Table Of ContentA 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
