Supporting Students' Development of Measuring Conceptions: Analyzing Students' Learning in Social Context (Journal for Research in Mathematics Education, Monograph, N.º 12)

MONOGRAPH NUMBER 12

Journal for Research

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Mathematics Education Supportin0g Students' Development of Measuring Conceptions: Analyzing Students' Learning in Social Context

NATIONALCOUNCILOF

NCTM

TEACHERS OF MATHEMATICS

A Monograph Series of the National Council of Teachers of Mathematics The Journal for Research in Mathematics Education (JRME)MonographSeries is published by the National Council of Teachersof Mathematicsas a supplementto the JRME.Each monographhas a single theme related to the learning or teaching of mathematics.To be considered for publication,a manuscriptshould be (a) a set of reportsof coordinatedstudies, (b) a set of articlessynthesizing a large body of research, (c) a single treatise that examines a majorresearchissue, or (d) a reportof a single researchstudy that is too lengthy to be published as a journalarticle. Any person wishing to submit a manuscriptfor consideration as a monographshould send four copies of the complete manuscriptto the current monograph series editor. Manuscriptsshould be no longer than 200 double-spaced, typewrittenpages. The name, affiliations, and qualificationsof each contributingauthorshould be included with the manuscript. Complete information about submitting a manuscript for consideration as a monograph and the current monograph editor can be found at www.nctm.org/jrme.

Series Editor NEIL PATEMAN,University of Hawaii, Honolulu, HI 96822 JRME Editorial Panel BETSY BRENNER, University of California, Santa Barbara;JINFACAI,Universityof Delaware;GUERSHON HAREL, University of California, San Diego; LENA LICON KHISTY, University of Illinois at Chicago; ' Chair; CAROLYNKIERAN, Universite du Quebec Board NORMA Illinois Liaison; PRESMEG, Montreal, State University; SHARON ROSS, California State University, Chico; MARTIN SIMON, Penn State University; SHARON WALEN, Boise State University, Idaho; ANDREW ZUCKER, Center for Educational Policy, Arlington,Virginia

Supporting Students' Development of Measuring Conceptions: Analyzing Students' Learning Social

Context

in

Supporting

Students'

Developmentof Measuring Conceptions: Analyzing Students' Learning Social

Context

Michelle Stephan Purdue UniversityCalumet JanetBowers San Diego State University Paul Cobb VanderbiltUniversity with Koeno Gravemeijer FreudenthalInstitute

Neil Pateman,Series Editor Universityof Hawaii

NCTNI NCTM

NATIONALCOUNCILOF TEACHERSOF MATHEMATICS

in

Copyright02003 by THE NATIONALCOUNCILOF TEACHERSOF MATHEMATICS,INC. 1906 Association Drive, Reston, VA 20191-1502 (703) 620-9840; (800) 235-7566; www.nctm.org All rights reserved

Libraryof CongressCataloging-in-PublicationData

Supportingstudents'developmentof measuringconceptions: analyzing students'learningin social context / by Michelle Stephan,JanetBowers, and Paul Cobb. p. cm. - (Journalfor researchin mathematicseducation. Monograph,ISSN 0883-9530 ; no. 12) ISBN 0-87353-556-1 1. Length measurement-Study and teaching (Primary)-Evaluation. 2. Mathematics-Study and teaching-Research-Evaluation. I. Stephan, Michelle. II. Bowers, Janet.III. Cobb, Paul. IV. Series. QA465.$78 2003 372.7--dc22 2003023612

Thepublications of theNationalCouncilof Teachersof Mathematics presenta variety of viewpoints. Theviewsexpressed orimpliedinthispublication, unlessotherwise noted, shouldnotbe interpreted as officialpositionsof theCouncil.

Printedin theUnitedStatesof America

Table of Contents Acknowledgments ................................................ Chapter1 InvestigatingStudents'ReasoningAbout LinearMeasurementas a ParadigmCase of Design Research ..................................... Paul Cobb, VanderbiltUniversity Chapter2 ReconceptualizingLinearMeasurementStudies: The Developmentof ThreeMonographThemes ............ Michelle Stephan,Purdue UniversityCalumet Chapter3

Chapter4

Chapter5

Chapter6

Chapter7

The MethodologicalApproachto Classroom-BasedResearch ............................ Michelle Stephan,Purdue UniversityCalumet Paul VanderbiltUniversity Cobb, A HypotheticalLearningTrajectoryon Measurement and Flexible Arithmetic............................... Koeno Gravemeijer,FreudenthalInstitute Janet Bowers, San Diego State University Michelle Stephan,Purdue UniversityCalumet CoordinatingSocial and IndividualAnalyses: Learningas Participationin MathematicalPractices......... Michelle Stephan,Purdue UniversityCalumet Paul Cobb, VanderbiltUniversity Koeno Gravemeijer,FreudenthalInstitute Continuingthe Design ResearchCycle: A Revised Measurementand ArithmeticSequence ......... Koeno Gravemeijer,FreudenthalInstitute Janet Bowers, San Diego State University Michelle Stephan,Purdue UniversityCalumet Conclusion ........................................ Michelle Stephan,Purdue UniversityCalumet

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Acknowledgments The investigation reported in this monograph was supportedby the National Science Foundation under grant number REC 9814898 and by the Office of EducationalResearchandImprovement(OERI)undergrantnumberR305A60007. The opinionsexpresseddo not necessarilyreflectthe views of eitherthe Foundation or OERI. The projectteamthatengagedin the researchreportedin this monographincluded Paul Cobb, Beth Estes, Kay McClain,Maggie McGatha,Beth Petty, andMichelle Stephan.ErnaYackel andKoenoGravemeijeralso collaboratedon this projectbut did not participateon a daily basis.

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Chapter 1

Investigating Students' Reasoning About Linear Measurement as a Paradigm Case of Design Research Paul Cobb

The issues discussed in this monographare situatedin a descriptionof a design experimentthatfocused on linearmeasurementandwas conductedin a first-grade class. Because the focus on this particularmathematicaldomain might appearto be relatively narrow,I want to clarify at the outset that the reportof this experiment is framed as a paradigmcase in which to illustrategeneral featuresof the design-experimentmethodology. Design experiments involve both developing instructionaldesigns to supportparticularforms of learning and systematically studying those forms of learning within the context defined by the means of supportingthem. This monographalso emphasizes that design experimentsare conductedwith a theoreticalas well as a pragmaticintent, in that the goal is not merely to refine a particularinstructionaldesign. In the example at hand, the experimentwas conductedwith the intentof contributingto the developmentof a domain-specificinstructionaltheoryof linearmeasurement.A theoryof this type specifies both a substantiatedlearningtrajectorythat aims at significant mathematicalideas andthe demonstratedmeansof supportinglearningalong thattrajectory. This concernfor domain-specifictheoriesreflects the view thatthe explanations andunderstandingsinherentin themareessentialif educationalimprovement is to be a long-term,generativeprocess. This attentionto methodologicalissues is timelygiven the increasingprominence of the Design Research methodology in mathematicseducation. For example, SuterandFrechtling(2000) recentlyidentifiedDesign Researchas one of the five primaryresearchmethodologies in mathematicsand science education.As they note, about 20% of the research projects funded by the National Science Foundation's (NSF's) Division of Research, Evaluation, and Communication between 1996 and 1998 were design experiments.Not surprisingly,this increasing activity finds expression in the growing number of articles written by mathematicseducatorswho reportdesign experiments(e.g., Bowers & Nickerson,2001; Lehrer,Jacobson,Kemney& Strom,1999;Moss & Case, 1999;Rasmussen,1998; Simon, 1995). However,as SuterandFrechtlinggo on to observe,a numberof rela-

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tively commonmisconceptionsaboutthe methodologyareapparent.Althoughthe methodology is dynamic and open for adaptation,several centraltenets serve as the basis for Design Research,namely,thatinstructionshould(a) be experientially real for students,(b) guide studentsto reinventmathematicsusing theircommonsense experience,and(c) provideopportunitiesfor studentsto createself-developed models. One of the goals of this monographis to illustratethesetenetsby grounding them in the concrete situationof a specific experiment.In this regard,the monographcomplementsthree chapterson Design Researchin a recent handbookon researchdesign in mathematicsand science education (cf. Cobb, 2000; Confrey & Lachance,2000; Simon, 2000). All the chaptersin this monographexplore three themes that concern design researchers:(1) the relationbetween the developmentof instructionaldesigns and the analysis of students'learning,(2) the relationbetween communalclassroom processesandindividualstudents'reasoning,and(3) the role of tools in supporting development.The first of these themes stems directly from the dual emphasis in Design Researchon the developmentof domain-specificinstructionaltheoriesand the formulation,testing,andrevisionof instructionaldesigns. The relevanceof the second theme is indicatedby theoreticaldebates that have appearedin leading educationaljournalsconcerningthe relationbetween collective communicational processes and the reasoningof the participatingstudents(e.g. Anderson,Greeno, Reder, & Simon, 2000; Anderson, Reder, & Simon, 1996; Anderson, Reder, & Simon, 1997; Cobb & Bowers, 1999; Greeno, 1997; Lerman,1996, 2000; Steffe & Thompson,2000a). These exchanges indicatethatno clear consensus has been reachedon how the relationbetweenthese two types of processesmightbe productively conceptualized.We pursuethatquestion and elaborateour contributionto the debate by exploringthe interplaybetween individualstudents'reasoningand collective processesof communicationandinteractionas theyplayedout in the case study in this monograph. The importanceof the third theme of mathematicaltools becomes apparent once one notes that the development of tools to support students' learning is centralto instructionaldesign.The realizationthatstudents'mathematicallearning is profoundly influenced by the tools they use, however, underpinsa range of different approachesto instructionaldesign. For example, Doerr (1995) distinguishes between expressive and exploratoryapproachesto design. In expressive approaches,studentsareencouragedto invent andtest the adequacyof tools, such as notation systems, that express their developing understandings.In effective designs of this type, students develop increasingly sophisticatedmathematical understandingsby incrementallyreformulatingtheir informal knowledge (e.g., Bednarz, Dufour-Janvier,Portier,& Bacon, 1993; diSessa, Hammer,Sherin, & Kolpakowski,1991). In contrast,exploratoryapproachesinvolve the development of computerenvironmentsin which students can investigate the links between conventionalmathematicalsymbolsystemsandselectedeverydayphenomena.For example,Kaput(1994) andNemirovsky(1994) have bothdevelopedenvironments in which distance-timegraphsareintroducedat the outsetandstudentsexplore the

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use of these graphsas a means of representingvarioustypes of motions. In effective designs of this type, the computerenvironmentsspanthe initial gulf between students' authenticexperiences and mathematicalways of symbolizing, such as graphing.The comparisonof the expressiveandexploratoryapproacheshighlights a tensioninherentin instructionaldesign betweenthe ideal of buildingon students' contributionsand the need to decide in advancethe tools and symbolizationsthat studentsshould eventually come to use. This monographaddressesthis tension when describingthe instructionaldesign developed and revised in the course of the measurementexperiment. In the following pages, I firstsituatethe threethemesof the monographby giving a brief historicaloverview of Design Research.In the remainderof this introductory chapter,I focus on two main aspects of the Design Researchmethodology. These aspects concern the instructionaltheory thatundergirdsthe formulationof instructionaldesigns and the interpretiveframeworkthat guides the analysis of classroomevents. EXPERIMENTINGIN THE CLASSROOM As I have indicated,the intimaterelationbetween theory and practiceis one of the defining characteristicsof the Design Research methodology. On the one hand,instructionaldesign serves as a primarycontextfor researchandthe development of theory. On the other hand, analyses of students'learning and the means by which it was supportedinformresearchersin theirrevision of the instructional design. As a consequence,the purposewhen experimentingin the classroomis not to try to demonstratethat the initial design formulatedin advance of the experimentworks.Instead,the purposeis to test and improvethe initialdesign as guided by both ongoing and retrospectiveanalyses of classroomactivities and events. Methodologiesin which instructionaldesign serves as a contextfor the development of theoriesof learningandinstructionhave a long history,particularlyin the formerSoviet Union (Menchinskaya,1969). The termDesign Research,however, was coined relatively recently and is most closely associated with Ann Brown (1992). In her formulation,Design Researchemerged as a reactionagainsttraditional approachesto the study of learningthatemphasizethe control of variables. In a more recent discussion of the methodology,Collins (1999) also takes traditional psychological methodology as his point of reference. For example, he follows Brown in contrastingstudies of learningconductedin relatively artificial laboratorysettings with design experimentsthatfocus on learningas it occurs in complex andmessy settings,such as classrooms.The purposeof Design Research for Brown and Collins, however, is not merely to study learning in situ. They emphasizethatthe methodologyis highly interventionistandinvolves developing designs that embody testable conjecturesabout the means of supportingan envisioned learningprocess. In addition,they underscorethat the methodologyhas both a theoretical and a pragmatic intent by drawing an analogy with design sciences, such as aeronauticalengineering.As they note, an aeronauticalengineer

