A Journey in Mathematics Education Research Mathematics Education Library VOLUME 48 ManagingEditor A.J.Bishop,MonashUniversity,Melbourne,Australia EditorialBoard M.G.BartoliniBussi,Modena,Italy J.P.Becker,Illinois,U.S.A. M.Borba,RioClaro,Brazil B.Kaur,Singapore C.Keitel,Berlin,Germany G.Leder,Melbourne,Australia F.Leung,HongKong,China D.Pimm,Edmonton,Canada K.Ruthven,Cambridge,UnitedKingdom A.Sfard,Haifa,Israel Y.Shimizu,Tennodai,Japan O.Skovsmose,Aalborg,Denmark Forfurthervolumes: http://www.springer.com/series/6276 · · Erna Yackel Koeno Gravemeijer Anna Sfard Editors A Journey in Mathematics Education Research Insights from the Work of Paul Cobb 123 Editors ErnaYackel KoenoGravemeijer PurdueUniversityCalumet EindhovenUniversityofTechnology 730RoySt EindhovenSchoolofEducation Dyer,Indiana,46311 CampusTU/e,Traverse3.27 USA 5600MBEindhoven [email protected] TheNetherlands [email protected] AnnaSfard UniversityofHaifa DrorStreet4 68135TelAviv Apt.11 Israel [email protected] ISBN978-90-481-9728-6 e-ISBN978-90-481-9729-3 DOI10.1007/978-90-481-9729-3 SpringerDordrechtHeidelbergLondonNewYork ©SpringerScience+BusinessMediaB.V.2011 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Pauldedicatesthisbooktohismother,tothe memoryofhisfather,andtoJenny,his partnerinlifeforover35years. Foreword PaulCobb’scontributionstoresearchinmathematicseducationinthepastquarter century have been outstanding and exemplary. The progression of his understand- ingrepresentedinthisveryvaluableandenjoyablevolumecorrespondsinimportant waystoaprogressionofconceptsandmethodsthatdevelopedinthefieldofmath- ematics education research and, more generally, in the learning sciences. Cobb’s trajectoryisaprototypeofthefield’sprogress.Hehasplayed,andcontinuestoplay, a major role inshaping the problems, methods, and explanations that the field has developed,and,inturn,theproblems,methods,andexplanationshehasdeveloped havebeenshapedbythosethatothersinthefieldhavedeveloped. Inthisforeword,Ihopetomaketwopoints,bothofwhichrelateCobb’sextraor- dinary contributions to the advancement of the learning sciences, especially in mathematics education research, during the time that he (and I) have been active members of these communities. First, I describe my understanding of some ways in which the field has progressed. Cobb’s progression is prototypical and seminal inthefield’sconceptualtrajectoryduringthisperiod,asIunderstandit,althoughI believethefield’sprogresshasbeenlesslinearthanCobb’shasbeen,atleastasitis presentedinthereconstructionhehasgivenusinthisbook.ThesecondpointItry tomakehereisprimarilymethodological,relatingtorecentandcurrentdiscussions aimed to strengthen the scientific quality of educational research. By reflecting on the progression of methods in Cobb’s research program, I believe that we gain in our understanding of what makes research scientific, conceptually productive, and potentiallyvaluableforeffortstoimproveeducationalpractice. Cobb’s,andtheField’s,LearningTrajectory The view of learning that Cobb developed (explicitly, in the middle four papers of this collection) focuses on progressive changes in the practices of a group and changes in the participation of individuals in those changing practices. A strong methodologicalclaim,statedbyCobb,isthatanyeventinaclassroomcanbecon- sideredproductivelybothbyfocusingonthepracticesthatareenactedcollectively andontheparticipationofindividualsinthosepractices.Ibelievethatthesamecan besaidaboutthekindoflearningthatconstitutesscientificprogress.Thematerials vii viii Foreword inthisbookfocusappropriatelyonadvancesinconcepts,findings,andmethodsthat wereachievedinthecontributionsofCobbandhiscollaborators.Ofcourse,during the time that Cobb has been working, the field of mathematics education research hasalsoadvanced,forwhichCobbandhiscolleaguesbearaconsiderableshareof theresponsibility. The metaphor of a journey is apt. Cobb and his colleagues have traveled along a conceptual and methodological pathway, and the opportunity this book gives us to travel their route provides valuable insights into important features, structures, andresourcesofthedomain ofmathematics education research.Cobbencourages aspatialmetaphorwhenhestates,morethanonce,thathehad“modified[his]the- oreticalposition.”Therearedifferentkindsofjourneys,ofcourse,andthejourney that Cobb and his colleagues have traveled has been more than a tour. It has