaycontexts?

44 Laveet al. (1984) andMurtaugh(1985) foundthatarithmeticpracticestaughtin schools wereignoredby U.S. groceryshoppersin favorof methodssuchas approximation andestimation,whichbettersuitedthe goals andpurposesof the groceryshoppers.Unit priceinformationwas not generallycalculated,norwas it usedif providedby the store. Quantitativeinformation,suchas best-buycalculations,wereused only afterthe options hadbeen narroweddownto two or three.Initialselectingwas basedon brand,quantity, size, andstorageconsiderations.Theemphasiswas thuson efficiencyandtime-saving. Whenbest-buycalculationswereperformed,all shoppers,regardlessof theirlevel of test of schooling,wereequallysuccessful.However,theresultsof a pencil-and-paper arithmeticproblemsshowedthatthe shoppers,who hadcarriedout virtuallyerror-free arithmeticin the supermarket, averagedonly 59%on the arithmetictest. The numberof on the test butnot with the frequency yearsof schoolingcorrelatedhighlywithperformance of calculationsin the store.In everydaysituationsin the grocerystore,school-taught arithmeticseemedratheruseless. In the studyby de la Rocha(1985)withparticipants in a WeightWatchersdieting of foods, the emphasiswas also on program,a programthatrequiresprecisemeasurement time savingandefficiencyandnot on employingschool-learnedarithmeticskills. De la Rochafoundthatthe dietersinventedstrategiesto simplifytheirmeasuringtasksin the kitchen.Forinstance,a measuredamountof food was placedin a containerwith a decorativepatternandthe levels notedrelativeto the pattern.The samecontainerwas then used repeatedlyto eliminatethe needfor continualmeasuring. Scribner(1984, 1985)reportedthatdairyworkersdevelopedefficient,labor-saving like the layoutof the dairyandthe structureof strategiesbasedon physicalcharacteristics productcontainers.Forinstance,insteadof collectingitemsfromshelves in the warehouse in the sequencelaid out by the orderforms,dairyworkersorganizedandcoordinatedtheir activitiesin such a way as to minimizethe distanceto be traveledaroundthe huge warehouse.

45 In the dairywarehouse,quartsof milkwerepacked16 unitsto a case. On the order form,ordersfor 17 to 24 quartswereexpressedas 1 case + 1 unit,up to 1 case + 8 units. Ordersfor 25 to 31 quartswereexpressedas 2 cases - 7 unitsup to 2 cases - 1 unit. However,in makingup milkorders,the workersdid not adhereto the way in which orders were writtenon the orderformbutwereobservedto havelargerepertoiresof solution strategies,dependingon whatconfigurationof cases was immediatelyavailablein the storeroom.In this way, mentalenergywas usedin orderto save the physicaleffortof movingextracases. The workerswerefoundto rely on visualization,ratherthanon arithmetic,to solve problems. Examplesof researchof thiskindarecompellingbecausethey raisefundamental questionsaboutthe usefulnessof school-taughtformalmathematicsfor people's lives. Whatis it aboutacademicmathematicsor aboutthe way in whichwe teachmathematicsin schools thatleaves peopleunwillingor unableto applythatknowledgeusefullyin everyday settings?Peoplearefully capableof devisinginnovativemathematicalmethodsto solve the problematicsandto accomplishthe goals thatthey set for themselves.Perhapspartof the problemwith school mathematicsis thatstudentsareseldomgiven the opportunityto definetheirown problematics.Solvingwell-definedacademicmathematicsproblemsin the classroomdoes not adequatelyportraythekindof problemsolvingrequiredin everyday settings,wherethe situationmaybe anythingbutclear-cut. Ethnomathematics Althoughtermslike ethnobiology,ethnoscience,ethnomedicir.,andethnohistory are well establishedin the anthropological has a literature,the term"ethnomathematics" muchmorerecenthistory(Brodie,1988).The descriptor"ethnomathematics" was first introducedinto the mathematicseducationliteratureby D'Ambrosioof Brazil(1985, p. 44), who statesthatthe field of ethnomathematics "lieson the borderlinebetweenthe historyof mathematicsandculturalanthropology." He claimsthatresearchin this areais important for recognizingthatdifferentformsof mathematicsmayresultfromdifferentmodesof

46 thoughtacrossculturalgroups.The "culturalgroups"to whichhe refersarevery broadly conceptualized.D'Ambrosio(1985) states: In contrastto this [i.e., themathematicsthatis taughtandlearnedin schools] we the mathematicswhichis practisedamongidentifiable will call ethnomathematics culturalgroups,suchas national-tribal societies,laborgroups,childrenof a certainage bracket,professionalclasses,andso on. (p. 45) "Ethno"... includestheirjargon,codes, symbols,myths,andeven specific ways of reasoningandinferring.(D'Ambrosio,1987, p. 2) of mathematicsthat D'Ambrosio(1985, 1987)calls for a widerconceptualization would allow legitimacyfor nonacademicpractices.He statesthatthe purposeof his researchprogramis to identifya structured body of knowledgewithinethnomathematics. The researchprogramis beingcarriedoutby collectingexampleson the mathematical groupsandlinkingthesepracticesto patternsof practicesof culturallydifferentiated reasoningandmodesof thought. The conceptof ethnomathematics presentedby D'Ambrosioremainsunclearas a is numberof alternativedescriptionsareprovidedto the reader.Ethnomathematics portrayedsimultaneouslyas the studyof mathematicswithina cultural,historical,and framework,as well as "themathematicswhichis practicedamong anthropological identifiableculturalgroups"(1985, p. 45) (i.e., the mathematicsitself is also called D'Ambrosiogives few practicalguidelinesas to whatspecifically ethnomathematics). shouldcountas "mathematical practices,"andthereis no discussionof the analytical processthatwouldbe requiredto uncoverthe patternsof reasoningmentionedabove. was coinedindependentlyby AscherandAscher The term"ethnomathematics" as "thestudyof (1986, p. 125) of the UnitedStates.The Aschersdefineethnomathematics mathematicalideasof nonliteratepeoples,"by whichthey meanthose peoplewho do not is distinctlydifferentfrom"illiterate," have any systemof writing.This use of "nonliterate" which refersto thoseindividualswho do not havethereadingandwritingskills possessed by othermembersof theirsociety.It shouldbe noted,however,thatthe Aschersagreewith wherestudywouldnot be restrictedto the a broaderuse of the term"ethnomathematics,"

47 mathematicalideas of only nonliteratepeoples(MarciaAscher,personalcommunication, October,1988). The workof AscherandAscherrefutesthe widely held views thatthe is confinedto number,lacks mathematicalthoughtof nonliteratepeoplesis "primitive," analyticthought,anddoes not leadto abstractionandsymbolism.Descriptionsof spatial ordering,numbersystems,andkinshipstructuresfromdiversenonliterateculturespointto complexmathematicalideasthatareimplicitin theseactivities(Ascher& Ascher, 1986). An accountof the logical-numerical systemembodiedin the quipusdevelopedby the Incas, an ancient,nonliterateSouthAmericancivilization,is given by AscherandAscher(1981). Therehave beenrecentattemptsto arriveat a cleardefinitionof "ethnomathematics" (Franco,1989).Thereis also a move to come up with a more acceptablenamefor this areaof research,since the prefix"ethno"conjuresup the idea of race,whichis neitherintendednordesirable.Thisis not the only etymologicalproblem since withoutthe correctsuffix to denote"thestudy faced by the word"ethnomathematics," of," this meaningintendedby bothD'Ambrosioandthe Aschersis sureto remainobscure. Attemptsto clarifythe definitionor alterthe termhavenot met with success so far (maybe wouldwork?). "anthromathology" Forthe purposesof this research,I modifyAscherandAscher's(1986) definitionto is the studyof the mathematical read"ethnomathematics ideasof any culturalgroup."I for the mathematicalideas disagreewith the use of the descriptor"ethnomathematics" themselves,since this may lead to distinctionsbeingdrawnbetween"ethnomathematics" I shallsimplyuse the wordsmathematics,mathematizing,or and "realmathematics." mathematicalideas/practicesto referto the contentmatter.The primarypurposeof investigatingthe differentkindsof mathematicsgeneratedby differentculturalgroupsis to broadenourconceptionof mathematicsandnot to separateandlabelculture-specific categoriesof mathematicalthinking. Gerdes(1985, 1986, 1988a,1988b)expressesgreatconcernaboutthe mathematics thathas been generatedwithinthe Mozambicancultureandthatis in dangerof being lost.

48 He asks, "Howcan this 'pushingaside'and 'wipingout' of spontaneous,natural,informal, and/orhidden(ethno)mathematicsbe avoided?" indigenous,folk, implicit,non-standard (1988b,p. 3). He addressesthis concernby illustratingthe manyways in whichcultural elements,suchas artsandcrafts,canbe usedto investigatemathematicalideas. He asserts thatthereis "hidden"or "frozen"mathematicsin the traditionalworkdone by artisans.For example,the geometricalformsof traditionalMozambicanbaskets,mats,pots, houses,and fishtrapsoftendemonstratethe optimalsolutionto problemsof construction.These objects reflectnot only the accumulatedexperienceandwisdomaboutthe physicalmaterialsused butalso revealthe resultof mathematizing by the originalcraftsmenwho developedthe techniques.Gerdes(1986, p. 12) claimsthatby "unfreezingthis frozenmathematics," peoplecan be encouragedthattheywerecapableof developingmathematicsin the pastand can continueto do so. Gerdes(1988b,p. 6) suggeststhatpupilscould be given opportunitiesto "reinvent"traditionaltechniquesof, for instance,housebuildingand and"explore"the frozenmathematics. weavingto "rediscover" ideas "unfrozen"by Gerdesare As pointedout by Brodie(1988),the mathematical ideas (e.g., the theoremof Pythagorasunderliesthe basicallyacademicmathematical methodsof the basketweavers;the propertiesof a rectangleunderliepeasants'housebuildingpractices).Whilethis workis undoubtedlyveryvaluable,we shouldbe carefulto acknowledgethatthereareotherformsof mathematicsthatmay not in any way resemble academicmathematics.Forinstance,Fasheh(1988) observedthathis illiterateseamstress motherhada richandcomplexrelationshipto mathematicsandthatthe mathematicsshe used was integratedinto herworldin sucha way thatit remainedout of his grasp. Unfortunately,Fashehdoes notelaboratemuchbeyondthis, nortryto find a way to describehis mother'smathematicsthathe recognizedas beingso vastlydifferentfromhis own academicexperience. was The paradoxicalsituationembeddedin the conceptof ethnomathematics pointedout in Chapter1. The vitalquestionthathas not yet been satisfactorilyaddressedby

49 is how to recognizeanddescribethe creationof researchersinterestedin ethnomathematics mathematicalideasthatdo not fit the traditionalmould.As suggestedby Gerdes, mathematizingmay be "frozen"in traditionalhandcrafts.The carefulstudyof the everyday actionsof peopleas they engagein traditionalpracticesandday-to-dayworkactivities remainsan intriguingpossibilitythroughwhichto find a way of articulatingthe tacitor "frozen"mathematicsthatguidessuchactions. Harris(1987, p. 27) suggeststhata metaphorthatimplies"hatchingor germination of undefinedpotential"as well as "defrosting" andmore may be moreappropriate powerful.By sayingthis, she seeks to highlightthe dangerthatresearcherswho have hada standardWesternmathematicseducation,withits attitudesandprejudices,may consciously or unconsciouslylimit whatis to be "defrosted" fromthe artifactscreatedby personswho have not hadthe standardWesternmathematicseducation.AscherandAscher(1986) explainthatthe mathematicalideasof nonliteratepeopleswill have to be foundimplicitin differentareasandactivities,since thereis no reasonto expectthatthe categoryof mathematicsshouldexist as a labelfor all peoples. In summary,it maybe productiveto discusssome salientpointsthatemergeon closerexaminationof theresearchreviewedin thelast two sections. 1. Manyof the researchersinterestedin themathematicalideas of different culturalgroupsworkin contextswherethereis or has beencolonization, culturalconflictandsuppression,andresistanceto oppression(Brodie,1988). The impetusfor thesestudieshascome froma concernaboutthe failureof mathematicseducationfor groupssuchas ethnicminorities,oppressed majorities,women,girls, andworking-classchildren(Brodie,1988). 2. The mathematicscreatedby diverseculturalgroupsandcommunities, includingthe communitiesof fishermen,candysellers,basketweavers, carpenters,dairyworkers,clothmerchants,andtailors,can be studied productively.

50 3. All peoplefromall culturesperformsome formsof the everydayactivitiesof counting,sorting,measuring,comparing,designing,building,andcomputing. 4. Differentformsof mathematicsgeneratedby differentgroupsmay display varyinglevels of sophistication(fromthe pointof view of someoneschooled in formalmathematics); however,it is importantto noticethatthe mathematicswill servea usefulandpowerfulpurposeto its creators. 5. The mathematicsof thegroupis trulyownedby its creators,whetheror not Thus,studying they arepartof theelite groupof academicmathematicians. the mathematicsof culturalgroupscouldgenerateculturalreaffirmationas well as individualempowermentthroughthe feeling of ownership. 6. Some attemptshavebeenmadeto describethe mathematicsinvented"in practice,"butthusfarmostdescriptionshavebeenmadewithinthe narrow confinesof academicmathematics. Statementof the Problem Academicmathematicseducationhas failedfor the majorityof people.This failure is due in partto the conventionalportrayalof mathematicsas a prizedbody of knowledge Researchconducted thatis the propertyof an elite groupof peoplecalledmathematicians. in settingsoutsideof schoolsshowsconvincingevidencethatpeopleare ableto construct mathematicalideasin orderto solve problemsthatthey deemto be significant.Thereis also evidencethatschool-taughtmathematicspracticesareignoredin favorof strategiesthatare inventedas partof everydaypractices. Ratherthandismissingmathematical practicesandstrategiesthatdevelopout of daily activitiesas lackingin authenticityandrigor,mathematicseducatorsneed to study such practices,acknowledgingtheirstrengthsandseeingtheirweaknessesas opportunities in the classroom.Severalmathematicseducatorshave to negotiatebroaderunderstandings thatthe social,cultural,andpoliticalaspectsof mathematics called for an acknowledgment of these factors andof mathematicseducationcannotbe ignored.An acknowledgment

51 wouldencouragea broaderconceptualization of mathematicsandmay begin a process wherebymathematicscouldbe seen as an activeexperience,accessibleto all people. of whatcountsas legitimatemathematicsand Workingwithina broaderconceptualization of the teaching/learning situationof apprenticeship incorporatingsome of the characteristics into classroominstruction,it is arguedthatthe usualpassivesubmissionto mathematics could be replacedby the activeexperienceof mathematizing. ResearchObjectives 1. To documentevidencethatvalidmathematizing is embodiedin the everyday woodworkingactivitiesof a groupof carpenters(who havehadlittle formalschooling)in CapeTown, SouthAfrica.Therearetwo maintasksembeddedin this goal: a. To understandor recognizethe mathematizing in the observedactivities. b. To describethis mathematizing to the mathematicseducationcommunityandto anyoneelse interestedin exploringthe developmentof mathematicalknowledge. 2. To studythe teachingandlearningof mathematicalideas thattakeplace modelin the settingof the carpentryworkshop.Thereare accordingto the apprenticeship two maintasksembeddedin this goal: a. To be acceptedas an apprenticeby a groupof carpenters. b. To participatein the practicalactivitiesof an apprenticeover an extendedperiod of time andto describethe mathematizing thatis bothexplicitandimplicitin this learningexperience.

Chapter3 IN THE WORKSHOP Introduction As stated,the presentstudyimplementsthe methodologyof ethnography.Therefore, this chapterbeginsby providinga cleardescriptionof the contextof the researchsetting.A is also given. The chapterends with a briefhistoryof descriptionof each of the participants the workshop. Contextof the ResearchSetting The carpentryworkshopis the placewheremost of the interactionsto be described took place. It was a large,well-lightedroom,oftenbathedin sunlightthatstreamedthrough windowssituatedalongtwo sides of the room.Two big doorson eitherside of the room alloweda good crossflow of airwhenthe weatherwas warm.In cold weather,the doors were sometimesclosed anda fire was lightedinsidein a largemetaldrumandfed with wood chips.Good-natured complaintsaboutthe smokewouldthenbe heardfromthe neighboringworkshop,a smallmotorcarrepairbusiness.The ownerof the repairshopwas the uncleof one of the apprentices.He repairedthe deliverytruckfor the carpentry workshopas well as the owner'sprivatevehicles.He oftenforgotto send a bill for the repairsuntilmonthsafterthe workwas completed.The atmosphereof this smallindustrial area,whichalso housedtwo electricalbusinessesandan autobody shop,was relaxedand friendly. Therewere largestacksof wood piledup alongone wall of the workshop:a big consignmentof importedSpanishmahogany.A smallstockof preciousindigenouswoods like stinkwoodandyellowwoodwas also keptthere.A collectionof handtoolswas visible in a big woodencabinetagainsta wall. Machinesweredottedaroundthe room-band saws, an overheadtablesaw, electricplaners,thicknessers,a mortisingmachine,andsanding machines.Next to the long woodenworkbenchesweretwo largewoodencheststhat belongedto two of thequalifiedcarpentersandcontainedtheirown personalsets of tools.

53

54 Furniturein variousstagesof completioncouldbe seen everywhere;piles of turnedlegs and othercomponentswere stackedup andreadyfor assembly.Severalbeautifulantique pieces-tables, armoires,andchairs-awaited restoration. Thereweretwo adjoiningfrontoffices:one for Peter,the ownerof the workshop, andone for Emily,his administrative assistant.Bothoffices werefilled with newly made furniture,as well as restoredantiques,waitingto be collectedby customers.Peterwas seldomin his office for long. He couldusuallybe foundat workin the workshop,wearing a dustcoatover a pairof casualtrousersanda shirt.Emilywas keptbusyansweringthe telephoneanddealingwith accounts,orders,andthe weeklypayroll. The workingday beganat 8:00 a.m.By the morningteabreakat 10:00a.m.,the workshopwas an active,noisy place.Againsta backdropof machineclamor,storieswere being swappedandlively singingandjokingcouldbe heard.Workstoppedfor 10 minutes each morning,timefor tea anda snack.Themen gravitatedinto smallergroupsfor conversationduringthe break;the two majorracialgroups("coloured"6 andblack) separated,andthenwithinthatseparationtherewerefurthergroupingsaccordingto status andseniority.At the half-hourlunchbreakandafternoonteabreak,some similargroups were formed;however,manyof the mencontinuedworkingon privatejobs (called"spare" or "loose"jobs) at thesetimes.The two dailyteabreaksandthe lunchbreakwere designated by the blowingof a loud whistle,a signalthatall the menobeyedwithoutquestion.In warm to relaxoutsidein the afternoonsunshine, weather,some of the men tookthe opportunity sittingor lying downundertreesthatformedpartof a scrubbywoodedbufferbetweenthe industrialareaandthe trainline.At 5:30,the end of the workday,most of the laborerstook

6Peopleof mixedAfricanandEuropeandescentareclassifiedas "coloured"by the South Africangovernment.This labelis consideredto be derogatoryandinvalidby the majority of SouthAfricans.Forthis reason,the Britishspellingwill be maintainedthroughoutthis workto emphasizethatthisis a raceclassificationpeculiarto the apartheidsystemandis not a generallyacceptableterm.

55 the trainhome.Most of the apprenticesandthe artisansusedtheirown carsor sharedrides with one another. Threelanguageswerespokenin the workshop:English,Afrikaans,andXhosa.Most of the work-relatedtalkwas conductedin Englishor in a combinationof Englishand Afrikaans.Privateconversationswereoftenconductedin Afrikaansandin Xhosa. Most peopleunderstoodbothEnglishandAfrikaans,althougha few haddifficultyin expressing themselvesin theselanguages.Xhosawas the mothertongueof all the blackworkmen. Peter,the artisans,andthe apprentices(withtheexceptionof Leonard,whose mother tonguewas Xhosa)hadno knowledgeof this language.Englishis my homelanguage. the languageandcan carryon a simple While not fluentin Afrikaans,I can understand conversation.I haveno knowledgeof Xhosa. Cast of Characters Therewere 18 menin the workshop(see Figure3), includingthe owner,Peter.Five of the men were artisans,fourwereapprentices,andtheremainingeight were laborers. andhad"papers."The Only one of the artisans,Clive, hadserveda formalapprenticeship otherartisanshadhadvaryingamountsof informaltrainingandwereappointedor promotedto the designation"artisan" by Peter.In thecase wherea carpenterhas no formal financialrewardfor good qualification,it is up to his employerto give the appropriate qualitywork.Basically,if someoneis paidthe wage of an artisan,thenhe is referredto as an artisan,whetheror not he is formallyqualified. The followingsectiondescribeseach person.I haveincludedthe raceof each workerbecausethis is importantfor the discussionof social interactionsandin the analysis of opportunitiesfor learning.I shalluse the terms"black,""white,"and"coloured,"as they are used in the apartheidsystem:"black"refersto theindigenousAfricanpeoples,whose mothertongueis an Africanlanguage(e.g., Xhosa);"white"refersto peopleof European origin,whose mothertonguemay be Englishor Afrikaans;"coloured"refersto peopleof mixed AfricanandEuropeanorigin,whosemothertonguemay be Englishor Afrikaans.An

56 CASTOFCHARACTERS Owner:

Peter

Artisans:

Mr.S

Clive

Jack

Apprentices:

Leonard

Patrick

PaulGarth

Laborers:

Sidney

Ernest

Winston

Anton

Durham

Singer

King

Michael

JimmyBrian

Figure3. The hierarchy:Peoplein the workshop. aptquotationis given for eachof the six peoplewithwhomI hadformalinterviews.No quotationis given for thosepeoplewhomI did not interview,andtheirdescriptionsare necessarilymuchbriefer.A shorthistoryof the workshopfollows the descriptionof the workers. The Owner of the Business Peter: "Thereis a skill or an aptitudeor an eye in visualizingwhatyou are building whichmakesyou betterthanothers." Peterwas the ownerof the business,a whiteman,fluentin bothEnglishand Afrikaans.He was in his late thirtiesandhadthreeyoungsons. Woodworkinghadbeen a partof Peter'slife for manyyears,beginningin elementaryschool andcontinuingthrough high school, wherehe took the subjectas an elective(not as partof his coursework). He beganhis tertiaryeducationstudyingcomputersciencebut laterqualifiedas an engineer.Afterworkingfor a big civil engineeringcompanyin the designoffice for a while, he beganrestoringhouses.By this timehe hadbegunto collect andrestoreantiquesandto designfurniture.His furniturebusinessbeganas a one-personfurniturerestorationbusiness, of classic or antiquefurniture.Since butover time he diversifiedinto makingreproductions

57 reproductionsarefar moreprofitablethanrestorationwork,this hadbecomethe mainline of workat the workshop.However,Peterwas particularly fond of antiquesandwas unwillingto dropthe restorationworkcompletely.He hadan enviablecollectionof antiquesat homeandwas well knownamongantiquecollectors,who frequentlyconsulted him. The workshophadbeenin operationat its presentsite for abouttwo yearsat the time the presentstudywas conducted. in carpentrybuthadlearnedinformallyfrom Peterhadno formalapprenticeship skilled artisansover manyyears.He was a talenteddesignerwho enjoyedphysically workingwith wood in the factoryon a regularbasis.Well liked by the carpenters,he was a good-humoredandkindmanwho was concernedaboutthe progressandwell-beingof his employees.He addedto the relaxedandhappyatmospherein the workshop. The Artisans Mr. S: "Ifoundthatmakingfurniturewas the only thingthatwouldappeasemy mind,and I becamequitegood at it. AlthoughI was theyoungestof thepeople he was teaching,whenmyfatherdied therewas not a lot thathe couldshow me. FromwhenI was a child,I stood by his side and watched;he designedhundredsofpieces.. .that is whyit is so easyfor me; I can lookat an articleand designmyselfa leg that'sgot to go underneath thatarticle...it just comes...if I thinkaboutit, itjust comes." "Geometry-that'stheone withthe circlesand things,isn't it? My daughterlearned thatat school." Mr.S, my primarycontactin the workshopandmy maincarpentryteacher,is the key figurein the presentstudy.Mr.S was a CapeMuslimman(classified"coloured"by the SouthAfricangovernment),fluentin bothEnglishandAfrikaans.He was the chief antique restorerat the workshop.He was the only personin the workshop,apartfromPeter,who was addressedas "Mr."by all the otherworkers,andwas supposedto be "incharge"when Peterwas absentfromthe workplace.Mr.S, however,did not enjoythis responsibility. Since he did not have an officialtitle of "foreman," he was not recognizedas havingany

58 realauthorityoverthe others.He preferredto remainuninvolvedin any disputesthatarose andsaw himselfas a loner,withan importantmeasureof independencenot possessedby the otherworkers.He frequentlytoldme thathe couldmakemuchmoremoney if he restoredantiquesat home,for his own customers,butthathe wantedto help Peterbecause he was an old friend. Mr. S was 57 yearsold andthe grandfather of two. All of his forefatherswere involvedin furnituremakingor boatbuildingor weresilversmiths:his fatherwas a furnituremaker;his grandfather, a boatbuilder;andhis father-in-law,a cooper.Mr.S's seven brotherswerealso involvedin the buildingtrade.Therootsof the Muslim communityat the Capego backto thelate 1600swhenskilledtradesmenandartisanseither arrivedas slaves or werebanishedto the CapefromJavaandIndonesiaas political dissidents. Mr.S beganhelpinghis fatherat homein the family'ssmallfurniturebusinesswhen he was a child about9 or 10 yearsold, andat age 13, he earnedhis firstincomefrom woodworking.At this stage,he decidedon a careerin furnituremakingandleft school. Workingat his father'sside for manyyears,throughperiodsof financialups anddowns, Mr. S learnedthe artof fine craftsmanship. The father-and-son teamoperatedtheirown privatebusinessat homewhentimesweregood andcustomersplentiful.At othertimes they workedfor a prestigiousantiqueauctioneer'sgalleryin the city. As a youngman,Mr. S delightedin examiningthe "lavishlybuilt"antiquefurnituredisplayedin museumsand artgalleries,andhis interestin restorationdeveloped.Together,he andhis fatherdeveloped a good reputationfor restorationandreproduction work.In his own words,Mr. S said, "We were sought-afterpeoplein this country-we couldhavemadea lot of moneyin Johannesburg."Of his father,he said,"Hewas one of the best-and he was a hardtaskmaster." Some yearsago Mr.S was askedto teachat the local technicalcollege, buthe refusedbecausehe was shy andfelt thathe was uneducated.He regrettedthe chancehe had missed.Forthis reason,I thinkthatteachingme to workwith wood andbeing involvedin

59 this researchprojectrepresenteda significantopportunityfor him. He was confidentand knowledgeableandhadmuchto tell abouthis experiencesas a carpenter.It seemedto be a good time in his life to reflecton all thathe hadachieved.He was approachingretirement age. His youngestdaughterwas in herfinalyearat school.His otherchildrenwere successfulacademicallyandwereall universityeducatedandhappilymarried.Therewere no majorfinancialworriesin his life. Giventhathe lovedreadingandbooks,the idea that he was being writtenaboutandthathis own knowledgewas beingvaluedandhis work describedwas probablyverygratifying.He stressedthatsome of the thingshe wouldteach me were secretandthatsometimesI wouldhaveto workawayfromthe othercarpentersso thatthey could not see whatI was doing.His knowledgewas presentedas being very specialandnot to be sharedlightlywithhis colleagues. Clive: "Without pride in yourwork,you'vegot nothing.Thekindof workthatyou producewill showwhetheryou are good or not." Clive was a "coloured"manin his latethirties,fluentin bothEnglishandAfrikaans. He was marriedandhadtwo daughters,one of whomwas bornduringthe time of this research.He studiedwoodworkingat high schoolandthenleft school at about16 yearsof age. Afteran unsatisfyingyearof workas a storeman,he decidedthathis ambitionwas to workwith his hands.He joineda constructioncompany,was endenturedas a carpentry apprentice,andqualifiedas a joinerin justunderfouryears.Duringthesefouryears,he spentthreemonthsof each yearat thetechnicaltrainingcollege wherehe attendedclasses, includinga mathematicsclass. Fortheremainingtime,he workedwith qualifiedartisans employedby the constructioncompanyin thecarpentrysection,in thejoinerysection,and also on buildingsites. Afterqualifyingas a joiner,Clive left the constructioncompanyandstartedboat he developedan allergyto teakwood building,whichwas still his firstlove. Unfortunately, andhadto give up his maritimeinvolvement.He returnedto the buildingtrade,wherehe remaineduntiltakingthejob as a furnituremakerwithPeter.

60 The aspectthathe enjoyedmostaboutthe workshopwas thathe createdfurniture "fromscratch,"by whichhe meantthathe startedwith a diagramandroughplanksof good qualitywood andendedwiththe finishedproduct.Tasksweregenerallynot fragmented in this workshopas they werein manyothersthatproducedcheaperfurniture.Clive sometimesworkedon the constructionof one articleof furniturefor a week or two with the help of his apprentice.He was extremelyenergeticandwas usuallyinvolvedwith a "spare" job at lunchbreakandafternoonteabreak.Duringweekendshe took on additional woodworkingprojectsto earnextramoney.PeterregardedClive as the best craftsmanin and the workshopandalso the personbest ableto planahead,budgethis time appropriately, solve problems.PeterremarkedthatClive alwaysseemedto workvery slowly and carefully,with meticulousattentionto detail,yet he oftenfinishedaheadof schedule. Clive becameanotherimportantteacherfor me whenI beganto constructarticles. He oftenwalkedoverto my workplaceto correcta smalldetailof my stancewhen I was tool, or to give some verbaladvicethat sawingor chiseling,to offerme a moreappropriate mademy taskeasier. Jack: "Nobodytaughtme nothing!You'vegot to steal withyour eyes!" Jack'scommentwas a verycommonphrasethatI heardfroma greatnumberof carpentersandotherworking-classpeoplefromthe samecommunity.It referrednot only to experiencesin the worldof workbutalso reflectedthe frustrationof beingdenieda fruitful learningexperienceat schoolbecauseof the politicalandsocioeconomicconstraintsof the apartheidsystem.Jackwas a good exampleof someonewho shunnedthe formaleducation systemat an earlyage andthenactivelyengagedin self-education,acquiringnumerous skills by "stealingwithhis eyes"fromexperiencedworkersin his community.Amongother practicalactivities,he was proficientat welding,braising,mechanicalwork,panelbeating, andspraypaintingandhadmadebuilt-inkitchenunitsandclosets. In all these activities,he was self-taught.