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createsa model, subjectsit to certainstresses,and generatesdatato test and revise theoreticalconjecturesinherentin the model. Similarly,a Design Researchteam createsa design to supportan envisionedlearningprocess,conductsan experiment to subjectthe design to certainstresses, and generatesdatato test andrevise theoreticalconjecturesderivedfrompriorresearchthatareinherentin the design. This analogy also clarifies thatin both aeronauticalengineeringand Design Research, generalizationis accomplishedby means of an explanatoryframeworkratherthan by means of a representativesample, in that the theoreticalinsights and understandingsdevelopedduringone or moreexperimentscan feed forwardto influence the analysisof events andthuspedagogicalplanninganddecision makingin other classrooms (cf. Steffe & Thompson,2000b). The concernsfor the developmentof domain-specificinstructionaltheoriesand for potential generalizability serve to differentiate Design Research from the activityof thoughtfulpractitionerswho continuallyseek to improvetheirpractices. The parallelsareindicatedby Franke,Carpenter,Levi, & Fennema's(2001) observation that mathematicsteaching is for some teachers a knowledge-generating activity in the course of which they elaborateand refine their understandingsof both their students' reasoning and the means of supporting its development. Whereasthe teachersdescribedby Frankeandothersworkin theirlocal classroom and school settings to improve their effectiveness in supportingtheir students' learning,Design Researchersexplicitlyframeaspectsof the learningprocessesthey areattemptingto supportas paradigmcases of broaderclasses of phenomena.Thus, whereaspractitionersareprimarilyconcernedwith the effectiveness of theirpractices in their local settings, Design Researchersalso focus on the developmentof theoreticalinsightswhentheyplanfor an experiment,generatedataduringanexperiment, and conductretrospectiveanalyses. Collins's (1999) and Brown's (1992) works have proved to be foundationalto the emergingfield of the learningsciences, itself a directdescendentof cognitive science (cf. DeCorte,Greer,& Verschaffel, 1996). As the increasingadoptionof theirterminologyindicates,theirargumentshave also hada considerableinfluence in mathematicseducation.AlthoughDesign Researchin the learningsciences and that in mathematicseducation are highly compatible, their histories differ. The emergence of the learning sciences from cognitive science signaled a relatively radicalchangeof priorities,whereasthe developmentof Design Researchin mathematics educationhas been moreevolutionaryandbuildson two existing research traditions: the constructivist teaching experiment and Realistic Mathematics Educationdeveloped at the FreudenthalInstitutein the Netherlands. The constructivistteaching experimentmethodology was developed by Steffe andhis colleagues(Cobb& Steffe, 1983;Steffe, 1983;Steffe& Kieren,1994;Steffe & Thompson, 2000b). The purposeof the teaching experimentas formulatedby Steffe andhis colleaguesis to enableresearchersto investigatetheprocess by which individualstudentsreorganizetheirmathematicalways of knowing.To this end, a researchertypicallyinteractswith a studentone-on-oneand attemptsto precipitate his or herlearningby posingjudiciouslychosentasksandby askingfollow-upques-

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tions, often with the intentionof encouragingthe studentto reflect on his or her mathematicalactivity.The primaryproductsof a teachingexperimentof this type typically consist of conceptualmodels composed of theoreticalconstructsdeveloped to account for the learning of the participatingstudent(s) (Thompson & Saldanha,2000). The intentin developingthesemodelsis thatthey will proveuseful when accountingfor the learningof other studentsand can thus inform teachers in theirdecisionmaking.Althoughthe researcheracts as a teacherin this methodological approach,the primaryemphasisof the constructivistteachingexperiment is on the interpretation of students'mathematicalreasoningratherthanon thedevelopmentof instructionaldesigns. An initial attemptto adaptthe constructivistteachingexperimentmethodology to the classroomsettinginvolved creatinga complete set of instructionalactivities for second- and third-grade classrooms (Cobb, Yackel & Wood, 1989). Nonetheless, the primaryfocus of these experimentswas consistentwith Steffe's (1983) emphasis on the development of explanatoryconstructsratherthan the formulationof instructionaldesigns. In particular,Cobb and his colleagues gave priorityto developing an interpretiveframeworkthatwould enablethem to situate students'mathematicallearningwithinthe social contextof the classroom(cf. Cobb & Yackel, 1996). In the course of this work, they came to view theirlack of positive design heuristicsthatcould guide the developmentof instructionalactivities as a severe limitation. In general, constructivismand related theories present a numberof negative heuristicsthatrule out a range of approachesto instructional design. The types of cognitive and interactionalanalyses that Cobb and others conducted at that time, however, did not provide an adequatebasis for design, becausethe analysesdid not focus explicitlyon the meansby which students'mathematicallearningwas supportedand organized. The secondresearchtraditionon whichDesign Researchis built,thatof Realistic MathematicsEducation(RME), offsets this weakness by focusing primarilyon design rather than the development of explanatory theoretical constructs (cf. Gravemeijer,1994; Streefland, 1991; Treffers, 1987). Guided by Freudenthal's (1973) notion of mathematicsas a humanactivity and building on his didactical phenomenologyof mathematics,RME researchershave developed, triedout, and modifiedinstructionalsequencesin a wide rangeof mathematicaldomains.Further, as Treffers(1987) documents,reflectionon this processof developing,testing,and revising specific instructionalsequenceshas resultedin the delineationof a series of heuristicsfor instructionaldesignin mathematicseducation.Gravemeijer(1994), for his part, has analyzed the process of the emergence of both these general design heuristicsand the domain-specificinstructionaltheoriesthatconstitutethe rationale for particularinstructionalsequences. As will become apparent,the developmentof the measurementinstructionalsequence that is the focus of this monographwas guided by RME design heuristics.In addition,the processes by which the initialdesign of the instructionalsequencewas testedandrevisedduring the first-gradedesign experimentis consistentwith Gravemeijer'sanalysis of the developmentof a domain-specificinstructionaltheory.

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Students'Reasoning Investigating

This briefhistoricalaccountof Design Researchshouldclarifythatmy colleagues andI attributegreatimportancebothto the instructionaltheorythatservesto orient the developmentof specific designs andto the interpretiveframeworkthat serves to organizeclassroomanalysesthatin turninfluencethe ongoingdesign effort.The remainderof this introductionfocuses on these two core aspects of classroom Design Research. DESIGNINGLEARNINGENVIRONMENTS Gravemeijer,Bowers, & Stephan(chapter4 of this monograph)set the stage for their discussion of the measurementdesign experimentby giving an overview of RME design theory.Against this background,they then describethe conjectured learningtrajectorythatwas formulatedin advanceof the experiment.As they make clear, the conjecturedmeansof supportingthe students'learningwere not limited to the instructionaltasks but included the students'use of several differenttypes of tools for measuring.They emphasize that these tools were not designed to serve merely as a meansby which studentsmight expresstheirreasoning.Instead, the design team conjecturedthatstudentsmight reorganizetheirreasoningas they used the tools. The approachtaken to instructionaldesign in the measurementdesign experimentis thereforecompatiblewith the propositionthatthe use of tools can serve not merely to amplify but to reorganizeactivity (cf. DbSrfler,1993; Pea, 1993). Stephan(chapter2 of this monograph)emphasizesthis point of agreement with distributedtheoriesof intelligenceby speakingof the students'reasoningwith the tools both when describing the conjectured learning trajectory and when analyzingclassroom events. In additionto focusing on instructionaltasks and associatedresources,such as tools, the design teamalso consideredpotentialmeansof supportthattypicallyfall beyondthe purviewof curriculumdeveloperswhenpreparingfor the measurement experiment.This work includedconjecturesaboutboth classroom discourse and the classroom activity structureas means of support.In the case of classroom discourse, these conjecturesconcernednormsor standardsfor what would count as an acceptableexplanationof measuringactivity. A distinctionthatThompson, Philipp,Thompson,andBoyd (1994) make between calculationalandconceptual orientations in teaching proved to be particularly relevant. The design team extendedthis distinctionby developinga contrastbetweencalculationaldiscourse andconceptualdiscourse.The standardsof argumentation in calculationaldiscourse are such thatan acceptableexplanationneed only describethe methodor process by which a resultis produced.With regardto measuring,a calculationalexplanation involves demonstratinghow a measuringtool has been used to find the length of an object. An importantpoint to clarify is that calculationaldiscourse is not restricted to conversations that focus on the proceduralmanipulationof conventionaltools andsymbolswhose use is a rule-followingactivityfor students.The methods for producingresults that studentsexplain as they contributeto such a conversation,might, in fact, be self-generatedandinvolve relativelysophisticated

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mathematicalunderstandings. As a consequence,the contrastbetweencalculational and conceptual discourse should not be confused with Skemp's (1976) wellknown distinctionbetween instrumentaland relationalunderstanding. The referenceto Skemp's(1976) workservesto emphasizethatthe definingcharacteristicof calculationaldiscourse concerns the norms or standardsfor judging what counts as an acceptable argumentratherthan for judging the quality of students'understandings.In calculationaldiscourse,contributionsare acceptable if studentsdescribehow they produceda result,andthey arenot obliged to explain why they used a particularmethod. By this criterion, examples of classroom discoursepresentedin the literatureto illustrateinstructioncompatiblewith current reformrecommendationsare,in fact, calculationalratherthanconceptualin nature. In contrastwith this emphasis on methods or solution strategies,the issues that emerge as topics of conversationin conceptualdiscoursealso include the reasons for calculatingin particularways. In measuring,these reasonsconcernthe way in which particularmethodsof measuringstructurethe space measuredinto units of length. I canbest illustratethe distinctionbetweenthesetwo typesof classroomdiscourse by foreshadowing Stephan, Cobb, & Gravemeijer'sanalysis (chapter5 of this monograph)of an incidentthatoccurredearly in the measurementdesign experiment.The teacherandobservingresearchersnotedthatdifferentgroupsof students developedtwo differentmethodswhen they measuredvariousdistancesby pacing heel to toe. In the ensuing whole-class discussion,the teacherasked selected childrento measurethe length of a rug in the classroomso thatshe could highlightthe differencebetweenthese two methods.In one method,the studentplaces one foot in line with the beginning of the rug but does not begin counting paces until she places the second foot heel to toe in frontof the first (i.e., from our standpoint,the child fails to count her first pace). We might reasonably speculate that these studentswere countingtheirphysical acts of placing down a foot per se insteadof attemptingto find the numberof pacesthey neededto taketo cover the linearextension of the rug. In the second method, the child counts her initial placementof a foot in line with the beginningof the rug as "one"andthen continuesby counting The project team conjecturedthat these successive paces as "two, three,. studentsmight have been partitioningthe linear extension of the rug instead of merely countingtheirphysical movementsper se. If calculationalnormshadbeen establishedfor argumentation,thenjustifications in which studentsrepeatedlydemonstratedto one anotherhow they had counted theirpaces would have been acceptable.Imaginefor a momenthow studentswho failed to count the first pace might have interpretedcalculationalexplanationsof the other method. Such explanationswould likely not have made sense to these students,becausethe methodis unrelatedto the task as they understoodit: to count physical acts of pacing. These students might, however, have realized that the methodbeing explainedis morevaluedthantheirown andadjustedtheirapproach accordingly.In doing so, they would have adoptedthe desiredmethodbut without reconceptualizingthe natureof their activity. The crucial point to note is that a