had thepropertyofaquest,inwhicheachnewlocationhaspresentedachallenge that hadtobeovercome.Butthismodernscientificquestlacksapropertyoftheclassi- cal quest narrative. Classically, thechallenge of each stage of a quest ismetif the protagonist survives, and the quest succeeds because the protagonist and (usually someof)hiscolleaguesarestrong,brave,pious,clever,orluckyenoughtoavoida catastrophethatwoulddestroythemall.Wescientistsusuallydon’triskourlivesto solve the problems we work on. In addition, Cobb’s, and our, quest has an impor- tantpropertythatclassicalquestslack:theachievementsthatweaccomplishalong the way are cumulative. By making progress toward understanding some aspects ofphenomenathatwestudy,wedevelopconceptualandmethodologicalresources thatwecarryalongandthatcanbeutilizedwhenweencounterthenextchallenge. Indeed,theresourcesthatwedevelopaswegoalongareinfluentialinshapingour understandingofwhatsubsequentchallengesare. Challenge #1: Explaining Elementary Understanding of Numbers. Cobb’s first paper in this collection presents part of the product of work done by Cobb with his mentors, Steffe and von Glasersfeld, while Cobb was a doctoral student. The challenge, as it was understood then, was to advance scientific understanding of children’s early understanding of number beyond the conclusions that Piaget and his associates had provided (e.g., Piaget, 1942). Steffe and Cobb, with von Glasersfeld (1988) provided a stunning analysis that focused on units that chil- dren constructed and used incounting operations, which progressed fromexternal objectstomotoractionstoentirelymentalentitiesastheirunderstandingdeveloped. Methodologically,thisstudywasanexemplaryadaptationofateachingexperiment, whichisthefocusofthereprintedpaper(Cobb&Steffe,1983). The understanding provided by research conducted at about this time included models of children’s understanding of additive and subtractive relations between quantities(Carpenter&Moser,1983;Nesher,1983;Riley,Greeno&Heller,1983), as well as Gelman and Gallistel’s (1978) contribution that showed that preschool children’sconceptualunderstandingofnumberincludesimplicitcognizanceofprin- ciples that are represented explicitly in mathematical formulations of the concept. Anderson (1983) constructed a computational model that simulates the informa- tionstructuresinvolvedinsolvinggeometryproofexercises,alongwithhypotheses about processes of learning. The theoretical and methodological achievements of Foreword ix this early cognitive period were carried forward into later work, including Cobb’s detaileddesignsandanalysesoflearningtrajectories,forwhichSteffeetal.’s(1988) developmentaltrajectoryprovidedakindofprototype. Whilecontributionsintheearly1980sthatdrewoninformationprocessingand cognitive developmental approaches were complementary and convergent, there wasanotherprogramthatprovidedanantithesistothecognitivethesis.Thespear- head of this counter-program was contributed by Lave, Murtaugh, and de la Rosa (1984),whoarguedthatreasoningwithandaboutquantitiesshouldbeunderstood as an achievement jointly produced by individuals and resources in the environ- ment, not just as operations on symbols in mental representations, an argument that Lave (1988) developed extensively. Bauersfeld (1980) also had raised issues thatchallengedtheadequacyofanalyzingmathematicslearningwithouttakinginto account aspects of interaction in the classroom. Lave, Smith, and Butler (1988) argued that problem solving in school mathematics should be understood as an everydaypractice,embeddedinschoolactivity.Moregenerally,Searle(1980)and Suchman (1985) presented critical arguments against the assumption that cogni- tion can be understood as occurring in a self-contained mental system, and Lave and Wenger (1991) reviewed analyses of apprenticeship learning and proposed a framingoflearningastrajectoriesofparticipationincommunitiesofpractice. Challenge#2:ExplainingLearninginInteraction.Theanalysesfocusedonindi- viduals’understandingsthatCobbandothersdevelopedwereevidentlyincomplete. Cobb’s response to this challenge included the papers reprinted in Parts II, III, and IV of this volume. Cobb and his colleagues concluded that students’ emo- tional acts depend (partly) on properties of classroom practices, not just on their individualemotionaltendencies.Theydesignedandstudiedcurriculumsequences, considering them as occasions