61 Jackwas a marriedmanin his late thirtiesandhada teenageson andanother youngerson. He was classifiedas "coloured"andwas fluentin EnglishandAfrikaans.As a very youngchild, fromthe age of aboutfive years,he beganto learnaboutwoodworking fromhis fatherandgrandfather, who hada smallfamilybusinessmakingfurnitureand woodentoys. Jackhelpedwith taskslike puttingwheelsonto pushcartsandwas latershown how to use handtools andhow to operatethe electricmachines.He maintainedan interest in woodworkuntilhe was 15 andhelpedhis brotherto continuethe familybusinessafter the deathof his grandfather. At the age of 16, Jack'sschoolingwas disrupted,andfor the next 10 yearshe workedas a movie projectionist.Duringthis time,he continuedto be involvedin woodworkingprojectsof variouskinds.Eventually,he returnedto woodworkingto make his living, constructingschoolbenchesandlaboratorytables.Laterhe diversifiedinto antiquerestorationandmakingotherfine furniture.By readingaboutantiquerestorationin librarybooks andby observingthemistakesthatotherpeoplehadmadeon the furniturehe restored,Jacktaughthimselfthe basicsof restoration.He describedhimselfas takingafter his father,who was also a "jack-of-all-trades." He beganworkingfor Peterwhenthefactoryopened,becausehe wanteda steady income.AlthoughJackhadcustomersfor the woodworkinghe did at home andusually workedfor his own profitat lunchtime,he preferredthe securityof a weekly wage to the prospectof relyingentirelyon his privateworkfor his livelihood.LikeClive, Jackenjoyed the kindof woodworkinghe did at Peter'sworkshop,commenting,"Whenyou workin any otherfactory,whichI haveseen andI know,if a manscrewsin screwstoday,he does that for the rest of his life; if he joins up drawerstoday,thenhe makesdrawersfor the rest of his life. Hereyou workfromthe floorrightup to the finishof thatparticularitem, andthatis whatI call a cabinetmaker...acraftsman." Jackwas the thirdartisanwho becamea significantteacherfor me as time went by. Jack,Clive, andMr.S werethe mostexperiencedartisansin the workshopandwere the

62 peoplewith whomI spentthe mosttime.I conductedseveralformalinterviewswith each of them. andwas classifiedas "coloured." Jimmy.He was about25 yearsold, unmarried, JimmyspokeAfrikaansandhaddifficultyexpressinghimselfin English.He hadnot served a formalapprenticeship forcarpentry,andtheexactcircumstancesof his informaltraining seemedto be a bit of a mystery.He madefrequentmistakesthatMr.S oftenquietlyhelped to putright.The otherartisansdid not valuehis workmanship highly.In fact, they citedhis workas an exampleof the workof a poorcarpenter.Oneof theircriticismswas thathe did not have his own set of tools, whichwas veryunusualfor an artisan.Accordingto Peter, Jimmy"hasa verynice personalityandworkshard,butdoesn'tgraspthingstoo easily." Petertold an amusingstoryabouthow Jimmyarrivedfor a job interview,very keen to have thejob, andtotinga toolbag.Peterdecidedto checkon whattools he had,as one of his criteriafor hiring,andaskedJimmyif he hada rebateplane.A rebateplaneis an expensiveitem,andif somebodyowns one it usuallymeansthathe has a very good set of tools. It turnedout thatJimmydid haveone, andit hada clean,sharpblade,whichimplied thathe lookedafterhis tools. Peterwas so impressedthathe hiredJimmyimmediately,then foundout laterthathe did nothaveany othertool besidestherebateplaneandstill does not own any othertools to this day! Brian: "Gee,thattooka long timefor me to teachit to you, butthatis becauseI had to teach it to myselffirst!" andnewly married.Peter Brianwas about21 yearsold, white,English-speaking, told an interestingtale abouthis meetingwithBrianandthe latter'sinauspiciousstartto a successfulcareerin woodwork.Brianwas about18 yearsold andhadno woodworking skills when Peterfirsthiredhim to helppainthis home.WhenPeterdecidedto startthe workshopandbeganlookingfor staff,he was told thatBrianwas interestedin joining.At the domainof "coloured"men in first,Peterwas suspicious,since carpentryis traditionally

63 SouthAfrica,andBrian,as a white,hadmuchbettercareeropportunitiesto choose from. However,Peterfinallydecidedto give Briana chance. Brianhadoverstatedhis high school qualifications At first,he was disappointed; somewhat,as becameevidentwhenPetertriedto enrollhim as an apprentice.He hadonly the equivalentof gradeseven,whichwas below the minimumrequiredfor embarkingon a he displayedno interestin anythingat the workshop, formalapprenticeship. Furthermore, often arrivinglate to workin the morning. At thattime,Peterusedto turnall the legs neededfor tablesandchairshimself.All this workwas doneon a lathe,usinghand-heldtools. Thiswas very time-consumingand neededspecialexpertise.Accordingto Peter,he realizedone day thathe could not keep up with all the turning,andso he turnedto Brian,and"Isaidto him,why don'tyou tryit? He immediatelytook to it...he has the eye for it...he has a very good eye. He turnedout to be tremendouslysuccessfulon the lathe!"On the strengthof Brian'shigh outputand profitability,Peterdecidedto payhim an artisan'swage, andthushe becamean artisanwith no formaltraining. The othercarpenterstold anotherstoryaboutBrianandhis skills on the lathe,a storythatrevealedsome of the workethicof theworkshop.Brianhadnot yet been madean artisan,althoughhis rapidlydevelopingprowessat turninglegs was impressive.He was able to workvery fast andaccurately.Peternoticedthis andjokinglytold Brianone morningthatif he couldcompletea certainnumberof legs by the end of the day, he would be paidan extrasum of money.Briancountered,sayingthatif he finishedthe legs by lunchtime,he shouldget the moneyas well as the restof the day off! Peteragreed,andthe wagerwas set. Some of the othercarpenterstriedto dissuadeBrian,sayingthatit was a bad idea to show the boss how muchhe couldreallydo if he triedhard;if he succeeded,then Peterwouldalwaysexpectsucha high output!Brian,however,couldnot be dissuaded,and workedat top speed,finishingeven morelegs by lunchtimethanPeterhadspecified.He was given his extramoneyandtherest of the dayoff, andshortlythereafter,he was made

64 an artisan.But as the menpredicted,his level of outputhas hadto remainhigh becauseof the expectationshe set. He seemedto enjoythework,despitethe pressureto produce,and singlehandedlyturnedeverycomponentthatneededto be workedon the lathe. The Apprentices Leonard:"WhatI makeis myown.I can do it accordingto myplan in myhead." Leonardwas Xhosa-speaking,an unmarried blackmanin his earlytwenties.He had a matriculation certificate(equivalentto grade12) withmathematics,science, andtwo languages,whichmeantthathe hadby farthe highestacademicqualificationsof all the employeesin the workshop.Leonardhada greatdealof ambitionandconfidence.Peter thoughthighlyof his abilitiesandwas especiallyencouragingtowardshim. He beganas a laborerat the workshop,sweeping,polishing,andscraping.After aboutsix months,he informedPeterthathe wantedto becomean apprentice.He began workingon his own woodworkprojectsduringbreaktimes,askingPeterfor help when was higher"(thanthatof the otherlaborers)and necessary.He claimedthathis "standard thushe deservedto be trainedas a carpenter.Peteragreed,buthe encountereddifficulties when he triedto enrollLeonardat the local technicalcollege. The apartheidsystemkeeps racegroupsstrictlysegregatedin the educationsystem-different institutionsareset up for each racegroup.As previouslymentioned,carpentryhas traditionallybeen the turfof "coloured"men, andtherearefew blackcarpenters.The technicalcollege wouldaccept only "coloured"students,andno facilityexistedin the city for blackstudentsto become carpentryapprentices.Finally,theeducationauthoritiesrelentedandacceptedLeonardas an apprentice.He left the workshopto attendthe technicalcollege full-timefor a threemonthperiodaboutfive weeks afterI startedworkat the workshop.Beforehe left, I spoke to him abouthis earlylife andhis ambitions. WhenLeonardwas a youngboy, he usedto maketoys andminiaturecarsout of wire andpieces of Cokecansthathe cut up to formsmallmetalsheets.He andhis friends once madea full-sizedcarfromscrapmetalandwood.It hadwheels takenfrom

65 wheelbarrowsanda steeringwheel thatoperateda steeringdevice.The groupof friends enjoyedpushingtheir"car"aroundandtravelingdownhillwheneverpossible! Leonarddescribedthe articleshe hadmadeduringthe breaksat the workshop, sayingthathe "worksfroma planwhichis in his head."His completedprojectswere two jewelryboxes anda cassettetapeholder,andhis mostrecentprojectwas a woodenbread box with a specialslidingdoor.The designfor the breadbox was copiedfromsomething similarthathe saw in a store.Using his outstretched hand,he was able to measurethe dimensionsof the box in the store,andan examinationof the way in which the slidingdoor worksenabledhim to makea replicaof the box in the workshop,using scrapwood. Leonardobviouslyenjoyedwoodworkingverymuchandparticularlycommented on the fact thathe ownedthe thingsthathe made.He told me thathe workedas a laborerin a flourmill for a few yearsandcameto therealizationthathe wouldneverown a mill himself.This realizationpromptedhis decisionto learna tradethatwouldallow him to makesomethingof his own, withhis own ideas. Patrick.Patrickwas about20 yearsold andunmarried. He spokeEnglishand Afrikaansfluentlyandwas classifiedas "coloured."It was said in the workshopthathe was Peter'sfavoriteandso was allowedto get up to all sortsof mischief.PeterregardedPatrick as exceptionallytalentedandpossessingan excellentgraspof spatialconcepts,buthe acknowledged,however,thatPatrickwas lazy andfull of youthfulrebellion.Peter anticipatedthatPatrickwouldsettledownas he got olderandthathe would developa more matureattitudetowardshis work. Like Brian,Patrickhadoverstatedhis highschoolachievementswhen he applied for thejob at the workshopandwas not allowedto enrollat the technicalcollege as an apprentice.Eventually,he was acceptedin the categoryof "learner"by the Industrial Council,which wouldallowhim to studyat the technicalcollege andto obtaina qualificationin carpentry.This careerpathhada ceiling,however,andhe would not be able to qualifyas a mastercarpenter.

66 Paul. Paulwas a yearor two olderthanPatrick;they wereclose friendsandshared an apartment.Paulwas classifiedas "coloured." He arrivedbackfroma three-monthstudy periodat the technicalcollege soon afterI arrivedat the workshop.Accordingto Clive, Paul was not reallydedicatedto his work,althoughhe seemedto do adequately.He sometimes spoketo me of his ambitionto teachcarpentryat the technicalcollege afterqualifying, becausehe did notrelishthe ideaof beingin a workshopfor therestof his life. His uncle ownedthe mechanic'sworkshopnext door.PaulandPatrickmadea formidableteamof prankstersin the workshop. Garth.Garthwas English-speaking, classifiedas "coloured,"unmarried,andin his earlytwenties.He was quietandreservedandwentabouthis workseriouslywithoutjoning in the generalhorseplay.Clive,the artisanwithwhomGarthworkedmost often,regarded him as the mostpromisingof the apprentices.Thepersonalprojecton whichhe spentmost of his lunchbreakswas a beautifulchess board,inlaidwith pieces of light anddarkwood. The Laborers Sidney.Sidneywas a verydignifiedXhosa-speakingmanapproachinghis seventies, who hada wife andchildrenin theCiskeihomeland.He hadknownPeterfor 20 yearsand hadworkedfor him for 18 of thoseyears,helpingto restorefurniture.Peterwent to Johannesburgfor a periodof 2 years,duringwhichtimeSidneywent to workfor anotherrestorer. WhenPeterreturnedto CapeTown,Sidneyrejoinedhis business.Sidneylived in a small apartmentthatwas attachedto Peter'shome. He actuallydidno woodworkingat all, althoughhe was extremelyknowledgeable aboutwood andwas an expertat identifyingdifferenttypesof wood. He was the workshop'sauthorityon agingthe newlyreplacedpartsof percentagerestorations(i.e., makingthe newly replacedportionlook as old as the original).He also threaded"riempies" (thinstripsof tannedhide)to makechairseats. Sidneywas well respectedby everyoneas an elderandwas seen as being"in charge"of the laborers.He kepttimeduringtheday,blowingthe whistleat the beginning

67 andend of tea andlunchbreaks,a signalthateverybodyobeyed.Sidneywas the uncleof Leonard,the apprenticepreviouslydescribed,andof Ernest(Leonard'sbrother),who is describednext. PeterrelieduponSidneyto recommendworkerswhen a new laborerwas needed.All of the blacklaborersat the workshopwereintroducedto Peterby Sidney. Ernest.Ernestwas Xhosa-speakingfromthe Ciskei,unmarriedandin his early twenties.Duringthe time I was at the workshop,he madetwojewelryboxes using a light wood with inlays of a darkerwood in the shapesof playingcardsuits.His workwas quite rough,andattractivein a rusticway. The artisanscriticizedhis workmanshiprathercruelly butmadeno effortto assisthim.He was muchmoreoutgoingandfriendlythanLeonardbut seemedto lack his brother'sconfidenceandresolve.He hadnot been singledout by Peter as beingespeciallycapableandwas still a laborer. WinstonandAntonwereXhosa-speakingmenfromthe Ciskeihomelandandwere brothers.Antonwas about20 yearsold, veryquiet,andworkedas a laborer.Winstonwas a little olderandhadbeen taughtto do Frenchpolishingby Mr.S. He usuallywalkedto the storeeverymorning,armedwith a long shoppinglist of everyone'sdaily food requirement, andreturnedin time for tea to distributethe food andcorrectchange. Durhamwas about50 yearsold andwas Xhosa-speaking.He was the truckdriver for the workshopanda good friendof Sidney.He sometimeshelpedSidneyto thread riempies,to age wood, andto polish. He was a Xhosa-speakingman Singerwas about25 yearsold andwas unmarried. fromthe Ciskei.Mr.S taughthimhow to Frenchpolish,andthis was his maintask.He was frequentlyexhortedby the othersto "geta wife,"becausehe washedhis clothesat workand hungthemout to dryon the fence outside!Singerwas a keen amateurboxerandhada great sense of humor.He sometimesdidthe morningshoppinginsteadof Winston. Kingwas about30 yearsold, Xhosa-speakingandunmarried.He hadpreviously workedfor Mr.S, andtheyjoinedPeter'sworkshoptogether,with King as Mr. S's assistant.He was a laborerandwas quietandreserved.Duringbreaks,he workedon his

68 personalwoodworkingprojectsandmadea coffee tablefromoff-cutsandlatermadea smallbedsidecabinetwitha mirror. Michaelwas a laborerin his mid-thirtiesandwas Xhosa-speaking.He was shy and was one of the mainbuttsof Mr.S's teasingandhorseplay.He workedextremelyhardin the workshopandwas regardedby Peteras the mostdiligentworker. Historyof the Workshop The businessbeganat Peter'shometwo yearsandsix monthsago. For six months, Peter,Sidney,andMr.S didrestorationworkonly;thenPeterbeganto look aroundfor premisesandto purchasemachines.Two yearsago, "TheJoinery"(notthe actualnameof the business)was establishedat its presentpremiseswitha workforceconsistingof the originalthree,plus PaulandPatrick,who hadhelpedpaintPeter'shouse,andtheirfriend Brian.The last threewereall about18 yearsold andhadno woodworkingskills. Patrick andBrianhadbothinflatedtheirschoolingqualificationsas previouslyexplainedandcould not be enrolledas apprenticesas planned. Brian'sspecialskill on the latheearnedhimrapidpromotionto artisanstatus,and Patrickbecamea learnercarpenter.Peteradvertisedin the newspaperandhiredClive, Jack, andJimmy.Clive hadgood credentials;he hadworkedfor one of the leadingbuildersin the countrywho haddecidedto close downhis businesswhenhe retired.Jackwas hiredon 10 days probation,butPeteraskedhimto stayafter2 daysbecausehe was impressedwith his work.As previouslyrelated,Jimmygot thejob on pureluck!Almostall of the laborers were membersof Sidney'sextendedfamilyor werefriendsof his. The men workeda five-dayweek anda nine-hourday, from8:00 in the morning until5:30 in the afternoon,withhalf an houroff for lunch.The atmospherewas happy,and therewas a lot of lightheartedbanterandgood humor.Apartheidhadputits seemingly inescapableandindeliblestampuponthe socialinteractions,andsome subtleandothernotwerepracticedby some of the workers. so-subtleformsof segregationanddiscrimination The social issues will be discussedin moredetailin Chapter6.

Chapter4 METHODOLOGY Introduction The presentstudywas conductedovera periodof six monthsin CapeTown, South Africa,whereI workedas an apprenticewitha groupof carpenters.AlthoughI learnedboth carpentryandnew ways to mathematizefrommanyof the carpenters,as mentionedearlier,I had one particularteacher,Mr.S, withwhomI workedon a dailybasis.He taughtme the basics of furniturerestorationandwoodworking,as he wouldhave done any apprentice. In Chapter1, I outlinedmy reasonsfor selectingan ethnographicapproachandfor the importanceof my positionas apprentice.In this chapter,I commentgenerallyon the use of ethnographicapproachesfor educationalresearch.The difficultiesof my entrdeinto the field aresketchedout, andmy experiencesin the workshopareelaboratedas I was initiated andgraduallyacceptedas a usefulworker.A descriptionis providedof the methodsused to collect data.Throughoutthe chapter,andparticularly in the second-to-lastsection,I point out some of the frustrations thata researcherencounterswhenshe is intenton doing naturalisticinquiryin a noisy, busyworkplace.I concludethe chapterby describingmy exit fromthe field. Ethnographyand EducationalResearch researchhas its rootsin culturalanthropologyandsociology. Ethnographic Researchershave foundethnographyto be a difficultconceptto describeprecisely. However,the themethatstandsoutclearlyin all discussionsof ethnographyis thatthe settingof interestfor theresearchconcernsthe interactionof humanstogetherandthe meaningsthattheycreatetogether. of the role thatethnography Interpretations playsin the contextof educationcan be Wolcott(1987) maintainsthatthe gleanedfromthe wordsof prominentethnographers. purposeof ethnographicresearchis to describeandinterpretculturalbehavior.Goetz and LeCompte(1984, p. 51) talkof ethnographyas beingthe "holisticdepictionof uncontrived

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70 groupinteractionover a periodof time,faithfullyrepresentingparticipantviews and meanings."SpindlerandSpindler(1987)claimthatethnographers attemptto makea coherentrecordingof how peoplein a communitybehaveandhow they explaintheirown behavior.Theyemphasizethe differencebetweenethnographyandinference,statingthat is to providereliablesourcematerialfor analysis, the ultimatepurposeof ethnography which is governedby systematicmodels,paradigms,andtheoryandout of whichinference will flow. Inferenceandspeculationarethereforegroundedin observationandtheory. is usuallyconcernedwiththe developmentof a theoryandnot with the Ethnography testingof existinghypotheses.GlaserandStrauss(1967) introducedthe idea of "grounded theory,"whichrefersto the discoveryof theoryfromdatathathas been systematically obtainedfromsocial research.An essentialpartof theresearchtaskis to discoverwhatis significantandwhatis importantto observe.Oneis thuscontinuouslyinvolvedin a process ratherthanbeing sequential aredoneconcurrently, of inquiry.Fieldworkandinterpretation steps in the researchprocess.GlaserandStrauss(1967) stronglyrecommendthatfor researchaimedat discoveringtheory,the threeproceduresof theorybuilding,coding,and dataanalysisshouldgo on simultaneously. Duringthe pastdecade,qualitativeresearchin educationhas becomeincreasingly acceptableas an alternativeto quantitativeandpositivistmodels.Educatorsoften turnto ethnographicapproachesto informtheirresearchquestions.The literaturecontainsmany criticismsof educatorswho presumeto call any attemptat descriptivework"ethnography" withoutpayingattentionto the philosophicandtheoreticrootsof the approach.Rist (1980, as a " 'hitandrun'forayinto the field,"while p. 9) describes"blitzkriegethnography" who provide of "would-beethnographers" Wolcott(1987, p. 40) writesdisparagingly wonderfuldescriptionsbutwho fail to analyzewhattheyhave observed,and"gonzo(1987, p. 45) who do not pay enoughattentionto the conceptof culture. ethnographers" Eisenhart(1988) gives an excellentdiscussionaboutthe ethnographicresearch traditionandmathematicseducationresearchwhereshe pointsout thatthe interpretivist

71 is very differentfromthe logical positivistideas that philosophyunderlyingethnography usuallyguide traditionaleducationalresearch.She identifiestwo broadgroupsof researchersdoingdescriptivestudiesof mathematicseducation:traditionalmathematics educatorsandeducationalanthropologists. Whereastraditionalmathematicseducators andtendto ask researchquestions usuallyhave goals thatare"descriptiveandprescriptive" in the vein of "Howcan mathematicsteachingandlearningbe improved?"educational anthropologistsinterestedin mathematicshavegoals thatare"descriptiveandtheoretical" andask "Whyis mathematicsteachingandlearningoccurringin this way in this setting?" (Eisenhart,1988, p. 100).It shouldbe madeclearthatalthoughthe outcomeof researchby educationalanthropologists withan interpretivist philosophymaynot directlyprescribea actionsbeing taken. of the problemscan lead to appropriate remedy,an understanding view tryto captureandsharethe Educationalresearchersholdingthe interpretivist in a socialsettinghave of theirknowledge,of whatthey are thatparticipants understanding is thatall humanactivityis teachingandlearning.An importantideain interpretivism fundamentallya social experiencewherepeopletryto makesense of theirworlds (Eisenhart,1988).This ideaconcurswiththe constructivistview thatpeopleconstructtheir own mathematicalknowledgethroughtheiractionsandreflectionon problematicsthathave particularsignificanceto them(Confrey,1985).The meaningof this knowledgeis in the social settingof a community. negotiatedandagreeduponamongparticipants ObtainingEntree The firstandmost difficultpartof conductingthisresearchwas to find andgain access to a suitableresearchsetting.Locatinga carpenterwho was preparedto acceptme as a learnerwas no easy task,andit was a full monthbeforemy searchingwas rewardedwith the sweet smell of success-sawdust! I lived witha familyin Athlone,CapeTown,in a neighborhoodthatis reservedfor "coloured"residentsin termsof the GroupAreasact, one of the foundationsof the apartheidsystem.It is an areawheremanyartisansreside.For variousreasons,thereis less of a stratificationalongsocioeconomiclines in this

72 neighborhoodthanwouldgenerallybe foundin the U.S. Blue-collarworkersmay have childrenwith professionalor academicpositionswho settlein the sameneighborhood. Throughmembersof thefamilywith whomI lived, I beganto builda social network of people who mightbe able to introduceme to carpenters.Forseveralreasons,I particularlywantedto workwithsomeonewho hadbeentrainedin an informal with familymembersandwho hadhis own smallbusiness.I thoughtthat apprenticeship someonewith a smallfamilybusinessmightbe moreflexible andthusmorelikely to accept me as a learner.I was not interestedin workingin a highlymechanizedworkshop,as some of the largerones tendto be. I did not wantto be taughtby somebodywho hadreceived formaltrainingat a technicaltrainingcenter,since academicmathematicsis takenas partof the coursework.Sincethe CapeMuslimpeoplearerenownedfor theircenturies-long in fine furnituremaking,I especiallytriedto meet people traditionof excellentcraftmanship fromthis culturalbackground. I met some verykindandhelpfulmembersof the communitywho triedtheirbest to anddead assistme in locatinga suitableresearchsetting.Thereweremanydisappointments ends, andon severaloccasions,I was temptedto give up my searchfor the elusive carpenters.It was duringthis timethatI learnedthe truemeaningof the words"frustration," and"tenacity"! I visitedseveralfurniturefactoriesandrestoration "perseverance," workshops,butnobodywas preparedto acceptme as a learner.In some cases, therewas resistanceto talkingto me becauseof my gender.I was told thatit wouldbe againstfactory regulationsfor me to workin a workshop.I also beganto realizehow noisy andrushedthe atmosphereis in a workshopwheremostof the workis beingdoneby machines.Sometimes it is absolutelydeafening.Conversationis impossible-in fact,the workerswearearmuffsto protecttheireardrums!Noise level was not mentionedin any of the articlesI hadreadabout doingresearchoutsideof an academicenvironment.It is definitelysomethingto be taken into accountwhenplanningresearchof this nature.

73 Manyof the carpenterswho hadlearnedtheirtradein the traditionalway fromtheir fathersandgrandfathers andwho hadoperatedsmallfamilybusinessesfromtheirhomes had died duringthe precedingfive years.I was oftentoldabouta memberof someone's family who wouldhave beenthe perfectteacherfor me butwho hadunfortunatelypassed away. I beganto thinkthatI hadstartedthis studyfive yearstoo late! Smallfamily businessesseemedto be a thingof the past.One"coloured"furniturefactoryowner whentryingto describedthe difficultiesfacedby blackand"coloured" entrepreneurs acquirefinanceto starta business.This particularfactoryownerhadstruggledfor many yearsagainstdauntingdiscriminatory practicesto set up his businessandto securepremises. The resultis thatmost of the businessesmakingtopqualityfurnitureareownedby whites, who employmostly"coloured"mastercraftsmenandblacklaborers.I decidedthatI would have to considerworkshopsof thislast typeas possibleresearchsites too. Aftertwo weeks of followingup everylead,sometimesat strangetimes of the night, andof makingcountlesstelephonecalls, I heardof a mastercraftsman,a CapeMuslimman, who was workingat a white-ownedrestorationandreproduction workshop.I called the ownerof the workshopandexplaineda littleaboutmy researchandaskedif I mightspeak to the masterrestorer.The owner,Peter,was opento the suggestion,andI arrangedto visit the workshopthe next morning.I spenta veryinterestinghourwith Mr. S, who hadexactly the backgroundthatI was seekingandwho obviouslyenjoyedshowingme roundthe workshopandtalkingabouthis life. He hadworkedwithPeterfor two yearsbuthadbeen acquaintedwith him for longer,andhe obviouslyhadseniorstatusin the workshop.It so happenedthathe lived nearsome of the peoplewho werepartof my social network.We arrangedthatI wouldvisit him at his homein 10 daystime so thathe could show me some of the furniturehe hadmadeby hand.He hada smallworkshopat his home. I was very excited to hearaboutthat,becauseI thoughttheremightbe a possibilityof my being able to workat his home.I was afraidto placetoo muchhopeon this,however,basedon my previousdisappointingexperiences.Duringthenext 10 days, I continuedto makecontact

74 with a wide varietyof people,meetingseveralinterestingcraftsmenwho were preparedto talkto me butnonewho were interestedin teachingme. My visit to Mr.S's homeexceededall my expectations.I was introducedto his We wentdirectlyintothe living roomandspenttwo wife, children,andgrandchildren. hoursin discussion.Mr.S was verytalkativeandrapidlyjumpedfromtopic to uninterrupted topic as if he wantedto tell me everythinghe knewall at once. His housewas filled with magnificentfurniturethathe hadmadeby handor restored.He told me thathe was "not educated"buthadreada lot of booksandarticlesaboutfurnitureandloved old English furniture.He had"stolenwithhis eye frombooks"andmademodificationsto suit his own taste.He haddone some beautifulhandcarvingon largebedroomcupboardsandtook great delightin showingme how differenthis designwas fromany other.He enjoyedmaking somethingunique.He drewmy attentionto tinydetailsthatI hadnot noticed. We hada deliciouslunchwiththe whole family,eatingtraditionalCapeMuslim dishes thathadbeenpreparedby his wife. Afterlunch,we againretiredto the living roomto continueourdiscussionaboutIslam,politics,history,education,andMr. S's work.He showedme his own research:folderscontainingcuttingsfroma greatvarietyof sources, with articlesaboutfurnituredesignandrestoration,wood carving,andreproductions. There were severaldiagramshe haddrawn,showingdesignsfor decorations.He clearlyhada greatinterestin andlove of his workandenjoyedreading.We lookedat his small workshop,whichwas overflowingwithwood andold furniture.Not muchroomfor a student,I notedwith disappointment! He showedme some carvedjewelryboxes thathe hadmadefor his daughters. Tentatively,I askedif he thoughtI wouldbe ableto makesomethinglike that.He was most enthusiasticandstartedto explainhow he hadmadethem.I told him thatI wouldlike to learnto do the practicalworkmyselfandthatI wouldlike himto teachme. To my amazement,he saidthatI shouldcome andworkwithhim at Peter'sworkshop.He proposedto startme off learninglike an apprentice,sandingandpolishing,andthengo on to teachme

75 how to makea jewelrybox andmanyotherthings.He advisedme to phonePeterin the morningto makesure,buthe seemedabsolutelycertainthatPeterwouldagree.FinallyI left, delightedwith the turnof eventsandexhaustedaftersix intensehoursof conversation. The next day,I decidedto allowsome of the morningto go by beforephoningPeter, hopingthatMr.S wouldbe ableto broachthe topicof my beingtakenon as his apprentice. WhenI telephonedPeterandaskedif I couldcome andtalkto him,he seemedmost amenable.MrS was at the entrancewhenI arrived,workingon a big table.He whisperedto me thathe had"laidthe foundationsfor me"withPeter,whichindeedhe had.I hadto wait a while for Peterto finishdealingwith a customer,andduringthis time, Mr. S took me round the workshopandintroducedme to severalpeople,saying,"Thisis Wendy-she'll be with us for a while."Needlessto say, I was mostencouragedby his confidencein his "foundations." I brieflyexplainedthe gist of my researchquestionsto Peter,emphasizingthe in workshopactivities.He agreedto allow necessityof my role as a learnerandparticipant me to workwith Mr.S. He seemedintriguedandslightlyamusedby the idea butofferedto help in whateverway he could.We arrangedthatI wouldworkat the workshopon a daily basis for the next five months,for approximately six hoursa day, startingthe next day. I decidedto reservethe latterpartof each dayfor writingfieldnotesat home.Petermadeit clearthathe intendedto leave the organizationof my workup to Mr. S andthatI couldstay as long as my presencedidnot hinderproductionin any way. We agreedthathe wouldask me to leave if he felt thatthe carpenters'workwas beingcompromisedby my research activities. Initiationand Acceptance Mr. S askedme whatI wantedto startwith,andI suggestedthathe give me whateverhe wouldusuallygive to a new apprentice.He lookedat my handsandremarked "Polishis going to spoil yourhands."I assuredhimthatI did not mind.WhenI arrivedat workthe next day, I foundmy firstjob neatlylaid out on a smallworkbench.My taskwas

76 to sanda polishedwalnutlampstanddownto barewood andthento Frenchpolish it again to high gloss. At firstmy workbenchwas on the outskirtsof the workshop.Duringthe firstweek it beganto inch forwardinto the mainworkarea.I was reservedanda bit anxiousabout approachingthe men unlessI hada specificquestionin mind.They approachedme at first-first Jack,thenPatrick,laterClive andthe others.On my second day at work,Patrick came over to my worktableandstoodandwatchedme for severalminutesas I laboredover the tall walnutlampstand.The lampstandhadinterestingcurveddetailsandwas difficultto sandevenly. It was a hot day.My backhurt,my face was flushedanddusty,andmy hair keptfallinginto my eyes. Patrickcontinuedto watchme closely. He said tentatively, "Wendy,wouldn't you ratherlearnto sew?""No" I repliedfirmly,"Iwantto learnto do woodwork."Patrickdigestedthis for a while.Finallyhe observed,"I supposeyou mustbe one of those tomboy-girls." Forthe firstfew weeks,Mr.S constantlyinquiredwhetheror not I was in a draft, agonizedoverthe fate of my hands,whichwerebecominghardenedandcoatedwith dark brownshellac,andhandedme smallclothsto wipe the dustfrommy face. The othermen also paidattentionto my handsandwarned,"Yourhandswill be harderthanyourfuture husband'shands!"Slowly theirinterestwanedas they becameaccustomedto my presence. One of my initialproblemswas to find a way to have tea andlunchwith the carpenters.Therewas a half-hourbreakat lunchtimeandtwo 10-minuteteabreaks,one at I workedfrom8:30 in the morning 10:00in the morningandthe otherin the mid-afternoon. to conversewith the workers"ontheir until3:00 in the afternoon,so I hadtwo opportunities own time."At all othertimestheyconveyedthe feelingto me thatwe were"stealingthe boss's time."Sidneykepta meticulouseye on the clock andblew a whistleto signify the beginningandthe end of eachbreaktime.Everybodyobeyedthis signal. Joiningthe workmenat breaktimeswas awkwardfor severalreasons.One unexpectedobstaclewas the friendlinessof the secretary,who was a youngwomaneagerto

77 have companyduringthe breaktimes.She sat alonein an office attachedto the workshop. She mademe tea andbroughtme good thingsto eat fromherhome.The moresignificant relatedproblemwas thatshe was not the only one who expectedme to keep hercompany. All the carpentersalso expectedme to havetea withher,since she was the only otherfemale memberof the staff.Mr.S commented,"Youcan havea breakfromus men andhave a womanto talkto." Peterexpressedhis opinionthatmy presencein the workshopduringbreaktime mightrepresentan intrusionof the men'sprivacy.He suggestedthatthe men mightfeel inhibitedif I sat with themwhile theywereeatingandtalkingandmightbe unableto relax adequatelybeforetheirnext periodof work.The clearsegregationpracticedby the men was easy to observe,andI did not wantto join any one particulargroup.The blacklaborerssat and ate togetheras a groupandhadlittleinteractionwith the others.The "coloured"artisans formedanothergroup,andthe apprenticesanothergroup.Interactionoccurredbetweenthe artisansandthe apprentices.Therewerelonersfromall threeof thesegroupswho sat apart fromthe others,one of thesebeingMr.S. The groupswereformedmainlyat morning teatime,whichwas whenmostpeopleate. At lunchtime,manyof the men were engagedin personalwoodworkingprojects. Afterabout10 daysI realizedthattheway to legitimizemy presencein the workshopat tea breaksandlunchtimesandto avoidall the problemsmentionedwas to have a "loosejob"or "sparejob"like everyoneelse. Thismeanthavinga job thatwas for oneself or for personalprofit.My engagementin a woodworkingtaskwouldprovidea clearly acceptablereasonfor spendingthe breaksin the workshopandwouldfurthermore put me in a positionto cultivateconversationswith whomeverI pleased.Underthe directionof Mr. S, I beganto workon a carvedjewelrybox madefromyellowwoodwith an inlaidstardesign on the lid madefromstinkwoodandbone.Bothwoods arepreciousindigenouswoods. Peterallowedeveryonein the workshopto use smalloff-cutsfor privateprojects.Mr. S designedthe box, makingsurethatI understoodeverythinghe was doingandincludingme

78 wheneverpossible.I enlistedthe assistanceof severalworkersbesidesMr.S when I began the construction. As time wentby, it becameeasiertojoin in the conversationsat breaktimesandafter a couple of monthsI becameMr.S's constantcompanionduringmorningteatimeand lunchtime.The secretaryleft aftermy firstmonththere.AlthoughI was sorryto see hergo, this event mademy presencein theworkshopat all breaktimesseem even moresensibleto the men, since I no longerhadthechoiceof a femalecompanion.Whenthe new woman secretaryarriveda week later,nobodysuggestedthatI shouldkeep hercompanyduring breaks. Aftersix weeks of scraping,sanding,andFrenchpolishing,I movedinto a different Mr.S toldme thatI haddoneenoughof thatsortof work,and stageof my apprenticeship. he beganto treatme morelike his personalassistant.My physicalpositionin the workshop changedtoo, frombeingon the outskirtsandcreepingin, to being,in Mr.S's words"inthe thickof things!"I workedclosely withhim,cuttingwood, fetchingthings,gluing,clamping, drilling,andchiseling.He stoppedworryingso muchaboutthe stateof my hands.I started workon replacingpartof a stinkwoodandyellowwoodinlaidtable,whereI hadto replacea missingdraweranddo the inlaywork.I latermadea seconddrawerwith dovetailjoints.At whentalkingto the other this stageMr.S beganto referto me as his "newestapprentice" men andto customersat the workshop.Some of carpentersnoticedandcommentedon the change.PaulgrumbledthatI was gettingfarbettercarethanhe hadas Mr. S's apprentice! WhenI spentthe day away,severalof the menwouldteasinglydemandan explanationfor my absence.Clive commented,"Weworkfive daysa week here,Wendy,not only four!"I learnedto tell severalof the men,andnot only Mr.S, aheadof time if I intendedto change my acceptedworkschedulefor anyreason.Mr.S commentedon how muchthey all missed me whenI was absent. Duringmy thirdmonthat the workshop,I completedtwojobs thatseemedto earn me a lot of recognition.A mahoganysideboardhadbeendesigned,built,andpolishedin the

79 workshop.Severalworkershadbeeninvolvedin theconstructionandfinishingof the sideboard.It was a specialpiece,custom-madefor someone.It hadfourdrawersfor storing silver cutlery.My taskwas to coverthecutleryholdersthatfit into the drawersandthe interiorof the drawersin greenbaize.The taskwas definitelysomethingcategorizedin the workshopas a "woman'sjob"andwouldhavebeendoneby Peter'swife at homeif I had not been available.It requireda lot of fine workwith a razorblade,tweezers,andglue. All the workerscame to look, to checkon the neatness,to admire,andto give advice.I am not surewhy this drewso muchapprovalfromthe group.Maybebecauseit was a specialitem of furnitureandthe greenbaizeliningputa fine finishingtouchto the whole piece. Since a numberof peoplehadbeeninvolvedin theproject,therewas pridein the workof the whole team.This generateda greatfeelingof camaraderie. The otherprojectthatdrewmanyfavorablecommentsfromthe groupwas the constructionof a stereostandthatI madefor a friend.Thisrepresenteda "sparejob" for which I was beingpaid.Althoughit was a "sparejob,"I got permissionfromPeterto work on it full-timefor a few days.Mr.S andClive directedmy workandgave assistancewhile PaulandPatrick,amongothers,watchedmy progressandmadecomments.Clive andI selectedthe roughplanksandthenworkingfroma diagramdrawnby my friend,cut the planksto size, thicknessedandsandedthem,madethejoints,gluedthe components together,andsquaredandclampedthe construction.I sprayedthe standwith sandingsealer andthenpolishedit. This was the firsttime I hadplayeda majorrole in a visible projectthat "startedfromthe floorup"(i.e., workedfromroughwood, constructedthe articleaccording to a specific design,andspentconsiderabletimeon the finishing).Althoughmy first personalprojectwas the constructionof thejewelrybox, muchof this workwas invisibleto the men, since Mr.S insistedon keepingthe box hidden,for fearthatthe othercarpenters would copy his design.I workedon the box at home,or keptthe componentsseparatewhen I workedon it at lunchtimein the workshop.Nobody,not even Peter,saw the finished jewelrybox.