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calculationaldiscussionof the two methodsprovideslittle supportfor substantive mathematicallearning.Instead,the studentslargely on their own would have to reconceptualizetheiractivity as one of covering a distance. A calculationalexchangeof this type can be contrastedwith a conceptualdiscussion in which the reasonsfor countingpaces in a particularway become an explicit topic of conversation.For example, a studentwho counted her placementof the first foot might explain thather paces completely fill the spatialextension of the rug withoutany overlapsand withoutleaving any gaps. Further,the studentswho used this method might argue that the studentswho did not count the first pace missed a piece of the rug. In the context of such an exchange, the way in which the two methodsstructurethe spatialextensionof the rugcan emerge as an explicit criterionfor comparingthem. The emergence of this criterion,in turn, serves to supportthe reconceptualizationof pacingas a distance-coveringactivity.Students' engagement in conceptualdiscussions of this type thereforeprovides them with resourcesthatmight enable themto reorganizetheirthinking.As Stephanandher colleagues demonstratein theiranalysisof the measurementdesign experimentin chapter5, these resourceswere not limited to what was said but include the notational schemes thatwere developedto recordthe ways in which differentmethods of measuringstructuredthe linearextensionof the objectbeingmeasured.The goals for students'learningthatwere formulatedwhen preparingfor the design experiment were not restrictedto methods of measuringbut included the meaning of measuring(i.e., the partitioningof the linearextension of objectsinto units). As a consequence, students'engagementin conceptualdiscussions in which differing ways of structuringdistancecame to the fore was viewed as a primarymeans of advancingthe pedagogicalagenda. To this point, I have discussedthreemeansof supportingstudents'learningthat were consideredwhen planningfor the measurementexperiment:(1) the instructional activities, (2) the tools that students would use to measure, and (3) the natureof classroomdiscourse.The classroomactivitystructureconstitutedthe final means of supportto which the design team attendedpriorto the experiment.The proposedorganizationof classroom activities had threemain phases: 1. The teacherwould develop with the studentsan ongoing narrativein which the charactersin the narrativeencounterproblemsthatinvolve eithermeasuringwith an existing tool or developing a new measuringtool. 2. The studentswouldengage in measuringactivitieseitherindividuallyor in pairs. 3. The teacherwould capitalize on the range of interpretationsand solutions that the studentsdeveloped duringindividualor small-groupwork to lead a wholeclass discussion in which mathematicallysignificant ideas that advance the pedagogical agendaemerge as topics of conversation. The design teamconjecturedthatstudents'engagementin the firstof these three phases would supporttheirlearningin two specific ways. Both these conjectures were premisedon the assumptionthat studentswould identify with the characters

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in the narrativeas they attemptedto resolve the variousproblemsthey encountered. The first conjecturewas thatstudents'measuringactivitywould take on a broader significanceandpurposewithinthe context of the narrativeso thatthey would not merelybe measuringat the teacher'sbehest. The design team's second conjecture was thatif studentsidentifiedwiththe charactersin thenarrative,the needto develop variousnew measuringtools as formulatedwithin the narrativewould seem plausible to them. Further,a discussionof possible solutionswould serve to clarifythe design specificationsfor the new tool. As a consequence,new tools wouldnot seem arbitraryto studentseven on those occasions when the teacherintroducedthem. Instead,studentswould view tools introducedin this way as reasonablesolutions to problemsthathad significance within the context of the narrative. These two conjectured ways in which the ongoing narrativemight support students' learning constitutedthe primaryjustification for the first phase of the proposed classroom-activity structure.The issues I raised when clarifying the distinctionbetween calculationalandconceptualdiscourseserve to justify the last two phasesof the classroom-activitystructure.Clearlythe intentof the whole-class discussions was not simply to provide students with an occasion to share their reasoning.Instead,the design team's overridingconcern was with the quality of the discussions as social events in which students would participate.From this perspective,the value of whole-class discussionsis suspectunless mathematically significantissues that advancethe instructionalagendaemerge as explicit topics of conversation. Conversely, students' participationin substantive,conceptual discussionsis viewed as a primarymeansof supportingtheirenculturationinto the values, beliefs, and ways of knowing of the discipline. This discussionof the classroomactivitystructureservesto clarifythe last of the fourmeansof supportconsideredwhen preparingfor the measurementexperiment: 1.The instructionaltasks 2. The tools the studentswould use to measure 3. The natureof classroom discourse 4. The classroom activity structure Because I have focused on each means of supportseparately,they might be construedas a largelyindependentset of factorsthatinfluencelearningseparately. An importantpoint to emphasizeis thatthe design team viewed the four means of supportas highly interrelated.For example, the instructionaltasks as actually realized in the classroom dependedon the extent to which the studentsidentified with the charactersin the narratives,the tools they used to measure,andthe nature of the classroomdiscussions. One can easily imagine, for example, how the tasks mightbe realizeddifferentlyif a conventionalmeasuringtool, such as a ruler,were introducedat the outset or if no whole-class discussions were held andthe teacher simply gradedthe accuracyof the students'measuringactivity. In light of these interdependencies,one can reasonablyview the variousmeans of supportas constitutinga single classroom activity system. This perspectiveis

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compatiblewith Stigler and Hiebert's (1999) contentionthat teaching should be viewed as a system. In making this claim, Stigler and Hiebertdirectly challenge analyses that decompose teachers' instructionalpractices into a number of independentmoves or competencies. They instead propose that the meaning and significance of any particularfacet of a teacher's instructionalpracticebecomes apparentonly when analyzedwithin the context of the entirepractice.In a similar manner,the fourmeansof supportconsideredwhen planningfor the measurement design experiment should be viewed as aspects of a single classroom activity system. Instructionaldesign from this point of view thereforeinvolves designing classroomactivitysystemsin whichstudentsdevelopsignificantmathematicalideas as they participatein theirexplorationand contributeto theirevolution. The orientationto designthatwe tookwhenpreparingforthe measurementexperiment sits uncomfortablywith approachesthatfocus exclusively on tasks andtools. In these formulations,tasksaretypicallycast as thecause andlearningas the effect. An importantpoint to emphasize is that when Gravemeijerand his colleagues discuss tasks and measuring tools while outlining the proposed instructional sequence in chapter4, they are describing aspects of an envisioned classroom activity system. Their overview of the proposedinstructionalsequence is made againstthe backdropof assumptionsaboutboththe classroomactivitystructureand the classroom discourse. As a consequence, their conjectures about students' learningas they use particulartools to complete particulartasks are metonymies for conjecturesaboutstudents'learningas they participatein an envisionedclassroom activity system. ANALYZING MATHEMATICALLEARNINGIN SOCIALCONTEXT As I noted in the first partof this introduction,the design of classroomlearning environmentsis one of the two principalaspects of Design Research.The second aspectconcernsthe analysisof mathematicallearningas situatedwithinthe social context of the classroom. In chapter3 Stephanand Cobb describein some detail the interpretiveframeworkthatwas used to organizethe analysis of the measurementteachingexperiment.Ratherthansummarizetheirpresentationby discussing the constructsthat compose this framework,I instead attemptto clarify its rationale. As should become apparent,the frameworkcan best be viewed as a potentially revisable solution to problems and issues encounteredwhile conducting design experiments. As a first step in outlining the rationale,a point worth reiteratingis that classrooms are complex, messy, and sometimesconfusingplaces. One of the concerns thatall Design Researchersneed to addressis thatof developingan analyticframework thatenables researchparticipantsto discernpatternand orderin what often appearto be ill-structuredevents. These concernsandinterestsgive rise to several criteriathatan analyticalapproachshould satisfy if it is to contributeto reformin mathematicseducation as an ongoing, iterative process of continual improvement. These criteriainclude the following:

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The results from the analyses should feed back to improve the instructional designs. " The methodology should permitdocumentationof the collective mathematical learningof the classroomcommunityover the extendedperiodsof time spanned by design experiments. " The analysis should permit documentation of the developing mathematical reasoning of individual students as they participatein communal classroom processes. "

The first of these criteriafollows directly from the emphasis on testing and revising the conjectures inherent in initial designs when conducting a design experiment.The second criterion,which emphasizes the importanceof focusing on the mathematicallearningof the classroomcommunity,stemsfromthe approach to instructionaldesign thatI have outlined.As noted,the designerdevelopsconjecturesaboutan anticipatedlearningtrajectorywhen preparingfor a design experiment.These conjectures,however,cannotbe aboutthe trajectoryof each andevery student'slearningfor the straightforward reasonthatsignificantqualitativedifferences are presentin theirmathematicalthinkingat any point in time. As a consequence, descriptionsof plannedinstructionalapproacheswrittento imply that all studentswill reorganizetheirthinkingin particularways at particularpoints in an instructionalsequence involve, at best, questionable idealizations. For similar reasons, analyses that speak of changes in students' reasoning are potentially misleadingin thatthey imply thatstudentswill all reorganizetheirthinkingin the same way. I should acknowledgethatI have, in fact, spoken in these termsto this point in this introductionfor ease of explication. Although this approachproved to be adequatewhen discussing the various means of supportingand organizing learning,it does not offer the precisionwe need to improveour designs. Againstthe backgroundof these considerations,an issue thatmy colleagues and I have sought to addressis that of clarifying what the envisioned learningtrajectories centralto our, and others', work as instructionaldesigners might be about. The resolutionthatwe proposeinvolves viewing a hypotheticallearningtrajectory as consisting of conjecturesaboutthe collective mathematicaldevelopmentof the classroom community.This proposal,in turn,indicatesthe need for a theoretical constructthatenables us to talk explicitly aboutcollective mathematicallearning, andfor this reasonwe have developedthe notionof a classroommathematicalpractice. Stephanand her colleagues (chapters3 and 5 of this monograph)define and illustratethisconstructwhen they presenttheiranalysisof the measurementdesign experiment.For the present,it suffices to note thata hypotheticallearningtrajectorythatis framedin thesetermsconsistsof an envisionedsequenceof mathematical practicestogetherwith conjecturesaboutthe means of supportingand organizing the emergence of each practicefrom priorpractices. The last of the threecriteriathatan interpretiveapproachshould satisfy focuses on qualitative differences in individual students' mathematicalreasoning. The rationalefor this criterionis again deeply rooted in the activity of experimenting

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InvestigatingStudents'Reasoning

in classrooms. The classroom activity structurethat I describedwhen discussing ourpreparationsfor the measurementexperimentis representativein that,in most of the design experimentsthat I and my colleagues have conducted,the students work either individually or in small groups before convening for a whole-class discussion of their interpretationsand solutions. During the individualor smallgroup work, the teacherand one or more of the projectstaff normallycirculatein the classroomto gain a sense of the diverseways in which studentsareinterpreting and solving the instructionalactivities.Towardthe end of this phaseof the lesson, the teacherandprojectstaff membersconferbriefly to preparefor the whole-class discussion. In doing so, they routinelyfocus on the qualitativedifferences in the students'reasoningto developconjecturesaboutmathematicallysignificantissues that might, with the teacher's guidance, emerge as topics of conversation.They might, for example, conjecturethata particularmathematicalissue will emerge if two specific types of solutionsarecomparedduringthe discussion.Giventhispragmaticfocus on individualstudents'reasoning,we requirean analyticapproachthat takes accountof the diverseways in which studentsparticipatein communalclassroom practices. In the hands of a skillful teacher,this diversity can, in fact, be a primaryresourceon whichthe teachercan capitalizeto supportthecollective mathematical learningof the classroomcommunity. This discussionof the threecriteriathatan interpretiveframeworkshouldsatisfy illustratesthe pragmaticorientationwe take in viewing theoreticalconstructsas conceptualtools whose developmentreflectsparticularinterestsandconcerns.From this perspective, the relevant concern when assessing the value of theoretical constructsis whetherthey enable us to be more effective in supportingstudents' mathematicallearning.Clearly,also, aninterpretiveapproachthatsatisfiesthethree criteria will characterizestudents' mathematicallearning in situated terms. In doing so, however, an interpretiveapproachwill take a differentview of the classroom activity system. The aspects of this system on which I focused when discussing the variousmeans of supportingand organizingmathematicallearning dealtwith resources(e.g., tasks,tools) andwithcharacteristicsof collective activity (e.g., the structureof classroomactivities,classroomdiscourse).Missingfrom this picture is an indicationof the specific processes by which increasinglysophisticatedmathematicalways of reasoningmightemergeas studentsparticipatein these collective activities by using the tools to complete instructionaltasks. The interpretive frameworkthat Stephanand Cobb (chapter3 of this monograph)employ when analyzingthe datageneratedin the courseof the measurementdesign experiment is designed to addressthis limitation. CONCLUSION My purposein thisintroductionhasbeen to situatethe measurementdesignexperiment within a broadermethodologicalcontext. To this end, I highlightedcycles of design and analysisas a definingcharacteristicof Design Research.I also noted the pragmaticemphasisof the methodologyin addressingproblemsof supporting