for the students and teacher to progress through a trajectoryofpractices,correspondingtoincreasinglysophisticatedunderstandingof mathematicalconceptsandprinciples.Andtheyanalyzedtheprogressofmathemat- icalunderstandingachievedbyaclassasthegroupadvancedthroughincreasingly sophisticatedpractices,butalsoanalyzedvariationsbetweenindividualstudentsin theirparticipationinthepractices,therebyshowingthatconsideringlearningatthe leveloftheclassroomandatthelevelofindividualstudentsarenotonlycompati- bleperspectives,butthatitisproductivetoexaminerelationsbetweenfindingsthat resultfromframinganalyseswitheachofthem. During the years that Cobb and his colleagues wrestled with the challenge of reconcilingindividualandsociallevelsofanalysis,othersinthefieldweresimilarly engaged. The result was that early in the twenty-first century, the field had devel- oped a strong beginning toward a body of concepts, principles, and methods for understanding, designing, and studying productive classroom environments, espe- ciallyinmathematics.ExamplesincludeLampert’s(1990,2001)andBall’s(Ball& Bass,2000)analysesoftheinteractionstheyorganizedintheirclassrooms,Boaler’s (1997/2002)findingsthatdifferentclassroompracticesresultedindifferent“forms of mathematical knowledge,” which she documented with assessments and inter- viewswithindividualstudents,BrownandCampione’s(1994)designandstudiesof thecurriculumandlearningenvironmentcalledFosteringCommunitiesofLearners, x Foreword andEngeström’s(2001)designandstudyoflearningbytwogroupsofmedicalpro- fessionals, interpreted as participants in two activity systems, through interaction organizedtoaddressandsolveaproblemofpractice. WhileCobb’sstudiesdocumentedandinterpretedvariationinthewaysthatstu- dents participated in classroom practices, this variation was unexplained by the conceptsCobbandhiscolleaguespresentedinthepapersreprintedinPartsII,III, andIV.Alsoduringthe1990s,though,othersinthefieldbegantodevelopconcepts of individual identity in perspectives of participation in activity systems. Holland, Lachicotte, Skinner, and Cain (1998) developed a concept of positional identity, characterizing patterns of participation in which individuals comply with or resist prevalent ways of participation in what they called figured social worlds. Wenger (1998)characterizedindividualidentitiesastrajectoriesofparticipationwithinand acrosscommunitiesofpractice. Challenge#3.ExplainingPersonalContinuitiesAcrossSituations.Theperspec- tive that emphasizes students’ activity and learning as participation clarifies many aspects of learning, but it also leaves many questions unanswered. Cobb and his colleagues’interpretiveframeworkforunderstandingaspectsofidentityandissues of equity provides one way to begin to account for sources of difference between ways of participating that different students develop. The idea that an individual’s identityisco-constructedandemergesininteractioninanactivitysystemwasalso developedinrecentstudiesbyGresalfi(2004;2009)andbyNasirandHand(2008). Thegeneralpropositionthatpeoplelearnbyparticipatinginpracticesappliesto teachers as well as to students. Research on teachers’ learning has been an active program, and a general finding is that teachers’ efforts to develop more effective practicesbenefitfrominteractinginacommittedprofessionalcommunityinwhich they reflect on their teaching aims, accomplishments, and challenges (e.g., Little, 1994). Challenge #4. Explaining Teachers’ Progress in Developing Practices. In their discussion of institutional contexts, Cobb and his colleagues contribute observa- tions and thoughtful reflections on two cases in which school leaders differed in supporting teachers opportunities to learn in interaction with each other (and with university-baseddesigner-researchers)tosignificantlydifferentextents. TheScientificQualityofCobb’sResearch InhisintroductiontoPartIofthisvolume,Cobbwrites,“...thereisnosubstitutefor sustained,first-handengagementwiththephenomenathatweseektounderstand.” ItakethistobeoneoftwofundamentalcommitmentsofCobb’sresearchmethod- ology.Theotheristheimportanceofconstructingexamplesofphenomenathatare worthy of close study and exemplary of learning resources that would strengthen ourinstitutionsofeducation. I believe that the test of a methodology is whether its use is productive in producing findings and especially (following Toulmin, 1972) in advancing the