80 Establishingmyself as a bonafide workerwas a slow process,butit was accelerated afterthe firstsix weeks whenmy workplacechangedfromthe outskirtsto the hubof the workshop.The natureof my workalso changedafterthe firstsix weeks, andI was expected to cut wood with the electricsaws (underMr.S's watchfuleye), use the electricdrillsand sandingmachines,use a handsawandchisel to makedovetailjoints, andbecomeinvolved in projectsrequiringinlay workandotherconstruction.Becauseof thesechanges,I had greateraccess to the men duringthe workingday,andit seemednaturalfor me to spend breaktimeswith themandto be includedin theirconversations.Peterpaidme for my work fromthe beginning,andthis also helpedto legitimizemy labor.Mr.S, Jack,andClive alwaysaskedme how muchI hadbeenpaidfor eachjob thatwentout to a customer. Judgingby the commentsof all themen,I was acceptedas a participantduringthe third month.As the weathergrewcolder,I was includedin the groupthatsat aroundthe fire drum at breaktimes.The workshopwas highceilingedanddrafty,andwe enjoyedtoastingour sandwichesandwarmingourhandsat the fireside. AlthoughI hadexplainedthe generalgoals of my researchto the workersat the outsetandhadtalkedaboutmy degreeprogramto themfromtime to time, it was interesting to hearthemconstructtheirown explanationas to why I was in the workshop.The timethat apprenticesspendin the workshop,betweenperiodsof studyat the technicalcollege, is referredto as "doingtheirpractical."Whena visitorexpressedsurpriseat seeing me (a white woman)workingin the workshop,or askedsomeonewhatI was doing there,s/he wouldbe told, "Wendyis studyingin America,butshe has come hometo do herpractical. Whenshe is finishedhere,she'll go backto college in Americaagain."In this way, my researchwas given meaningthatmadesense in the contextof the workshop. Afterworkingon a dailybasisforjust overthreemonths,I left the workshopfor threeweeks for a periodof writing,thinking,consolidation,andplanning.On my return,I receivedan enthusiasticwelcomeandwas putto workimmediately.Therewere many absenteesbecauseof illness, andI was neededas a worker.I wentalongwith this to a

81 certainextentbut felt thatmy researchwas beingcompromised.Also, I was consciousthatI would be leavingwithintwo monthsandfelt thatI wantedto phasemyself out slowly to avoid an abruptdeparture.However,therewas a lot thatI could do to help, andfor several days I playedthe partof a workerandperformedsolitarytasksthatdid not give me access to any of the othercarpenters. WhenMr. S went awayfor a few days,I askedJackif I couldhelp him construct tables,whichwas his specialty.I also beganto helpClive with the constructionof chairs, his chief taskat thattime.In this way, I was ableto keep two conversationsalive as I moved as a helperfromJackto Clive. My projectsled to numerousinterestinginteractionswith manyof the men, some of whichhavebeendescribedin Chapter5. I have triedto describemy initiationandacceptanceat the workshopin the context of the cultureof the workshop.The workers,althoughall bornandraisedin SouthAfrica, representedseveraldifferentlanguageandracegroupsandat least two differentreligious this workingcommunityhadspun groups.Despitethese differencesin backgrounds, definite"websof significance"(Geertz,1973,p. 5), withits ownjargon,habits,traditions, beliefs, expectations,patternsof behavior,andsystemsof symbols.Frommy pointof view as a SouthAfrican,conductingresearchin a countrylike SouthAfrica,wherethe political andeconomicdominanceof one groupoverall othershas beenentrenchedfor so long, the conceptsof powerandconflictareessentialin a discussionof culture.Withreferenceto Giroux(1981), the notionof whatconstitutesculturefor this studyencompassedalso the experiencesof thesepeoplein theirstrugglefor politicalpower,theiroppositionto economicinequity,andtheirabilityto shapetheirown lives in the face of severeeconomic and politicalrestrictions.It wouldnot be possibleto makesense of the teachingandlearning situationsin the workshopwithoutconsideringthe prevailingsocial, political,andeconomic constraintsandthe limitsthattheseconstraintsimposeon learningopportunities.

82 Focus on Problematics Throughoutmy timeat the workshop,I focusedon mathematicalproblemsthatarose spontaneouslyduringcarpentryactivities.Workingwithinthe theoreticalframeworkof constructivism,I claimthattheproblemsthatarosein the carpentryworkshopbecame for the carpentersthroughmy questioning.The "Episodes" problematicsor "roadblocks" describedin Chapter5 areexamplesof the engagementof the carpentersin the of the problematicby the constructivistcycle of actionandreflection.The acknowledgment carpenter(manifestedby a willingnessto continueourconversationsandto searchfor a solution),servedas a twofold"callto action."Froma methodologicalpointof view, it was a call to me as a researcherto pay specialattentionto whatwas happening,a signal thatI To the carpenter,acknowledginghis problematicwas a call to mightwitnessmathematizing. removethe roadbock,to becomeinvolvedin a cycle of actionandreflectionin an attemptto reachresolutionandrestorationof equilibrium.As a novice,I was expectedto ask questions to begindiscussionsthatI anticipatedas fertilegroundfor andso couldseize opportunities exploringthe mathematicalideasof the carpenters. Once I hadlocatedthis particular groupof carpenterswho werepreparedto teachme andhadshownthatI was preparedto workhard,my genderno longerappearedto be an obviousobstacle.As a woman,I was not expectedto know anythingabouta man'swork.I could thusask questionsoverandoveragain,encouragingthe carpentersto reflecton their the firsttime.It did not seem explanations,withoutbeingscornedfor not understanding strangeto themthatI couldnot,for example,calculatethe volumeof a piece of wood, despitethe fact thattheyknewthatI was a mathematicsteacher.Sucha calculationwas seen as a technicalmatter,meantto be solvedonly by someone(definitelya man)who had specialistknowledge.I was thusfreeto pursuethisquestionat greatlengthandable to encouragethe carpentersto talkabouttheirconceptsof volumeandthe measurement thereof.

83 Of courseit is not possibleto predictwhatwouldhave happenedif anyoneelse had askedthese questionsor whatthe discussionswouldhavesoundedlike if I hadnot been thereat all. My presenceas a whitewomanworkerin a carpentryworkshopwhereall the otherworkersweremen, andmostlyblackmen,is unusualfor SouthAfrica,where by raceandgenderabound.Did the men respondpositivelyto separationanddiscrimination my questioningbecauseI am whiteandthusin a categoryto be treatedwith deference Ordid theyenjoysharingtheirknowledgewith an accordingto apartheidindoctrination? "ignorant"woman?Orperhapsourinteractionsresultedout of the friendshipthatgrew as a resultof mutualrespect,admiration,andmanysharedjokes?I do not know the answersto these questions. Data CollectionMethods Fourmainmethodsof collectingdata,as outlinedby Eisenhart(1988), were used. These were (a) ParticipantObservation,(b) Interviewing,(c) Searchingfor Artifacts,and(d) ResearcherIntrospection.All fourmethodswereusedconcurrentlyduringfieldwork,since each methodgave a differentperspectiveon theresearchandeach one could be used to informthe other.Accordingto Eisenhart(1988), in ethnographicresearch,the more perspectivesthatarerepresented,the strongerthe researchdesign,since each perspective addsto the completenessof the pictureone is tryingto present.By employinga varietyof datacollectionprocedures,theresearcheris able to cross-checkobservationsrecordedin field notes. Participant Observation As Sharp(1989) pointsout,intensiveparticipation can occuronly when the activitiesinvolvedarecredibleto the "locals."It is the "locals"who mustconferthe status of "participant" on the researcher.This is not a statusthatone can conferupononeself, a fact thatenthusiasticethnographers usuallyomit whenadvocatingthe methodof participant observation.In manycircumstances,theresearcheris not given this status,andit is then actuallylong-termobservationthatprovidesthe data,with participationplayinga

84 secondary,supportiverole in theresearch(Sharp,1989).Withthis in mind,I was constantly alertas to how I was beingtreatedby thecarpentersandrecordedin detailthe changesI perceivedin theirattitudestowardsme in the previoussection,entitled"Initiationand of the developmentssuggeststhat Acceptance."As mentionedpreviously,my interpretation I becamea participantduringmy thirdmonthin theworkshop. Participantobservationis seen as the majortechniquefor remainingin the "marginal positionof simultaneousinsider-outsider" (Hammersley& Atkinson,1983, p. 100). The researcherhas to be simultaneouslyinvolvedin the activityas an insideranddetached enoughto allow reflectionas an outsider.I foundthis dualrole difficultto play. It became increasinglydemandingto weartwo hatsat once. At first,wheneverythingwas freshand new, it was easy to noticeeverythingfroma researcher'spointof view. As time went on, andI becamemoreof an "insider,"steppingbackwas sometimesvery hard.I hadto deliberatelyforcemyselfto takethatstep.Afterfive weeks at the workshop,I recordedin my diary: I have to persuademyselfto get into "researcher mode"at times.It is quite an effort, andI have to talkmyselfinto it. It is so mucheasierto stayin passive"carpenter mode,"casuallylisteningandobserving,andgettingon with thejob at hand.It is far morerelaxingfor me to spenda day like this, andI sometimesconsciouslychoose to do this if I am tired.I think,"Oh,tomorrowis anotherday. It is too muchof an effortto changethe subjectto fit in withmy 'researcher'agenda.I am rather enjoyingthis piece of familygossip,or topicaldiscussion,or talkaboutfood!"Or, "Iam so enjoyingthe peaceof workingwiththis wood or chairthatI don'tfeel like talkingto anyoneat all!" I participatedin the workshopas an apprentice,as a helper,as a worker,andin a social capacity.I triedto recordmy interactionswiththe carpenterson all these levels and to observeandrecordtheirinteractionswitheachotheras well. I wrotedetailedfieldnotes every evening,sketchingthe eventsof eachday,thusgiving a pictureof the day in the workshop.Therewereobviouslycertainkindsof eventsthatinterestedme morethanothers, andI havegiven a partiallist of importantcategoriesbelow. The categoriesarederivedfrom a combinationof the theoreticalcontextthatframesmy researchobjectives

85 (see Chapter2) andof my practicalexperiencein the field. The list is necessarily incomplete,since I did not spendthe day filling presetcategoriesbuttriedto absorb of whatI was experiencing. whateverI could thatwouldenhancemy understanding OccasionallyI wouldwritemyself a noteon a scrapof paperduringthe day, butgenerally therewas no time for this, andI dependedon my memory.The most successfulstrategywas to recreatethe day in my mindin minutedetaileacheveningandto recordmy observations directlyon a wordprocessor.Even on dayswhenI initiallyfelt thatnot muchhadbeen observedor accomplished,on reflection,manyinterestingandrelevantobservationsdid emergewhenI beganto write. My observationscoveredmanyareas.I recordeddescriptionsof1. the woodworkingprojectsin whichI was engaged; 2. the woodworkingthatI learnedandhow thatlearningoccurred; 3. the teachingmethodsof Mr.S andsome of the othercarpenters; 4.

problemsolving doneby thecarpentersalone,with me, or in a largergroup;

5. the mathematizingdemonstrated by the carpenters; 6.

the carpenters'ideasaboutmathematics;

7.

I learnedandhow thatlearningcame about; the mathematizing

8. links to academicmathematicsthatI couldrecognizein the contextualized I was learning; mathematizing 9.

for the workers; learningopportunities

10. the role anduse of tools in the workshop; 11. problemsthatthe carpentersencounteron certaintasks; 12. social interactionsbetweenmyselfandthe otherworkers; 13. the changingattitudesto my presence; 14. social interactionsbetweenthe otherworkers; 15. interactionsbetweenPeterandthe workers.

86 Interviews Interviewsrangedfrom(a) informalconversationsto (b) semi-structured conversationsto (c) formalstructured interviewswhereI hada scriptthatI wantedto follow. Informalconversationsoccurredthroughoutthe day while we workedandduring breaks.I wouldsometimestryto maneuverthetopicto somethingthatI was interestedin hearingmoreabout.Mosttimesthis wouldwork,butoccasionallyI wouldmeetresistance andbe unableto engagethecarpenters'attentionin my preferredsubject.Practicalfield experienceshowedme thatcertaintopicswouldoftengeneratediscussionof mathematical concepts.The kindsof topicsthatwerefruitfulin revealingmathematizingwere those that requiredthe carpenterto1. explainsomethingto me verballyor nonverbally, 2. takesome actionon a problematic, 3. engagein reflection, 4. offer some argumentfor verification, 5. appraisesome workcritically, 6. workwith a moreablepeer,or act as a moreablepeer. To this end, I continuallyaskedquestionswhile we worked,for instance:"Canyou explain why?""Howdo you know?""Howcan you be sure?""Whatdo you think?""HowshouldI do this?""Whydoes thatwork?""HaveI donethiscorrectly?""Howcould I have doneit better?""Howcan I correctthis mistake?""Canyou help me with this?""Whatwould happenif we do __

?""Isthereanotherway to do this?""Howcan I be surethis is okay?"

In this way I encouragedthecarpentersto talkaboutwhattheywere doingandto comment on my work. I was alwaysconsciousof the fact thatwe werein a businessenvironmentwhere productionwas of the utmostimportance.Therewas no timeto standaboutandtalk.Our informalconversationstook placewhilewe worked,andtherewere manyinterruptions.

87 Sometimeswhenthe machineswerebeingused,it was impossibleto hearanything,andmy researchagendafor the dayhadto be modified. Some of the informalconversationsthatI foundhighlysignificanthave been describedas "episodes"andarediscussedin detailin Chapter5. Theseconversationsare significantbecausethey arecenteredaroundvariousproblematicsthatarosespontaneously in the workshop.The episodesprovidevivid illustrationsof the mathematizingthattook place as the carpentersactivelyengagedwiththeirproblematics,andreflectedon their actions,finallyarrivingat solutionsthatsatisfiedthem. "Semi-structured" conversationswereaimedat the collectionof life storiesof some of the carpenters.I madepriorappointments withthe carpenters,andwe usuallysat in my car at lunchtimeto escapethe noise in the workshop.I beganthe conversationwith some generalopeningquestionsaboutthe life of the carpenterandallowedthe conversationto developfromthere.I askedcertainstandardquestionsif theseseemedappropriate(for generalscript,see AppendixA). In thisway I gleanedinformationabouttheircultural background,theirschool experience,theircontactwith andviews of academicmathematics, theircarpentrytraining,andtheirideason craftsmanship andexpertise.I recordeda history of the workshopby interviewingPeter,who was also ableto give some additional backgroundinformationaboutthecarpenters. interviewspresentedsome unexpectedhurdlesat Conductingthe "semi-structured" first.Aftera few weeks of persuasion,Mr.S consentedto speakto me abouthis life while I tape-recordedthe conversation.As usual,therewas not muchprivacyor quietto be foundin the workshop,so at lunchtimewe took oursandwichesandsat in my car,which was quiet, warm,andcomfortable.Whatfollowedwas mostamusing,if unproductive! We beganby talkingaboutMr.S's childhoodandwhenhe startedto learnabout woodworkingfromhis father.He was clearlya bit uncomfortableaboutthe taperecorderat first.Aftera while, he becamea littleless consciousof it andwas warmingto the subject when the secretary,who couldno longercontrolhercuriosityaboutwhatwe were doingin

88 the car,appearedat the doorwaywith a cup of tea for me. I explainedthatwe were tapingan interview.Nothingdaunted,she said,"Oh,well, thenI'll join you in the backseat!"Dolores was a very largewomanwho worebig flowinggowns in brightcolors. Herpresencefilled up the backof the car,andshe hadentirelyherown ideasaboutthe directionof the conversation.PoorMr.S's reverieswerecompletelyhaltedandthe interviewwas ruined! Structuredinterviews,whereI askedspecificquestionsaccordingto a scriptI had prepared,wereconductedwithClive andJackduringmy last week at the workshop(forthe for the interviews.Clive's interview script,see AppendixB) . Again,I madeappointments was conductedat my home,andJack'sinterviewtookplaceat his own home.Most of the questionsthatI askedrelateddirectlyto workandideaswith whichI knew they were familiar.I also includeda taskthatwas unfamiliarandthattheywoulddefinitelyregardas Theresultsof theseinterviewsarediscussedin detailin Episodes17, being"mathematics." 18, and20 in Chapter5. Whileinformalconversationswerewrittenup as partof my daily fieldnotes,all the interviewswere tape-recorded. conversationsandthe structured "semi-structured" Tapeas soon as possibleafterthe event andthe recordedinterviewsweretranscribed storedon a wordprocessor.In the workshopwe oftenreturnedto topics transcriptions of what previouslydiscussed,whichgave me an opportunityto check on my interpretation the carpentershadsaid. AdditionalMethods In additionto the two primarydatacollectionmethodsdescribedabove, two ancillarymethods,also consideredto be key datasourcesfor ethnographicresearch,were used.These werethe collectionof artifactsandresearcherintrospection. Thecollectionof artifacts.Thismethodcan assistin gaininga broaderunderstanding of the contextthancan be gainedby relyingsolely on personalexperience.Goetz and LeCompte(1984) pointoutthatanythingthatthecommunitymakesanduses

89 resultsin artifactsthatconstitutedataindicatingpeople'sexperiencesandknowledge.This may includephysicalobjectslike carpentrytools or everydayhouseholdarticles.The search for artifactsmay also involve archivalstudies,suchas the examinationof writtenrecords, historicalrecords,or graphicalmaterials.Eisenhart(1988) statesthatany information or others,whichis tangibleandhas bearingon the topic of producedby the participants study,may be useful. A varietyof artifactswas gatheredduringthe presentstudy.On firstenteringthe field, I studiedthe historyof the CapeMuslimpeople,since this was the groupwith whomI hopedto work.This informationwas usefulduringmy conversationswith Mr. S (theonly CapeMuslimat the workshop),who was well-informedaboutthe historyof his people.I collectedwrittencalculationsandsketches(ormadecopies of theselater,since in the workshopsketchesareoftenmadeon verylargepieces of wood!)thatthe carpentersmade while workingor duringourconversationsandinterviews.I took photographsof tools and furnitureunderconstructionto illustratewrittenexplanationsin my fieldnotes,andmade manysketchesdepictingproblem-solvingstrategiesdevisedby the carpenters.I was given two hand-madewoodentools, usedfor drawingoval shapes.Duringmy last week at the workshop,I hada video recordingmadeof the men at work.In the video, I briefly introducedeach carpenterandthenaskedhimto explainwhathe was doing,while the camerarecordedhis actions.Someof the menthoroughlyenjoyedgiving detailed of theirwork.The examinationof all these artifactsandthe explanationsor demonstrations of this abilityto sharethis tangibleevidencewith othershas enrichedmy understanding communityof carpenters. Researcherintrospection.Researcherintrospectionis the practiceof keepinga personaldiaryto recordreflectionson theresearchactivities(Eisenhart,1988). This method is peculiarto researchof an interpretivist nature. I kept a regulardiaryto tracemy emerginginsights,feelings, andintuitionsso as to documentthe evolving natureof the study.I recordedhow I was treatedin the factory,what

90 kind of workI was doing,andwhatI enjoyedaboutit. I lamentedmy failuresandmade hopefulschemesfor morefruitfullines of inquiryin the next stage of the researchprocess. The diaryis interestingandusefulto readin conjunctionwith the fieldnotes.It provideda muchneededvehicle for reflection.Self-preservation is a very realconsiderationwhen planningan extendedperiodof field work.I elaborateon this in the next section. The Emotional Demandsof EthnographicResearch:Some Coping Strategies As pointedout by Sanday(1983),participant observationdemandscomplete In additionto the lengthof timerequiredto be commitmentto the taskof understanding. can be emotionallydraining.The spentin the field, otherdemandsmadeon anethnographer keepingof meticulouslydetailednoteson a dailybasisandhavingto be constantlyalertand open to conversationday afterday,week afterweek, can becomeexhausting.The fieldworkeruses herselfas the principalandmostreliableinstrumentof observation, selection,coordination,andinterpretation (Sanday,1979).Onehas to cope with moral dilemmasthatariseas a resultof the complexweb of mutualdependenciesbetween It is essentialto maintainthe tensionbetweenbeingtotally researcherandparticipants. immersedin anotherrealitywhilestill operatingfromone's own frameworkas a researcher with colleaguesandan academiccommitteeto reportbackto. I foundit importantto developa networkof colleaguesoutsideof theresearch fromthe Universityof CapeTown who setting.I was befriendedby an anthropologist An old friendwho is invitedme to join a weeklyseminarwitha groupof anthropologists. bothan educatoranda skilledcarpenterprovedto be anothersignificantresource.I regularlydiscussedmy researchwitha mathematicseducatorat the Universityof Cape Town. Membersof my SpecialCommitteeat Cornellwroteto me regularly,giving me feedbackon the notes thatI mailedto Ithaca.Thiscorrespondence providedexcellent guidancefor the durationof my fieldwork.Theseopportunitiesto sharemy perceptionswith otherinterestedpeoplehelpedto keepsome balancein my life.

91 Everytwo weeks, I took a one-dayleave of absencefromthe field to review my fieldnotesandpreparethemto be mailedto my committeemembers,with a reporton my progress.This was essentialto allow some breathingspaceandto encouragereflectionon the totalprocess.Manuallaborwas new to me, andI oftenfelt physicallytiredandlacking in energy.OccasionallyI took a day off simplyto restandsleep. The difficultiesof entrdemayin some ways be matchedby the problemsof withdrawalfromthe researchsetting.Aftermonthsof dailycontact,friendships, expectations,anddependencieshaddeveloped,andI was awarethatI hadto handlethe terminationof my relationshipswithcareandsensitivity. Exit Duringthe last few weeks,Clive, Mr. S, andJack,in particular,mademany referencesto the fact thatI wouldsoon be leaving.Clive complained, Why areyou so late today?Onyourlast daysyou mustcome in very earlyso you can have moretimewithus! I am alreadyfeeling sad thatyou areleavingus. It is like giving you off (localway of talkingabouta father"givingaway"his daughter in marriage).Yourgroomhas come to fetchyou, andyou have to cut all yourold ties. You will forgetaboutus soon. As soon as you leave us, you'll wonderwho these funnyguys were. He was referringto the fact thatI wouldsoon be leavingfor the U.S. with my American fianc6.In fact, this paternalisticview mademy exit mucheasierthanI hadexpected.My last day at the workshopwas like a pre-weddingcelebration.They presentedus with a thick handmadeyellowwoodchoppingboardthateachpersonhadsigned.Duringan extralong lunchbreak,we all watchedthe video thatI hadmadeof the carpentersat work.We left in a flurryof good wishes andsage advicefor futurenewlyweds.

Chapter5 RESULTSAND ANALYSIS: MATHEMATIZINGBY THE CARPENTERS Introduction The aim of this chapteris to addressthe firstresearchobjectiveof the study.Data are describedandanalyzedto provideevidencethatvalidmathematizingis embodiedin the everydayactivitiesof the groupof carpenterswithwhomI worked. Chapter5 is separatedinto two parts.In thefirstpartof the chapter,the selectionof "episodes"as unitsof analysisis discussedandthecriteriafor selectionareprovided.The anda seriesof 20 episodesis described. mainparticipantsarebrieflyreintroduced, The secondpartof this chapterprovidesa comprehensiveanalysisof the nature of the mathematicalideasimplicitin the activitiesof the carpenters.The 20 episodes previouslydescribedarethe unitsunderconsiderationin this partof the analysis.The developmentof categoriesis documentedanddiscussed,andthe "formal,systematicand logical procedures"usedto generateconstructsaredescribed(Goetz& LeCompte,1984, p. 167). Part One: The Episodes The Establishmentof Units of Analysis Accordingto GoetzandLeCompte(1984), the establishmentof unitsof analysisis one of the firstmajortasksin processingethnographic data.Ethnographers arerequiredto describewhatthey observe,to divideobservedphenomenainto units,andto indicatehow unitsaresimilaranddissimilarfromeachother.Eisenhart(1988, p. 107) refersto the definitionof "meaningful" unitsof the material(meaningfulto the researcheror to the participants)as an importantinitialstep in the processof ethnographicanalysis. Precedentsfor the selectionof certainincidentsas vehiclesfor descriptionandanalysis can be foundin the workof Erickson(1982, p. 167),who uses the term"encounter" to describea "boundedsetting"on whichto focus the investigationof learningin

93

94 classrooms.An "encounter" is definedas a socialgatheringcharacterizedby distinctive foci of attentionandmeaningandincludesface-to-faceinteractionsbetweenpeople as well as interactionsbetweenan individualanda physicalobject.The pointof focusing is thatone has to findthe "crucialsites in everydaylife, situationsof uponan "encounter" time,place andaction"wherethe objectof researchis mostlikely to occur(Erickson, 1982, p. 163). Ericksonrecommendsa chronologicallyaccurate,analytic,narrativestyle of reportingfieldwork. Criteria for the selection of units of analysis. In the present study, the primary units

of analysisarecalled"episodes.""Episodes"aredefinedas those instanceswherethe carpenters(individuallyor in a group)engagedin a cycle of actionandreflectionon a problematicin an effortto resolvea dispute,to solve a problem,or to give a convincing only thoseinstanceswherethe carpenters explanationto a novice.Furthermore, demonstratedone or moreof the six environmental activities,claimedby Bishop(1988a)to be the foundationof all mathematical ideas,will be consideredto be "episodes."7An exceptionto this rulewill be seen in the inclusionof Episodes4, 6, and 12. These episodes were chosento illustratesome of the carpenters'conceptionsof mathematicsandof their own relationshipsto mathematics. The term"episode"was chosenbecausethe incidentsforma seriesof relatedevents duringthe courseof my continuousexperiencein the workshop,andeach incidentis a vignettethattells a coherentstoryin itself. Onegroupof episodescomprisesdetailed narrativesof a problematicor set of problematicsengagedin by one or moreof the carpenters.Anothergroupof episodesis includedto give the readeran idea of the conceptionof mathematicsthatprevailedin the workshopandof how the carpenterssaw themselvesin relationto mathematics.Manyof theepisodesdescribeproblemsthatarose spontaneouslyfromactivitiestakingplacein the workplacewith the carpenters' 7 A chart activitiesdescribedby Bishopare showingwhichof the six environmental representedin each episodecan be furnishedby the authoron request.

95 mathematizingunfoldingas they discussedtheirproblematicsandconstructedsolutions. Sometimesthe problem-solvingactivitieswereinitiatedby the questionsI posed in my role as novice. To providethereaderwithlinksto the latterpartof this chapter,a brief ideas andprocessesthatareimplicitin the activitiesof the commentaryon the mathematical carpentersis includedwitheachepisode. The detaileddescriptionsof encounterswith the carpentersas they discussand reflecton theirproblematicsareintendedto give the readerthe opportunityto "hear"the discussionsbeforeI categorizemy conclusionsaboutthem.This representsan attemptto let the participantswho madethis researchpossiblespeakfor themselvesandpresenttheirown cases to some extent.I do not wanttherichnessof the humaninteractionsto be reduced merelyto a set of abstractfindings. Introduction to the Episodes The episodesarearrangedchronologically,andthe dateon which each episodetook place is given to providea temporalperspective.Throughoutfive monthsof intensive contactwith the carpenters,my role in the workshopwas thatof a learner.My role as researcherwas bothas an observerandas a participant. As describedin Chapter4, my physicallocationin the workshopandthe natureof my taskschangedgraduallyover time. of events,I was acceptedby the carpentersas a bonafide Accordingto my interpretation workeror "participant" in the methodologicalsense duringthe thirdmonthof my time at the workshop. The episodesdescribedin this sectiontook placeacrossthe five-monthspan,with Episode 1 occuringnine daysafterI arrivedat the workshopandthe last episodeoccuring on my last day at the workshop.Duringthe monthof June,I was absentfromthe field for a periodof almostthreeweeks, hencethe long gap betweenEpisode9 andEpisode 10. In some of the episodesI am an observerandin othersI am bothan observerandan active participant.

96 The main characters in the episodes. The carpenters had had little contact with

formalschool mathematics.My principalteacher,Mr.S, a manapproaching60 yearsold, left school andbeganas a wage-earningcarpenterat the age of 13 years.He studied arithmeticat elementaryschool,butsince he hadnot beento highschool, he hadnot encounteredalgebraor geometry.Jack,in his latethirties,left school andstartedworkat the age of 16 years,so he hadone or two yearsof juniorhighschool mathematics.Clive, also in his late thirties,left schoolandbeganworkingat the age of 16 years,aftertwo yearsof high school mathematics.He was the only artisanin the workshopto have attendedclasses, includingtwo mathematicsclasses,at a technicalcollege thatprovidesformaltrainingfor artisans.Brian,21 yearsold, attendedhigh schoolfor one yearbeforebecominga wage earner.Mathematicssyllabifor the firsttwo yearsof highschool includeelementary algebraandEuclideangeometry.Life historyinterviewsof Clive, Jack,andBrianindicated thattheiryearsat high schoolhadbeendisruptedfor variousreasons,andit was difficultto assess exactlyhow involvedtheyhadbeenin theirmathematicsclasses. Mr.S, Clive, Jack, andBrianarethe maincharactersin theepisodesthatfollow. Episode 1 (March 9th) Drawinga star design and locatingit on a rectanglarbox lid My first"loose"job was the constructionof a jewelrybox. Jackhadmadean inlaid stardesignon a chinacabinet,whichMr.S showedme anddescribedas a "squarestar." See Figure4.

Figure4. The squarestar.

97 Mr. S took the rectangularlid of my jewelrybox, about12 inches by 7 inches,and drewthe diagonalsin pencil,sayingthatI shouldinlay a stardesign"rightin the center"of the lid, wherethe lines thathe drewcrossedeachother.I took the lid over to Jackandasked him to drawa starsimilarto the one on the chinacabinetbutto makethe horizontalpoints of the starlongerthanthe verticalpoints."Thatwill look stupid!"he muttered.Using his squarerule as a straightedge, he measuredthe diagonalsthatMr.S haddrawn.Apparently satisfiedwith these measurements (thelengthswereequal),he drewa smallcross at the pointof intersectionof the diagonals.See Figure5.

Figure5. Positioningthe squarestaron thejewelrybox lid. He beganto explainhow to drawa "square"star(see AppendixC). Finally,at my insistence,he drewthe staraccordingto my request(see Figure6), explainingeach of his actionsout loud (see AppendixD).

Figure6. Jack'smodificationof the squarestar.

98 It was clearlythefirsttimethatJackhadmodifiedthe shapeof the "square"star, andhe improviseda set of rulesfor the new shape(see AppendixD). He said thatit hadto be donethatway "tomakeit right."He claimedthathe hadlearnedto drawa "square"star simplyby lookingat the designon otherfurniture.In a discussionI hadwith Peterlaterthat day, he informedme thathe hadshownJackhow to drawthe squarestarthe previousday! This makesJack'sinitialresistanceto modifyingthe shapeof the starmoreunderstandable andthe modificationmoreinteresting. Episode 2 (March 13th) How tofind the centerof the box lid PeterandMr.S lookedat the stardesignandpronouncedit to be "tooheavy"for the size of the lid. I consultedwithJack,andwe decidedto makea squaredesignafterall. Jack obliginglysandedoff all thepencilmarks,drewthe diagonalsin pencil on the lid, and for the starshapewouldbe made.I asked markedthe cross on whichthe measurements Jackwhy this way of findingthe centerworked.He lookedat me in surpriseandsaid, "Becauseit is the center!"I persisted,"Buthow do you know that?" Gesturingto the cupboardhe was making,Jackexplainedthatwhenhe wantsto squaresomething,like a cupboardor the frameof a table,he measuresacrossthe frame fromcomerto comerandnotesthis length.He thenmeasuresacrossthe framethe other way, fromcorer to corer. Onelengthmaybe slightlylongerthanthe otherone. If this is so, thenthe cupboardor tableframeis not square,andthis has to be rectified.Gesturingto my block, he said thathe haddrawnthelines fromcorer to corer a few days ago andhad measuredthemandfoundthatthey wereequal;thusthe top was square.Thereforehe could be surethatthe placewherethe lines cut was thecenterof the lid. In fact, his usualpracticewas to placea long narrowplankacrossthe frame(e.g., of a box or a table),fromthe bottomleft-handcomerto the top right-handcomer, andto make a pencil markon the narrowplankto indicatethe lengthfromcorer to corer. See Figure 7.