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students'mathematicallearningsimilarto those addressedby practitioners.In addition, I discussedthe theoreticalintentof developing domain-specificinstructional theoriesthatenable the resultsof an experimentto be generalizedby means of an explanatory framework. This dual focus on theoretical issues and pragmatic concernsmakesthe methodologyparticularlywell suitedto the taskof investigating the prospectsand possibilities for reformin mathematicseducation at the classroom level. The two core aspectsof Design Researchon which I have focused arethe design theorythatguides the developmentof specific designs and the interpretiveframeworkthatguides the analysisof classroomevents. In discussingthe design theory, I emphasizedthatthe conjecturesdeveloped when preparingfor the measurement experimentwere not restrictedto curriculumdevelopers'traditionalfocus on tasks and tools. As I illustrated,the conjecturesalso took accountof both the natureof classroom discourse and the classroom activity structure.To accommodatethis broaderperspectiveon design, I introducedthe notion of the classroom activity system. The hallmarkof such systems is that they are intentionallydesigned to producethe learningof significant mathematicalideas as studentsparticipatein them and contributeto their evolution. In light of this shift from a focus on tasks andtools to a broaderconcernfor the activitysystemsthatconstitutethe social situations of students'learning,one might appropriatelyspeakof design experiments ratherthanof teachingexperiments.The term teaching experimentindicatesthat the methodologyis agendadrivenandhighlyinterventionist.The termdesignexperimentaddsa theoreticaldimensionby indicatingthebroaderperspectivefromwhich the interventionsare conceptualized. My discussionof the secondcore aspectof Design Researchfocused on the three criteriathatan appropriateinterpretiveframeworkshould satisfy. As I noted, the resulting accounts of students'mathematicallearningare necessarily situatedin that they are tied to analyses of the actual environmentin which that learning occurred.As Stephanandothers' analysis in chapter5 of the measurementexperiment illustrates,one can thereforedisentangle aspects of this environmentthat served to supportthe developmentof the students' reasoning.Doing so, in turn, leadsto the developmentof testableconjecturesaboutthe way in whichthosemeans of supportand thus the instructionaldesign mightbe improved.In this regard,the methodology employed by Stephanand her colleagues remainstrue to Brown's (1992) andCollins's (1999) vision of educationalresearchas a processof ongoing, iterativeimprovement. Given my purposein this chapter,I have focused on generalcharacteristicsof the design experimentratherthanon concreteaspects of the methodology.These aspects are discussed in the remainderof this monographand include the specification of the relationbetween classroom-basedresearchand instructionaldesign, the interpretiveframeworkusedto analyzelearningin the socialcontextof the classroom,andthe role of tools in bothlearninganddesign.In chapter2, Stephansynthesizes the literatureon students'measurementconceptions to furtherdevelop the threemonographthemes. In chapter3, Stephanand Cobb discuss the interpretive

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InvestigatingStudents'Reasoning

frameworkand methodthatguided the analysis of students'learningin the social context of the classroom. In chapter4, Gravemeijer,Bowers, and Stephanillustratethe centralinstructionaldesign heuristicsof RME and presenta hypothetical learningtrajectorythat served as the basis for the measurementexperimentation. In chapter5, Stephan,Cobb, and Gravemeijeroffer an analysis that coordinates the learningof the classroom communitywith the reasoningof the participating students.Gravemeijer,Bowers, and Stephanthen use this analysis in chapter6 to influence the revision of the prior hypothetical learning trajectory.Finally, in chapter7, Stephanrevisitsthe threemonographthemesin light of themeasurement experiment.The contentof the chapterswas organizedin this mannerto illustrate the cyclic processof instructionaldesign andclassroom-basedresearchthatis characteristicof Design Research. REFERENCES Anderson, J., Greeno, J., Reder, L., & Simon, H. (2000). Perspectives on learning, thinking, and activity. EducationalResearcher,29(4), 11-13. Anderson,J., Reder,L., & Simon,H. (1996). Situatedlearningandeducation.EducationalResearcher, 25(4), 5-11. Anderson, J., Reder, L., & Simon, H. (1997). Situative versus cognitive perspectives:Form versus substance.EducationalResearcher,26(1), 18-21. Bednarz,N., Dufour-Janvier,B., Portier,L., & Bacon, L. (1993). Socioconstructivistviewpointon the use of symbolism in mathematicseducation.TheAlbertaJournal of EducationalResearch, 39(1), 41-58. Bowers, J., & Nickerson,S. (2001). Documentingthe developmentof a collective conceptualorientation in a college-level mathematicscourse. MathematicalThinkingand Learning,3, 1-28. Brown, A. L. (1992). Design experiments:Theoretical and methodological challenges in creating complex interventionsin classroom settings.Journal of the LearningSciences, 2, 141-178. Cobb, P. (2000). Conductingteachingexperimentsin collaborationwith teachers.In A. E. Kelly & R. A. Lesh (Eds.), Handbookof researchdesign in mathematicsand science education(pp. 307-334). Mahwah,NJ: Erlbaum. Cobb, P., & Bowers, J. (1999). Cognitiveand situatedperspectivesin theoryandpractice.Educational Researcher,28(2), 4-15. Cobb, P., & Steffe, L. P. (1983). The constructivistresearcheras teacherand model builder.Journal for Research in MathematicsEducation,14, 83-94. Cobb, P. & Yackel, E. (1996). Constructivist,emergent,and socioculturalperspectivesin the context of developmentalresearch.EducationalPsychologist, 31, 175-190. Cobb, P., Yackel, E., & Wood, T. (1989). Young children'semotionalacts while doing mathematical problem solving. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematicalproblem solving: A new perspective (pp. 117-148). New York: Springer-Verlag. Collins, A. (1999). The changing infrastructureof educationalresearch.In E. C. Langemann& L. S. Shulman(Eds.), Issues in educationresearch (pp. 289-298). San Francisco:Jossey Bass. Confrey, J., & Lachance,A. (2000). Transformativereadingexperimentsthroughconjecture-driven researchdesign. In A. E. Kelly & A. Lesh (Eds.), Handbookof researchdesign in mathematicsand science education (pp. 231-266). Mahwah,NJ: Erlbaum. DeCorte,E., Greer,B., & Verschaffel,L. (1996). Mathematicslearningand teaching. In D. Berliner & R. Calfee (Eds.), Handbookof educationalpsychology (pp. 491-549). New York:Macmillan. diSessa, A. A., Hammer, D., Sherin, B., and Kolpakowski, T. (1991). Inventing graphing: Metarepresentationalexpertise in children.Journal of MathematicalBehavior, 10, 117-160. Doerr,H. M. (1995, April).An integratedapproachto mathematicalmodeling:A classroomstudy.Paper presentedat the AnnualMeetingof the AmericanEducationalResearchAssociation,San Francisco, CA.

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Ddrfler,W. (1993). Computeruse and views of the mind. InC. Keitel & K. Ruthven(Eds.), Learning from computers:Mathematicseducationand technology.Berlin, Germany:Springer-Verlag. Franke, M. L., Carpenter,T. P., Levi, L., & Fennema, E. (2001). Capturingteachers' generative change: A follow-up study of teachers' professional development in mathematics. American EducationalResearchJournal, 38, 653-689. Freudenthal,H. (1973). Mathematicsas an educationaltask. Dordrecht,Netherlands:Reidel. Gravemeijer,K. (1994). Developing realistic mathematicseducation. Utrecht, Netherlands:CD-B Press. Greeno,J. G. (1997). On claims thatanswerthe wrongquestions.EducationalResearcher,26(1), 5-17. Kaput.J. J. (1994). The representationalroles of technology in connectingmathematicswith authentic experience.In R. Biehler, R. W. Scholz, R. StriBer,& B. Winkelmann(Eds.), Didactics of mathematics as a scientific discipline (379-397). Dordrecht,Netherlands:Kluwer. Lehrer,R., Jacobson,C., Kemney, V., & Strom,D. A. (1999). Building upon children'sintuitionsto develop mathematicalunderstandingof space. In E. Fennema& T. R. Romberg(Eds.),Mathematics classrooms thatpromote understanding(pp. 63-87). Mahwah,NJ: Erlbaum. Lerman,S. (1996). Intersubjectivityin mathematicslearning:A challenge to the radicalconstructivist Journalfor Research in MathematicsEducation,27, 133-150. paradigm'? Lerman,S. (2000). A case of interpretationsof "social":A responseto Steffe andThompson.Journal for Research in MathematicsEducation,31(2), 210-227. Menchinskaya,N. A. (1969). Fifty yearsof Soviet instructionalpsychology.In J. Kilpatrick& I. Wirzup (Eds.), Soviet studies in the psychology of learning and teaching mathematics(Vol. 1). Stanford, CA: School MathematicsStudy Group. Moss, J., & Case, R. (1999). Developingchildren'sunderstandingof the rationalnumbers:A new model and an experimentalcurriculum.Journalfor Research in MathematicsEducation,30, 122-147. Nemirovsky. R. (1994). On Ways of Symbolizing:The Case of Lauraand Velocity Sign. Journal of MathematicalBehavior, 13, 389-422. Pea, R. D. (1993). Practicesof distributedintelligenceanddesigns for education.In G. Salomon(Ed.), Distributedcognitions (pp. 47-87). New York:CambridgeUniversityPress. Rasmussen,C. (1998). Reformin differentialequations:A case study of students' understandingand difficulties.Paperpresentedat the annualmeetingof the AmericanEducationalResearchAssociation, San Diego, CA. Simon,M. A. (1995). Reconstructingmathematicspedagogyfroma constructivistperspective.Journal for Research in MathematicsEducation,26, 114-145. Simon, M. A. (2000). Researchon the developmentof mathematicsteachers:The teacherdevelopment experiment.In A. E. Kelly & R. A. Lesh (Eds.), Handbookof research design in mathematicsand science education (pp. 335-359). Mahwah,NJ: Erlbaum. Skemp, R. (1976). Relationalunderstandingand instrumentalunderstanding.MathematicsTeacher, 77, 20-26. Steffe, L. P. (1983). The teachingexperimentmethodologyin a constructivistresearchprogram.In M. Zweng, T. Green, J. Kilpatrick, H. Pollak, & M. Suydam (Eds.), Proceedings of the Fourth InternationalCongress on MathematicalEducation.Boston: Birkhauser. Steffe, L. P., & Kieren, T. (1994). Radical constructivismand mathematicseducation. Journalfor Research in MathematicsEducation,25, 711-733. Steffe, L. P., & Thompson,P. (2000a). Interactionor intersubjectivity?A replyto Lerman.Journalfor Research in MathematicsEducation,31(2), 191-209. Steffe, L. P., & Thompson,P. W. (2000b). Teachingexperimentmethodology:Underlyingprinciples and essential elements. In A. E. Kelly & R. A. Lesh (Ed.), Handbookof researchdesign in mathematics and science education (pp. 267-307). Mahwah,NJ: Erlbaum. Stigler, J. W., & Hiebert,J. (1999). The teaching gap. New York:Free Press. Streefland,L. (1991). Fractions in realistic mathematicseducation. A paradigm of developmental research. Dordrecht,Netherlands:Kluwer. Suter, L. E., & Frechtling, J. (2000). Guiding principles for mathematics and science education research methods:Reportof a workshop.Washington,DC: National Science Foundation.

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Thompson,A. G., Philipp,R. A., Thompson,P. W., & Boyd, B. (1994). Calculationalandconceptual orientationsin teachingmathematics.In Douglas B. Aichele (Ed.), 1994 Yearbookof the National Council of Teachers of Mathematics(pp. 79-92). Reston, VA: National Council of Teachers of Mathematics. Thompson,P. W., & Saldanha,L. A. (2000). Epistemologicalanalysesof mathematicalideas:A research AnnualMeetingof theNorth methodology.In M. Fernandez(Ed.),Proceedingsof the Twenty-Second American Chapterof the InternationalGroupfor the Psychology of MathematicsEducation (pp. 403-407). Columbus, OH: ERIC Clearinghousefor Science, Mathematics,and Environmental Education. Treffers,A. (1987). Threedimensions:A modelof goal and theorydescriptionin mathematicsinstruction-The WiskobasProject. Dordrecht,Netherlands:Reidel.

Chapter 2

Reconceptualizing Linear Measurement Studies: The Development of Three Monograph Themes Michelle Stephan Purdue UniversityCalumet

Measuringlengths of objects is a common practicein a wide range of out-ofschool situations.Forexample,some families keep trackof theirchildren'sgrowth by measuringtheirheights andeitherkeeping a writtenrecordor makingphysical markson a wall. In addition,learningto measureforms a foundationfor investigating other mathematicaltopics, such as proportion,decimals, and fractions,to name a few (cf. NationalCouncilof Teachersof Mathematics[NCTM],2003). For these reasons, instructionin linear measurementis included in most elementary curricula.The NCTM Principles and Standardsfor School Mathematics(2000) emphasizes the importanceof establishing a firm foundation in the underlying concepts and skills of measurement.The documentemphasizesthatchildrenneed to "understandthe attributesto be measuredas well as what it means to measure" (p. 103). A large body of literatureon children'sconceptionsof measurementhas amassedover the past three decades, with undoubtedlythe most influentialwork being that of Jean Piaget and his colleagues (e.g., Piaget, Inhelder,Szeminska, 1960). Piaget and his colleagues identified developmentalstages throughwhich they claimed childrenpass as they learn to measure.As a result of this analysis, many researchershave tried to isolate the ages at which childrendevelop certain measurement concepts. Other researchers have devised training programs to increase the acquisition rate of measurementconcepts. Whereas each of these studiestook a primarilyindividualisticapproachto learningandtraining,few studies have been conductedthataddresssocial' influences on children'sdevelopmentof measuring abilities. As such, I argue that currentmeasurementstudies should include social aspects as well as tool use as an integralpartof learningto measure.

I We use the termsocial throughoutthe monographto referto the interactionsbetween the teacherand the studentsat the local level of the classroom, not to any particularsocietal characteristic,such as a student'sracialbackgroundor SES status.

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The purposeof this chapteris to identify anddevelop threeresearchthemesthat drive the organizationand substanceof this monographin regardto supporting students' developmentof linear measurementconceptions. To provide rationale for these three research themes, I first include an extensive literaturereview describing prior research on children's measurementconceptions. This review includes descriptionsof studies conducted by Piaget and his colleagues and of trainingprogramsthatweredesignedto supportchildren'sdevelopmentof measurement skills. Second,I use this literaturereview as backgroundto elaboratethe three researchthemes that serve as the basis of inquiryfor this study.Third,I elaborate our theoreticalposition on each of the researchthemes to arguethat new investigations of children'smeasurementdevelopmentshould attendto social aspects of learningto measure,the use of tools to supportdevelopment,and learningas it is closely tied to the designer's in-actiondecisions. MEASUREMENTINVESTIGATIONS Most investigations into children's conceptions of measurementhave been based on studies that were published in the 1950s and 1960s by Piaget and his colleagues(i.e., Piaget& Inhelder,1956;Piagetet al., 1960).Otherresearchershave set out to either substantiateor disprove the claims that Piaget made about children's development.Because the majorityof the literatureon children'sconceptions of measurementfocused on Piaget's stage theory, I begin this section by discussing Piaget's investigationson measurement.Further,I draw connections between our findings and those of Piaget throughoutthe monograph,in particular in chapter5. Piaget's Studies Piagetdefinedmeasurementas a synthesisof changeof positionand subdivision (Piaget et al., 1960). More precisely, measurementfor Piaget involved (a) understandingspace or the length of an object as being partitionable,or as being able to be subdivided(subdivision),and(b) partitioningoff a unitfroman objectanditerating thatunit withoutoverlapor empty intervals(change of position). Piaget and his colleagues arguedthat the coordinationof these two notions along with the understandingthat these continuous units form inclusions (i.e., the first length measuredis included in the length that comprises two units, etc.) leads to a full understandingof measurement.The claim thatconservationof length develops as a child learnsto measureis implicitin this definition.Conservationof lengthmeans thatas a child moves an object, the object's length does not change.Conservation of length is not equivalentto the concept of measurementbut, rather,develops as the child learnsto measure(Inhelder,Sinclair,& Bovet, 1974). For Piaget, knowledge of measurementrequiredthata child be able not only to apply the proceduresand skills of measuringbut also to carryout these activities without merely following specific, formalized measuring procedures.In other

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words, measurementfor Piaget did not consist of simply performingthe steps requiredto measure.Rather,childrenwho could measurein his termswere almost immediatelyable to use the standardmeasurewith insight, not by trial and error (Piaget et al., 1960). Consequently, a child who could use a ruler correctly to measurewould not necessarilyhave an operationalunderstandingof measurement accordingto Piagetandothers'definition.Piagetandhis colleagues' analyseshave particularvalue because they focused on the mathematicalmeanings and interpretations ratherthan solely the methods of measuringassociatedwith particular measurementunderstandings.These types of analyses are also consistent with current mathematicalreform efforts. For example, the NCTM Principles and Standardsdocument(2000) emphasizes that mathematicsshould be meaningful activity ratherthan a set of learnedrules and proceduresfor calculating. Piagetet al. (1960) categorizedchildren'sdevelopmentof measurementconceptions into eitherthree or four stages, dependingon which aspect of measurement was being investigated. For example, the development of the conservation of lengthfell into threedistinctstages with varioussubstages.Whatfollows is a brief, integrativedescriptionof some of the mainstages in Piaget's accountof children's development of measurementconcepts. For a more detailed inspection of the variousstagesof development,see Piagetet al. (1960), Carpenter(1976), Copeland (1974), and Sinclair(1970). At the initial stages of development, including various substages, children conserve neitherlength nor distance.Piaget used the termdistance to referto the empty space between two objects, such as two trees. He used the term length to denote the physical extension of an object, such as a stick. Childrenwho areclassified as being at early stages of developmentrely strictlyon visual perceptionfor judgmentsaboutlength.Forinstance,Piagetaskedseveralchildrenin clinicalinterviews abouttwo stripsof equal length (see Figure 2.1). He placed them in direct alignmentwith each other, as shown in situation(1) in Figure2.1, and asked the child which stripwas longer. Once the child acknowledgedequalityof length, the interviewermoved the bottomstripa few centimeters,as shown in situation(2) in Figure2.1, and again asked which one was longer. Those childrenclassified as in the early stages of developmentconcluded that the bottomstripwas now longerthanthe top stripbecausethe bottomstripextended beyondthe top one. Piagetconcludedthatthese childrenwere reasoningby relying

(1)

(2)

Figure 2.1. Interviewtask focusing on conservationof length.

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on perceptualjudgmentscenteringon the position of the endpoints.He arguedthat operationalmeasurementis not possible for childrenclassified as in early stages, because space is not viewed as a common medium containingobjects with welldefined spatialrelationsbetween them.Accordingto Piaget,childrenwith such an uncoordinatedview of space do not understandthatthe distancefromA to B is the same as the distancefrom B to A. Piaget arguedthat childrenwhom he classified as in the early stages were not able to reasontransitively,an abilitythatinvolves, for instance,using a stick as an instrumentto judge whethertwo immovabletowers are the same size. Being able to reason transitivelyis essential for operationalmeasurement.A child who can reason transitivelycan take a thirdor middle item (e.g., the stick) as a referentby which to comparethe heights or lengths of otherobjects. Duringthe early stages of development, however, when children do not conserve length, transitive reasoning is impossible because once childrenmove the middle object (e.g., the stick), the length of thatmiddle object, in their view, can change. Piagetassertedthatchildrenclassifiedas in the earlystagesdo nothave an understandingof either subdivisionor displacement.This lack of understandingcan be seen as childrenmeasurethe lengthof an item with a smallerunit.Typically, these childreneitherrunthe smallerunit alongside the length of the item withoutproperly partitioningthe length or they make iterationsof unequalspaces, sometimes neglecting gaps or overlappingareas.Their thinkingat these early stages is said to be intuitive and irreversible(i.e., they are unableto go backward;for instance, A to B is not the same as B to A). Piagetidentifiedtransitionalstagesbetweentheseearlystagesandthe final stages of operationalmeasurement.Childrenin these transitionalstages(about6 to 7 years old) begin to reasontransitivelyby using theirbody as a middlereferent.Later,at about7 to 8 years old, they can reasontransitivelywith otherobjects andbegin to conserve length. Piaget asserted,however, that childrenclassified as in a transitional stage have not yet coordinated change of position and subdivision but possess each of the notions separately.These childrenunderstandthat changing the position of an item does not alterquantityand thatthe whole is the sum of its parts,but they do not coordinatethese two concepts. Finally, the child attainsthe last stage, operationalmeasurement,when he or she has synthesizeddisplacementwith subdivision(about8 to 10 years old). Implicit in this acquisitionfor Piagetwas thatmeasuringconsistedof a seriesof nestedrelations. In otherwords,as a unitis iterated,the distancecoveredby the firsttwo units is nested in the distancecovered by threeunits, and so on. Differentiatingthis last stagefromthe mereabilityto performthe skills of conventionalmeasuringis important. To be operational,the actions of the measurementprocess must be interiorized into conceptual acts that are an integral part of an organized structure (Carpenter,1976). In otherwords,the child's activityof measuringcorrectlydoes not indicateoperationalmeasuring;rather,the activitymust be accompaniedby a conceptualreorganization.Childrenat this stage can compareunits' lengths and discover that a small unit is a thirdor half of another.Similarto Piaget's percep-

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tion, we characterizeacting in a spatial environmentas coordinatingunits with differentmeasures,an ability that includes structuringpartsof units in fractional units. In addition, the results of measuring should come to signify a series of nested distances (what we call accumulationof distance). In Piaget's developmentalstage theory,childrenpass from one stage to the next by making shifts in how they reorganize their previous experiences. Children proceed throughthese stages in a relatively fixed order.Piaget's developmental stage theory seems to characterizelearningas a series of clear-cutcognitive reorganizationsthat signal thatan individualhas passed from one stage to the next. This characterizationis individualisticin thatPiaget explainedlearningas occurring solely in the mind of the child. Piaget would not deny that conceptual reorganizationsoften occur as a consequenceof social interaction,but his analysis of children's learningto measurewas castprimarilyas an individualaccomplishment. In contrast,currentstudies have demonstratedthe social natureof learning (Lave, 1988; Rogoff, 1990; Saxe, 1991; Vygotsky, 1978). Piaget et al.'s (1960) analysisthatchildrenpass througha fixed set of stages did not accountfor social aspectsof the learningprocess.The studythatwe reportin this monographcan be viewed as drawingon Piaget and his colleagues' descriptionof what counts as a deep understandingof measurement(e.g., operationalmeasurementas subdivision anddisplacementandinclusive),yet it recastscognitionas occurringin situ.In other words,we find Piaget andothers'psychological analysisto be extremelyvaluable for determiningthe instructionalintentof the measurementdesign experimentbut insufficientfor analyzingthe learningof studentsas they participatein the social context of a classroom community. Therefore,we draw on Piaget's conceptual distinctionsregardinglearningto measureto make sense of children'slearningin social context. Reactions to Piaget Researchers,sparkedby Piaget's analysisof children'sconceptionsof measurement,focused on two differentsets of issues. One interpretationof Piaget's developmentalmodel was thatcertainmeasurementconcepts developed in fixed order. For example, some researchersarguedthat the conservationof length developed before transitivereasoning.Also, some researchersinterpretedthe ages at which children construct particularmeasurementconcepts as being relatively fixed. Researcherscommonlytestedthe orderin which measurementconceptsdeveloped or debatedthe ages at whichchildrenreasonedtransitively.As a consequence,many researcherssought to substantiateor discreditPiaget's earlierclaims. Along this same line, more recent studies have used Piaget's epistemology to detail the conceptualschemas that childrenconstructas they learn to measure.Thus, I call the first collection of measurementliterature"cognitiveanalyses." A second set of investigationsbased on Piaget's analysesfocused on increasing the acquisitionrateof certainmeasurementconcepts (Beilin, 1971). Forexample, researchersattemptedto train children to conserve length by using a variety of