99

a box. Figure7. "Squaring" Afterthis, he wouldplacethe sameplankin positionfromtop left-handcorer to bottom rightcomerandindicatethislengthwitha pencilmarkon the plank.If the markscoincided, thenthe box wouldbe consideredto be square.If the pencilmarksdid not coincide,the plankwouldbe cut off at the pointmidwaybetweenthe two marks.The plankwouldthen be wedgedinto the positionof the shorterdiagonalafterglue hadbeen appliedto all the joints. Clampswouldusuallybe usedto helpholdthe framein positionwhile the glue dried. Jack'sexplanationusedthe idea thatunderliesthe familiarphysicalactionof usinga narrowplankto wedge diagonallyacrossthe frameof a box underconstructionto ensure thatthe frameremainssquare.Thiswas an actionthathe performedas a matterof routine on a daily basis while makingtableframes.His actionswereclearlynot performedin a rote, of this idea, thatis, meaninglessfashion,since he was ableto use a differentrepresentation drawingfromcorer to corer witha pencilon the lid andthenmeasuringandcomparing the lengthsof theselines to decidewhetheror not the lid was square. AlthoughJack'sexplanationhingesalso on the fact thatthe sides of the lid should be equivalent,he did not makethis partof his argument.This is perhapsbecauseof the way in which the saws areset andthe way in whichthe boardsareflippedover duringthe cuttingprocess.This ensuresthatif the anglesaresquare,thenthe sides will be equivalent.

100 Jackused a conceptof "center"thatled to the choice of a pointthathadimportant symmetries.Straightlines drawnthroughthe pointdividedthe lid into balancedportions. This encounteris a good exampleof thekindof explanationI becameusedto hearingin responseto my questions.Physicalandtangibleexamplesweregiven, and explanationsusuallyinvolvedactionof some sort.Jackhadshownme how to squarea table on a previousoccasionandthereforeknewthatI was familiarwith the practice. Episode 3 (March 13th) Thestar is crookedand two squarerules are different Mr. S was watchingus while Jackusedhis squareruleas a straightedgeto drawthe set of axes andto markoff lengthsalongtheseaxes,explainingeach step to me in a loud voice. Clive andJimmycameoverto watch.Aftersome struggle,the "square"starshape was drawnin pencil.It did not look quiterightto me, andI mentionedthis to Jack.I noticed thatthe triangleswerenot all congruentas I knewthey shouldbe, butwas puzzledby the practicalimplicationof this anddidnot voice this thought.As we all studiedthe drawingof the star,it was clearthatJackandMr.S werealso puzzled. withthe straightedgeandthensaid, Jackactedfirst.He tooksome measurements "Thestaris not straight,thatis the problem-I can use my squareruleto check."Mr.S had alreadygone to fetchhis own squarerule,havingquietlyfiguredout the problem.Jack checkedthe positionof the starwithhis squarerule,usingthe edges of the lid as frames of reference.Mr.S did the sameusinghis squarerule.Sureenough,the north-south orientationof the starwas crooked.It was dulyerased,butmoreimportantfor this discussion,Mr. S foundthathis squareruledidnot matchJack'sexactly! Jackputthe two tools on top of eachother(a directmethodof checkingfor congruence)andcouldfeel withhis fingertipsthatthe squareruleswerenot at the same angle.He held themup, andit was clearto see thatthey did not matchexactly.Mr.S started to claim thathis tool was morereliablethanJack's,butJackwas quickto pointout thatif his were not true,thenall the furniturehe constructedwouldbe off-square.He held bothof

101 the squaresaloft andsaid,"Thesearebothsquarerulesandthey aredifferent-you can't tell which one is true!Why do you wantto believehis?"I agreedto settle for Jack'ssquare rule, since the new sketchof the star,whichhe hadorientedusinghis squarerule,looked good to me. Episode 4 (March 15th) Jack expressessome opinionsaboutmathematics Abouttwo weeks afterI hadstartedworkin the workshop,JackandI had a conversationthatrevealedsome of his ideasaboutmathematics.He asked,"Now,Wendy, whatareyou reallydoinghere?Whatareyou going to do whenyou leave us?"I said, "I am going to writea thesisabouteverythingthatI amlearninghere.""Andthenwhatareyou going to do?"he persisted."ThenI amgoing to teachmaths,"I replied.A look of astonishmentcrossedhis face. "Butthenyou mustbe veryclever!"he exclaimed."Whydo you say this?"I asked."Becausepeoplewho do mathsareall ..." andhe grimacedand rolledhis eyes upwardsin a rathergraphicparodyof someonewho is brilliant,butlooks idiotic andis sociallyincompetent!I inquired,"Don'tyou do any mathsin this workshop?" The responsewas quick."Ohno! Maybea bit of addingandmultiplying,butthat'snot real maths!"andhe walkedoff. Althoughit was Jackwho actuallymadethesestatements,his attitudewas a good of the generalattitudeof thecarpenters.Mathematicswas seen as being representation outsideof theirsphereof experience,an activityof moreintelligentandeducatedpeople thanthey themselveswere. Episode 5 (April 6th) Whenis a squarerule square? Threeweeks afterthe two squareruleshadbeenshownto be differentfromeach other,I askedMr. S how he woulddecidewhichsquarewas truein a situationsuch as the one thatarosewith Jack.He answeredthathe woulduse a spiritlevel anddemonstrated whathe meantimmediately.Placingthe spiritlevel in a doorwayhe checkedthatthe floor

102 was horizontalandthe wall vertical.Whenhe was surethatthey were,he placedthe square rulein the corer at the foot of the doorwayandtoldme to look for any gaps betweenthe wall andthe squareor the floorandsquare,wherethe lightmightbe comingthrough.There were none;thushe was convincedthathis squarewas true. This methodshowedthathe drewa usefulanalogybetweenthe functionof the spirit level andthe functionof the squarerule.The spiritlevel was not designedto check for squareness,andI did not ever see anyoneelse usingit for thatpurpose.It was usuallyused in the workshopto checkwhethera horizontalsurfacewas level (like the flat bed of a thicknessingmachine)or whetheran uprightstructurewas plumb(like the side of a doorframe).Neitherof theseusualactivitieslinked"level"to "plumb"( i.e., linked horizontalto vertical)to implysquareness.ThissuggeststhatMr.S did not use his square of whatits use meant.He describedthe rule by rotebutthathe hadan activeunderstanding squareruleas consistingof two parts;namingthelong straightedgethe "plumb"andthe shorterstraightedgethe "level"(see Figure8). Thusthe nameshe gave to the two partsof the tool werenot emptylabelsbutwereinfusedwith actionandmeaning.

Plumb

?

'Level

Figure8. Squareruleor carpenters'square. I askedif he couldthinkof a way to solve the problemwithoutusing the spirit level-to imaginethathe was on a desertislandwithtwo squarerulesthatdid not match andthathe hadto decidewhichone was true.Aftersome reflection,he generateda second solutionthatalso derivedfromthe way in whichhe describedthe componentsof the square of the functionsof the two rule.This methodwas a literaltranslationanddemonstration partsof the tool.

103 Pantomiminghis actions,he describedtakingtwo straightsticksof equallength, puttingtheminto the sandsome distanceapartandplacinga thirdstick acrossthe top of the actionof usinga stonehangingfroma stringto check that these two. He demonstrated the two supportingstickswerestandingstraightup (i.e., vertically).By mimingthe action of using the lengthof string,he indicatedthatit was importantto makesurethatthe distancebetweenthe groundandthe topsof thetwo stickswas identical.Then,he claimed, the shapemadeby the top crossbarandone of the stickswas exactlywhata squarerule shouldlook like. See Figure9.

stone hanging | from string

[

]

)

distance from ground

3

to tops of sticks is identical

Figure9. Firstversionof Mr.S's modelto check whetherhis squareruleis true. It was clearthathe was assumingthe groundto be level. In an effortto challengethis conjecture,andto checkwhetherhe wouldproducea moregeneralsolution,I explained thatI did not understandexactlywhathe meantandneededsome furtherexplanation.We decidedto talkagainat lunchtime. Mr. S broughtto my attentionhis belief thathe hadactuallyinventeda way to make a squareruleandnot simplyto checkif any squarerulewere true.It was clearthathe felt a sense of proudownershipof the ideahe haddescribedandthatdesigninga squarehad becomehis problematic,as will be seen froman exchangethathe hadwith Jacklaterin the day. Independentlyof my interactionswithMr.S, I askedJackhow he wouldcheck which one of the two differentsquareruleswas true.Jackclaimedthatit would be

104 impossibleto decidebetweenthe two squarerulesunlessone hadaccess to somethingthat was knownto be trueandagainstwhicheachsquarecould be checked.I said, "But somebodyhadto inventthesquarein thefirstplace.Wheredid it come from?"Jackreplied, "Well,wheredo straightlines come from?"Mr.S enteredthe conversation,"Thereis a straightline wherethe sky meetsthe sea, at the horizon.It is the straightline thatGod gave us. You can see if you look at thetopsof buildingswhenyou arehalfwayup the mountain. If you checkthe tops of buildings,whicharestraight,you can line themup with youreye along the horizon."Jacklookedunconvinced. Mr.S's explanationis an indicationof how closely his mathematicswas integrated andthe worldof his personalexperience.His homewas built into his physicalsurroundings high up on a mountainside,overlookingtheocean,the tall buildingsin the city, andthe CapeTownharbor. I askedJackto thinkaboutthe problemof the squarerule.He queried,"Do you know the answer?"andI replied,"No,I don'tknowhow to solve it, butMr.S thoughtof a way. Maybetherearelots of ways.But don'taskMr.S for his idea! I wantto know your own idea!" Now andthenwhenJackpassedby my workbench,he said, "Doesit have anything "Doesit haveanythingto do with a 1-2-3 triangle?"He clearly to do withcircumference?" andwas searchingfor the "right"answerto give identifiedmy questionas "mathematics" me, couchedin "realmathematical" jargon.Abouthalf an hourlater,he called, "Hey Wendy!You could do it like this,"andhe drewFigure10 in chalkon the workshopfloor. I did not thinkthatthis wouldwork.In Figure11, I providea sketchthatdemonstratesthe flaw in Jack'sargument. I askedJackto explainwhy his ideaworked.He beganto explain,thenchangedhis mindandsketchedanothersolutionin pencilon a piece of wood. See Figure12.

105

2. Flips square rule and traces bottom line

1. Traces line around square rule

-I

-

I.

E

-1

-NEM

3. Moves square rule to right and traces again

-

I

-

I

mm

4. Jack claims that the distances between these two lines must be the same

Figure10. Jack'sfirstattemptto checkwhether his squarerulewas true.

106

greater than 90 degrees

1. Traces around "square"rule

2. Flips square rule and traces bottom line

3. Moves square to the right and traces again. Jack's claim still holds: these two lines are the same distance apart.

the flaw in Jack'sargument. Figure 11. This demonstrates

107

1. Traces line around square rule

2. Flips square rule and traces bottom line

3. Picks up square and replaces along line Jack claims that this line must be straight

Figure12. Jack'ssecond(andcorrect)attemptto check whetherhis squarerulewas true.

108 AlthoughJack'sfirstsolutionwas flawed,since a squarelike the one shownin Figure11 will pass his test (exaggeratedto makethe effect clearer),he was nevertheless followingup on his firstidea thata straightline was somehowinvolved.He triedto use the reflectionsymmetryof a straightline, makinga connectionacrosshis ideas. In this case, he realizedthathe was unableto defendhis solutionandgive a convincingreasonas to why it shouldwork,so he carriedon lookingfor an alternativeway to solve the problem. Aftersome reflectionandexperimentalactionwith the squarerulethathe hadin his hand,he arrivedat his secondsolution,whichusedthe idea of the reflectionsymmetryof a straightline successfully.He was confidentthathe hada defensiblesolutionandwas able to give me a convincingargument.He drewa diagramto show whatwouldhappenif the andtoo wide (i.e., if the angleweregreaterthan90 degrees).See squarewere "offsquare" Figure13.

Figure13. Inverted"V"insteadof a straightline whenthe angle is greaterthan90 degrees. He drewanotherdiagramto show whatwouldhappenif the squarewere "offsquare"and too narrow(i.e., if the anglewereless than90 degrees).See Figure14.

Figure14. "V"shapeinsteadof a straightline whenthe angle is less than90 degrees. WhenMr. S lookedat Jack'ssolution,he remarked,"Butyou need a squareruleto do that!"I remindedhim thatthe problemhadbegunwith two squarerules.Payingno

109 regardto my reminder,he challengedJackwiththe remark,"Yes,buthow can you makea squareif you don'thave one?"Jack,however,hadsolved his own problematicanddid not wish to engagein the problematicthatMrS haddeveloped. At lunchtimeMr.S andI sat at the roadsidewhile he workedwith little bits of straw and wood, tryingto physicallyconstructhis ideain orderto explainit to me. Finally,I took two sticksof the samelength,putone up on the sidewalk,andone downon the road,to grosslyexaggeratethe hill I wantedhimto takeinto consideration.He responded immediately,saying,"Iwill use the horizonto makesurethatthe top plankis level. There is a straightline wherethe sky meetsthe sea. I will taketheseplanksup to the top of a hill, anduse the horizonas my naturallevel." The whistleblew to signify theend of lunchbreak,andwe went backinside.Mr. S beganto constructhis idea,nailingpiecesof wood togetherto formtwo unevenlegs witha plankacrosstop. He "builtup"underone of the legs, usingscraps,until"hiseye told him" windowin the farwall servedas a mock thatthe top plankwas level. The long rectangular horizon.As a furtherway to convinceme, he useda spiritlevel to verifythatthe top plank was level. Using a piece of stringanda castor( a smallheavywheel shapedlike a ball), he fashioneda plumbbob andusedit to ensurethatthe legs of the structurewere vertical. Thenhe puthis squarerulein positionandshowedme thatit was true.See Figure15.

plumb bob "build up" under here until level Figure 15. Secondversionof Mr.S's modelto check whether his squarerulewas true. His initialassumptionthatthe groundwouldbe level did not standup to closer scrutiny,andon reflection,the assumptionwas modified.At all times,his explanationwas accompaniedby actionandhe carefullyplacedhis solutionin context,describingin detail

110 how to makethe plumbbobandhow to use the horizonto checkfor levelness. Everyday objectslying abouton the workshopfloorwereusedto illustratehis ideas.ThusMr. S's conceptof squarenessandhis actsof cognitionas he exploredthis conceptseemedto be deeplyembeddedin the environmentof the workshopandstronglylinkedto his actions. As a probeto investigatewhetherMr.S wouldbe able to constructanothersolution, I asked,"Do you thinkthereis a way to do this withoutusingthe horizon?"He repliedthat he coulduse the 3-4-5 triangleto checkwhetherhis squarerulewas true.8 Pointingto the squarerule,he indicatedthe side alongwhichhe wouldmeasurethreeinchesandthe side alongwhichhe wouldmeasurefourinches.He drewa linejoiningthe two sides in the air with his finger(i.e., indicatingthe hypotenuse),sayingthatthe lengthof this line shouldbe the idea of the five inchesif the squareruleweretrue.It appearedthathe hadincorporated Pythagoreantripleas anotherfacetto his conceptof squareness.Thus,in a finaleffortto help me understandhow to checkthata squareruleis indeedsquare,he gave me an explanationthathadits rootsin somethingthatI hadtold him in the past.

Episode 6 (April 7th) Jack and the "right"answer I foundout fromPeterthatJackhadaskedhimhow to show thata squareis square. Peterhadgiven him some assistance.WhenI askedJackaboutthis, he was somewhat 8This questionrevealedwhatI regardas a methodologicalblunderon my part.Although

it was not my intentionto "teach"Mr.S anything,this hadoccurredinadvertantlyat our firstmeeting,whichhadtakenplacesix weekspriorto Episode5. As describedin Chapter4, on the occasionof thatmeetinghe showedme thejewelryboxes he hadmade a smallbox beforegluingit together. for his daughtersandexplainedhow he "squared" In manyof the pilotinterviewsI hadconductedwith carpenterswho workedon building sites, Pythagoreantripleshadbeenusedto squarethe cornersof buildings.I askedMr.S if he hadever usedthe idea of a 3-4-5 triangleto squareanythingandhe appearednever to have heardof the concept.Not realizingthathe was to becomemy teacherfor the next how he couldsquarethejewelrybox usingthe idea of a 3-4five months,I demonstrated 5 triangle.We hadnot talkedaboutthe topicsince thattime.

1ll abashedandreplied,"Well,he didn'tactuallytell me, we workedit out together,andtwo headsarebetterthanone. That'show they do it at school!"SinceJack'ssolutionclearly evolved while I was watchinghim,it wouldnot be fairto say thatit was not his own idea. However,the fact thathe wouldaskPeteris significant,becauseit highlightshis effortsto find the "right"answerto give me. Later,whenI askedwhetherhe hadgivenMr.S's problemof how to designa squareruleany morethought,he retorted,"Youknowthe answer[noticethe singularnoun] andthatis why you keep askingme!"WhenI protestedthattheremustbe manydifferent ways to solve the problemandthatI was interestedin his ideas,he replied,"Well,you know somethingaboutit, andthatis why you keepcomingbackto it!" His commentwas, of course,legitimate,andhighlightsthe difficultpathI hadto treadas a mathematicseducator/novice carpenter.I triedto avoidsituationslike this becausetheyencouragedJackto concealfromme the fact thathe was unsureof whatto do andfurtherconvincedhim thatthereexistedonly one correctanswerto my question. Episode 7 (April 18th) "Do it byproportion"-The table withthe inlaidpattern Mr. S assignedme to the taskof restoringa veryold yellowwoodtable.Severalof the originalpartsweremissing,includingthe tabletop andthe frontapronwith a central drawer.The tablehadbeautifulinlaiddesignson the backandside aprons,madeof dark stinkwood.See Figure16.

Figure 16. The inlaypatternon the backapronof the table.

112 The firststepwas to constructa new frontapronfroma piece of old yellowwood. We beganby cuttinga plankthe samelengthas the backapron.The backapronwas laid on a long plankof the samewidth,anda pencilline tracedalongits edge. A saw cut was made along this line. Mr.S tookcareto showme how to ensurethatthe backandfrontaprons were exactlythe samelength,placingone on top of the otherandfeeling the ends with his fingertips.This methodof checkingfor congruenceis similarto thatused by Jackto comparethe squarerules. The size of the drawerthathadto be replacedwas deducedby inspectingthe partof the runnersthatremainedon the backapron,andthecutoutfor the drawerwas markedon the new frontapron.The inlaypatternon thebackandside apronsneededto be scaled andI beganto down to fit the drawerfront.Mr.S instructedme to "doit by proportion," measureall therelevantdimensionsof the side apronandto use a calculatorto scale these to fit the drawerfront.Briancame overto see whatwas going on, all downproportionally andgot involvedin the task.He suggestedtakingmoremeasurements.Soon we hada long list of measurements,andourmethodbeganto feel awkwardandroundabout.We were not makingmuchprogressin drawingthesmallerdesign. Mr.S returnedto see how we weregettingalong.Ignoringourlist of figures,he pickedup the backapronof the tableandstudiedit. Using comparisonsmadeinitiallywith his eye andthenwithhis fingers,he estimatedthatthreeflowerheadscould fit alongthe base triangleof the design.By makinganothervisualcomparison,he madea conjecture aboutthe relationshipof the lengthof the backapronto the lengthof the draweron which the scaled-downversionof the designhadto be put.He physicallycheckedthe lengthof the drawercut fromthe new frontapronby puttingit ontothe old backapronandconfirmed werekept to an absolute thatthe drawerwas half the lengthof the apron.Measurements minimum;in fact, only thebase andheightof the trianglewere measured. Mr.S pointedout thatthe drawerwas exactlyhalfthe size of the frontapron,and the frontapronwas exactlythe samesize as the backapron.Thereforewe could simply

113 halve the size of the designon the backapron!Brianbeganto measureevery componentof the designon the backapron,includingthe lengthof individualpetals,the distancefromthe tip of the upperpetalto the tip of thelowerpetal. In the midstof his measuring,Mr. S stoppedhim. "Theflowershapeis madeof threeidenticalpetals.Youjust need to cut the templateof one petalandturnit around!Andlook at the grainof the wood on each petalit runsthe lengthof the petal.You needto cut eachone out separatelyandputthreetogether to makeup the flowerdesign."He continued,"Youdon'thaveto measurethe floweronce you havemeasuredthe trianglebase.It is just necessaryto makesurethatthe sizes agreethreeof the flowerheadsfit alongthe lengthof the trianglebase in the biggerdesign,so you wantthis same thingin the smalldesign."Brianagreedwith this, but arguedthatit did not help him to drawthe shapeof the petal.Then,pickingup a pencilandignoringall his measurements,he proceededto drawa perfectflowerdesignfreehandby repeatingand rotatingthe single petal!Mr.S checkedtheproportionsusingthe guidelineshe had suggested. Mr S's solutionillustrateshis understanding of proportional size andhis appreciationof the geometryof the designandof the importanceof the internal andrelies on the moredirect relationships.His solutionavoidsthe use of measurements methodof comparison.His solutionis simple,direct,andconcise andrelies on the of the piece of furniture.Ratherthanthe fragmentationof geometricinterrelationships design andtasksthatBrianandI weretryingto perform,Mr.S's practicalsolutiontakes cognizanceof the broaderpicture.

Episode 8 (May 15th) Clive,Mr. S, and the dovetailjoints Buildinga replacementdrawerfor the inlaytablerequiredthe constructionof dovetailjoints. See Figure17.

114

Figure 17. Dovetail joints.

Clive came overto helpme sketchthejointsin pencilon the drawercomponents. He broughta slidingbevel, a smallmetaltool withtwo movingarmsjoined by a screwthat can be tightenedto holdthe positionof the two arms.See Figure18.

Figure18. The slidingbevelis usedfor markinganglesotherthan90 and45 degrees. Whileexplainingthatthe "angle"of a dovetailjoint is "threeup andone along" (usingan arbitraryunit),he set the angleof the slidingbevel andinstructedme to choose the lengthfor the baseof thejoint andthenuse the slidingbevel to drawthe sides of the method,I succeededin locatingthejoints evenly along the joint. Using a trial-and-error backedge of the drawer.

115 Mr.S hadwatchedme usingClive's slidingbevel with some disapprovalandcame over to show me his own methodof drawingandplacingdovetailsevenly along an edge. He beganby drawinga lightpencilline parallelto the edge of the plankalong which the dovetailswouldbe placed.The line indicatedthe depthof the dovetailneededto accommodatethe thicknessof the plankfromwhichthe otherhalf of thejoint would be cut. See Figure19. edge of the plank

pencil line

Figure19. Mr.S's methodof preparingto place dovetailjoints. Mr. S's methodof placingthejointsevenly withinthese parallellines was as follows: Step 1. Findthe middleof the plankanddrawa line usinga squarerule. See Figure20.

Figure20. Step 1: Drawa line at the midpointof the plank. Step 2. Choosehow far"in"fromthe edge the firstjoint shouldstartandmarkthe point. See Figure21.

Figure21. Step 2: Marka pointat a chosendistancefromthe left edge.

116 Step 3. Place a pointto theleft of the line drawnin Step 1, the samedistancefromthatline as the distancechosenin Step2. Similarly,placea pointto therightof the line. See Figure 22.

Figure22. Step 3: Markpointsto theleft andrightof the centerline, the same distance fromthe centeras the firstpointis fromthe left edge. Step4. Divide the left half of the plankin half again,drawingthe centerline with a square rule,thenrepeatStep3. Do the samewiththerighthalfof the plank.Continuethese steps untilyou canjoin up the pointsto give a seriesof even dovetailjoints. See Figure23.

Figure23. Step4: Dividethe left halfandtherighthalf of the plankin half againandrepeat step 3. In this way, Mr.S useda repetitivepatternbasedon symmetryto ensurethe even placementof regularlyshapeddovetailjoints.His methodwouldresultin the dovetails havingdifferent"angles"dependingon the widthof the plankandthe thicknessof the

117 wood. He did not, however,indicateany interestin the angleof the dovetailand emphasizedinsteadthe even placementof thejoints andthe choice of a sensibledistance fromthe edge of the plankto ensurea "strongjoint." WhenI startedworkingon thematchingjointcutoutson the secondplank(which was to be joined to the firstone), Mr.S handedme a "markinggauge,"saying,"Itdoes the same as the tapemeasureandthesquare-it is like a shortcut."See Figure24.

Figure24. Markinggauge. He did not namethe tool. The smallblockcanmove alongthe shaft,towardor awayfrom the nail at the end of the shaft.Theblockcan be tightenedinto any positionalongthe shaft using the butterflynut.The tool is usedlike a caliper,as a comparisontool. The distance betweenthe blockandthe nail is set by movingtheblockandtighteningthe butterflynut. This distancemaybe takenfromthe thicknessof a plank,for instance.Whenthe nuthas been tightened,the tool can be pulledalongthe edge of a plank,with the block pushed firmlyagainstthe edge, andthe nailwill scratcha line parallelto the edge of the plank,at the chosendistancefromthe edge. Fromhis comment,I assumedthathe meantthatusingthe markinggauge was more directthantakingthe two actionsof (a) usinga tapemeasureto measurethe required line acrossa plank.The tool length,and (b) usinga squareruleto "pull"a perpendicular seemedto representto Mr.S a way of allowingthosetwo actionsto be condensedinto one directaction.

118 Episode 9 (May 19th) Mr. S's ruler I noticedthatalthoughMr.S usuallyusedhis woodenruler(whichis markedin Imperialunits),he also owneda metaltape(whichis markedin metricunits)thathe occasionallyused.The woodenrulerwas brokenin half;one side was marked1-18 inches andthe otherside was marked19-36 inches.I askedhimif he hadany preferencefor one tool over the other.He repliedthathe foundeachone usefulfor differentjobs. Since most of the antiquefurniturewas builtaccordingto Imperialunits,it mademoresense to use the old woodenrulerfor restorationwork.He saidthathe mightchoose to use the tapemarked of the new furniturethatwas constructedin the in metricunitsto takemeasurements workshopor to measurethe new wood.It was clearfrommy observationsthathe favored the Imperialwoodenruler,whichwas usuallykeptin the pocketof his coveralls.He used Imperialunitsfor boththejewelrybox andthe smallchestthathe helpedme to make.He fromthe pointmarked"36 inches"on his hadan intriguinghabitof measuring"backwards" woodenruler.Despitemanypreviousquestionsaboutthis practice,it was not untilthis occasionthathe decidedto beginto explainhis reasonto me. He showedme thatthe ruler is dividedinto sixteenthsof an inch on the side thatdisplaysfrom 19 inchesto 36 inches andis dividedintoquartersof an inchon the side thatdisplaysfrom 1 inch to 18 inches. Mr.S held up his brokenwoodeninchrule,remarking,"25 millies is aboutan inch in the old Imperialmeasure."He toldme thatplanksused to be sold in thicknessesof 7/8 of an inch andthatit hadbeenverydifficultfor him to get usedto the metricmeasurements when the metricsystemwas firstintroducedin SouthAfrica(almost30 yearspreviously). The personfromwhomhe purchasedwoodhadhelpedhim by tellinghim that7/8 of an inch is about22 mm.Mr.S hadalso beentold thatone whole inch was equivalentto 25 mm, so he reasoned,"Thosemissing3 milliesareequivalentto the 1/8 of an inch neededto makeup a full inch. So if I look at my ruleron the side thatshows 16thsof an inch (halfof

119 1/8 ),

I can see approximately11/2 millies.Onehalf of 1/16 is almostone millie."(In fact,

1/32 = 0.8 mm.)

He clearly wanted a visual way in which to think about the new system of metric

unitsandalso wantedto continueto use his old ruler,so he designeda "conversionsystem" suitedto his needs.His systemenabledhim to "see"the new systemof measuring,not to performnumericalcalculationsandconversions.Thereareobviousinaccuraciesin his conversionsystemdue to roundingerrors.Mr.S was seldomcalleduponto use the metric systemduringthe courseof his work,andso the shortcomingsof his systemhadnot yet becomeevidentto him. He oftenimpresseduponme the fact thathe did not need a tape measureor ruler.Instead,he preferredto use his handor fingeror a piece of wood to in Episode7. comparesizes, as demonstrated Episode 10 (July llth) The rattanchair I was given the taskof gluingtogethersome loose portionsof a couple of old chairs with rattanseats. Realizingthatthe shapeof the seatprovidedme with a good opportunity to probefor Mr.S's ideason how to get thecenterof a shapethatwas not a rectangle,I askedhim how he wouldfind thecenterof the seat.He did not answermy questiondirectly. He pointedout thatthe frontedge of the frameis longerthanthe backedge andthen proceededto addressthe issue of the correctplacingof the holes in the seat framefor threadingthe rattan.See Figure25.

Figure25. The chairseatframein whichholes areplacedfor threadingrattan.

120 He explainedthatthe holes wouldbe placedalongthe frontedge of the framefirst. This wouldbe doneby makingthe firsthole at the midpointof the frontedge andthenplacing the rest of the holes at equaldistancesapartalongthe frontedge, usinga compass.The holes wouldthenbe placedalongthe backedge of the seatframe.Again,the firsthole would be placedat the midpointof the backedge, andthenthe samenumberof holes as on the frontedge wouldbe placedat equaldistancesapartalongthe backedge. Mr.S emphasized thatthe numberof holes on the frontandbackedges hadto be the same, since the ratHe pointedoutthatthe holes alongthe backedge tan patternhadto remain"square." would,of course,be closertogetherthanthosealongthe frontedge. In analyzingMr.S's actionsof findingthemidpointsof the oppositesides, I recognized thatperhapshe hadtriedto answermy questionaboutthe centerof the seat. The line joiningthe midpointsof the frontandbackedges splitthe chairseatinto two equivalent halves,so his explanationwas consistentwiththekindof symmetryhe hadrequiredfor the centerof a rectangle. The placementof holes for the threadingof "riempies"(tannedleatherthongs),on a chairwith a seat similarto thatshownin Figure25, requiredholes to be madeon all four edges of the seat. Mr.S describedplacingtheseholes in the samemanneras for the rattan I did not ask him aboutthe centerof seat, findingthe midpointof eachside. Unfortunately, the "riempie"chairseat,so whetheror not Mr.S's conceptof centeris linkedto the equivalenceof oppositesides remainsunclear.

Episode 11 (July llth)

The Tulbaghchair A coupleof Tulbaghchairsstoodnext to thechairwith the rattanseat. Partof the top of the seat backof one of theTulbaghchairs,withits distinctivecarvedshape,hadto be replaced.See Figure26. Theright-handside of the carvedbackhadbeen damaged.

121

Figure26. Thebackof a Tulbaghchair. Mr. S gave me a piece of cardboardanda pencilandinstructedme to tracearound the intactleft-handside of the carving.I askedhow this wouldhelp us to replacethe part thatwas missingfromthe otherside, andMr.S responded,"Youcut aroundthe lines you have traced,thenturnthe paperoveranddrawthe shapeon the block of wood you wantto use. The shapeis the sameon each side of themiddleof the chair.The left side corresponds to the rightside." This strategyis clearlybasedon an understanding of the conceptof symmetry,and of reflectionaboutan axis of symmetry. Episode 12 (July 13th) Jack avoidsa question All the starsusedfor decorationsin the workshophadan even numberof points, usuallyfour.Mr.S told me thathe hadseen manystardesignswitheight andeven sixteen points.I decidedto tryto find out how Mr.S andJackwouldproducea stardesignwith an odd numberof points. FirstI askedJackto show me how to drawa starwithmorethanfourpoints,andhe sketcheda designfor an eight-pointedstar(see AppendixE), with Mr.S looking on. ThenI askedhim how he woulddrawa starwith five points,or any odd numberof points.He looked at me andsaid,"Haveyou everseen sucha thingbefore?"I nodded."Well,the personwho did thatmustbe stupidor something-I thinkthatwouldlook very funny!" Withthat,he walkedaway.His reactionto situationslike this hadby now becomefamiliar, andI was not surprisedby his reluctanceto becomeengagedin the questionI posed.

122 Episode 13 (July 17th) Mr. S drawsmultipointedstars A few dayslater,I askedMrS how to drawa starwith manypoints.He showedme how he drewan eight-pointedstar(see AppendixF) andcriticizedJack'sway for requiring "allthatmeasuring."Mr.S's methodhadbuilt-inchecksso thatthe starwould"comeout right"(i.e., wouldbe symmetrical),whichJack'smethoddid not include. I also askedMr.S to show me how to drawa starwith five points.He hadnot done this beforebutsaid thathe wouldworkit out.He startedwith a roughdiagram.See Figure 27.

Figure27. Mr.S's firstattemptat drawinga five-pointedstar. I asked,"Howcan I makethis neaterso thatI can use it as a templatefor an inlay?" He respondedby drawinga five-sidedfigurethatlookedlike a naive pictureof a house. See Figure28.