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trainingtechniques. The primaryfocus of the "trainingstudies"was to increase the rate at which childrenlearn the conservationand transitivityof length using various training techniques. Therefore, many different experimental methods were being used. In the following sections, I synthesize the majority of the studies based on Piaget's analyses. Because the researchersmade a varietyof differentand sometimes conflicting theoreticalassumptions,however, definitive conclusions about theirfindings arevery difficultto draw.Thus, I describebriefly the most common theoreticalperspectives that guided the research.My intent is not to present an exhaustive accountof the researchbut ratherto providean overview of the topics thatwere studiedand debated. Cognitive analyses. In this first set of studies, what I have termed "cognitive analyses,"manyresearcherstestedthe orderof Piaget's measurementconcepts by using differentexperimentalmethods and tasks (e.g., Carpenter,1975; Gelman, 1969;McManis,1969;Murray,1965, 1968;ShantzandSmock, 1966).Forinstance, ShantzandSmock conducteda studyinvestigatingwhetherthe developmentof the conservation of length occurs before the development of a spatial coordinate system. Their findings supportedPiaget and Inhelder's (1956) theory that the developmentof the conservationof length,which occursaroundage 7, precipitates the development of a spatial coordinate system, which occurs at about age 9. Generally,earlierresearchon the age and orderof the developmentof measurement concepts supportedPiaget and Inhelder's claims. In more recent studies, however, researchersarguedthatconservationand transitivitydo not necessarily need to be constructedfirstafterall. In fact, they contendthatchildrencan construct the inverserelationbetweennumberandunit-size beforeconservation(Clements, 1999; Hiebert,1981;Petitto,1990)but thatconservationis animportantnotionthat leads to operationalmeasurement(Petitto, 1990). A secondbody of researchfocused on the age at which studentslearntransitivity. Transitivitystudies were the subject of so many methodologicalvariationsthat researchersarguedthatthe ages at whichtransitivitydevelopedin childrenranfrom as early as 4 years to as late as 8 years old. As a result, a debateensued between Braine(1964) and Smedslund(1963, 1965) concerningassessmenttechniquesand the need to define more exactly what was meantby evidence of transitivity.Both BraineandSmedslundaccountedfor learningusingPiaget's developmentalmodel, but they disagreedon the ages at which transitivityoccurred;Brainesuggestedthat transitivityoccurredat 4 to 5 years old, and Smedslundsuggestedthatit occurred at 7 to 8 years old. A more recent study by Kamii (1997) confirmedPiaget and Inhelder's (1956) and Smedslund's findings. She also found thattransitivitymust be developedbeforeunititeration.As to Piaget's earlierclaims thattransitivityand conservationdevelopsimultaneously,some researchersperformedexperimentsthat negated this claim (Brainerd, 1974; Lovell & Ogilvie, 1961; McManis, 1969; Smedslund, 1961; Steffe & Carey, 1972). With regardto constructinga full, operationalunderstandingof measurement, Lovell, Healey, andRowland(1962) generallysubstantiatedPiagetandInhelder's

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(1956) stage theory categorizations.Also, the ages at which full measurement develops in childrenwere found to be close to Piaget's estimate of 9 years old. Kamii (1997) obtainedsimilarresults in her study, suggesting that full measurement understandingdevelops around9 to 10 years old. Lehrer,Jenkins,& Osana (1998) substantiatedthis age range when they found that childrenbelow fourth gradeincorrectlymeasuredthe length of an objectusing two different-sizedunits. Lehrerandhis colleaguesconcludedthatthese studentshadnotconstructedthe relation betweenunitof measureandthe attributebeing measured(see also Barrettand Clements, 1996; Clements,Battista,& Sarama,1998). Although Piaget and other researchersoften disagreed, all these studies share one assumption.In these analyses, learningis primarilyanalyzedas an individualistic achievement.Learningconsists of cognitive reorganizationsthat occur in the mind of the child, sometimes as a consequence of social interactionor tool use. In otherwords,the researchersrealizedthatthe processof learningcould have social aspects, but analyses were cast solely in termsof an individual's cognitive development. Trainingstudies. The second body of literature,trainingstudies, came aboutas a reactionto Piaget'snaturalisticslanton learning.Piagetdevelopeda generaltheory of conceptualdevelopmentfor transitionsbetweenstages,butmost of this research focused on children'sperformingtaskspresentedin clinical interviews.Very little empiricalevidence supportedhis theoriesaboutthe processandways of supporting conceptualdevelopment.Consequently,trainingstudies centeredon the acquisition of cognitive structuresby consideringthe effect of differenttypes of training procedures. Piaget described trainingstudies as an attemptto rush development (Rogoff, 1990), andhe is generallyinterpretedas doubtingthattraininghas an effect on the development of children's conceptions. Nevertheless, many researchersbased their trainingtechniqueson the basic componentsof Piaget's equilibriummodel. Equilibrationis best described as a dynamic, self-organized balance between assimilationand accommodation.Assimilation is the process of organizingnew experiences in terms of prior understanding.When new experiences cannot be adequately organized in this way or when they give rise to contradictions,an accommodationtypically occurs. An accommodationinvolves reflective, integrative processes that create a new understandingto restoreequilibrium.Piaget conjecturedthat disequilibriumis brought about by experiences that generate cognitive perturbations(von Glasersfeld, 1995). Many of the trainingprogramsused aspects of the equilibriummodel as the assumptionsbehindtheirtrainingmethods.Researcherstriedto induce cognitive conflict in childrento bring about disequilibrium(Bailey, 1974; Brainerd,1974; Murray,1968; Overbeck& Schwartz,1970; Smedslund,1961). Otherresearchers arguedthatreversibilitymightplay an importantrole in decreasingthe age at which conservationof lengthdeveloped in children(Brison, 1966;Murray,1968; Smith, 1968). Forinstance,one of the experimentaltreatmentsin Smith's (1968) training

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programinvolved askingchildrento makejudgmentsaboutthe lengthof an object after adding and subtractingpieces from the end of it. In this way, the child was "shown"thatreversingoperationscould help determineequalityof objects. Like the cognitive analyses, trainingstudies also focus primarilyon individualcognitive growthandhence do not include descriptionsof any social aspectsor tool use involved in the trainingsessions. Although these trainingstudies were based on aspects of Piaget's equilibrium modelto determinetrainingmethods,theirtraininggoals seem to have differedfrom Piaget's goals. In my view, Piaget was concerned with identifying children's understandingof, and their constructedmeanings for, conservation.In contrast, analyses of the trainingstudies focused only on whetherchildrencould conserve lengthby the end of the trainingsessions. Theirprimaryconcernwas on children's acquisitionof the conservationof length, and this attainmentwas determinedby correct answers to conservation tasks. In other words, evidence that a child conservedlengthwas determinedby his or herperformanceon conservationtasks, not by detailed analyses of his or her thinking.One exception to this techniqueis a studyconductedby Inhelderet al. (1974). This studydifferedin thatpreliminary experimentswere conductedto understandchildren'sconceptionsof the relationship between numberand length and to identify stages of developmentpertaining to this relationship. After the stages were identified, training sessions were conductedandseveralof the children'sresponsesduringthe session were analyzed to identifythe psychologicalprocessesinvolvedin coordinatingdiscreteunitswith continuouslength.The mainproductof these trainingsessionsconsistedof psychological analyses of children'ssolutions, notjust scores on tests. On the one hand, this kind of studyreflects Piaget's theoryof learningmore thanthe othertraining studies. On the otherhand, such analyses were still cast in terms of cognitive reorganizationsdevoid of social context. Another set of trainingstudies focused on whether children transferredtheir measurementconcepts to other task situations(or domains). For example, many researchersinvestigated whether children who were trainedto conserve length would transferthose skillsto conserveweightor area(Beilin, 1965;Brainerd,1974; Brison, 1966;Gruen,1965;Tomic, Kingma,& Tenvergert,1993).Trainingsuccess was determinedby comparingscoreson pretestsandposttestsinsteadof attempting to explainchildren'smeaningfulactivityon a varietyof interviewtasks.Generally, researchersfound thattrainingfor the conservationof length using various techniques transferredto otherdomains(cf. Beilin, 1965; Brainerd,1974). For a more in-depth review of the literatureconcerning transferof training, see Osborne (1976). A final groupof trainingstudieswas conductedfroman information-processing perspective.Studieshadbeen conductedin the 1970s by researcherstryingto identify the information-processingdemandsof tasks thatdealt with the conservation of length (Baylor, Gascon, Lemoyne, & Pothier, 1973; Hiebert, 1981; Klahrand Wallace, 1970). The computerwas the dominantmetaphorwith whichto describe children's mathematicallearning,as in the following:

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Itwasbecausetheactivitiesof thecomputer itselfseemedin somewaysakinto cognitive processes.Computersacceptinformation, symbols,storeitems in manipulate "memory" andretrievethemagain,classifyinputs,recognizepatternsand so on. 1987,p. 119) (Gardner, Humanthoughtwas conceptualizedas an information-processingsystem, and the goal was to develop computational models whose input-outputrelations matchedthose of children's observedperformance(Cobb, 1990). In some cases, researchersactually attemptedto write computer programsthat matched their observationsof children'sstrategies(e.g., Bayloret al., 1973). Accordingto these researchers,many mathematicstasks requirethe ability to process several pieces of informationsimultaneously,and studentswith low short-term-memorycapacities were assumed to perform less well than children with high capacities. Therefore,researcherstook into accountthe information-processingrequirements of a taskas theyanalyzedstudents'solutionsto problems.Hiebert(1981) andBaylor et al. (1973) studiedthe relationshipbetweeninformation-processing capacitiesand children'sabilityto learnconceptsof linearmeasurementandfoundthatthe information-processingcapacity had no detectable influences. Nonetheless, Hiebert arguedalong with Baylor and his colleagues that the usefulness of this perspective in educationalsettingsrequiresmore research(see also Inhelder,1972; Klahr & Wallace, 1970). A more recent study outlinedthe sequence in which measurement concepts should be taught and based their ordering on the informationprocessing demand of the tasks (Boulton-Lewis, 1987). This approach,as other computationalmodels of the information-processingandtrainingtheoriesdo more generally,focuses primarilyon cognitive aspectsof training.As such,the approach influencedthe work of researchersattemptingto learnwhat types of interventions work,butit did not generateinsightsregardingwhatsocial aspectsor tools affected the observedoutcomes. Summaryof MeasurementInvestigations Throughoutthis review, I have alluded to several common themes among the various studies. In this section, I summarizethe findings from the literatureto describe three main issues that appearto run throughoutthe literatureregarding Piagetian development:(1) learning was analyzed primarilyfrom an individual perspective,(2) most studies involved documentingthe criteriafor the success of trainingstudies,and(3) very few studiesemphasizedthe role of tools in supporting students'development.Of these threemain issues, I identifythetheoreticalthemes thatrunthroughoutthe monograph. The firstissue runningthroughoutthe literatureinvolved characterizinglearning as an individualaccomplishment.In most cases, social aspectswere seen as serving as catalystsfor cognitive reorganizations,but theirrole as such was not explicitly described. For example, consider the debate between Smedslund (1961) and Brainerd(1974) regardingthe age at which studentslearntransitivity.Smedslund interpreted Piaget as giving very little importance to social processes (i.e.,

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SmedslundarguedthatPiaget would claim thatpositive feedbackfrom the interviewer was not the cause for conceptualreorganizations,becausethey were social processes;conceptualreorganizations,in Smedslund'sinterpretation,were caused by the individual).BrainerdcounteredthatPiaget would arguethatpositive feedback could cause cognitive perturbations.Both interpretationsexplainedlearning in termsof cognitive reorganizations(at times causedby social events) andplaced little emphasis on learningas a social process. At the beginning of this chapter,I pointedout thatcurrenttheoriesemphasizeplacing more thana secondaryrole on the social processes involved in learning.As a consequenceof these currenttheories, furtherinvestigationsof children'sdevelopmentof measurementconceptions mightfocus on students'activityin social context.Thus,one theoreticalthemethat follows naturallyout of this literaturereview concernspayingexplicit attentionto learningas it occurs in the social contextof the classroom. A second issue illuminated in the literaturereview concerns the criteria for success of the training studies. A training technique was judged successful if posttest scores were significantly higher than pretest scores. The means of supportingchildren's development,however, were not describedin the analysis of trainingsessions. In otherwords, if trainingtechniquesdid, in fact, increasethe likelihood that students could conserve length, then trainingwas considered a success. Analyses of students'learningduringthese trainingsessions were not tied to the ways the researchersengagedwith themthatmighthave led studentsto make their conceptual reorganizationsduring training sessions. Currentinstructional design theories call for analyses of students' learning to be tied to our (i.e., teachers',trainers',or researchers')role in supportingthatlearning(Cobb,Stephan, McClain,andGravemeijer,2001). In otherwords,the goals of a researcherarenot only to improvestudents'learningbut also both to reflect on the processby which studentsdeveloped and to locate themselves as designers in that process. Thus, analyses of students'learningin social context can feed back to informdesigners of the ways in which instructional sequences (or training techniques) can be revised to bettersupportstudents'learningin the social context of the classroom setting.The second theme,therefore,thatrunsthroughoutthis monographfocuses on ways of proactively supporting students' developmentfrom the designers' perspective. A thirdissue runningthroughoutthe literaturereview centerson the role of tools in supportingor constrainingstudents'mathematicaldevelopment.Withtheexception of a few studies(e.g., Barrett& Clements, 1996;Clementset al., 1998;Nunes, Light, & Mason, 1993; Piaget et al., 1960), very few of the prior studies on measurementfocused on the role thatmeasuringtools play in supportingstudents' mathematicaldevelopment.Whentools wereconsideredas partof the child's development,the natureof these tool analyseswas individualisticas well. Forexample, Piaget and his colleagues ascribeda child's assimilationor accommodationof a tool to previousmeasuringexperiences.Again, it was as if the tool servedas a catalyst for cognitive reorganizationsbut was not an integral aspect of what was learned. In contrast,currenttheories suggest that furthermeasurementanalyses

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considertools as an intimatepartof students'mathematicaldevelopment.Currently in the field of education,many theoristshave emphasizedthat learningdoes not occur apart from tool use (cf. Cobb, 1997; Kaput, 1994; Meira, 1995, 1998; Nemirovsky & Monk, 2000; Pea, 1993; van Oers, 1996; Vygotsky, 1978). Some researchersgo furtherby arguingfor an extremelystrongrelationshipbetweentool use and learning.They contendthatsymbol creationand meaningmakingare not separate endeavors but occur simultaneously (Meira, 1995, 1998). Because measuring clearly involves using measurementtools, a third and final theme concerns supporting students' mathematicaldevelopment as they reason with culturaltools. RECONCEPTUALIZINGMEASUREMENTINVESTIGATIONS Theoreticalgrowthin thefield of mathematicseducationallowsus, as researchers, to look back on the tremendous work of our predecessors and build on their measurementfindings to learn more about supportingstudents' developmentof fromnew perspectives.Currenttheorieson social and measurementunderstandings culturalprocesses and new methods of proactively supportingstudents' mathematicaldevelopmentprovidenew lenses with which to view old problems.In this section I develop our theoreticalposition on each of the three literaturereview themes discussed above: (1) social aspects of learning,(2) proactivelysupporting students'learning,and (3) culturaltools. Social Aspects of Learning All the studies discussed in the literaturereview involved a primarilyindividualistic view of learning. An increasing number of researchers,however, have adoptedthe premisethatlearningoccursin a socially situatedcontextandthatthese social and culturalaspects greatly influence the way individualsconstructtheir understandings(cf. Cobb & Bauersfeld, 1995; Cole, 1996; Lave, 1988; Rogoff, 1990; Saxe, 1991; Vygotsky, 1978; Wenger, 1998). One of the consequences of adopting such a position is that whereas most researchersacknowledgethatlearninghas both social andindividualqualities,the emphasis they place on each perspectivemay differ significantly. For example, whereasPiaget and his colleagues (Piaget et al., 1960; Piaget and Inhelder,1956) cast analyses in terms of cognitive reorganizations,with social aspects serving merely as a secondarycatalyst,some theoriststake a highly contrastingviewpoint (cf. Lerman, 1996). For example, some socioculturalistshave maintainedthat learningis first and foremosta social endeavor,with individualprocesses consideredto be secondary.This argumentis often supportedby the following commonly cited quote by Vygotsky: twice,orontwoplanes.Firstit appears appears Anyfunctionof thechild'sdevelopment betweenpeople onthesocialplaneandthenonthepsychological plane.Firstit appears as an interpsychological categoryandthenwithinthechildas anintrapsychological category.(Vygotsky,1978,p. 63) (italicsadded foremphasis)

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The authorsof this monographattemptto transcendthese two extremepositions andview learningas a processthatinvolves bothindividualandsocial aspectswith neither takingprimacy over the other. This way of viewing learningis a version of social constructivismcalled the emergentperspective. In this view, the relationship between social and individualprocesses is extremely strong, in that the two cannotbe separated-the existenceof one dependson the existenceof the other. The result of this assumptionis thatresearchersassumingthe emergentperspective attemptto accountfor individuals'mathematicaldevelopmentas they participatein the social andculturalpracticesof the classroomcommunity(Cobb,2000; Yackel and Cobb, 1995). On the one hand, individual students' development is analyzed in terms of their participationin, and contributionto, the emerging, communalmathematicalpractices.On the otherhand,mathematicalpracticesare the taken-as-sharedways a communitycomes to reasonandarguemathematically (Cobb et al., 2001; Stephan, 1998) and often involve aspects of symbolizing. (Chapters3 and5 of this monographfurtherelaboratethe meaningof the construct of a mathematicalpractice.)Thus, we can say that the relationshipbetween individual and social processis strongbecause althoughpracticesconstitutethe takenas-sharedlearningof a community,we takeit as equallyimportantto acknowledge thatstudentsare seen to contributeto the evolution of the classroommathematical practices as they reorganizetheir mathematicalactivity. In other words, mathematical practices and individual students' conceptual development are seen as reflexivelyrelated.Whereaspracticesareestablishedandevolve as studentsparticipate in, and contributeto, them, these acts of participationconstitute,for us, acts by individualstudentsof reasoningand possible conceptualreorganizations. Although a strong dichotomy seems to exist between situated and individual perspectives at the extremes (consider, e.g., the recent debate between Lerman (2000) and Steffe & Thompson(2000)), the emergentperspectiveattemptsto coordinatethe two positions(cf. Confrey, 1995). The classroomteachingexperiment reportedin this monographis an example of an investigationthatplaces students' understandingsof measurementin the local mathematicalpracticesof theirclassroom community. Despite differentperspectivesregardingthe role of social andculturalprocesses in learning,most researchersin this areaagree that students'developmentcannot be adequatelyexplainedin cognitivetermsalone;social andculturalprocessesmust be acknowledgedwhen explainingmathematicaldevelopment.These social theories, of course, do not discount psychological analyses conductedin interviews. Theorists from both sociocultural and emergent perspectives, however, would arguethat traditionalpsychological analyses of interview situationscharacterize students'conceptualunderstandingindependentlyof situationandpurpose(Cobb & Yackel, 1996) andthusmustbe complementedby social analysesto gain a more comprehensiveview of the learningprocess as it evolves.

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Proactively SupportingStudents'Learning A second theme that was illuminatedby the priorliteratureconcernedthe role that designers and researchersplay in supportingstudents' mathematicaldevelopment. The prior trainingstudies showed that students' scores on pretests and posttests determined the success of the training techniques and thus whether studentsconstructedthe targetedconceptualstructures.The analyses of students' progress,therefore,focused on neitherthe process of their developmentnor the researchers'means of supportingconceptualreorganizations. In contrast,we have been in the process of refining an approachto classroombased investigations that both situates analyses of students' learning in social context andties these analysesto ourmeans of supportingdevelopment.This type of research,calledDesign Research(Brown,1992;Cobbet al., 2001; Collins, 1999), involves a reflexive relationshipbetween classroom-basedresearchand instructional design (see Figure2.2).

Instructionaldesign guidedby domain-specific Instructional theory

Classroom-basedresearch guidedbyinterpretive framework

Figure2.2. Phasesof theDesignCycle.

Gravemeijer(1994) has written extensively about the process of design. In brief, he explains that designers engage in an anticipatorythoughtexperimentin which they anticipatehow the taken-as-sharedmathematicallearninganddiscourse might evolve as proposed instructionalactivities are enacted. This conjecturing amountsto developing a potentialtaken-as-sharedlearningroute and the means by which designers might supportsuch mathematicalactivity. These conjectures

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are provisional and are tested and modified in the course of classroom-based experimentation.As Cobb and his colleagues (2001) note, the process of testing and refining is guided by classroom-based research, the second aspect of the Design Cycle. These analysesfeed back to informthe designerof both the role of the instructionalactivitiesin supportingthe learningof the communityandpossible changesin the instructionalsequence.This way of proceedingdiffersfromprevious studies conductedon measurementin that a more detailed accountof children's learningas it occurredin social context is documented.Further,the success of the measurementtrainingstudieswas determinedby pretestandposttestscores. Thus the process of supportingstudents' learning was left unexamined. In contrast, Design Researchis one viablemodel thataccountsfor boththe processof students' measurementconceptionsand the ways of supportingthatdevelopment. CulturalTools For the most part,the measurementanalyses discussed throughoutthis chapter did not take into account students' activity with cultural tools, which, for our purposesinclude physical materials,tables, pictures,computergraphsand icons, and both conventionaland nonstandardsymbol systems. Interestingly,those four studies thatdid considertool use in students'developmentdid so only in a limited way. Forexample,the analysesconductedby Piagetandhis colleagues(1960) characterizedindividualdevelopmentas a consequenceof assimilatingculturaltools with prior experiences. The analyses conducted by Barrett& Clements (1996), Clements et al. (1998), and Nunes et al. (1993) all focused on the ways in which tools facilitated conceptualdevelopment and served as catalysts for developing thought processes. Currenttheories contend that conceptual development and symbolcreationarereflexivelyrelated(Meira,1995, 1998;van Oers,2000). These theories hold that symbols and meanings are constructedsimultaneouslyand are consistent with Vygotsky's (1978) view emphasizingthat semiotic mediation is integrallyinvolved in students'development. One theoreticalperspectivethathas been developinga socially situatedposition on tool use is the socioculturalview. Socioculturalismdeveloped in reactionto mainstreampsychology's attentionto individualactivity.Socioculturalistsemphasize the complexity of learningwithin a social environmentand suggest thattools have a directinfluenceon how individualslearn.The notion of distributedintelligences, which drawsheavily on socioculturaltheories,suggests thatstudentsmay use culturaltools as scaffolds to theirinternalizationprocess.Pea (1993) describes tool use as a reorganizer,ratherthanamplifier,of understanding.Socioculturalists characterizetool use as a process of internalizinga culturaltool so thatit becomes a tool for thinking(Davydov& Radzikhovskii,1985) or of appropriating it to one's own activity(Newman,Griffin,& Cole, 1989). These andothersocioculturaltheorists often argue that tools are the primaryvehicles for enculturationinto the culturalpracticesof the wider communitybecause tools are carriersof meanings from one generationto the next (van Oers, 1996). In this view, studentsare said