B\ Figure28. Lookingfor a pentagon.

123 He triedvariousways of extendingtheselines to give a starshape.Whenthis did not give the desiredresult,he stoppedandthoughtfor a bit. Thenhe drewthe diagramreproducedin Figure29.

29. MrS drawsa pentagon. Figure Figure 29. Mr S draws a pentagon.

He said, "Itwill havethis shapein the middleandit musthave five sides, but each mustbe the same.Eachside will havea point.It looks like a starfish!"He sketchedthe diagram reproducedin Figure30.

Figure30. Mr.S's secondattemptat a five-pointedstar. I askedhow he wouldensurethatthe sides of the five-sidedfigurewouldbe the same length,andhe replied,"Ican use a compass,"andsketcheda circlefreehandroundthe pentagonhe haddrawn.See Figure31.

Figure31. Mr S suggestsusinga compassto ensurethateach of the five sides will be the samelength.

124

Using his eye, he drewin some pie slices, saying,"Ican dividethis circleup into five equal parts."See Figure32.

Figure32. Dividingthecircleintofive equalparts. I askedhow he woulddrawa starwith ninepoints,andhe beganby tryingto sketch a nine-sidedfigure.His drawingkeptendingup witheight sides andhe commented,"This is an octagon. It is hardto drawone withninesides."He finallydrewthe nine-sidedfigure, thenenclosedit in a circle,andputon the ninepointsto forma star. He went backto his diagramof the eight-pointedstar,whichhe haddrawnat the startof ourconversationanddrewoverthe lines again."Youcan also use a circle for stars with fourandeightpoints,buta crossis all right-it is easier.But you cannotuse a cross for starswith five andseven andninepoints.You haveto use a circle. So actually,for odd numbersyou mustuse a circleto get it, andfor even numbersyou can use a cross."(Note thathis commentis not altogethercorrect-crosses workfor drawingstarswhen the numberof pointsrequiredis a multipleof four.Forexample,a cross wouldnot be helpful for drawinga six-pointedstar.) I hadexpectedMr.S to beginthe nine-pointedstarby drawinga circle andmarking However,he struggledfirstto drawa nineoff nine equalarcsalongthe circumference. sided figurefreehand,andthenhe enclosedit in a circle.It is not clearwhatfunctionthe circle playedin his method,norhow he wouldhaveusedthe compassto dividethe

125 circumferenceof the circleintoequalparts.WhenI askedClive to drawa starwith an odd numberof points,he suggestedbeginningwitha circleandusing a compassto dividethe circumference.He indicatedthathe would"usehis eye" to decideon the appropriate compassspan,andthentryit out,changingthe spanif it did not workout the firsttime. I methodin his use of the suspectthatMr.S wouldalso havesuggesteda trial-and-error of the solutionuntilhe reached compass.Mr. S graduallyimproveduponhis representation a pointwherehe articulateda generalprinciplefor drawingstarswith an odd numberof pointsandstarswith an even numberof points. Episode 14 (July 19th) Replacinga brass escutcheon Mr. S askedme to replacea brassescutcheonthatwas missingfromthe right-hand doorof a sideboard(buffet).See Figure33.

II

B

i

12I~~~~~~~1

Figure33. The brassescutcheonon the buffetdoor. I askedhim to explainhow I shouldbegin.Handingme a piece of paperanda blunt pencil,he told me to placethe paperoverthe escutcheonon the left-handdoorandto rub

126 the pencillead over the paperuntiltheshapewas visible.He theninstructedme to cut out the shapeandglue it onto the piece of brassfromwhichthe replacementescutcheonwould be cut. Using his hand,withthumbandlittlefingerextendedin imitationof the shapeto be how the new escutcheonwouldbe rotatedthrough180 degrees traced,Mr.S demonstrated to matchthe old one on the left-handside of the cupboard. In contrastto the episodeabouttheTulbaghchair,he did not tell me to turnthe templateover beforetracingaroundit. Thereasonfor this becameevidentlater,afterthe new escutcheonhadbeencut out.Theglue withwhichI hadstuckthe papertemplateto the metalprovedto be ratherresistantto removal.Mr.S toldme not to botheraboutremoving the stickypaper,since it wouldnot show once the escutcheonhadbeenglued in place.By not flippingthe template,the paperhadof coursebeenstuckto the backof the new brass escutcheon. Episode 15 (July 24th)

Tablelegsfrom a plank Brianwas sittingon a big pile of mahoganyplanksthathadrecentlybeen unpacked andstackedandappearedto be thinkingdeeply.I wentover to ask whathe was doing,and he describedhis problematicandthe solutionhe hadworkedout.Brianhadused a meter tapeto measurethe dimensionsof a plank.On the plankhe hadwrittendownthe length, width,andthicknessandsome calculations.He wantedto laminatethe wood to formblocks 110 mm square.Theseblockswouldbe usedto turntablelegs. The problematicas he describedit was to figureout how manylegs he couldget fromthe plankthathad dimensions2.4 m x 0.6 m x 0.075 m, andto come up with a planof how to cut themfrom the plank.Withobviousprideandsatisfactionandby gesturingto the plankwhile pointing to the calculationshe hadwrittendown,he explainedto me how he hadsolved his problem. Althoughhe did not drawa diagram,his explanationmadeclearto me whathe had visualized.

127 As can be seen frommy diagramshowinghis method(see Figure34), he had envisageddividingthe plankinto 15 pieces,each0.8 m long by 120 mm wide by 0.075 m thick.

,4

2.4 m

0.075 m 14-0.8

m-1

120 mm

Figure34. My diagramof whatBrianvisualized. He explainedhis strategyto me by referringto the calculationsreproducedbelow. Since 0.075 m was too thinfor the blockBrianwanted,the firststep in the strategywas to join threeof the fifteenpieces (whichhe hadvisualized)together.This madefive new pieces, each 0.8 m long by 120 mm wide by 225 mmthick,whichwas abouttwice the thicknessof the desired110 mm squareblock.He thenimaginedcuttingthese five pieces in half, whichwouldresultin ten new pieces,each 0.8 m long by 120 mm wide by 112.5 mm thick.Fromthese he wouldbe ableto turnten tablelegs accordingto his original specifications,each 0.8 m long and 110 cm by 110 cm square. Brian'scalculations(performedwithouta calculator)wereas follows: Step 1.

0.6 m/5 = 120 mm.

Step 2.

I can cut 15 pieces fromthis plank.Eachof the pieces will be 0.8 m x 120 mm x.075 m.

Step3.

0.075 m x 3 = 0.0225 m. Thisis the sameas 225 mm.

Step4.

If I glue 3 pieces together,I will have5 blocks,each 0.8 m x 120 mmx 225 mm.

128 Step5.

If I cut each of these5 blocksin half,I will get 10 blocks, each 0.8 m x 120 mm x 112.5mm (since 225 mm/2= 112.5mm). I wantblocksthatare 110 cm square.Theseblockswill be slightlybigger,whichis okay because I can cut theextraawaywhenI turnthe leg on the lathe. Briandevelopeda clearstrategyfor how to cut the plankandglue the partsto form

blocksthathe couldturnon the lathe.He also figuredout how manylegs could be cut from a plankof certaindimensions,so he hada way to estimatehow manysuch plankshe would need.His solutionwas basedon visualizingthe plankin space,imaginingcuttingthe plank intoconvenientunits(i.e., unitsthatdividedtheplankevenly),andmentallyreconstructing new shapesuntilhe reachedthe desiredoutcome.In orderto performthese mentalspatial manipulations,he hadto use his knowledgeof the metricsystemandof arithmetic. Introductionto Episodes 16, 17, and 18 Peterhadtold me thatin his opinion,the carpentersdid not understandthe concept of volume.He supportedhis observationby statingthatthey wereunableto calculatethe cost of wood, whichis pricedin cubicmeters.I wantedto explorethis topic andfelt thata good opportunityhadpresenteditself withthe conversationI was havingwith Brian.It seemedclearto me thatthe solutionBrianhadmappedout couldnot have been conceived of the conceptof volume.In the next threeepisodes,involving of withoutan understanding Brian,Clive, andJackrespectively,I explorethe ideathatthe carpentershaddevelopedan of the conceptof volumethatwas notconnectedwith numbers. understanding Theseepisodesdiffermethodologicallyfromthe precedingepisodes,since the problemdid not arisespontaneouslyfromsomethingwithwhichthe carpenterswere involved.I posedthe problemto the carpentersandaskedthemto solve it, specifically becauseof Peter'scommentsabouttheirinabilityto solve suchproblems.It was thusclear beforehandthatthey wouldhavedifficulties.I decidedto takethis methodologicalriskfor severalreasons.First,I felt surethatthis problemwouldbe seen as a technicalmatterand somethingthata novice like me mightbe curiousabout.Second,I hadformedgood

129 relationshipswith the carpentersby thattimeandknewthatthey felt comfortablewith me, andlast, since it wasjust two daysbeforeI was due to finishmy fieldworkandleave the workshop,if the encounterswereawkwardor failedin any otherway, they could not result in muchdamageto my datacollection. In the nextepisodewithBrian(frommy own pointof view), the topic seemedto flow somewhatnaturallyfromwhatwe hadbeendiscussing.AlthoughI saw the questionas a logical progressionfromourconversationaboutthe tablelegs, I am surethathe did not perceiveit as such.He hadnot expressedanyinterestin knowinganythingaboutthe cost of the legs beforeI askedandI was awareof thefact thathe wouldprobablyhavedifficultyin answeringmy question.This mademy role as naivequestionerdifficultto maintain,andthe encounterwas morelike a teachinginterviewfollowingthe formof a Socraticdialogue. ForClive andJack,the problemwas posedas partof a formalinterviewthatI taperecorded,so it was even furtherremovedfrompractice.Theseinterviewswereconducted afterI hadwrittenup the encounterwithBrian,so my questionsweremorecircumspectin an effortto avoidreproducingthe situationof a teachinginterview.I explainedto themthat BrianandI hadtriedto solve the problemin the workshopandthatI was interestedin hearinghow they wouldapproachit.

Episode 16 (July 24th) How much does thisplank of mahoganycost? AfterBrianhadfinisheddescribinghis strategyfor cuttingup the plankto make tablelegs, I askedhim if he knewhow to estimatethecost of wood. He repliedthathe did not. I suggestedthatwe tryto figureout thecost of the plankfromwhichthe tablelegs would be cut. Brianlookedat the measurements of the length,width,andthicknessof the plank thathe hadwrittenon its roughsurface.He addedthemtogether:2.4 m + 0.6 m + 0.075 m andcame up with a totalof 3.075 cubicmeters.

130 He exclaimed,"Mahoganycosts R3000percubicmeter(over$1,000 percubic meter).This plankcannotcost morethanR9000-there mustbe a mistake!"He pondered the measurementsfurtherandseemedto drawa blank;thenhe muttered"Nowwhatare He looked squaremetersandwhatarecubicmeters?Theremustbe threemeasurements." aroundthe roomandsuggestedthatwe call Peterto help. I replied,"Peterlooks very busy.Let's trya bit harderourselves.Maybeit will help if we thinkaboutthe areaof somethingfirst.Say we havea room3 m x 4 m thatwe wantto carpet.How manysquaremetersof carpetingdo we need?"I sketcheda rectangleand markedone edge 4 m andthe other3 m. See Figure35.

4m

3m

room. Figure35. Calculatingthe areaof a rectangular He said, "Itmustbe seven squaremeters!" Wendy:"Why?" Brian:"Ohno, thereis another3 andanother4 (lookingat the side oppositeto the ones I hadlabeled),so it mustbe 7 + 7 =14 squaremeters." Wendy:"Canyou show me how this works?I don'tunderstand." He staredat the diagram,thenslowly markedoff threeequalunitsalongone edge, andfourequalunitsalongthe other,saying"Theseareeachone meter,so I have fouralong here...."He drewa 3 x 4 gridandcounted12 blocks."Itis 12 squaremeters!"This visual madesense to him andhe was clearlysatisfiedwith whathe haddone. See representation Figure36.

131

Figure36. Brian'sdiagramrepresentingarea. Wendy:"Andwhatif the carpethadheightor depth?Whatcan we workout from that?" He turnedthe rectangleinto a three-dimensional shapewith a heightof threeblocks andtriedto countthe blocks,buthe remarkedthathe couldnot see all the blocks (see Figure37). Aftersome thoughtandcounting,he arrivedat a totalof 12 x 12 cubic meters.

\f\\\t\ Figure37. Brian'sdiagramrepresentingvolume. Wendy:"Howdid you get that?" Brian:"Well,thereare 12 here(pointingto the top face), andthereare 12 here (pointingto the side face). So it makes12 x 12. But it soundslike too much.I can't do it. Let's ask Peterto help us." Wendy:"No,he still seems too busy.I am surewe can do it. Let's figureit out. Whataboutthinkingof theseas kids' buildingblocks,andthenimaginelifting off the top layerof blocksandputtingthemaside.How manydo you see whenyou look down on what is left?" Brian:"Another12. So thatis 24 cubicmeters."

132 Wendy:"Andif you hadanotherstackunderneath?" Brian:"Thenanother12, so 36 cubicmeters!Oh,so you multiplythese together. 3 x 4 x 2 = 24 and3 x 4 x 3 = 36. Yes, thatis right.I remembersomethinglike this." Wendy:"Socan we thinkaboutthis plankagain? How can we use whatyou have just done?" Brian:"Wecan say the 2.4 m is like the 4, andthe 0.6 m is like the 3, andthe 0.075 m is like the 2." (Brianmultiplied2.4 x 0.6, andgot 14.4.) I said, "Isn'tthattoo much?"He did not thinkso. I persisted,"Isn't0.6 aboutlike 0.5, whichis a half, andwhatwouldhalf of 2.4 be?" Brian:"Itwouldbe 1.2....But herewe aremultiplying,so thingsmustget bigger!" Wendy:"Butwhatif I said, 'Giveme halfof the legs you have turnedby teatime.' And you hadturned20. How manywouldyou give me?" Brian:"I'dgive you 10 legs." Wendy:"Andif I said, 'Giveme 0.5 of whatyou have?"' Brian:"Thenagain 10." Wendy:"Andif I askedfor 0.6 of whatyou hadturned?" Brian:"Thenit wouldbe 11 or 12...,butit is gettingsmallerbecauseI am giving themaway,takingthemaway!Oh....Okay, I won't give themaway,I'll polishthem.If I polish0.5 of themit will be 10, andif I polish0.6 it will be 12. So it does get smaller...." He went backto the problemandtriedto rewriteit so thatit gave him somethinghe believedin, butto no avail. He pondereda while longerandthenremembereda pairof gates thathe had recentlydesigned,costed,andbuiltfor his brother.He becameextremelyexcitedand animatedandranto the office to fetcha calculator.I was consciousthatPeterwas awareof us, andI felt uncomfortableaboutthe lengthof time we hadspentin discussion,since Brian shouldhave beenconstructingthe tablelegs. Brianrushedbackanddrewa diagramof one

133 of the gates on a plank.As he sketched,he labeledthe dimensionsof each componentof the gate, convertingeasily fromcentimetersto meters.See Figure38.
'ri"

r 4

-

w 0.2nm ,?0,2nm

--

turned.05x.05 square .05x.05 spindlquare

^

-

* thickness of all planks is .05 m

Figure38. Brian'ssketchof the gateshe built. He wrotedownthreerowsof calculations,rememberingthe answersby rote.The calculationshows the totalfiguresfor bothgates. long long top andbottomplanks:1.8 x 0.6 x uprightplanks: 0.6 x spindles:

wide 0.2 x 0.2 x 0.05 x

thick no. 0.05 x 4 = 0.072 m3 0.05 x 6 = 0.036 m3 0.05 x 12 = 0.018 m3 0.126 m3

He checkedthe answersusinga calculator,so the positionof the decimalpointpresentedno problem.Pointingto the diagram,he describedwhereeachmeasurementcame fromand exactlyhow the calculationswereset out.I askedif he coulduse this exampleto help solve the problemof the cost of the plank.Using the calculator,he triedmultiplyingthe dimensionsof the plankagain:2.4 x 0.6 = 1.44. Brian:"So it does get smaller!" Using the gate problemas a model,he generalizedthe solutionandmultipliedthe dimensions2.4 x 1.6 x 0.075, usingthe calculator.He finallyarrivedat a pricethat appealedto his commonsense: R324 for the plank. He said,"Gee,thattook a long timefor me to teachit to you, butthatis becauseI had to teachit to myself first!"

134 Thisepisodedifferssignificantlyfromthe previousone. In Episode16, Brianhada planthathe designedandexecuted.The gateexample,whichhe used as a modelin this episode,also suggeststhathe haddesignedandcarriedout a successfulstrategy.In this episode,however,it is clearthatI playedan activerole in the solutionwith which Brianwas finallysatisfiedandthatmy questioningled him to follow my plan.Interestingly,as can be seen fromhis last comment,Briandidnot appearto be overlyawareof beingled by my comments.Duringthis episode,Brianmadetwo appealsto Peteras an outsideauthorityand appealedalso to the authorityof the calculatorto dealwiththe decimalpoint.In the previousepisode,he haddiscussedhis solutionwithconfidenceandwithoutappealingto any externalauthority. Thisencountergives a good indicationof theextentof Brian'slack of familiarity with basic arithmeticconventionsandalgorithms.However,by relyingon otherstrengths like his use of spatialvisualizingandby checkinghis solutionagainsthis commonsense knowledge,he engagedin repeatedactionandreflectionon his problematicandfinally arrivedat an answerthatsatisfiedhim.A key partof his successwas the use of a previously solved problematic,whichhe usedas a modelto solve the novel problem. Episode 17 (July 25th) Clivedesigns a strategyfor calculatingthe cost of a plank I posedthe questionto Clive duringa lunchbreak,asking"Ifmahoganycosts R3000 percubic meter,whatis the priceof a plank2.4 meterslong, by 0.6 meterswide, by 0.075 metersthick?"I hadgiven hima pencilanda piece of paperon whichto work. Clive's initialresponsewas thathe hadno ideahow to estimatethe cost of a piece of wood andthatonly peoplewithspecializedknowledgecoulddo such a calculation.He suggestedthatI shouldask Peterif I reallywantedto knowhow muchthe plankwouldcost. Clive claimedthathe didnot knowwhata cubicmeterlookedlike andmadeno attemptto explorethe idea. I persistedas I hadwithBriananddrewa rectanglarroom,labelingthe sides 3 metersand4 meters,andaskedif he couldcalculatethe floor areaof the roomI had

135 sketched.He was reluctant,buthe madean attempt,addingup the lengthsof the sides and arrivingat the answerof 7 meters.Clearlyhe realizedthatthis was not correct,butwas not sureabouthow to calculatethe area.I remindedhimthathe hadexpertlyestimatedhow manyplanksI neededto purchaseto replacepartsof the floor of my housethathadbeen damaged(he hadreplacedpartof the floorof my homeone weekend),andhe responded with an embarrassedsmile. Aftersome morethought,he calculatedthe areato be 12 square metersandthenbecameinvolvedin tryingto sketcha diagramthatrepresentedone square the whistleblew meter.He becamequiteinterestedin his problematic,butunfortunately signalingthe end of lunchtimeandwe hadto go backto work. Two dayslater,I askedhimthe samequestion,at a formalinterviewat my home afterwork. Clive: "Oh,okay,I don'tknow...letme see now...forargument'ssake, say it is R160 permeter,percubicmeter." Wendy:'That is verycheap!" Clive: "Thenyou will havetwelveof two...four,(thinkingout loud)...letus say 150 wide and...(pause)25 thick...sothatmeansyou will havetwelve of these...."In his mind's eye he hadvisualizedthe schemeshownin Figure39. He did not drawthis diagram,butI was able to constructit fromhis explanationof his solution.Fromthe diagram,it can be to millimetersandthenvisualizeddividingthe seen thathe convertedall the measurements plankinto 12 new planks,each2400 mm by 150 mmby 25 mm.

4

600 mm

242400mm

I

150 mm (4 planks)

Figure39. My diagramof whatClive visualized.

136 Clive: "Soyou haveto multiplyall of these.So 24 times 150 times...butfirstyou mustget it into meters,so it will be noughtpointtwo four,timespointnoughtone five, times pointnoughttwo five (0.24 x 0.015 x 0.25). So you multiplyall of these andthen once you get thatanswer,you multiplyby twelve,andonce you get thatansweryou multiplyby 160. The woodcosts R160 percubicmeter." I pointedto a figureof 2400 he hadwrittendownandasked,"Arethesemillimeters? Two fournoughtnoughtmillimeters?" Clive: "Yes,thesearemillimeters.Yes, it is 2 meters400, butyou bringit to...so it's two pointnoughtfour...(hewrites2.40)...yes." centimeters?" Wendy:"Andthis is 150 millimeters...or Clive: "Thisis 150 millimeters,andthis is about25 millimeters."He puzzledover the conversionsto metersthathe hadpreviouslywrittendown."Thisis pointone five." Wendy:"Pointone five meters?Whydon'tyou writeit all downin millimetersso we can see?" He wrotedown 12/2400mmx 150 mmx 25 mm. Clive: "Sonow I changeit to meters."And he wrote 2.4 m x 0.15 m x 0.025. He lookedat it for a while, thenchangedit to 2.4 m x 0.015 m x 0.0025 m. Afterstudyingthis andmurmuring quietly,he finallychangedit backto 2.4 m x 0.15mx 0.025 m, saying,"So you multiply2.4 by 0.15 by 0.025 andget youranswer.Thenmultiplythatby 12. Thenonce againmultiplyby 160.And thenof courseyou putthe decimalin andthat will give you whatthe wholethingwill cost."He didnotexplainwhy he changedthe rand amountpercubicmeterfromR3000percubicmeterto R160 percubic meter.

137 Clive was somewhatunsureaboutplacevaluewhenconvertingfrommillimetersto meters.However,he hadno problemconvertingfrommetersto millimeterswhen he began his solution.The problemwas posedusingmeters.He told me thatafterI hadfirstasked him how to calculatethecost of the plank,he haddiscussedmy questionwith Peter.Peter had obviouslygiven himthe standardalgorithmfor calculatingvolume,andthey hadalso discussedvisualizinga cubicmeterof wood as beinga pile of timberone meterhigh by one meterlong by one meterwide. Clive incorporated the standardalgorithmfor calculating volumeinto his own strategyof dividingthe plankintocomponentsthathe could visualize. He imaginedcuttingtheplankintosmallerunits,aboutthe size of the plankswith whichhe usuallyworked. Clive commentedon the helpthatPeterhadgiven him:"Itmadesense in a certain way, but I haven'tgot his brain!He is a civil engineerandis used to workingwith figures. He has been (involved)his wholelife withquantities.Herewe get only the simplethings... we do not workout the materials...wedo not workout the cost." As in the case of Brian,Clive appealedto Peteras theexternalauthoritywhen asked to solve the problem.Costestimationwas seen as requiringspecialskills andknowledge thathe did not possess.His actionsandreflectionon the problematicwere constrainedby withstandardmathematicalconventions. uncertainty,lack of confidence,andunfamiliarity Episode 18 (July 26th) Jack inventsa convenientunit to help solve an unfamiliarproblem The firsttime I askedJackhow to calculatethe cost of a piece of wood was duringa lunchtimeinterview.We weresittingon a pile of planksin the workshop,andI hadgiven him a pencil anda piece of paperfor diagramsandcalculations.I asked,"Ifmahoganycosts R3000 percubicmeter,whatis the priceof a plank4 m long by 0.6 m wide by 40 mm thick?" Jackclaimedthathe hadno experiencewiththis typeof work.He emphasizedthat only Peterdid thesecostingcalculationsin the workshop.However,he enjoyedpuzzling

138 over the problemandstudiedthe dimensionsof the plankthathe hadwrittendown.Aftera few minutes,he wrote5/4m x 150 x 40, saying,"Fromthatone plank,you can get 4 planks, each 4 meterslong by 150 wide by 40 thick."At my request,he drewa labelleddiagramof one of the narrowerplankshe hadenvisaged.See Figure40.

40 mm

/40

mm:

150 mm

150mm

V t^ 4m

<

C>

Figure40. Jack'sdiagramof the narrowerplankhe visualized. Afterdrawingthe diagram,Jackreiteratedthathe did not knowhow to calculatethe cost of wood. Again,I introducedthe topicof areaanddrewa rectangulardiagramwith sides marked9 m and3 m. Jack,usingthe sameincorrectmethodas Clive, calculatedthe areato be 12 squaremeters.He thenchangedhis mind(like Brian),addedtogetherthe lengthsof all the sides, andgave an areaof 24 squaremeters.Finally,he abandonedthe 9 m by 3 m rectangulardiagramI hadsketched,andsketcheda 3 x 4 grid,whichhe extendedto blockshape.See Figure41. a three-dimensional

\

\

\

\

N

Figure41. Jack'sdiagramrepresentingvolume.

139 It was clearthathe was tryingto extendhis idea of areato a thirddimension.Lunchbreak came to an end andthe whistlesounded,so we couldnot explorefurtherthatday. The next evening,I interviewedJackagainat his homeandaskedthe same question aboutthe cost of the plankthatI hadposedto Clive andto Brian:"Ifmahoganycosts R3000 percubic meter,whatis the priceof a plank2.4 meterslong by 0.6 meterswide by 0.075 metersthick?" Jacksaid thathe hadthoughtaboutthiskindof problemsince the previousday and had an idea of how to solve it. Using the pencilandpaperI hadgiven him, he started performingall sortsof calculations,addingtogetherall the measurements,multiplyingby andwentrapidlyfromone three,seeminglyat random.He was clearlyuncomfortable attemptedcalculationto the next.I stoppedhim andaskedwhatwas going on. He said that he hadaskeda friendhow to do the problemafterI hadaskedhim at lunchtimeandthathe was tryingvery hardto rememberwhathis friendhadsaid.He realizedthathe had forgottencompletely.I persuadedhim to stop attemptingto rememberthatparticular solutionbutto try to thinkit out for himself.He finallyconstructedthe following argument thatsatisfiedhim. Jackchose unitsthatallowedhimto estimatethe cost of mahoganyfrommemory. He told me thatstandardplanksareabout23 mmthick.He visualizeddividingthe 2.4 metersx 0.6 metersx 0.075 metersplankinto a stackof threethinnerpieces of 25 mm each. Figure42 shows my diagramof the schemevisualizedby Jack(he did not drawa diagram). <---

2.4 m

r

300 'mm (2 planks)

\

,z5 @@ Figure42. My diagramof whatJackvisualized.

140

He said thatthe standardwidthof a plankis about300 mm andthata mahoganyplankof thatwidthandlengthandthicknesswouldcost R80. He wrotedownthe followingcalculations: 80 x3 240 x2 R 480 is the cost of the plank. This is probablyan appropriate figurefor thecost of mahoganyplanksbought approximate in a smallquantityat a retailstore.The figurethatBrianworkedon was the wholesaleprice for wood boughtin greatquantity.Jackgave a solutionthatmadesense in the light of his personalexperienceas a carpenterwho hadneverboughtwood in greatbulk.Since he did not know the standardalgorithmfor calculatingthe volumeof a piece of wood, he hadto rely on visualizationandcommonsense to constructa solutionto the problemthatI posed. Commentson Episodes 17 and 18 The problem-solvingstrategiesdevisedindividuallyby bothClive andJack dependedon the selectionof suitableunits.Jackvisualizedcuttingthe plankinto unitsof which he knew the price,andthis simplifiedhis calculations.He selectedthe unit because he knew its price.It is not clearwhathe wouldhavedonehadthe planknot split fairly neatlyinto an integralnumberof his selectedunit.Clive chose a unitthathe could visualize andthatwas similarto the size of plankhe was accustomedto handling.He imagined dividingthe plankinto twelve of theseunitsandthenappliedthe standardalgorithmto the unitandmultipliedby twelve. On analyzingtheirresponsesin moredetail,I concludethatthe difficultylies not of theconceptof cubicmeasurebutwith (a) a misplacedsense of with a misunderstanding inferiority-a feeling thatonly certainpeoplearequalifiedto do calculationslike this, that this falls beyondtheirsphereof expertise;(b) a confusionaboutdecimalplaces,converting betweenmillimetersandmeters;(c) a lackof experiencewithmathematicalconventions;

141 and (d) the fact thata cubicmeteris too big a measureto be usefulto a manwho builds tablesandchairs.The amountof wood neededfor a chairis alwayssome minutefractionof a cubic meterthatis very difficultto picture.As Peterpointedout,cubic feet are so much easierto visualizeandto talkabout.Oneneedsroughlyone cubicfoot of wood to makea chair,andthreecubicfeet to makea table. Althoughtheseepisodesseem to confirmthatthe carpentershave developeda visual of the conceptof volume,theepisodesalso servedto uncovermany understanding misconceptionsandto highlightthe severelimitationsthatareimposedon the abilitiesof the carpentersto performsimplenumericalcalculations.The demeanorbothJackandClive displayedwhile they were tryingto solve this problemwas totallydifferentfromtheirusual confidentmanner.They stumbledandstruggledandmadeit clearto me thatthis calculation was not in theirdomainof work-it was a specialcalculationfor whichPeterwas andthereforebeyondtheirreach.In the past,Jack responsible.It was seen as "mathematics" hadpurposelyavoidedany situationswith me wherehe mightnot be able to answera question.This time,however,he was trappedin an interviewsituationandwas clearly somewhatuncomfortableat times. Episode 19 (July 26th) Whyisn't "straightup"a full 100 degrees? Jackhadused his squareruleon countlessoccasionsto help me check thatjoints were square,to "pull"a line alongthe edge of a plankto be cut to ensurethatthe cut would be square.In his day-to-dayworkconstructingtables,the squarerulewas oftenused to check for "squareness" alongthe way. He hadconstructedan innovativesolutionto the question"Whenis a squarerulesquare?"describedin Episode5. I was thereforesurprised by this exchangethattookplaceduringa formalinterviewwith him at his home. I hadaskedhim, "Whatdoes it meanto be square?"The followingexcerptis taken fromhis discussion:

142 Jack:"Nowyou get anothersquare...it'snot a squareactually,butyou can makeit into a square.It is calleda slidingbevel.It lets you cut in any angleyou want."See Figure 18 (page 114). Wendy:"Thatis whatClive uses for drawingdovetailjoints." Jack:"Yes,you can use it on a dovetailjoint.It can reachfromnoughtto a 90-degree angle." Wendy:"Sonoughtwouldgive you what...justa straightedge?" Jack:"Yes,just a straightline." Wendy:"Anda 90-degreeanglewouldgive you?" Jack:"A 90-degreeanglewill give you that"(he drewan angle of about80 degrees). See Figure43.

Figure43. Jackcalledthis a "90-degreeangle." Jack:"Thisis 100 degrees...(hedrewa 90-degreeangle)...anda 90-degreeanglewill give you that" (he drewoverthe line at about80 degreesthathe hadalreadydrawn).See Figure44.

Figure44. Jackcalledthisa "100-degreeangle." Wendy:"Whichis 100 degrees?The line going straightup?" Jack:"Theline straightup is 100 degrees.Peoplewill say thatthis is a 90-degree line),...butI do notcall this a 90-degreeangle." angle (pointingto the perpendicular

143 Wendy:"Don'tyou?Why not?" Jack:(hesitating)"Umm...thenI am out of square...Idon'tknow." Wendy:"Yoursquarerule,the metaltool, is that100 degrees,or 90 degrees?" Jack:"It'sa 90-degreeangle...Yes...because theysay the 45 is halfway.Splitthis in half. They say that's45 degreesandthat's45 degrees.I wantto know why you don't get 100 degrees." Wendy:"Whenyou addthemup?" Jack:"Yes,whenyou addup the square.Squaremeans"straightin line."Thatis on its full amount.Whenit reachesits full point,the maximumpoint...whatis the degrees? This is the sign for degrees,isn't it?"(He drewa smallcircle.) Wendy:"Yes,that'sthe rightsign. Andthat(pointingto the verticalline) is 90 degrees." Jack:"Butwhy is it 90? Whyis it not a full hundred?" Wendy:"Mmmm...Are you askingme?" Jack:"I am tryingto figureit outmyself...that'swhatI am tryingto figureout. Basicallyit startsfromnoughtto 90 degreesangle.Onyourmachine(thetablesaw) you alwayssee noughtto 90 degreesangle.The 90 degreesangleis the full angle,the full size." Wendy:"Whatif this wentall the way down.Whatif you bentthis all the way down,thenit wouldbe...."See Figure45.

Figure45. Movingthe verticalarmclockwisethrough90? ("bendingit all the way down"). Jack:"Ifyou bentthis all the way down..." Wendy:"Allthe way downuntilit was level again."

144 Jack:"Thatwill be calleda nought.Yes, thatis nought.You startfromnought...to nought." See Figure46.

45 degrees

45 degrees

0 degrees

0 degrees

Figure46. My diagramof Jack'sdescriptionof angularmeasure. Wendy:"Witha 90 in the middle." Jack:'That will be your45 on yourrightside andthatwill be your45 on yourleft side. And all the otherdegrees,30 or whatever,70 come in betweenhere." Wendy:"Soit lookslike you neverget to 100?" Jack:"Yes,I wantto knowwhy is it thatyou neverget to 100!" In combinationwiththe previoustwo episodes,this conversationwith Jackforced a concept.Thereappearsto an examinationof my ideasaboutwhatit meansto understand or knowingthe conventionsthatsurrounda concept be a differencebetweenunderstanding a conceptin a (theseconventionsareveryseldom,if ever,questioned)andunderstanding way thatallows its active,practicaluse. Jackunderstoodthe conceptof measuringangles, knew whata rightanglelookedlike, andknewthata rightanglewas said to measure90 deandthe propertiesof a rectanglesuccessfullyin grees. He used the conceptof "squareness" his work.However,he was not familiarwiththe conventionsof anglemeasure,thatit was agreedthattherewouldbe 360 degreesin a circle,thatan angleof 180 degreesrepresentsa straightline. On the tablesaw thatJackused,degreesweremarkedon a dial as shownin Figure47.

145

0 degrees

I 0 degrees Figure47. The dialon the tablesawin the workshop.

Therewas no evidenceof any anglethatmightbe assigneda numbergreaterthan90 degrees. of volume,Jackseemedto have an As in the case withthe carpenters'understanding of angles,straightness,andsquarenessthatwas richin spatialawarenessbut understanding poorin its link to numericalvalues. Episode 20 (July 26th) Cliveand Jack talkaboutstraightness The followingareexcerptsfromformalinterviewsthatI conductedwith Clive and Jack,respectively.I hadaskedClive, "Whendo you call a line straight?How do you check thatsomethingis straight?" Clive: "Okay,you get a piece of timber.Firstthingyou do is pick it up andlook at the edge of it or the side of it, or whateveryou wantto checkis crookedor not. Whenyou look down a piece of timberandcheckit withyoureye, the momentyou can see a dip,you know it is not straight.You can also checkit by puttinga straightedgealongit andlooking for any gaps. Oryou can lay the piece of timberon a flat surfaceandcheck for gaps. Anotherway to checkit is usinga 'trysquare'(see Figure48), puttingthe trysquareon a flat edge andseeing if it is openingat one end or at the other.Orwith surfaceplaningor by handplaning."

146

Figure48. Trysquare:The woodenhandlesits flushagainstthe edge of the surface,andthe metalbladeindicatesa 90-degreeangle. Wendy:"Howwouldyou use the trysquareto checkif a piece of timberis straight?" Clive: "Ifyou wantto checka wide surface,you putthe trysquareagainstthe edge so thatit restson the wide surface.Say, for argument'ssake,thatyourtimberis 3 inches thickand20 incheswide. Thenyou putyourtrysquarerestingon the 20 incheswide, and hardup againstthe edge. If you see a gap betweenyourtimberandyoursquare,thenyou know yourtimberis off. It is not straight."See Figure49.

0 o

Figure49. Using a trysquare. Wendy:"Howwouldyou use surfaceplaningor handplaning?" Clive: "Say,for instance,you havea piece of roughtimberthatyou wantto straightenout andyou arein a joiner'sshopor any othermachineshop. You surfacethe widestsectionfirstwith a surfaceplanerandget thatstraight.Eachsurfaceplanerhas a long flat surface,andyou can alwayscheckthe timberon thatby layingit flat down on thereand lookingfor gaps. Oryou can checkit by eye. Whenyourflat surfaceis finished,thenyou turnit andputthe flat surfaceagainstthe fence of yoursurfaceplanerandplanethe bottom

147 edge of the plank.By doingthatyou will get yourtwo edges 90 degrees.You firstmakethe surfacestraight,andthenthe edge." Wendy:"Okay,I see thatnow. How wouldyou checkthata rounddowel is straight?" Clive: "Youcan neverget a dowelperfectlystraight.You can look downthe length of a dowel, andwhenyou can see it go up anddown,up anddown,andon the side you can see in andout, in andout, thenit is not straight.You can also turnit andlook downthe lengthof the dowel. You will see one pointandeverytimeyou turnit, you miss one point.It seems to dip away andyou can'tsee it, thenall of a suddenyou see it again.Whena dowel is made,the wood goes througha machineandit goes throughso fast, turningat the same time.Whenit comes out of the machineit is hot!It has to cool downandit twists andturns. If you do get it straight,it's just luck." Clive's practicesarebasedon an implicitunderstanding of straightness.He it to forma smoothplank,thus imaginestakinga piece of timberandstraightening contextualizingmy questionandturningit intoa familiarproblematic.His methodsand explanationsarebasedon actionsthatoriginatefroman underlyingfoundationof the symmetriesof a straightline. I askedJacksimilarquestionsin his interview:"Jack,whenwouldyou say thata line is straight,andhow do you checkthatsomethingis straight?" Jack:"Icoulduse my squareto check,butif I haven'tgot that,I could use my eye. Say this is a piece of plankhere(he makesa horizontalplaneusinga piece of paper,holdsit up at eye level, andsquintsalongits top surface).Onceyou see the frontpart,you have to see the backpartof the wood. If you don'tsee that,thenyou knowthatthereis a hump somewhere,or a dip.Youreye falls rightinto thatdip or hump.If youreye falls on nothing, thenyoureye will fall righton theend of thatpiece of wood andthenyou will see thatit is straight."

148 "Tomakeyourwood straight,you can planeit on a planingmachine.Say you have a roughplankthatis bumpy,you haveto putit throughthe surfaceplanerto get thatside smooth.Thenyou takeit to thethicknesser,andset themachineto any thicknessyou want. You putthe side thatyoujust planedontothethicknessertable,on thatflat surface,and pushit throughthe machine." Wendy:"Howdoes the thicknessermachinework?" Jack:"Thethicknesserhas a flat tableon whichyou resta plank.Thereareblades abovethe table.You can lift the tableup (closerto theblades)or down(furtherfromthe blades)by turninga handle.Bothof the gearshaveto shiftevenly to keep the balanceof the tableeven. The otherdayit was 'out.'I measuredone side of my plankandit read 23.5 mm, andthe otherside read22 mm." Wendy:"Howwouldyou knowif a rounddowel,like a curtainrod, was straight?" Jack:"A rounddowel?I wouldroll it ontoa smoothsurfaceandif it is 'out,' it will flop around.Youjust pushit along,andas it rolls,you can see whereit is high andwhereit is low." Like Clive, Jackimaginedhimselfhandlinga plankof wood, solving a problematic thatoccursfrequentlyin the workshop.He set my questionin context.He also used the methodof "sighting"alongthe plank,whichdependson the propertyof light travelingin straightlines. He describedusingthe planingmachineto straightenone face of the plank, visualizingparallelhorizontalplanes.The use of the thicknesserextendsthe idea of horizontalplanes.He notedthe importanceof the levelnessof the tableto ensurethatthe planesremainedhorizontal,pointingouttheconsequences(uneventhicknessof the plank) if the tablewerenot level. His commentsaboutthe dowelrevealan implicitunderstanding of the rotationalsymmetryof a straightline. Part Two: Analysisof Episodes As has beenpointedout by GoetzandLeCompte(1984), datacollectionanddata becausetheethnographer may not know analysisareinextricablylinkedin ethnography

149 exactlywhatquestionsto ask andwhatactivitiesto observeuntilinitialperceptionshave been analyzedandtentativeconclusionshavebeenformulated.Analysisis thusnot relegatedentirelyto the periodfollowingdatacollectionbutis a processthatcontinues throughoutthe researchendeavor. Grouping of Episodes Followingthe patternof the analysisprocessas laid out by Goetz andLeCompte (1984) andEisenhart(1988), the next step (afterdefiningthe unitsof analysis)involves An inspectionof the data groupingthe unitsof analysisaccordingto similarcharacteristics. led to the identificationof threegroupsof episodes.All of the episodes,exceptfor Episode 20, took place in the workshop. GroupOne.GroupOneincludesthoseepisodesthatdevelopedas a resultof problematicsthatarosespontaneouslyfor thecarpentersandthatwere stronglylinkedto theirusualactivities.Mostof the encounterswereencouragedor covertlyengineeredby my questionsas a novice. This groupcomprisesEpisodes1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, and 19. GroupTwo.GroupTwo includesthoseepisodesthatdevelopedfroma problem thatI posed andthatwas less directlyrelatedto customaryworkshopactivities.These encounterswere in the formof interviews,ratherthanbeingpromptedby the questionsof a novice in the contextof the workshop.Threeof theseepisodestook place awayfromthe workshop.This groupcomprisesEpisodes16, 17, 18, and20. GroupThree.GroupThreeincludesthoseepisodesthathighlightedthe attitudesand the relationshipsof the carpentersto theirconceptionsof mathematics.This group comprisesEpisodes4, 6, and 12 andcertainpartsof Episodes1, 5, 16, 17, and 18. CategorizationWithin Groupsof Episodes Eachepisodewas scannedthroughsystematiccontentanalysis,in orderto generate descriptionsfor the propertiesandattributestheycontained.By comparing,contrasting,and synthesizingthesepropertiesandattributes,threecategorieswerefinally derived.The

150 derivationof the categorieswas also linkedto the facilitationof the primaryobjectiveof the The three presentstudy-to show thatthe carpentersengagein validmathematizing. categorieswere1. conventionalmathematical conceptsthatwereembeddedin carpentrypractice in the workshop, as demonstrated 2. the carpenters'conceptionsof mathematicsandfeelings aboutmathematics, of the carpenters'mathematizing. 3. significantcharacteristics Eachepisodewas scannedfor examplesor otherevidenceof these threecategories, anda 20 x 3 gridwas drawnup to recordthelinksbetweenepisodes;the frequencyof occurenceof ideas,practices,andattitudes;andthe elementsmakingup the characteristics In the next threesections,the resultsof this phaseof the of the carpenters'mathematizing9. analysisprocessareelaboratedfor eachof the threecategories. ConventionalMathematicalConceptsThat Were Embeddedin Carpentry Practices in the Workshop The firsttwo groupsof episodeswereexaminedto identifyany conventional mathematicalconceptsthatmayhavebeenimplicitin the actionsor in the discourseof the carpentersor thatmayhavebeenexplicitlydiscussedandusedin the workshop.Recurrent uses of the same mathematical conceptsas well as othercommonthemesweresearchedfor withineach groupof episodesandacrossthe two groupsof episodes.Similaritiesand differencesarepointedoutacrossthe two groupsof episodes. Not unexpectedly,thereis very littlediscussionof mathematicsfor its own sake in the workshop.This does not implythatmathematicsis not discussedor usedin the conceptsandconventional workshop;it merelymeansthatconventionalmathematical mathematicalterminologyareoftenconcealedbehindphysicalactivities,handgestures, silent acts of visualization,andthecarpenters'own terminology.

9 Tablesdocumentingthe analysisof eachepisodecan be furnishedby the authoron request.

151 Theconceptof "straight." In Episode20, Clive articulatedseveralinstancesin that of the conceptof somethingbeing straight.He his practicesshow an implicitunderstanding gave me the one exampleof how he would"lookdowna piece of timber"(sometimesthis practiceis called "sightingalonga line"),andsee "dips"if the plankwas not straight.This methodworksbecauselighttravelsin straightlines. Threeothermethodsthathe suggested for checkingwhethera piece of timberis straightimplicitlyusedthe propertythata straight line has reflectionsymmetrythroughitself.Thesethreemethodswere puttinga straightedge along a plank,layinga plankon a knownflat surface,or puttinga trysquareagainstthe surfaceof a plankandcheckingfor gaps.Usingthe surfaceplaneris basedon the same propertyof reflectionsymmetry. Clive's firstexplanationof how to checkthe straightnessof a rounddowel was basedon "sightingalong"the lengthof the dowel, similarto his firstresponseaboutthe plank(see Episode20). He also gave anotherway, basedon the propertyof rotational symmetrypossessedby a straightline, describinghow to turnthe dowel while looking down its length."Youwill see one point,andeverytimeyou turnit, you will miss one point It seems to dip awayandyou can'tsee it, thenall of a suddenyou see it again"(see Episode 20). In Episode20, Jackdemonstrated the act of "sighting"alongthe surfaceof a piece of paperandmentionedthe "dipsor humps"intowhichhis eye would"fall"if the edge was not straight.His explanationof usingthe planingmachineandthe thicknesserwas basedon the visualizationof two horizontalplanesurfaces.His responseto checkingwhethera round dowel was straightinvolvedthe propertyof rotationalsymmetry.In Episode5, Jack's solutionwas basedon the reflectionsymmetryof a straightline. Theconceptof parallel lines.Jackrevealedan understanding of the conceptof parallellines whenhe usedthe sides of the box lid as framesof referencein correctingthe orientationof the star(see Episode3). Sincehe knewthatthe box lid was rectangular,Jack realizedthatin orderto drawa correctlyorientedset of orthogonalaxes, he shoulddraw

152 lines parallelto the sides of the box lid usinghis squarerule.In Episode20, he visualized using parallelhorizontalplanesto explainhow he wouldproducea straight,even plank froma roughpiece of timber. In Episode5, Mr.S usedthe conceptof parallellines implicitlyin his explanations of how to use threestraightsticksor threeplanksto provethathis squarerule was true. Based on his first(andfaulty)hypothesisthatthe groundwouldbe level, he claimedthatif the distancefromthegroundto the top plankwerethe sameat bothsupportingplanks,then the top plankwouldalso be level (i.e., thenthe topplankwouldbe parallelto the level ground).In his morepowerfulandgeneralversionof thatsolution,he explainedhow he would"lineup"the top plankwiththe horizon,usingthe horizonas a natural"level"and againimplicitlyusingtheconceptof parallellines. Parallel lines intersectedby a transversal.Parallellines wereagainused implicitly by Mr.S in his methodof placingdovetailjointsevenly alongthe edge of a drawer(see Episode8). The repetitivepatternthathe employedcan be viewedin termsof a seriesof transversalscuttingtwo parallellines.Thushe ensuredthatthe "angle"of his dovetail remainedthesame alongthe drawerfront(sincealternateanglesareequal).See Figure50. Note thatneitherJacknorMr.S usedthe term"parallel"whendiscussingtheiractionsand explanationsduringtheseepisodes.

Figure50. AnalyzingMr.S's methodfor placingdovetailsin termsof parallellines and transversals. Theconceptof "center."As discussedin Episodes1 and2, JackandMr. S used a conceptof centerthathadimportantsymmetries.Theirconceptof the centerof a

153 rectangularfigurebeingat the pointof intersectionof the diagonalsof the rectangle dependeduponthe fact thatthe oppositesides of a rectangleareequivalentin length.In Episode 10, I triedto probefor Mr.S's conceptof the centerof a four-sidedfigurethatdid not have oppositesides of equivalentlengths.His explanationsuggestedthathis conceptof centerin this case was tied to symmetry,consistentwith his conceptfor a rectangle,butmy remainsunclear. interpretation The concept of symmetry and the axis of symmetry. The concept of symmetry was

often implicitin the actionsof the carpenters.Theywouldsometimesverbalizetheiractions while usingsymmetrybutneverusedthatconventionallabel.Jackused the reflection symmetryof a straightline in his explanationof how he couldshow thathis squarewas square(see Episode5). Mr.S usedthe conceptof symmetrywhenexplainingto me how the carvedbackof a Tulbaghchairshouldbe traced(see Episode11). He told me to trace half of the designon a piece of paperandthenturnthe paperover to get the otherhalf of the design.He said, "Bothsides mustbe the same,theycorrespond"(i.e., he referredto reflectinghalf of the designaboutthe axis of symmetryof the whole design).Clive made similarcommentswhenusinga tool thatenabledthe userto drawoval shapes(notreported as an episode), "Youonly haveto drawhalfof the shape,becauseyou can get the other half by foldingthe paperover." In Mr.S's systemfor theeven placementof dovetailjoints,he beganby findingthe centerof the piece of wood anddrawinga line perpendicular to the edge alongwhich the joints will be placed(see Episode8). He usedthis line as an axis of symmetryto guide the placementof two points.His systemworkedby focusingon differentsectionsof the plank and using the axis of symmetryof eachsuccessivesectionto guidethe placementof points. ComparingMr.S's instructionsto me for tracingthe shapeof the backof the Tulbaghchair(Episode11) andfor tracingthe shapeof the brassescutcheon(Episode14) highlightsa significantpoint.In contrastto his directionsfor the Tulbaghchair,wherehe specificallytold me to turnthe templateoverbeforetracingthe shapeonto a piece of wood,

154 he did not tell me to turnthe templatefor the brassescutcheonoverbeforetracingit. As describedpreviously,by not flippingthe template,the paperwas stuckto the backof the new brassescutcheon,leavingthe frontface unblemished.Since the paperwas difficultto removecleanly,this strategyresultedin theoptimumsolution.The choice of strategyshows an understanding of the distinctionbetweenreflectinga shapethroughan axis of symmetry in two dimensionsandrotatinga shape180 degreesaroundan axis of symmetryin three dimensions.This distinctionremainsblurredin formalmathematics-theresultsof these two actionslook identicalin a textbook.In practice,however,the distinctionis clear,as exemplifiedby Mr.S's differentstrategies. of a "square"angle Theconceptof a rightangle. Mr.S revealedhis understanding (i.e., a rightangle)as the intersectionbetweenthatwhichis "level"andthatwhich is "plumb"(i.e., horizontalandvertical)in Episode5. In Episode20, Clive mentioneda firsttheface of a plankon a surfaceplaner similaridea whenhe describedstraightening (i.e., makingit a horizontalsurface)andthenrotatingthe plankinto an uprightpositionwith the horizontalsurfaceagainstthe fence of the planein orderto straightenthe edge of the plank.He said by doingthat,"youwouldget yourtwo edges to 90 degrees"(i.e., you would createa rightanglealongthe edge of theplank). The diagonals of a rectangle are equal. The common practice of "squaring" the

of the fact thatthe apronsof a tableduringconstructioncarriesan implicitunderstanding diagonalsof a rectangleareequal(see Episodes1 and2). The explanationgiven by Jackin of the Episode2 supportsmy claimthatthepracticeis basedon an activeunderstanding and"diagonal"were not usedby conceptandis not performedrotely.The terms"rectangle" the carpenters.All rectangularshapeswerereferredto as "square,"andcarpentersreferred fromcornerto corner." to the "length/distance The diagonals of a rectangle bisect each other. The practice of drawing the

diagonalsof a rectangleandusingtheirpointof intersectionto locatethe centerof the rectangleis basedon the fact thatdiagonalsof a rectanglebisecteach other(see Episodes1

155 and2). Again,I claimthatJackdemonstrated an activeunderstanding of this conceptin his explanationin Episode2. Theconceptof triangle.Mr.S useda 3-4-5 triangleto supporthis argumentthathis squarerulewas square(see Episode5). Spatialvisualization.In Episode14, the methodthatMr.S instructedme to use for replacingthe missingbrassescutcheonwas basedon theconceptof symmetry.In this encounter,he usedhis hand,withthumbandlittlefingerextendedto imitatethe shapeof the escutcheon,to show me how rotatingtheexistingleft-handescutcheonwouldgive the correctshapefor the replacementright-handescutcheon.By mentallyreorientingthe shape in space,he knew thatthe templateshouldnotbe flippedoverbeforegluingit to the metal becausethis wouldensurethatthe stickypaperwouldend up on the backof the template and wouldthusnot have to be removed. Similarly,Jackmentallymanipulated(flipped)the imaginaryshapeof an "offsquare"squarein Episode5, whenhe drewa sketchof whatthe outcomeswouldbe if a squarewere "off squareandtoo wide"or "offsquareandtoo narrow."In Episode7, Mr.S mentallymanipulated(rotated)a shapein spacewhenhe recognizedthatthe flowerdesign on the inlaidtablewas madeup of the rotationsof a single petal. In Episode 15, Brianperformedthe followingmentaloperations.He visualizedthe plankfromwhichhe wantedto cut the leg components,dividedit into sections,reasssembledthese sectionsinto new structures,anddividedthese againinto sectionsthat matchedhis mentalimageof the legs he wantedto turnon the lathe.Briandid not make any diagramswhile solving theproblem.Althoughincludedin this section,entitled "Conventionalmathematicalconceptsthatwereembeddedin carpentrypracticesin the workshop"(see page 151),it is arguedthatthekindof visualizingthatBrianused to obtain the optimumsolutiongoes beyondthe kindof visualizationusuallyrequiredby formal mathematics.This pointwill be discussedfurtherin the sectionentitled"Constructing functionalvisual units"(see page 166).In Episode16, he constructeda visualconceptof

156 volumewhenhe sketcheda three-dimensional blockshapeandimaginedmovingunitsof the blockasidein orderto see andcountall the unitscomposingthe block. In Episodes17 and 18, Clive andJackeachmadementalconstructionsof unitsin orderto solve the problemof calculatingthe cost of a plank.Clive imaginedsectioningthe plankinto congruentunitsthatwerethe size he was usedto handling,whereasJack imagineddividingthe plankintocongruentunitsof whichhe knew the price.They visualizedthe acts of sectioningandcountedtheimagesof the smallerunits. Theconceptof congruence.Theconceptof congruencewas implicitin Jack's placingof his squareon top of Mr.S's squareto show thatthey were not the same (see Episode3) andwas also implicitin the customarypracticeof placingone plankon top of anotherandfeeling thatthey werethe samesize withouttakingany measurements(see Episode7). The commonpracticeof creatingtemplatesby tracing,in orderto reproduce shapes,is also basedon the conceptof congruence(see Episodes11 and 14). The term was not usedin the workshop. "congruence" was specificallymentioned Theconceptof proportion.Theconceptof "proportion" by Mr.S whenhe instructedme to makean inlaydesignfor a drawerfrontthathadto be similarto butsmallerthanthe designinlaidon otherpartsof the table(see Episode7). Mr. S used proportionnot only to scale thedesignon the backapronby one half, he also used proportionto checkthatthe internalrelationshipsof the designremainedin harmonywith those of the original. In Episode9, Mr.S describedhis methodof convertingbetweenthe metricsystem andthe Imperialsystem.He said,"Thosemissing3 millies areequivalentto the 1/8inch neededto makeup a full inch."It is clearthathe was visualizingandcomparingthe relative sizes of the two lengthsrepresentedby thosenumbersandthathe hadconcludedthatthey were equivalentin length.He continuedby reasoningthatthe lengthrepresentedby 1/16 inch wouldbe equivalentto the lengthrepresentedby 1.5 mm, andfinallythatthe length representedby 1/32inch wouldbe equivalentto the lengthrepresentedby "almostone

157 millie."Thusby usingproportional reasoning,he constructeda visualimage of the approximatelengthrepresentedby 1 mm. Theconceptof volume.Episodes15, 16, 17, and 18 supportthe claim thatthe of the conceptof volume.This understanding is carpentershave an understanding characterizedby a strongsense of the spatialandvisualmeaningof volumeratherthanby a knowledgeof how to calculatevolumenumerically. In Episode15 andin Episode18, neitherBriannorJacknumericallycalculateda volume,butbothusedthe conceptof volumevisuallyto sectionplanksinto unitsthatwere useful for theirseparatepurposes.Brianmentallysectionedandreassembledunitsuntilhe achieveda particulardesiredshapeof wood. Jacksectionedthe plankinto unitsof whichhe knew the priceandso was ableto estimatethe cost of a plankwithoutknowingthe numericalsize of its volume. Theconceptofplace value.In Episodes15, 16, 17, and 18, the carpenters demonstrateda limitedunderstanding of placevalue.Theirattemptsat conversionswithin the metricsystemwere not consistentlycorrect.Convertingfrommetersto centimetersor millimeterspresentedfewerdifficultiesthanconversionsin the oppositedirection. Theconceptofproof Theconceptof proofis illustratedby the explanationsof both JackandMr. S in Episode5. Jackclearlyrealizedthathe neededto producea reasoned argumentin orderto convinceme thathe hadshownthathis squarerulewas square.When he realizedthathis firstmethoddid not convinceme (and,I suspect,noticedthe flaw himself)he succeededin constructinganotherargumentthatwas convincing,bothto me andto himself.He also providedtwo counterexamples (the"off-square"squares)to further bolsterhis argument. Mr. S was also intenton constructinga convincingargumentthatwouldimpress uponme the fact thathis squarerulewas "true."The processby which his argument developedandimprovedunfoldedaroundhis conjecturesandhis responsesto my awkward questionsandto the restrictionsI puton which"tools"he was allowedto use.

158 Summary.Frommy pointof view as a mathematicseducator,I foundexamplesof conventionalmathematicalideasin manyof the explanations,actions,andprocedures describedanddemonstrated by the carpentersin the episodes.By comparingthe mathematicalideas acrossGroupOne andGroupTwo episodes,it can be seen thatthe carpentersfavorgeometricimageryas a way in whichto act uponproblematicsandthat they avoidencounterswith algebraiccalculations. In all episodesfromGroupOne (i.e., wherethe problematicsarosespontaneously), geometricmethodswerechosenandusedin actionto arriveat a solution.Even Mr.S's designof his conversionmethodfromImperialto metricunitsdependedon the relative lengthsrepresentedby the numbers(see Episode10). In episodesfromGroupTwo (i.e., whereI posedthe problems),whenI poseda problemthatalmostseemedto force an algebraicapproach(see problemposedin Episodes16, 17, and 18), the carpenters developedmethodsthatweredesignedeitherto minimizetheneed for calculations(see Episode 18), or to incorporatecalculationsas secondaryto a visualsolution(see Episode 17). In Episode17, Brianfinallyperformedtheconventionalcalculationfor volume,after using a previouslysolved problematicas a modelthathe appliedvisually. The Carpenters' Conceptionsof Mathematicsand FeelingsAbout Mathematics was specificallymentionedon rareoccasionsin the workshop. "Mathematics" However,some very clearattitudesconcerningmathematicsandthe connectionof the carpenters'workto mathematicswerearticulatedanddisplayedin otherless obviousways. Based on an analysisof the thirdgroupof episodes,two mainthemesareelaborated.First, the carpentersdid notconsidertheirworkto havemuchto do with "real"mathematics.Jack claimedthatthey did "abit of addingandmultiplying,"whichhe did not countas genuine mathematics.Second,in thoseepisodeswheretheproblemarosespontaneouslyin the workshopandthe carpenterswereableto frametheirown problematics,they weremore successful,moreconfident,andmorewillingto becomeinvolved.

159 Jackwas by farthe mostoutspokenof thecarpentersandthe mostreadyto articulate his opinionon anymatter.Consequently,it is mostlyJack'svoice thatcan be heardin the episodes.He gave a clearindicationthatas faras he was concerned,no "real"mathematics was used in the workshop(see Episode4). Furthermore, he portrayedmathematicians as being highlyintelligentindividuals(if somewhatdeficientin otherways!). In Episodes1, 6, and 12, Jackmadeattemptsto avoidbeinginvolvedin an activitythathe seemedto identify as "mathematics." His motivationappearedto be the fearof beingmadeto feel foolish and he was surethatthere inadequate.WhenJackidentifieda problemas being"mathematical," was only one correctanswer(see Episodes5 and6). As can be seen fromepisodes16, 17, and 18, therewerestrictboundarieson the sortof mathematicalknowledgethatthe carpentersallowedthemselvesto own. The "specialized"taskof estimatingthe cost of wood was seen as beingPeter'sdomainand inaccessibleto the carpenters(probablypartlybecauseof the mathematicsinvolvedand partlybecausethe problemconcernedmoney).Theyportrayedthe mathematicalknowledge of which theydid claimownershipas trivialor "simple."Forexample,Jackstatedthatthey do "justa bit of addingandmultiplying,butno real maths"(see Episode4), andClive remarked,"Herewe get only the simplethings"(see Episode13). A comparisonof thoseepisodesthatdescribeencountersfocusingon spontaneous problemswith thosethatdescribeencounterswhereI deliberatelyposeda problem highlightedseveraldifferences.Whenthey wereableto definetheirown problematics,the carpenterswere morelikely to drawon theirown resourcesto find a solutionandless likely to seek an outsideauthority.Forexample,Brian'sbehaviorin Episode15 conveyed confidenceandpridein his solution,buthis initialbehaviorin Episode16 conveyed uncertaintyanda lack of confidence.A furtherexampleis providedby consideringJack's behaviorin Episode2, wherehe conveyeda beliefin his own resources,whereasin Episode 18 he appealedto Peteras the externalauthority.Therewas also morewillingnessto engage with a self-framedproblematic,anda moreconfidentattitudewas evident.

160 Comparedto Jack,Clive andMr.S weremuchmorecircumspect,more accommodatingof my questions,andpossiblysomewhatmoreconfidentabouttheir seemedto be verycomfortablewith his knowledgein general.Mr. S, in particular, knowledge.He stronglyvaluedhis skills,experience,andwisdomas a carpenter,andwas not intimidatedby beingaskedto engagein novel problems.Althoughhe told me thathe andclaimedto knowonly a "bitof arithmetic," not havingstudied was "uneducated" mathematicsat school,he clearlyknewthathe was ableuse his experienceandhis of his environmentin a powerfulway. His understanding of geometrywas a understanding partof his life andhadfirmlinksto naturalphenomena.He did not deferto authoritybut thingsandof creatingorderandsense in his preferredto find his own way of understanding world(e.g., his use of the horizonas a "naturallevel"andhis creationof a way to "see" metricunits). was generallynot seen to be somethingownedand Summary.While"mathematics" used by the carpentersin theirwork,theydisplayeda rangeof attitudesaboutbeing intellectuallychallengedby questionsthatdrewupontheirmathematicalexperiences.While Mr. S welcomedthe intellectualstimulation,BrianandClive werequietlyparticipative. Jackoften felt threatenedby my questions.Judgingby the questionthathe raisedin Episode19, I thinkthatwithmorepractice,he wouldhavebegunto enjoytakingmore risks. SignificantCharacteristicsof the Carpenters'Mathematizing The workof Krutetskii(1976) was recognizedas beinga valuablemodelfor helping Krutetskiideclaredas one of thecarpenters'mathematizing. to describethe characteristics of his researchgoals "toreflectthe basicspecificcharacterof mathematicswithinthe frameworkof the demandsit makeson a person'smentalactivity"(1976, p. 86). In the preliminaryanalysisthattookplaceduringthe earlystagesof datacollectionfor the present of mathematical thought"(p. 87) were seen to be a study,Krutetskii's"basiccharacteristics of good matchfor the datathatI hadcollected.His categoriesdescribingthe characteristics

161 mathematicalthinkingbecamea usefultool for generatingquestionsandfor collecting furtherdata. Krutetskii(1976, p. 87) lists nine "componentmathematicalabilitiesthatarisefrom the basic characteristics of mathematical thought,"basedon the resultsof a 12-yearstudy concerningthe abilitiesof Russianschoolchildren.The followingis a summaryof thatlist. The pointsarenumberedK1 throughK9 for purposesof referencein the text thatfollows. material,to isolateformfromcontent,to K1. An abilityto formalizemathematical abstractoneselffromthe concretenumericalrelationshipsandspatialforms, andto operatewith formalstructure. K2. An abilityto generalizemathematical objects,to detectwhatis of chief importance,andto see whatis commonin whatis externallydifferent. K3. An abilityto operatewith numeralsandothersymbols. K4. An abilityfor sequential,properlysegmentedlogical reasoningthatis related to the needfor proof,substantiation, anddeductions. K5. An abilityto shortenthe reasoningprocess,to thinkin curtailedstructures. K6. An abilityto reversea mentalprocess. K7. Flexibilityof thought-an abilityto switchfromone mentaloperationto another-and freedomfromthe bindinginfluenceof the commonplaceandthe hackneyed. K8. A mathematicalmemory(i.e., a memoryfor generalizations,formalized structures,andlogicalschemes). K9. An abilityfor spatialconcepts. Krutetskiiworkedin the contextof schoolsandso confinedhis investigationsto the mathematicstaughtin the standardRussianschoolcurriculum.It shouldbe noticedthat Krutetskiidoes not believethatabilitiesareinnatebutthatthey are"developedby living andworking"(1976, p. 3). It shouldalso be notedthatKrutetskiiconductedhis researchin

162 a laboratorysettinganddid not takeintoaccountanyculturalor social factorsthatmay have influencedthe students. The episodesin GroupOneandGroupTwo wereexaminedto identifythe characteristicsof the mathematizing engagedin by the carpenters.Commonthemeswere searchedfor withineachgroupof episodesandacrossthe two groupsof episodes.Referring to Krutetskii'sworkhelpedto provideme with a vocabularyfor describingthe thatI discernedin the explanations,actions,and characteristicsof the mathematizing thatI describeare customarypracticesof the carpenters.Some of thecharacteristics developedinductivelyfromthe data.Otherswererecognizedbecauseof the observed similaritiesto Kruteteskii'sfindings.Thesewill be referencedto Krutetskii'snine componentsusingthe numbersKl to K9. Developmentof concreteproblematicsand the use of physicalactions.One of the is the way in whichthey of the carpenters'mathematizing most significantcharacteristics developedconcreteproblematicsin orderto generateexplanations.Problemsthatwere form(e.g., "Whendo you call a line "straight"? posedin a deliberatelydecontextualized and"Howcan you show thatsomethingis straight?"See Episode20.) evoked similar responsesfrombothJackandClive.Bothcarpentersdevelopedproblematicsinvolving pieces of timberanddescribedthe physicalactionsnecessaryto ensurethatthe plankswere straight. In Episode2, whenI askedJackhow he knewthata particularpointwas the center of a box lid, he linkedhis explanationto a familiarproblematicanddescribedthe actionshe would taketo solve thatproblematic.In Episode5, my questionabouthow to show thata squareruleis squaregeneratedthe posingof severaldifferentproblematics,all linkedto physicalactions. Ideas areframedby context,shapedby tools. Anothersignificantcharacteristicof the mathematicalideasdevelopedandusedby the carpentersis the way in which theirideas were framedby the contextof the workshopandshapedby familiartools. Theirphysical

163 actionsandverbalexplanationswereuniquelythosebelongingto a carpenter,but the ideas expressedwereuniversal.Forinstance,the conceptof straightnesswas elaboratedusing the planingmachine,trysquare,thicknesser,andstraightedgeas imaginedprops(see Episode 20). Mr. S used planks,a castor(smallswiveledwheel),string,a doorframe,anda window frameto constructconvincingargumentsaboutsquarenessandhorizontalandverticallines. Toolsand mathematicalformulae. Mr.S appearedto view a tool as somethingthat providedhim with a way to replaceseveralindirectactionswith one directactionneededto performa task.The tool couldbe usedas a one-stepaction,ratherlike a mathematical formulais one step thatcan replacea sequenceof steps. In Episode5, Mr.S drewan analogybetweenthe functionsof two differenttools. The analogyrevealedthe actionsthatareabbreviatedby the use of those tools. In his solutionshowingthata squareruleis square,Mr.S elaboratedthe actionsthatarereplaced by the use of a squarerule.The actionsreplacedby usingthis tool directlywere (a) to find a "level"(horizontal)line-say, a plank-and (b) to find a line-say, anotherplank-that is with the firstplank. "plumb"(perpendicular) In Episode8, Mr.S referredto the marking-gauge as being a "shortcut." I assumehe meantthatusingthe tool was moredirectthantakingthe two actionsthathe described. Tools are obviouslydesignedby carpentersto maketheirtaskseasier,to shortenthe labor is parallelto thatreferredto in K5. process.Thischaracteristic Mathematizingis implicitin actions.Manyexamplesof the mathematicalideas implicitin physicalactivitiesaregiven in the sectionthatdescribessome of the conventionalmathematicalconceptsunderlyingthe practicesof the carpenters.Fora detaileddiscussion,referto the heading"Conventional mathematicalconceptsthatwere embeddedin carpentrypracticein the workshop." Abstractionof mathematicalideasfrom actions.Most often,the mathematicalideas remainedimplicitin the actionsandwerenot articulated.However,in Episode2, Jackused an abstractionof his customaryphysicalactivityof squaringa tableto explainhow to find

164 the centerof a boxlid.In his explanation,he useda differentrepresentation of the idea that underliesthe physicalaction,thus"isolatingformfromcontent"(see K1). He represented the physicalactionof placinga narrowplankfromcorer to corer in a frameby referring to the lines drawnon the boxlid,thususingthe pencillines as "symbols"of the narrow plank(see K3). Generalizationto novelsituation.Jackgeneralizedthe solutionof the familiar problemof drawinga squarestarin orderto constructa new solutionfor drawinga starwith an unfamiliarshape(see Episode1). Althoughthe two starslooked"externallydifferent," he developeda set of rulesbasedon whathe saw as being"common"in the shapesof the stars(see K2). Briangeneralizedthe solutionof a familiarproblematicandused it as a modelto solve a novel problem,noticingthe elementsthatwerecommonin the two differentsituations(see K2 andEpisode16). In Episode13, afterconstructinga strategyfor drawinga five-pointedanda nine-pointedstar,Mr.S madea statementgeneralizingthe methodsfor drawingstardesignswith oddandeven numbersof points,extrapolatingfrom his particularsolutions. Thedevelopmentof convincingarguments.The logical andconvincingarguments thatbothJackandMr.S providedme in theirexplanations(see Episodes2 and5) showed the sequential,logical reasoningthatis relatedto the needin mathematicsfor proofand substantiation (K4). Innovativesolutionsandflexibleideas. Mr.S was able to solve the problematicsin whichhe was engagedin Episode5 in a numberof differentways. All his methodswere innovativeandcertainly"freefromthe bindinginfluenceof the commonplaceandthe hackneyed"(K7). Jackconstructedan alternativesolutionto his firstflawed attemptin Episode5 whenhe realizedthathe didnot havean adequatedefensefor the firstsolution. JackandClive, in Episodes17 and 18, constructednovel solutionsto an unfamiliar problem.

165 Theuse of spatialconcepts.In orderto overcomea lackof knowledgeof the conventionalmathematicalalgorithmsandpooralgebraskills, the carpentersrely on spatial visualizationandtheirown commonsense to solve problems(see Episodes16, 17, and 18). Thereis a generaltendencyto use spatialvisualizationeffectivelyto solve problems (see Episodes15, 16, 17, and 18). The carpentershavedevelopeda conceptof volumethatis richin its visual and spatialaspectsbutweakin the numericalcalculationof size (see Episodes16, 17, and 18). of the conceptof angles,althoughhe is Similarly,Jackhas a strongvisualunderstanding unfamiliarwith the conventionsof anglemeasure(see Episode20). Since the spatialand visual aspectsof understanding volumeandanglesareclearlymoreimportantthan measuringandattachingnumbersto the volumesandanglesin the workshop,it is not surprisingthatthe visualaspectsarewell developed,whereasthe numericalaspectsarenot. Constructing functional visual units. A significant characteristic of the carpenters'

spatialvisualizingwas theirconstructionof functionalvisualunitswhen solving problems. Brian'ssolution(Episode15) for cuttingup a plankintopieces suitablefor turningtable legs illustrateshis visualstrategyof sectioningthe plankandmentallymanipulatingand reconstructingunitsto the size andshaperequiredfor the tablelegs. He producedthe optimalresultof 10 legs, whichwouldnot be obtainedwithformalprocedures,such as using the standardalgorithmfor calculatingthe volumeandthenusing the division algorithm.His strategywas morethansimplya decompositioninto blocks,as he did when tryingto visualizethe calculationof areaandvolume(Episode16). In this case, the of units,the size andshapeof whichwere decompositionwas followedby a reconstruction chosen to suit a particularfunction. Similarly,Jack(Episode18) mentallydecomposedthe plankinto unitsthathe chose to suit a specific function-that of estimatingthe totalcost of the plankfroma unitof which the cost was known.

166 Comparingversusmeasuring.Thecarpentersfavoredthe use of comparisonrather thanmeasuring.Comparisonemployedtheirsensesof touchandsight andoftenresulted in a moredirectandaccuratemethodthanwouldhaveresultedfrommeasuring.This characteristicwill be discussedin moredetailin Chapter6.

Chapter6 RESULTSAND ANALYSIS: THE WORKSHOP-A CLASSROOMWHERE MATHEMATICS IS TAUGHT AS AN ACTION Introduction The aim of this chapteris to addressthe secondresearchobjectiveof the study-to ideasthatoccurredin the contextof my studythe teachingandlearningof mathematical in the settingof the workshop.In this chapter,the teachingmethodsof the apprenticeship carpentersaredescribedfrommy standpointas a learnercarpenter.New insightsinto my mathematicalknowledge,whichresultedfrombeingtaughtto mathematizelike a carpenter,areexplored.Thereis an examinationof the social andpoliticalissues involved in viewing the workshopas a classroom.Comparisonsaredrawnbetweensituationsin the workshopandsimilarsituationsthatexist in mathematicsclassrooms. In comparisonto the style of the previouschapterof results,whichgives a detailed descriptionof the analysisprocess,thischapteris writtenin the style of an essay that conveys a descriptionof my experiencesas a learner.The essay style was chosenfor two reasons.First,the datawerecollectedless systematically,relyingnot on the recordingof of my experiencesin the specific encountersbutratheron a historicalreconstruction generaleducationalmilieu.Second,theessay developedfromexaminingmy field notesin conjunctionwith my personaldiary,andso a narrativeformseemedmost appropriate. The TeachingMethodsof the Carpenters I shall be commentingon the teachingmethodsthatthe carpentersemployedin my own instructionandnot in the teachingof otherapprentices.Since teachingoccurredas a partof ongoingdailytasksandI was fully engagedin suchtasks,I was not able to observe any instructionthatthe otherapprenticesreceived.It shouldalso be notedthattechnically speaking,I was not reallyan apprenticeat all, althoughMr.S referredto me as "his apprentice."The title of "apprentice" designatesa certaintype of trainingthattakesplace

167

168 over a periodof aboutfouryearsandentailsattendinga technicalcollege for coursework for a three-monthperiodduringeachof the fouryears. This was the safestexperienceI haveeverhadas a novice. Mr.S was a wonderful teacherwho hadan uncannyknackof arrivingat therightmomentto renderexactlythe rightkindof assistance.He watchedmy actionsveryintently,even if he was not working nearby,andseemedto knowwhensomethingwas going wrong.His methodof dealing with this was to proceedwithaction-sometimes he did not say a word!He wouldtakethe tool or sandpaperfromme anddeftlyperformthe actionsneededto improvewhathadnot gone so well andthenwouldcontinueuntilI couldsee progressbeingmade.Sometimeshe would putmy handson the tool andthencoverthemwith his own andcontinueto workso thatI could"feel"how to use the tool correctly.He wouldpointout detailslike my wrist movements,tensionin my forearm,my feet pointingin the wrongdirection,andthe height at whichI was working(sometimestoo high).Muchof his adviceto me was a messageto be awareof my whole bodyandof therhythmof my movements.He stressedmy gripon the handtoolsoften,encouragingme to "tryto squeezeso hardthatyou will breakthe handle."Whenhe was satisfiedthatI couldproceedalone,andthis he wouldcheck on by standingbackandwatchingme for a while, he wouldleave. He gave me the impressionthattherewas nothingthatcouldgo wrongso badlythat it could not be rectifiedsomehow.Therewas no "wronganswer"in the eyes of Mr.S. Perhapssomethingwouldnot turnoutas intended,andmodificationsmightbe necessary. At worstone wouldhaveto startagainfromthe beginning,buteven this wouldbe seen in a positivelightbecauseone wouldhavehadmorepracticeat the task!His favoriteresponse to my anxiousquerieswhenI fearedthatI hadcut somethingtoo long or too shortor I had scrapedtoo hardwas to say, "That'snothing!"He wouldthentakeover anddemonstrate why I shouldnot be worried. Jackwas also a vigilantteacherwho kepta close watchon my progress.He gave me the firstlesson on how to use a chisel,showingme firsthow to hold the chisel,

169 how to cut awaythe starshapeon my describinghis actionsout loud as he demonstrated jewelry boxlid.He handedme the hammerandchisel andwatchedmy firstattempt critically,stoppingme afterone knockwiththe hammer.He warnedthatI was holdingthe chisel too tightlyandwouldhurtmy hand.I triedagain,andagainhe stoppedme almost immediately.I was crouchingoverthe chisel in a tenseposition-I shouldrelax.He took the tools anddemonstrated again.I triedto copy his stanceandmovements.This time he was satisfied,andI tookthe hammerandchisel awayto my own workbenchandcontinued to hew awaythe shapeof the starto be inlaidon theboxlid. On anotheroccasion,I was helpingClive assemblesome chairsandhadto use the electricdrillto makesome holes intowhichwoodendowelswouldbe drivento holdthe componentstogether.It was importantthattheholes be drilledat rightanglesto the seat components.Clive watchedthe positionof my handsandthe drillclosely andcorrectedany slantthatI inadvertentlyintroducedby pushingthe drillinto the correctpositionwhile I was working.Thisgave me the opportunity bothto feel andto see theresultof the correction. Therewas a greatdealof supportfor any artisanor apprenticewho neededit. The men watchedeach otherandpresentedthemselves,unasked,to give help or to offer a more convenienttool. Towardsthe end of my timeat the workshop,I was workingon a small chest, cuttingout dovetailjoints witha handsaw. Mr.S stoodandwatchedas I began, correctingthe positionof my feet andmy gripon the saw handle.Afterhe left, Clive came over andtook the saw fromme, advisingme to watchthe angleat whichhe held the saw to ensurethatthe cut was straight.I triedto copy the anglewhile he observedme makingthe next cut. A few minuteslater,Jackarrivedat my side andtook the saw. He cut along the side of one of the dovetails,urgingme to guidethe saw bladeby applyingpressurewithmy forefingerandthumb,as he was doing.Eachpersonhada differentpiece of adviceto offer anddemonstratedtheefficacy of thepracticethathe recommendedby demonstrating the

170 actioninvolved.As a learner,I was expectedto copy theseactionsandto understandtheir importancein the productionof workof an excellentquality. The Carpenters' Tools It is throughteachinga novicethe correctuse of tools thatthe carpentersteachthe mathematicalideasimplicitin theircraft.Throughthe actionsof usingsaws, drills,planing machines,squarerules,straightedges,slidingbevels,marking-gauges, sandpaper,andother of theconceptsof "straight," tools, an implicitunderstanding "horizontal," "vertical," and"proportion," "angles,""parallel,""symmetry," "congruence," amongothers,may develop.Forinstance,whenI askedJackandClive aboutthe conceptsof "straight"and "square,"theirdiscussionswerebuiltaroundhow to use the plane,the thicknessing machine,the trysquare,andthe carpenters'squarecorrectly. Theirtools wereextremelyimportantto the carpenterswith whom I worked.Only the artisanshadtheirown sets of tools, andthesewerekeptsharp,clean, andneatlypacked away whennot in use. Clive andJackhadlockablewoodentrunksthatthey left in the workshopovernight,while Mr.S broughthis tools alongdailyin a metalbucket.It was my responsibilityto watchMr.S's set of tools for two hourson Fridayswhen he went to mosqueat noon.He allowedme to use anythingI neededbutfrequentlywarnedme not to leave articleslying aboutunattended.SometimesI wouldfind thathe hadputsomething away thatI was still using,simplybecauseI hadmomentarilymovedawayfromthe place whereI hadbeenworking. I was given a small"toolbox"aftermy firstfew days at the workshop.Mr.S ingeniouslymadethe tool containerfromtwo plasticmilkbottleswith handleson the sides. He cut off the bottomsof the milkbottlesandpressedthemtogether,makingan airtight case thatwas usefulfor storingpolishingmaterials.Insidethe container,I storedall the tools necessaryfor my occupationas a sanderandpolisher-a largenail, two small nails, anda flat sharpenedstick,for cleaninginsidecarvings,a metalscraper,sandpaper,steel wool, a bunchof horsehair,andthe padof cottoncloth,tightlywrappedin plastic,thatI

171 used for Frenchpolishing.Aboutsix weekslater,I was given a woodenmalletthatMr. S carvedspeciallyto fit my grip.Whileobservingtheinlay workI was engagedin, he had noticedthatthe smallesthammerwas too heavyfor me to use comfortably,so he designed a moreeffective tool thatwas lightenoughfor me to handlewith ease. of his tools thatI did not see evidenceof richunderstanding Mr.S hada particularly amongthe youngerartisans.As discussedin the previouschapter,he appearedto view a tool as somethingthatprovidedhimwith a way to replaceseveralindirectactionswith the one directactionneededto performa task. Learningto MathematizeLike a Carpenter I noticedwithgreatinterestthatI was learningnew ways to mathematizefromthe carpenters.Therewerefertilegroundsfor learningmathematicalideasin the activitiesin which I took partas an apprentice.I was fortunateenoughto be includedin a wide rangeof activitiesduringthe monthsI spentin the workshop.In manycases, the mathematizingto which I was introducedin thecontextof the workshopled me to recognizeideas with which I was alreadyfamiliarandthatI hadpreviouslylearnedin an academicsituation. The practicalexperiencewas freshandexcitingandled me to perceivethe worldin new ways. The Developmentof My Sense of Touch The firstjob I was given was to sandan antiquewalnutstandinglampdown to bare wood andthento Frenchpolishit. This tookseveraldays. On finishingthe task,I reflected on whatI hadlearnedandnotedin my diary: I wonderedwhatI hadlearnedfrommy firstjob. First,an appreciationof the lovely curvesin the design.WatchingMr.S workwith the wood andstrokeit like he does, I have triedto copy him andnow knowshapein a slightlydifferentway thanI did before.I have alwaysknownshapewith my eyes, andoften only two dimensionally. Now I knowit withmy palmsandfingertipsin threedimensions.It is a qualitativelydifferentway of knowingshape.I also knowmoreaboutchoosing something(in this case a smallpiece of wood to wrapsandpaperaroundto get to the groovesandcurves)whichfits snuglyinto anothershape,like a practicalversion of some engineeringaptitudetests.It is essentialwhensandingsomething which is carved,to choose a shapewhichfits evenly into the gaps, otherwisethe varnishwill be rubbedoff unevenly.

172 Forthe firsttime,my sense of touchwas trainedto checkfor congruence,to comparelengths,andto exploreshapes.Mr.S madefrequentattemptsto enhancemy awarenessof my sense of touch.Whenteachingme how to Frenchpolish,he carefullytold me how thingsshouldfeel. Forinstance,the shellac-soakedinnerball of soft cotton coveredwith an outercloth thatis usedfor polishing"shouldfeel like the body of a live pigeon in yourhand."In betweencoatsof shellac,he taughtme to sandpaperthe piece of furniture,"becauseall the separategrainsof wood arestandingup-you can feel themwith yourfingers." Sandingandpolishingtabletopsgave me the opportunityto developa tactilesense of whatis bothsmoothandlevel. This sense helpedme to understandthe idea of being "straight"as elaboratedby the carpenters.The conceptsof beinglevel andsmoothwere relatedto the conceptof beingstraightandwereusedby thecarpentersto get a straight edge on a plankas describedin Episode20 in Chapter5. Whencuttingplanksto the samesize, Mr.S urgedme to "feel the edges with your fingers,it is the best way to tell if they arethe samelength."The sense of touchseemed to me to be intimatelyrelatedto the abilityto visualizeandwas thuslinkedto the developmentof spatialconceptsandthe abilityto design.I hadfurtherinsightinto this idea throughClive's explanationof how to use a smallplaneto get rid of some "bumps"on a chaircomponent.The parton whichI was workingwas the long piece of wood thatforms the side of the chairbackandcontinuesdownto formone of the backlegs. I askedhim how I wouldknow whenthe taskwas finishedsatisfactorily.He replied,rubbingthe wood reflectively,"Itmustbe a smoothflowingshape,it mustflow evenly, like this (movinghis handdownthe lengthof the wood)...itmustbe one continuousline." The furniturein the workshopwas alwaysbeingtouchedandstroked.On several occasionswhenI askedone of the carpenterswhetheror not he liked a piece of furniture, he wouldcommenton its tactilequalitiesas well as otherissues of designandconstruction. It occurredto me thatmy formalacademicmathematicseducationcouldhave been greatly

173 sense of enrichedby developingmy touch, not only in the elementarystageswherethe value of concretemanipulativesis now well recognized,but,for instance,in the studyof calculus,to "feel"a smoothcurve,or a pointof inflection.Exploringthe shapeof the hull of a yachtwith my fingertipsalongwith anexplanationof how calculusis used in the of calculus. designwouldhaveaddedconsiderablyto my understanding Learning How to "Use My Eye" Learninghow to "usemy eye"is anextremelyvaluablelesson for a novice carpenter."Havinga good eye" is synonymouswith beinga good carpenter,andbeing told thatone "hasan eye" for a tasklike designing,carving,or latheworkis a great compliment.Althoughno directallusionsweremade,it was clearthatthe carpenterstried to trainmy eye while theytaughtme the basicsof woodworking.In the mathematics classroom,I hadbeendrilledin the moreconventionaland"accurate" practicesof measuringlengthsof lines witha rulerandsizes of angleswith a protractoron the twodimensionalpage.In the workshop,my abilityto makejudgementsby sight was developed by beingencouragedto comparelengthsor to checkwhethera line was straightor a surface was horizontal"withmy eye."It was in this way thatthe carpentersbecameadeptat visual The conceptsof beingstraightandlevel weregiven concretemeaningin approximations. threedimensionsandwerelinkedto actions. On one occasion,I was helpingMr.S reconstructa largeantiquetabletop thathad brokenapartinto fourseparateplanks.Theplanks,whichwere slightlywarped,did not have paralleledges anddid not fit togethersnuglyany longer.Thecurvedlines andcracks hadto be preservedwhile the plankswerebeingfittedtogether,since creatingstraightline of the table.A kindof slow jigsaw joins wouldnot have preservedthe agedappearance puzzle was createdas Mr.S decidedwhereto alterthe shapeof the planksusing a small handplane.Sometimeshe chose to smoothawaya bumpthatwas situatedfar away froma troublesomegap. To my surprise,theremovalof a seeminglyunrelatedbumpcauseda snug fit wherethe gap once was! I couldnot discernhow he did it, norcould he tell me

174 whatto look for. The enigmatic"Isee it withmy eye" hadto suffice.He frequentlyasked me for "advice"on whereto plane.Sometimeshe wouldfollow my suggestionandwould be able to demonstratepracticallythatit did not work.He wouldthentell me his choice, thathis strategydidhelp. In this way, he triedto train performthe actions,anddemonstrate my eye. On anotheroccasion,Mr.S askedfor my assistancewhenhe hadto drillinto a tableleg to makea housingfor a verythickdowelthathe intendedusingto replacethe bottomof the leg thathadbrokenoff. He lay thetableon its side andaskedme to help him by guidingthe drillingso thatthe dowelhole wouldbe straight.He instructedme to squat down so thatmy eye was at the samelevel as the drillbit. My taskwas to "usemy eye" to line up the drillbit parallelto thefaroppositeleg andthenalerthim if the drillwas dropped down or pulledup in sucha way thatit was no longerin line with the oppositeleg. When he hadfinishedthe drilling,he invitedme to "sight"the leg fromthe bottom,to check that the hole was locatedin thecenterof the roundend face. He thenfixed on the bottompartof the leg andinvitedme to "sight"it bothfromthe bottomandthe side again,to check thatit had been fixed on straight.This skill is obviouslyveryimportantto him andhe spenta lot of time makingsurethatI was "eyeing"everythingcorrectly. Learningto use one's eye in theseways developsthe abilityto visualize,to "see in the mind'seye," andis an importantpartof designingto ensurebalanceandharmonyin a finishedpiece of furniture.Accordingto Peter,"Thereis a skill or an eye in being able to visualizewhatyou wantto buildthatmakesyou betterthansome"as a designer.On severaloccasions,Mr.S tookme roundthe workshopandcriticizedsome furnitureas a learningexercisefor my eye. He pointedout a carverwith big curvedarmsandsaid that"to his eye" the armsweretoo heavyfor the thinturnedlegs. The chairlooked top-heavy.He pointedout tablelegs thatweretoo lightandnarrowor too heavyfor the size andthickness used for the tabletop.He paidat leastas much,if not more,attentionto the aestheticquality of designfor furnitureas he did to practicalconsiderations.

175 Learning When to Measure and When to Compare As describedpreviously,aftersix weeks of sandingandpolishing,I moved into a new phaseof learningandbeganto helpin constructiontasksusingthe electricsaws and otherequipment.It was at this timethatI beganto noticethatI hada tendencyto use a measuringtapemorefrequentlythanMr.S considereddesirable.Mr.S madea clear and"measuring." distinctionbetweenthe two operationsof "comparing" By "comparing," I mean,for example,the act of checkingwhethertwo geometricor physicallengthsare congruent,or one is shorterthanthe other,or one is twice the lengthof the other.The two lengthscould be checkeddirectlyor couldbe comparedto a thirdlength.Checkingcan done by the eye andby feeling with thefingertips,anda high degreeof accuracycan be I meanassigninga numberto a geometricor physicallength. attained.By "measuring," The numberassignedto indicatethe lengthwill dependon the unitsused. My firstimpulsewas to reachfor a tapemeasurewheneverI hadto cut a piece of wood, comparelengths,or copy a design.In mostcases, however,Mr.S did not use a tape measureor a rulerbutpreferredto use his eye or to markoff the lengthrequiredwith his handsor fingers.Sometimeshe useda stickandmarkedoff a lengthusinga pencil andthen used this as a comparisontool. In manycases this practicewas muchmoreconvenientthan measuringwith a tapeor ruler.He tookgreatcareto tell me whenI did not need to use a tapemeasureandencouragedme to use his methods. Comparisonoftenturnedout to be the mostdirectandaccurateway of completinga task.Frequently,the assignmentof a numberwouldbe an unneccessarystep. Forexample, when squaringthe apronsthatsupporta tabletop,it was not necessaryto know the lengths of the diagonals.It was enoughto knowwhetherthey werethe sameor different.If they were different,the very stick beingusedto comparethe diagonalswouldbe cut andused as a rigidbarto pushthe apronsinto the correct"square" position.Thusmeasuringto the nearestfractionof an inch or millimeterwitha tapewouldsimplybe extrawork,as well as less accurate.

176 As anotherexample,whencuttingplanksof wood to matchthe size of one thathad alreadybeen cut, a comparisonoftendecidedwherethe new cuts shouldbe. Sometimesthe plankthatwas alreadythe correctsize wouldbe laid on top of a longerone anda pencil markmade.A squarerulewouldbe usedto "pull"the line acrosswherethe new saw cut shouldbe made.If thereweremanynew planksto be cut, a thickchunkof wood wouldbe nailedonto the woodentableof the tablesaw as a stopperat the left-mostend of the originalplank,while theright-mostend was againstthe saw blade.The new plankswould pushedup againstthe stopperas a frameof referenceandcut with no measurementstaken. Fingertipswouldbe usedto checkfor congruence. On one particularoccasion,describedin Episode7 in Chapter5, measuringactually got in the way of the taskat hand,andI realizedwith greatclaritywhatMr. S hadbeen tryingto teachme aboutthe differencebetweenmeasuringandcomparing.Mr S's solution illustrateshis understanding of proportional size andhis appreciationof the geometryof the designandof the importanceof the internalrelationshipsof the design.His solutionavoids the use of measurementsandrelieson a moredirectmethodof comparison.The elegance of his solutionlies in its simplicityandconcisenessandits relianceon the geometric of the piece of furniture.BrianandI fragmentedthe designandthe tasks, interrelationships while Mr.S took a holisticapproach. WhenMr.S did decideto measure,he knewthatthe unitwas his own choice. Sometimeshe selectedImperialunitswhenrestoringold Englishor colonialfurniture, becausethatis whatthe originalcraftsmanhadused.On otheroccasionshe used metric units.The metricsystemwas obviouslyless familiarto him,buthe hadworkedout his own conversionsystemfromthe Imperialsystemin orderto makesense out of it. Makingthe distinctionbetweencomparingandmeasuringdrewattentionto the idea of choosinga unit.I recognizedfrommy academicmathematicstrainingthatthese basic ideas underlietheconceptsof vectorspaces.Abstractvectorspacesdo not have units or coordinatesassigned,andtwo lengthsareaddedby juxtaposition(i.e., like puttingtwo

177 next each to out the to other figure lengthof a thirdplank).Scalarmultiplicationis planks done by magnifying(e.g., the inlaydesignwas magnifiedby one half).Choosinga unit (e.g., Imperialor metric)is the sameas pickinga basisfor a vectorspace.In the vector These arethings space,the coordinateswithrespectto the basisarethe "measurements." thatI have knownfor some time,butit was not untilI was encouragedto reflectuponmy experiencein the workshopthatI was ableto attacha practicalmeaningto my knowledge. Routine Activities in the WorkshopThat Did Not EncourageMathematizing I do not wish to claim thatall the activitiesin the workshoprequiredthe carpenters to do so. Muchof the productionworkwas to mathematizeor even gave themopportunities streamlinedandperformedin a rote,mechanicalfashion,wherespeed was moreimportant thanreflectingon the taskat hand.The carpenters'problematicthenwas to get the componentscut as quicklyandwith as close a matchas possibleto the templatesthatPeter made. The men were obedientto a set patternof action.The patternhaddevelopedover time, andwhile I am surethatthe workerscontributedto its development,thereseemedto be a set rhythmandstyle for somejobs thatdiscourageddeviation.Forinstance,once Peter had designeda chair,the artisanswouldcut outtemplatesof the componentpartsandmake andtest one prototype.If it was decidedthatproductionshouldproceed,the templateswere used to cut the componentsin bulk.Cuttingwas doneby one or two men (usuallythe apprentices),usinga manuallyoperatedelectricsaw. The chairswerejoined with mortiseand-tenonjoints (see Figure51).

Figure51. Mortise-and-tenon joint.

178 One of the apprenticeswouldmakethe mortisecuts requiredon certaincomponents using the electricoverheadmortisecutter.This was a mechanicaltaskwherethe depthof the cut could be set on the machineandthe lengthof thecut madein accordancewith two pencilmarks.Later,a thirdpersonwouldmakethe tenoncuts on the components,andit was thenmainlythatthirdperson'sresponsibilityto assemblethe chairs. The fragmentation of thesetasksmayhaveresultedin some advantagefor productionbutcertainlyled to otherdifficulties,since the firstgroupdid not have to take responsibilityfor the assemblyof the chairs.Theydid not participatein solving the problemof how the chairsshouldfit together.Thustheirproblematicdid not extend beyondthe cuttingof thecomponents.Thepersonmakingthe tenoncuts oftenfoundit difficultto decidewhathis fixed pointof referenceshouldbe beforestartingto cut andso maderepeatederrors.Whenthechairswerefinallyputtogether,a lot of time-consuming andlabor-intensivehandworkhadto be doneto get the chairsto meet the standard required.All in all, the cuttingof chaircomponentsseemedto be donerotelyandwithout muchcognitiveengagementon thepartof thecarpenters,ratherlike a studentmightapply algorithmsanddefinitionsrotelywhensolvinga routinealgebraproblem. The Workshopas a Classroom In exploringthe analogyof the workshopas a classroom,the social andpolitical issues come to the fore.I recognizedthatmanyof the barriersto learningthatarepresentin classroomswere also evidentin the workshop.Unlikethe structuredteaching-learning situationsthatone expectsin a classroom,thesituationin the workshopseemedat first glanceto be muchmorerelaxedandinformal.I hadexpectedto find the knowledgeof the carpentersaccessibleto everyonein theworkshop. However,a hierarchyexistedthatresultedin the poordistributionof learning opportunitiesacrossmembersof theworkshopgroup.A situationsimilarto the labelingof studentsas "moreable"and"lessable"thatoftentakesplacein mathematicsclassrooms was foundin the workshop.The laborerswerethe "silentmembers"(Volmink,personal

179 In the of this particularworkshop,the restrictions case communication,September1989). placedon the behaviorof the individualandhis access to knowledgewere exacerbated becauseof the pervasiveandperniciousatmosphereof apartheid. Externalauthorityfor certainideaswas seen to be vestedin Peter,or in a calculator, or in a better-informed friend.In some circumstancestherewas evidenceof a searchfor the "rightanswer."As in classrooms,productswere offeredfor the scrutinyof the groupandof the authorityfigureandfoundto be lacking.Criticismwas not alwaysconstructive. Traditionally,carpentryhas beenthe preserveof "coloured"men in SouthAfrica. Womenareneverhired,andblackmen in a carpentryworkshoparealmostalwayshiredto sand,clean, andfinishfurnitureandto sweepthe floor.They aregenerallyseen as a laboringandunskilledgroup,unlikelyto ever havethe desire,need,or abilityto learn woodworkingskills. An exampleof someonewho fell into this groupwas Sidney,who had been involvedwith restorationandconstructionof bothhousesandfurniturefor at least 20 years, 18 of these yearsin affiliationwithPeter.Sidneywas an expertat identifyingwood andat workingon percentagereproduction (whennew partsareaddedandare "aged"to matchthe originalpieces),buthe hadnevermadeany articleof furnitureof his own. A majorpartof his workinglife tookplaceduringthe heydayof "grandapartheid,"whenjobs for blackswere even moreseverelyrestrictedthanthey aretoday,andit would have been unthinkablefor him to havetriedhis handat woodworking.At the time of this research,he was an old manandwas unlikelyto startanythingnew duringhis last yearsbefore retirement. Peterhadtriedin his workshopto encourageeveryoneto workon his own woodworkingprojectsat lunchtimes.The menwere allowedto use wood offcutsto make whateverthey could andwereencouragedto use the machineryat lunchtime.Manyof the youngerlaborerstook advantageof this, andI saw severalsmallcoffee tables,hinged boxes, a breadbox,anda cabinetbeingbuilt.However,therewas no organizedplanto familiarizethe laborerswith the basicsof woodworking,andthey receivedvery little

180 encouragementandonly grudgingassistancefrommostof the artisans.Workdone by the laborerswas regardedwith some derisionby theartisans,andtheireffortsreceiveda lot of unconstructivecriticism. Therewas a sense thateverybodyknewhis placeandthattherewas little chanceto changethat.Eachpersonknewwhatthe expectationwas of his performanceby the group as a whole as well as by Peter.All the laborerswereblackmen.The laborerswerenot expectedto do woodworkconstructionbutwereinvolvedin sanding,polishing,spraying, andfinishingfurniture.Therewas an unspokendelegationanddemarcationof tasksthat resultedin a poordistributionof learningopportunities for the blackworkers. The artisansandapprentices(withone exception)werewhiteand"coloured"men. The one exceptionto this was Leonard,a youngblackmanwho was Sidney'snephew.As describedin Chapter3, it was clearthatLeonardachievedhis goal of becomingan apprenticethrougha combinationof assertivenessandhardwork,backedup by a set of unquestionablyexcellentacademicresults.Peterwas obviouslyconcernedwith the welfare andprogressof all his workersandclearlyactedin Leonard'sbest interests.However, Leonardwas not soughtoutandofferedan opportunitybutseized thatresponsibility himself. no ambition given to Brian,who haddemonstrated Comparethis to the opportunity or interestin his job, hadhabituallyarrivedat worklate, andhadan abysmalschool record. Petersaid thathe hadbeensuspiciousof Brianat first,since he was a youngwhiteman reservedfor "coloured"people.As a white wantingto join a tradethatwas traditionally opento him. Peterobviouslyexpectedmore person,Brianhadso manyotheropportunities to achievemore.As it turnedout, Brian fromBrianandso gave him a goldenopportunity did have a particulartalentwiththe lathe(thatmayhavebeenlatentin othersnot given similaropportunities)andwas soon promotedto artisanwithoutany formaltechnical training.

181 This storywas not intendedas a criticismof Peter,who hadcreateda particularly harmoniousworkplaceandwho stroveto give opportunitiesto all the workmenwherenone hadexistedin the past.The pointis thatSouthAfricansocietyhas differentexpectationsof its people,basedon race,andthis to a largeextentdeterminesaccessto knowledgeand opportunity,even in an informaleducationalsettinglike the carpentryworkshop. As a whitewoman,I was accordeda privilegedstatusandwas allowedan opportunityto studythe craftandcultureof carpentry.However,the positionof bonafide workerwas by no meansautomaticallyattained.I hadto workhardto get beyondthe stage of being treatedpolitelyas a visitor,to earnthe confidenceof the carpenters,andto be includedin generalworkshopactivities.Carpentryis consideredroughandheavy work, suitableonly for a man.My genderclearlyraisedsome doubtsaboutthe sortof workthatI wouldbe capableof whenI firststartedat the workshop.As time went by, I demonstrated not only thatmy handscouldget dirtybutalso thatI was ableto do fairlyheavymanual labor,andI was acceptedas a temporarymemberof the team.However,wheneverany workcame up thatdid notrequireheavymanuallaborandthereforelookedvaguely "feminine,"it was automaticallyassumedthatI wouldbe the best personto do the task.For instance,I spenta week liningdrawersandcutlerytrayswith greenbaize,replacedthe greenbaize on an antiquecardtable,andwas askedto do the fine gilt paintingon some chairs.As in Leonard'scase, it took assertivenessanddeteminationto ensurethatI did not remainstrictlywithinthe narrowconfinesof society'sexpectationof tasksthataresuitable for a woman. In the workshopsetting,wherequality,profit,andproductivityarethe watchwords, anyonewho can enhancethesein any way will be rewarded.Ostensibly,all the experienced craftsmenwerewilling to sharetheirskilledknowledge.Groupdiscussionsof woodworkingproblemswerefairlycommon,andin generalthe carpenterspreferredto try to solve problemstogetherbeforetakinga questionto Peter.The instructionof apprentices, whetheras a formalor informalrequirement, is partof thejob of a mastercraftsperson.

182 However,therewerecomplexsocial obstaclesat all levels. I foundthe artisans hesitantaboutany tasksthatthey designatedas beingin a "specialist"categoryand "above"them,even whentheywerecapablecompletingthe task.At the same time, these artisanswere oftenunhelpfulandcriticalof laborerswho neededassistancewith their privatewoodworkingtasks.Therewas a prevalentperceptionthatcertaintasksbelongedto certaingroupsof people,andtherewas resistanceto any changesin this fixed structure, even thougheveryonewouldbenefitin the long runfromsuchchanges. The issues of powerandaccessto knowledgearethornytopics,andno quickand easy solutioncan be expected.This is as truein the academicworldas it is in the carpentry workshop.Theseobservationshavetemperedmy optimismaboutthe "unfreezing"of mathematicsin traditionalcrafts,or the introductionof curriculathatfocus on "people's of Westernmathematics. mathematics" leadingto an automaticdemocratization Summary The accountof my learningexperiencesin the workshophighlightsthreemain themes.First,I commentedon the teachingmethodsof the carpenters,who were operating modelwill be further modelof instruction.The apprenticeship withinthe apprenticeship discussedin Chapter7. Second,workingas an apprenticeprovidedme with the practical experienceof learningmathematicsthroughactions.The carpenterscoachedme in the use of woodworkingtools andin the use of traditionalpracticessuch as "usingmy eye." In the processof learningtheseactions,I also learnednew mathematicalideas. The thirdthemeconcernsthe widereducationalmilieuin the workshop.As in classrooms,thereexistedhiddenbarriersto learningthatresultedin a poordistributionof opportunitieswithinthe group.Somememberswererelegatedto fragmentedandrepetitive to mathematize.They wentthroughthe motionsof tasksthatrobbedthemof opportunities the activitiesbutwere not providedwith theexposurethatencouragedfruitfuloutcomes. Sometimesit was a subtlenuancethattookawaythe opportunity,sometimesa more deliberatestep.

Chapter7 DISCUSSION Introduction This chapteris organizedintofive sectionsthatdiscussthe variouspractical andtheoreticalimplicationsof the presentstudy.I will beginby discussingsome epistemologicalimplicationsfor mathematics.The secondsectionfocuses on the educationalimplicationsof the study.Thethirdsectionexaminessome of the methodologicalissues relatedto the study,whichmay be of benefitto those interestedin In the fourthsection,I presentsome suggestionsfor further exploringethnomathematics. research.The chaptercloses with a sectionentitled"Epilogue." EpistemologicalImplications:Knowingin Action A questionthatI have askedmyselfcontinuouslythroughoutthe presentresearch, and one thatI thinkshouldbe askedmoreoftenby mathematicseducatorsis, "Whatis mathematics?" Whileno definitiveanswerswereexpectedto emergefromthe present research,the accountof the carpenters'activitiesmayperhapscause a few dedicated formaliststo ponderthe abovequestionandthe questionof whatit meansto do mathematics. I have arguedthatthecarpenters"domathematics" in the workshop,accordingto the criteriathatI set up in Chapter1. Theirmathematicshas some uniquecharacteristics thataredifferentfromthe mathematicsthatwe areaccustomedto seeing in textbooksand thatshow how the cultureof the workshopplacedan indeliblestampon the mathematical ideas thatwere developedthere. First,actionis vital in theirmathematics.Thereis tacitmathematicalknowledgein theirphysicalactions.Physicaldemonstrations formedpartof activeexplanations,andthere was movementandan involvementin the environment.Forexample,Mr. S's handactions when demonstrating how the new escutcheonwouldbe flippedover to matchthe existing one showeda tacitknowledgeof reflectivesymmetry,andhis actionsof turningthe

183

184 of rotational separatepetalsin the designon the inlaidtableshoweda tacitunderstanding symmetry.Jack'sactionswhensquaringa box carriedimplicitknowledgethatthe diagonalsof a rectangleareequal. Second,reflectionon actionoftenleads to the articulationof tacit knowledge.For example,Jack'sreflectionson the actionshe took whensquaringa tableled to his boxlid,while Clive's reflectionson explanationof how to findthe centerof a rectangular the actionshe took whenplaningthe edge of a plankandusinghis eye to check thatthe edge is straightled to his discussionon the conceptof straightlines. The resultsof the presentresearchsuggestthatphysicalactions,reflection,and knowledgeof mathematicsarelinkedin the practiceof carpentry.Otherresearchershave linkedaction,reflection,andknowing,in variouspracticalsettings.In particular,the present researchwith carpenterscan be seen as anextensionof the workof Schon(1983, 1987), who arguesfor a new epistemologyof practice,claimingthatsuccessfulprofessional practitionerslike architectsandpsychologistsdisplaya competencethatdoes not depend andwho engagein uponbeingable to describewhattheyknow("tacitknowing-in-action") at work.The presentstudydiffersfromthe workof Schonin its focus "reflection-in-action" on thephysicalactionsandreflectionsof manualworkersat work,while Schonexamined the less obviouslyphysicalactionsof, for example,architectsat work. The workof Gerdes(1986), who arguesthatmathematicsis "frozen"in traditional craftsandculturalartifacts,is also extendedby the presentstudy.One of the shortcomings of the workof Gerdesis thatfew guidelinesaregiven as to how one practicallygoes about ideasto whichhe refers.The examination,in accordance "unfreezing"the mathematical with the theoreticalframeworkI sketchedout,of the actionsandthe reflectionsof people relatedto woodworkingpracticesthathave who arephysicallyengagedin problematics been "handeddown"(notethe physicalactionimplied)fromfatherto son, representsa proposition. practicalsuggestionfor Gerdes's"unfreezing"

185 I claimthatthe evidenceprovidedin thepresentstudysuggeststhatmathematical of this claimrequiresa knowledgecan be implicitinphysicalactions.Acknowledgement shift in the acceptedepistemologyof mathematics. The thirduniquecharacteristic of the carpenters'mathematicsis thattheir explanations,discussions,andproblem-solvingactivitieswere deliberatelylinkedto concrete,contextualproblematicsthatwereusedto providephysicalprops,mentalmodels, andanalogies.QuestionsthatI posedin decontextualized formswerere-interpreted by the formswith"real-life"problematicsto solve carpenters,who replacedthe decontextualized instead.Forexample,whenI askedJackhow to find thecenterof a rectangularplank,he derivedhis explanationfroma discussionon how he would"square"a table,andwhenI askedMr. S how to checkwhichof two differentsquarerulesweretrue,he developedthe problematicof designinga methodto makea squarerule.I claimthatthe carpenters' contextualization of problematicsdoes not implythattheirmathematicsremainedat the concretelevel and was neverabstract.Examplesshow thattheywere able to generalize solutionsto new situations,andto discussabstractideas,butthattheyused different symbolsfrom thoseconventionallyacceptedas mathematicalsymbols.Contextualizing problematicsfor explanationsandteachingresultedin a rich andvivid portrayalof concepts frommy pointof view as a learner. The fourthuniquecharacteristic is thata significantrole was playedby the ideas.Manymathematicalideas were carpenters'tools in the shapingof theirmathematical expressedthroughthe mediumof a woodworkingtool. I claimthatthe carpenters'tools were used as importantsymbolsin theirmathematics.As mathematicalsymbolsare manipulatedto illustrateexplanationsin formalmathematics,so the carpentrytools were physicallymanipulatedto communicateexplanationsin the workshop.Some of the tools "stoodfor"actions:for example,Mr.S gave a demonstration of the actionsabbreviatedby the carpenters'square.

186 describea mathematicsthatdoes not fit As a collection,theseuniquecharacteristics the conventionalmold.I arguethattheconventionalepistemologicalview of mathematics shouldbe broadenedto admittheactivitiesof the carpentersas an authenticexperienceof makingand usingmathematicsin thecontextof theworkshop. An exampleof how a broaderview was takenof whatshouldbe allowedto countas valid mathematicsmaybe foundin the workof Confrey(1990). As a resultof the schemesthatdid not fit the conventional examinationof studentlearning,multiplication modelof repeatedadditionwerenoticed.Insteadof theschemesbeing dismissedas deviant andin need of remedy,theywereexploredfor theirstrengths.As a result, Confrey(1990) putforwardan alternativeconceptof multiplicationdescribedas "splitting."The "splitting" modelis claimedto be a moreappropriate explanatorymodelfor exponentialand to, the customarymodel logarithmicfunctionsandis independentfrom,butcomplementary of multiplicationas repeatedaddition.Broadeningourview of mathematicswill thus requirea toleranceof ideasthataredifferentfromthe norm. Severalexamplesof the carpenters'mathematizing (e.g., Mr.S's methodof situatingdovetailsevenly alongthe edge of a plank,andhis methodof placingholes along the frameof a chairseatfor the threadingof rattan(andfor the threadingof riempie)were foundto be consistentwith the "splitting"modelof multiplication.I believe thatthe argumentsfor challengingthe acceptedconceptionsof mathematicswill be strengthenedif links continueto be foundbetweenstudents'mathematicsandmathematicsas it is practiced in everydaysettings. As a furtherchallengeto theepistemologyof mathematics,I have suggestedusing (attributedto Weyl, see the term"mathematizing." By usingthe verb"mathematizing" Footnote3) in this work,I havetriedto emphasizethe experienceandprocess of mathematics.Mathematicsis somethingdynamic,alwaysbeingaddedto or subtracted The noun"mathematics" usuallyimplies from,somethingin whichpeoplecan participate. somethingunchanging,set, andstatic;it is not anexperience;it does not invite

187 it is a cut-and-dried bodyof knowledgethatcan be learnedaboutbutnot participation; experiencedin action. Creatinga new wordin theformof a verbfromthe conventionalformof a nounhas been done in otherareasas well, suchas in the disciplineof linguistics,where"languaging" has beencreatedfrom"language"(Maturana & Varela,1987),andin the disciplineof law, where"lawyering"has beencreatedfrom"law"(Schon, 1987).In bothcases, the invention of the new wordrevealsa desireto go beyondthe strictlimits of a highlystructuredand rule-bound,abstractdomain.The factthatthe new wordsareverbsformedfromnouns signifies the desirefor actionanda rejectionof the sense of an "accomplishedfact" conveyedby the noun.Becker(1989), in discussinglanguage,makesthe following statement: Linguistsandothershavetendedto see languageas an abstractstructureand/or system;a set of generalrulesof constraintsoperatingon or withina lexicon, i.e. a grammaranda dictionary.In short,as an "accomplishedfact"... grammarsand dictionariesarethingsmadeby linguistsandhavelittleto tell us aboutwhathappens when we "language."Theshiftfromnounto verb,fromlanguageas an abstract thingof some sortto languagingas an act, ... make(s)a cleardistinctionbetween language,a descriptionproducedby an observer,andlanguaging,an act experiencedin statunascendi.(p. 6) It is with intentionssimilarto thoseimpliedby the writingsof Beckerandof Schon thatI have suggestedthe shiftfromthe conventionalnoun"mathematics" to the verb to signify a shiftfrommathematicsas an abstract"accomplishedfact" to "mathematizing" mathematicsas an action,somethingthatpeopledo, somethingnot completed,butstill in process.On the basis of the resultsof thepresentstudy,I arguefor an epistemologyof mathematicsthatis able to accommodatetheword "mathematizing." EducationalImplications Threemaineducationalimplicationsof the presentstudyarediscussed.First,I a conceptandreflecton the ways in which explorethe meaningof understanding competencein a conceptis oftenjudgedin classrooms.Second,I commenton the possible use of the apprenticeship modelas a viablemodelof instructionfor mathematicseducation

188 in schools, andlast, I exploresome of my emergingconceptionsaboutthe significanceof consideringmathematicseducationas praxis. The Meaning of Understandinga Concept A significanteducationalquestionthatmy experiencewith the carpentersraisedin a concept?"We often describea conceptby my mindis, "Whatdoes it meanto understand invokingthe standardalgorithmfor its calculation,so "volumeis lengthtimes widthtimes height"seems an adequateexplanationfor the concept.It is easy to confusethe two separateissues of the meaningof theconceptof volumeandof knowinghow to calculate volume,andto thusacceptthatknowledgeof thealgorithmis sufficientto provean of the concept. understanding The presentstudyprovidesevidencethatunderstanding theconceptof volume meansmuchmorethanknowingthe conventionalalgorithmusedfor calculating V=LxWxH, whereV representsvolume,L representslength,Wrepresentswidth,andH represents height.Peterwas mistakenwhenhe toldme thatthecarpentersdid not understandthe conceptof volume.The fact thatthe carpenterswereunfamiliarwith the standardalgorithm for calculatingvolumesimplyimpliedtheirlack of expertisewith the acceptedconventions of theconceptof volumeitself. surroundingthe conceptandnot a lackof understanding By school standards,thecarpenterswouldnot havebeenconsideredto have an of volume,since the algorithmwas unfamiliarto them.Yet they understanding of the concept,basedon spatialvisualization, demonstrateda richandrobustunderstanding andnot on numbers.Theywereableto use theirconceptof volumein verbalandin to generateexplanationsandto solve problemsrelevantto their physicaldemonstrations purposesas carpenters. I wonderhow the problemthatBriansolved,concerninghow manylegs he would be able to makefroma certainplankof wood, wouldhavebeensolved by thosepeoplefor whomthe algorithmfor calculatingvolumeleapsto the fore at the mentionof the word

189 "volume."An attemptcouldbe madeto solve the problemby workingout the volumeof the blockrequiredfor each leg anddividingthis into the volumeof the whole plank.The answer(11.15 legs) producespracticalproblems.First,this methoddoes not providea schemefor cuttingthe planks,andsecond,althoughthe volumeof the offcutsmay be enoughfor an eleventhleg (Brian'sschemeallowedhim to cut ten legs), it wouldbe undesirableif not impossibleto makea leg fromscraps.I am not disclaimingthe importanceof beingableto calculatevolumenumerically.Rather,I am claimingthatthe the conceptof volume(i.e., usingthe usual demonstrationof competencein understanding algorithmto calculate)looksratherthinin comparisonto the knowledgedisplayedby the carpenters. As pointedout by Schoenfeld(1988, p. 2), mathematicseducatorsneed to use than'masteringsymbolmanipulation "broaderdefinitionsof mathematical understanding procedures',"since thereare"dangersto the narrowassessmentsof competencythatare currentlyemployed."In manyclassrooms,competencecan be adequatelydisplayedsimply by imitatingthe proceduresof the teacherandby repeatingsymbolicmanipulationsin learnedalgorithms.Studentsareoftennot encouragedto buildmultiplerepresentations of concepts.The examplegiven by Saxe (1988) (see Chapter2), who exploredthe problem solving of carpentryapprenticesandmastercarpenters,furtherillustratesmy point.The carpentryapprentices,who hadall learnedthe algorithmto calculatevolumein formal mathematicslessons, wereunableto solve the problem"Howmuchwood neededfor five beds?"andofferedabsurdanswers.The mastercarpenters,who hadhadconsiderablyless formalschoolingandhadlearnedthroughapprenticeships andexperience,were all able to solve the problemwithease andwithoutrecourseto the algorithm.This suggeststhat mathematicsteachersoughtto providecontextsfor studentsthatdemandmorestrengthand thatsuffices in most classroomsat versatilityfromconceptsthanthe symbolicmanipulation present.Theresultsof thepresentresearchunderscorethe recommendations of the

190 constructivistliteraturethatstudentswouldbenefitfrom being encouragedto constructand use multiplerepresentations of theirconcepts. Apprenticeshipas a Model for Instruction Manystudentsfind it difficultto be courageousaboutlearningsomethingnew in mathematicsclassrooms.Often,theirobjectiveis to get the rightansweras quicklyas possibleandto move on. The product(i.e., the rightanswer)in this case is moreimportant thanthe process.Theapprenticeship system(accordingto mypersonalexperience) providesa contextin whichit is safe to be daringaboutlearningnew things.Thereare no rightor wronganswers,since theprocess is part of thesolution.The aim is for faultless performance,butthis is not expectedto occurwithoutmistakesalongthe way, without effortsthatfall shortof the desiredgoal. Practiceis valuable;the buildingof experience, with success andfailurealongthe way, is the way in whichlearningoccurs.Collaboration forjointproblemsolving,for sharingnew ideas, for is important:for encouragement, teachingnovices, forreachingnew levels of developmentwith the help of a moreexpert colleague.Throughcoaching,the masterteacherprovidesa modelof a competent practitionerfor apprentices.Apprenticesareencouragedto appreciatethe productsas well as the processof creation. since it is traditionallythe carrieswithit the idea of craftsmanship, Apprenticeship crafts(thoughtof as nonacademic)thataretaughtvia apprenticeships. (Note, of course,that butthis has beenre-labeledas "internship"). medicaldoctorsservea kindof apprenticeship, in carpentrythatmademe reflect Jackmadean interestingcommentaboutcraftmanship moredeeplyabouttheeffects of fragmentingtasks,formalizingcertainsteps, and encouragingrepetition.He said: Herewe areworkingwith thesolid woods,butthereis a completedifference betweenus andotherfactories.Theretheyworkwith laminatedchipboard.A completedifferentstory.Whenyou workin anyotherfactory,whichI have seen andI know,if a manscrewsin screwstoday,thenhe screwsscrewsfor the restof his life. If he doesjoiningup of drawers,thenhe makesdrawersfor the restof his life. Hereyou workfromthe floor,rightto the finish of thatparticular craftsman. item, andthatis whatI call a cabinetmaker...a

191 Jackimpliedthata genuinecraftsmanperformsa taskthathas integrityandthatis neither fragmentednorcompletelyrule-bound.The craftsmanwoulddesigna planof action.The taskmay have differentparts,buteachpartwouldbe integratedinto the whole. Above all, the processwouldbe satisfying,meaningful,andengagingto the carpenter. It is instructiveto look at theseideasin the workshopsetting,farremovedfroma conventionalclassroom,andto reflecton whatwe, as mathematicseducators,can learn fromJack'sdescriptionof whattranspiresin differentworkshops.Whenfragmentation, formalization,andrepetitiondeprivecarpentersof the opportunitiesto workas craftsmen, to mathematize?Fragmentation, are they simultaneouslydeprivedof the opportunities formalization,andrepetitionarethe hallmarksof mathematicsinstructionin many classrooms.Whatwouldbe analagousin the classroomto the studentsworkingas "craftspeople"in mathematics,to working"fromthefloor, rightto thefinish"? The problemsthatstudentscurrentlyaregiven to solve in class wouldprobablybe describedas "laminatedchipboard."Whatwouldcountas "solidwood"in the realmof mathematics problems? Mathematics Educationas Praxis The experienceof conductingmathematicseducationresearchin a practicalwork contextin whichto considerthe meaningof praxis.Lefebre settingcreatedan appropriate (1968) definespraxisas comprisingtwo mainideas.Firstis the notionthatpraxisinvolves actionas opposedto philosophicalspeculation,andsecondis the view thatthe fundamental characteristicof society is materialproductionto meetbasicneeds.He suggeststhatman primarilyacts on the naturalworldandonly secondarilythinksaboutit. In an educationalcontext,Fasheh(1988, p. 4) has arguedthat"educationperceived as praxisis the oppositeof hegemoniceducation."He describespraxisas the combination of the social, cultural,andmaterialconditions,in constantinterplaywith reflectionand action(Fasheh,1988).Fasheh(1988) continues: Hegemonyis understoodhereas a formof domination.... Hegemonyis always linkedto an ideologywhichreflectsthe mannersandinterestsof the invadersand

192 theirculture ... Crucialto the hegemonicrelationshipis the invaded'sbelief that the lifestylesandvaluesof thehegemonicgroupareinherently,naturallyand objectivelysuperior.Hegemonyis successfulwhenthe invader'sideology is ... assumedto be universalandsuperior;when,like the mathematicsI valued,it is believedto transcendclass, gender,culture,andnationalboundaries.(p. 3). Fasheh'sreferenceto the mathematicshe valuedformspartof his argumentthathegemonic mathematicseducation"rendersmarginal,deemsinferiorandmakesinvisible"(1988, p. 3) mathematicslike thatpracticedby his illiterate,seamstressmother(i.e., mathematicsthat does not fit the dominantview of mathematics). Theperspectiveof mathematicseducationas praxis is consistentwith the shiftin the epistemologyof mathematicsthatis arguedfor in theprevioussection.Mathematics educationneedsto extendthe invitationto studentsto participatein the experienceof In contrastto thenotoriousreputationthatmathematicshas earnedfor being mathematizing. an obscure,difficult,andirrelevantsubjectto be avoidedat school,mathematicseducators need to convincepeoplethattheexperienceof mathematizing can be a vibrant,exciting, andempoweringfacet of theirlives. If mathematicseducatorscontinueto ignorethe mathematicsthatis generatedin everydaysettingsandcontinueto workstrictlywithinthe narrowframeworkof formal mathematics,mathematicswill continueto be uninvitinganddisempoweringto the majority of students.Weneed to bringnonconventional mathematicsinto classrooms,to valueand to buildon the mathematicalideas thatstudentsalreadyhavethroughtheirexperiencesin theirhomesand in theircommunities. MethodologicalImplications The presentstudyoutlineda methodologicalprocedurefor conductingpractical ethnomathematics research.Formathematicseducationresearch,particularlyfor exploring the approachof ethnographyprovidesa useful questionsaboutethnomathematics, approach,with an extendedperiodof methodologicalframework.An ethnographic fieldwork,allowedme the timeto exploretheideasof the carpentersas they were engaged in theirdailytasks.Becauseof the lengthof timeI spentat the workshopandbecauseI

193 participatedas an apprenticeandbonafide worker,I was able to observespontaneous I was affordedthe opportunityof developing problemsolving in context.As as participant, of thecontextof the workshop,whichassistedin the a thoroughunderstanding I observedandin whichI took part. of the mathematizing interpretation The master-teacher-apprentice relationshipthatwas nurturedbetweenMr. S andme provedto be a fruitfulsourceof data.Mr.S felt responsiblefor my educationas a learner carpenterandstroveto provideme with a wide rangeof woodworkingexperiencesduring the time I spentat the workshop.As his pupil,I was in a privilegedpositionin thathe taughtme certainthingsthathe keptsecretfromthe othercarpenters.He guardedsome of his knowledgeclosely andinstructedme clearlyon whatcouldandcould not be mentioned in frontof the otherworkers.His acceptanceof me as his pupilcontributedsignificantlyto my credibilitywith the othercarpenters,in view of his seniorposition.Consequently,my abilityto collect datawas enhancedin a way thatmay not have occurredthroughotherdata collectionprocedures. The negativeside of this intenseteacher-apprentice relationshipwas thatit was sometimesdifficultfor me to workwith the othercarpentersat a time when they were doing somethinginteresting.I hada responsibilitytowardsMrS as my specialteacherand respectedhis allocationof tasksto me. My experienceswith the presentresearchpermitme to soundsome warnings, however.Ethnographic researchis particularly demandingandtime-consuming,since it requiresextendedperiodsof fieldwork.Locatinga suitablesite for conductingfieldworkis no simpletask,since theremayexist unanticipated barriersthathinderentranceto the field. anda suitablesite aretwo of the firsthurdlesto be Findingan accommodating"gatekeeper" researcha risky and cleared;thereareseveralmorehurdlesthatmakeethnographic challengingmethodologicalchoicefor a dissertation.The questionof whetheror not one will becomea participantcannotbe judgedaheadof time.If one is not designatedas a participantby the groupbeingstudied,themainmethodof ethnographicdatacollectionwill

194 be observation.Althoughobservationdatamay sufficefor some researchdesigns,the presentstudywouldnot havesucceededhadI not been acceptedas a participantby the carpenters. Limitationsof the Study and Suggestionsfor Future Research The resultsof this studyhavegenerateda numberof questionsworthfurther research.The perennialquestionsaboutwhatmathematicsis andwhatit meansto do mathematics,remain,of course,unanswered.Askingthesequestionsin nonacademic settings,however,will producean intriguingset of answers.It may be interestingto replicatethis studyin a differentcontextandto comparethe characteristicsof mathematizingacrosscontexts. Furtherresearchon the findingthatphysicalaction,reflection,andmathematical knowledgeareinterlinkedshouldbe conducted.Of particularinterestherewouldbe the developmentof curricularmaterialthatexploitsthe actionsassociatedwith mathematical concepts.Forexample,studentscouldbe taughtgeometrythroughinvolvementin practical projects. In reflectingon the significanceof educationas praxis,I came to realizethe significancealso of researchas praxisandto considerthe limitationsof the presentstudy. PattiLather(1986) has defined"researchas praxis"as the act of researchersinvolvingthe people whomthey researchin a democratizedprocessof inquiry,characterizedby negotiation,reciprocity,andempowerment. Afterfive monthsof intensivestudyat theworkshop,as trustdevelopedbetween some of the carpentersandme, opportunities beganto arisewhereit seemedthatI could explorethe carpenters'mathematicswiththemin a way thatwouldbe empoweringto them. In particular,Jackhadbecomecuriousaboutsome ideasthathe hadandhadovercomehis initialdesireto hide whathe was unsureof fromme. However,I resistedevery opportunity thatarosebecauseI felt thatI hadto remainfaithfulto my methodologicalframework,andI did not wantto compromisemy positionas an apprentice.My role as an apprenticewas

195 centralto the argumentI hadputforwardfor beingable to describethe mathematizingI was learning,andI wantedto remainthe learnerandnot takeon any partof a role as teacher. Thusmy researchcouldnot be describedas praxis.My own actionsandreflectionsin the contextof the workshopandthe natureof the interactionsthatI could have with the carpenterswere severelycurtailedby my adherenceto my chosenmethodologicalstance. Similarly,in thecase of the laborers,I notedthe inequitiesof theiraccess to learningin the workshopbutcould not use my researchto tryto raisetheirawarenessof this. to anybodyinterestedin pursuingpracticalresearchin My recommendations wouldbe to designa studythatcouldconsciouslybe used to help ethnomathematics participantsunderstandandchangetheirsituations. Epilogue Once in a while, one meetssomebodywho has a genuinelyheartfeltregardfor knowledgeanda love of teaching.Everygood educationalinstitutionhas at least a few teacherswho wouldbe describedthus. Mr. S was such a person.His classroomwas unconventional-a carpentry workshop.His educationalbackgroundwas unusualfor a teacherof such stature-after finishingelementaryschool, he discontinuedhis formaleducationandbeganan informal with his father.His topic-carpentry,or was it mathematics?(I learneda lot apprenticeship of bothfromhim.) For44 yearshe hadbeendesigningandconstructingfurnitureandrestoring Mr.S lived his mathematics.It was a antiques.And he was an insightfulmathematician. partof his actions,a partof his environment.Fromhis homeon the mountainside,he looked out overthe city andthe harbor.He saw the horizonas the straightline thatGod gave us, wherethe sea meetsthe sky. Continuingthe discussionaboutstraightlines, he talkedaboutthe rays of the sun, the verticalupwardpathof the bubblesthatemergefrom the mouthsof the goldfishin his fishtank....His mathematicsis unfortunatelynot yet found in children'stextbooks.

196 He liked to surroundhis knowledgewithsome mystery.Thatwhichhadgreatvalue was not sharedlightly.Mutualtrustandrespectwerepartof the unspokenbondbetween teacherandstudent.Secrecyshroudedsome of his teaching.He toldme of the legendary wormsthathangupsidedownin caves in SaudiArabia,whose excretais collectedand turnedinto shellacfor Frenchpolishing!He mademysteriouspotionsfor coloringwood andfor cleaningantiques.His remediesweretrusteddespitethe fact thattheircontentswere unknown. Along withhis love of mystery,Mr.S coulddemonstratehis insightinto a complex idea with a handgesture,or he couldbuilda physicalmodelto explaina concept.His teachingwas vivid, active,inviting,andengaging.My timeas his studentprovidedme with uniqueopportunitiesto gain profoundinsightsinto boththeexperienceof mathematicsand the artof teaching.

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APPENDICES

Appendix A: Script for "Semi-structured" Conversations A. General,open-endedquestionsto startthepersontalkingabouthis life. B. Morespecificquestions(notall of thesewereaskedeachperson): 1. Whendid you leave school? 2. Did you learnmathematics at school? hasmuchto do withwhatgoes on in the workshop? 3. Do you thinkthatmathematics 4. How did you beginlearningcarpentry? 5. Canyou rememberwhatit was like to be an apprentice? well andthatyou 6. Canyou rememberanythingyou madethatturnedoutparticularly areespeciallyproudof? Why?How wasit differentfromotherthingsyou have made? 7. Canyou thinkof anycarpenters you havemetwho do excellentwork? 8. Canyou thinkof anycarpentersyou havemetwho do poorwork? In whatways does he workor thinkdifferently 9. Whatmakesa "good"carpenter? fromsomeonewho is not good? 10. Whenyou havea problemto solve, do you ask theothersfor help? 11. Whatdoes it taketo be a goodapprentice? 12. If you couldchoosean apprenticeto teach,whatwouldyou look for? 13. Whatwas it aboutyourfather's(ortheteacher's)teachingthatmadeit special?What does it taketo be a goodmasterteacher? 14. Whenyou look at a pieceof work,whatparticular thingsdo you look for when formingan opinionaboutits quality?

205

206 Appendix B: Script for Structured Interviews 1. Give intervieweea pieceof papercutthesamesize andshapeas the smallbox ("wakis")thattheyhadall seenme making.Ask howto placethe dovetailjointsalongthe side. Lookfor use of proportion,symmetry,measuring,comparing,angles,argumentsfor practicalityand/oraesthetics.Askthemto commenton my work. 2. Begin a discussionaboutstraightlines.Whatis a straightline?How do you check thata line is straight?Lookforuse of symmetries,innovativeexplanations. 3. Whatis the meaningof "square"? Whenis somethingsquare?How do you check this? 4. PeterpaysR3000percubicmetrefor mahogany.How muchdoes a plankwith dimensions2 metersby 0.6 metersby 0.075 meterscost?

207 Appendix C: How to Draw Jack's "Square" Star

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208 Appendix D: Jack's Modification of the Steps for a Square Star

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209 Appendix E: Jack's Eight-pointed Star Note:Thisis my interpretation anddiagramof Jack'sverbalexplanation.I havepresentedhis explanationas a seriesof steps. 1. Sketcha four-pointedstaras before. 2. Findthe centerof AB, markthis midpointD. 3. Findthe centerof BC, Markthis midpointE. 4. ExtendXA and YC.JoinXY. 5. Withstraightedgeparallelto XY,move straightedgeoutwardsuntilyou have reachedthe placewhereyou wanttheextendedstarpointto reach. 6. Drawa line thereparallelto XYandwhichcutsXAand YC. 7. Findthe midpointof thisline. CallthismidpointF. 8. JoinDF andEF. This makesthe firstextrastarpoint.Repeatthis procedure betweeneachpairof pointsof thefour-pointed star. PROBLEM:It is easy to "gooutof square"withthismethod.

210 Appendix F: Mr. S's Eight-pointed Star anddiagramof Mr.S's verbaldescriptionandhastysketch Note:Thisis my interpretation on a piece of wood. 1. Drawa four-pointedstaras before. 2. Drawanotherset of orthogonalaxesthroughthe origin,intersectingthe fourpoints. pointedstarat four"inward-pointing" 3. Findthe midpointsof eachof theeightsides of the star. 4. Decideon thelengthof thenew fourarmsto be added. 5. Markthe positionsof theendpointsof the fournew arms(of the eight-pointed star)alongthe secondset of orthogonalaxes. 6. Withstraightlines,join eachof thefournewlymarkedpointsto the midpointsof starthatareclosestto eachnew point. the sides of the four-pointed NOTE:In contrastto Jack'smethod,Mr.S's methodensuresthatthe eight-pointedstar remains"square."

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