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to inheritthe culturalmeaningsinvolved in using a particulartool. Fromthe socioculturalperspective,the tools and their associated,culturallyapprovedmeanings are objects to be internalizedby the individual. Nunes et al. (1993) took this approachwhen analyzingtheir students'developmentof measuringconceptions. In the emergentperspective,tools are also viewed as a constituentpartof individuals' activities. Further,emergent theorists would argue that the meanings associatedwith tools arecreatedandevolve as studentsreasonwith physicalmaterials,pictures,andso forth(CobbandYackel, 1996). This perspectiveis in marked contrastwith socioculturalviews in which establishedculturalmeaningsare said to be internalizedby the individual.Along with Meira(1995), emergenttheorists contend that the meanings involved in acting with a particulartool evolve as studentsare engaged in goal-directedactivity with the tool. From the emergent perspective,the idea thatlearningoccursas studentsact with culturaltools is based on the assumptionthatthe symbolic and conceptualmeanings that evolved from students'activity with tools do not develop apartfromthe mathematicalpractices in which studentsparticipate.The teacher'srole, then, is to initiate and guide the developmentof individuals'mathematicalactivity-in particular,tool activityas well as the mathematicalpracticesso thatthese practicesbecome morecompatible with those of the wider society (Cobb, 1994). Althoughways of analyzinganddescribingtool use are still being investigated, attending to the role of semiotic mediation in measurementclearly is of great importance. Culturaltools were used to a large extent in training students to conserve length, to reason transitively, and to learn to measure. The question remainsas to how students' activity with these tools changed the way they came to view measuring.With this question in mind, we hope to build on the important work of those researcherswho have alreadybegun to highlight tool use in students' measurementdevelopment (e.g., Barrett& Clements, 1996; Clements et al., 1998; Nunes et al., 1993). CONCLUSION In this chapter,I used the priorliteratureas backgroundto develop three theoreticalthemesthatlay the foundationfor this monograph.The first of these themes concernedoutliningthe theoreticalperspectivefor documentingstudents'mathematical (i.e., measuring)activity as they participatein the social context of their classroomcommunity.The second themefocused on issues of instructionaldesign andways of supportingstudents'measurementdevelopment.The thirdthemedealt with the evolutionof the collective meaningsassociatedwith tools used to support students'measuringactivity. Each of these themes reflects an orientationtoward learningas both a social and individualprocess.The view I have portrayedis one in which social and individualprocesses are reflexively related.In other words, social and individualprocesses coexist, and learningin the context of classroombasedinstructioncannotbe adequatelyexplainedin termsof eitherone or the other. Also, an argumenthas been madethatindividualas well as collectiveactivitycannot

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be separatedfrom activity with culturaltools. Therefore,in specifying the mathematicalpracticesandindividualmathematicaldevelopmentwithinthese practices, activitywith culturaltools becomes a centralfocus. Inthe nextchapterof the monograph, we expand more generally on our instructionaldesign theory while integratingthe othertwo researchthemes into the discussion. REFERENCES Bailey, T. (1974). Linearmeasurementin the elementaryschool. TheArithmeticTeacher,21, 520-525. Barrett,J. E., & Clements, D. H. (1996). Representing,connecting and restructuringknowledge:A microgeneticanalysis of a child's learning in an open-endedtask involvingperimeter,paths, and polygons. Paperpresentedat the EighteenthAnnualMeeting of the NorthAmericanChapterof the InternationalGroupfor the Psychology of MathematicsEducation,PanamaCity, Florida. Baylor,W., Gascon,J., Lemoyne,G., & Pothier,N. (1973). An informationprocessingmodel of some seriationtasks. CanadianPsychologist, 14, 167-196. Beilin, H. (1965). Learningand operationalconvergence in logical thoughtdevelopment.Journal of ExperimentalChildPsychology, 2, 317-339. Beilin, H. (1971). The trainingand acquisitionof logical operations.In M. Rosskopf, L. Steffe, & S. Taback(Eds.),Piagetiancognitive-development researchand mathematicaleducation(pp. 81-124). Washington,DC: NCTM. Boulton-Lewis,G. (1987). Recentcognitivetheoriesappliedto sequentiallengthmeasuringknowledge in young children.BritishJournal of EducationalPsychology, 57, 330-342. Braine, M. (1964). Development of a grasp of transitivityof length: A reply to Smedslund. Child Development,35, 799-810. Brainerd,C. (1974). Trainingand transferof transitivity,conservation,and class inclusion of length. Child Development,45, 324-334. Brison, D. (1966). The accelerationof conservationof substance.Journal of GeneticPsychology, 109, 311-332. Brown, A. L. (1992). Design experiments:Theoretical and methodological challenges in creating complex interventionsin classroomsettings. Journal of the LearningSciences, 2, 141-178. Carpenter,T. (1975). Measurementconceptsof first- and second-gradestudents.Journalfor Research in MathematicsEducation,6, 3-13. Carpenter,T. (1976). Analysis and synthesis of existing researchon measurement.In R. Lesh & D. Bradbard(Eds.), Numberand measurement.[Papersfrom a researchworkshop].(ERICDocument ReproductionService No. ED 120027) Clements, D. (1999). Teaching length measurement: Research challenges. School Science and Mathematics,99(1), 5-11. Clements, D., Battista,M., & Sarama,J. (1998). Development of geometric and measurementideas. In R. Lehrerand D. Chazan(Eds.), Designing learningenvironmentsfor developingunderstanding of geometryand space (pp. 201-226). Hillsdale, NJ: Erlbaum. Cobb, P. (1990). A constructivistperspective on information-processingtheories of mathematical activity. InternationalJournal of EducationalResearch, 14, 67-92. Cobb, P. (1994). Theories of mathematical learning and constructivism:A personal view. Paper presented at the Symposium on Trends and Perspectivesin MathematicsEducation,Institutefor Mathematics,Universityof Klagenfurt,Austria. Cobb, P. (1997). Learningfromdistributedtheoriesof intelligence.In E. Pehkonen(Ed.), Proceedings of the Twenty-FirstInternationalConferencefor the Psychology of MathematicsEducation(Vol. 2, pp. 169-176). Finland:Universityof Helsinki. Cobb, P. (2000). Conductingteachingexperimentsin collaborationwith teachers.In A. E. Kelly & R. A. Lesh (Eds.), Handbookof researchdesign in mathematicsand science education(pp. 307-334). Mahwah,NJ: Erlbaum. Cobb, P., & Bauersfeld,H. (1995). The coordinationof psychological and sociological perspectives in mathematicseducation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interactionin classroom cultures.Hillsdale, NJ: Erlbaum.

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Cobb, P., Stephan,M., McClain, K., & Gravemeijer,K. (2001). Participatingin mathematicalpractices. Journal of the LearningSciences, 10(1, 2), 113-163. Cobb, P. & Yackel, E. (1996). Constructivist,emergent,and socioculturalperspectivesin the context of developmentalresearch.EducationalPsychologist, 31, 175-190. Cole, M. (1996). Culturalpsychology. Cambridge,MA: BelknapPress of HarvardUniversity Press. Collins, A. (1999). The changing infrastructureof educationalresearch.In E. C. Langemann& L. S. Shulman(Eds.), Issues in education research (pp. 289-298). San Francisco:Jossey Bass. Confrey,J. (1995). How compatibleareradicalconstructivism,socioculturalistapproaches,and social constructivism?In L. Steffe & J. Gale (Eds.), Constructivismin education(pp. 185-225). Hillsdale, NJ: Erlbaum. Copeland,R. (1974). How childrenlearnmathematics:Teachingimplicationsof Piaget's research(2nd ed.). New York:Macmillan. Davydov, V., & Radzikhovskii,L. (1985). Vygotsky's theory and the activity-orientedapproachin psychology. In J. Wertsch(Ed.), Culture,communication,and cognition: Vygotskianperspectives (pp. 35-65). New York:CambridgeUniversityPress. Gardner,H. (1987). The mind's new science: A history of the cognitive revolution.New York:Basic Books. Gelman, R., (1969). Conservationacquisition:A problemof learningto attendto relevantattributes. Journal of ExperimentalChildPsychology, 7, 167-187. Gravemeijer,K. (1994). Developingrealisticmathematicseducation.Utrecht,Netherlands:CD-P Press. Gruen,G. (1965). Experiencesaffecting the developmentof numberconservationin children.Child Development,36, 963-979. Hiebert,J. (1981). Cognitive developmentand learninglinear measurement.Journalfor Research in MathematicsEducation,12(3), 197-211. Inhelder,B. (1972). Informationprocessing tendencies in recent experimentsin cognitive learningempiricalstudies. In S. Farnham-Diggory(Ed.), Informationprocessing in children(pp. 103-114). New York:Academic Press. Inhelder,B., Sinclair,H., & Bovet, M. (1974). Learningand the developmentof cognition.Cambridge, MA: HarvardUniversityPress. Kamii, C. (1997). Measurementof length:The need for a betterapproachto teaching.School Science and Mathematics,97(3), 116-121. Kaput,J. J. (1994). The representationalroles of technology in connectingmathematicswith authentic experience. In R. Biehler, R. W. Scholz, R. Strasser,& B. Winkelmann(Eds.), Didactics of mathematics as a scientific discipline (pp. 379-397). Dordrecht,Netherlands:Kluwer. Klahr,D., & Wallace, J. (1970). An informationprocessing analysis of some Piagetianexperimental tasks. CognitivePsychology, 1, 358-387. Lave, J. (1988). Cognition in practice: Mind, mathematics,and culture in everydaylife. Cambridge, England:CambridgeUniversityPress. Lehrer,R., Jenkins,M., & Osana, H. (1998). Longitudinalstudy of children's reasoningabout space and geometry. In R. Lehrer& D. Chazan(Eds.), Designing learning environmentsfor developing understandingof geometryand space (pp. 137-167). Hillsdale, NJ: Erlbaum. Lerman,S. (1996). Intersubjectivityin mathematicslearning:A challenge to the radicalconstructivist paradigm?Journalfor Research in MathematicsEducation,27(2), 133-168. Lerman,S. (2000). A case of interpretationsof "social":A responseto Steffe and Thompson.Journal for Research in MathematicsEducation,31(2), 210-227. Lovell, K., Healey, D., & Rowland, A. (1962). Growth of some geometrical concepts. Child Development,33, 751-767. Lovell, K., & Ogilvie, E. (1961). A studyof the conservationof weight in thejuniorschoolchild. British Journal of EducationalPsychology, 31, 138-144. McManis, D. (1969). Conservationand transitivityof weight and length by normals and retardates. DevelopmentalPsychology, 1, 373-382. Meira,L. (1995). The microevolutionof mathematicalrepresentationsin children'sactivities.Cognition and Instruction,13(2), 269-313.

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Meira,L. (1998). Makingsense of instructionaldevices:Theemergenceof transparencyin mathematical activity. Journalfor Researchin MathematicsEducation,29(2), 121-142. Murray,F. (1965). Conservationof illusion distortedlengths and areas by primaryschool children. Journal of EducationalPsychology, 56, 62-66. Murray,F. (1968). Cognitive conflict and reversibilitytrainingin the acquisitionof length conservation. Journal of EducationPsychology, 59, 82-87. National Council of Teachersof Mathematics(NCTM) (2000). Principles and Standardsfor School Mathematics.Reston, VA: NationalCouncil of Teachersof Mathematics. NationalCouncil of Teachersof Mathematics(NCTM). (2003). Learningand TeachingMeasurement (2003 Yearbook).Douglas H. Clements (Ed.). Reston, VA: NCTM. Nemirovsky, R. C., & Monk, S. (2000). If you look at it the otherway... . In P. Cobb,E. Yackel, & K. McClain(Eds.),Symbolizing,communicating.and mathematizing:Perspectiveson discourse,tools, and instructionaldesign (pp. 177-221). Mahwah,NJ: Erlbaum. Newman, D., Griffin,P., & Cole, M. (1989). Theconstructionzone: Workingfor cognitivechange in school. Cambridge,MA: CambridgeUniversityPress. Nunes, T., Light, P., & Mason, J. (1993). Tools for thought:The measurementof length and area. Learningand Instruction,3, 39-54. Osborne, A. (1976). The mathematicaland psychological foundationsof measure.In R. Lesh & D. Bradbard(Eds.), Numberand measurement.[Papersfrom a researchworkshop].(ERICDocument ReproductionService No. ED 120027). Overbeck,C., & Schwartz,M. (1970). Trainingin conservationof weight. Journal of Experimental Child Psychology, 9, 253-264. Pea, R. (1993). Practicesof distributedintelligence and designs for education.In G. Salomon (Ed.), Distributedcognitions (pp. 47-87). New York : CambridgeUniversityPress. Petitto,A. (1990). Developmentof numberline andmeasurementconcepts. Cognitionand Instruction,

7(1),55-78.

Piaget, J., & Inhelder,B. (1956). Thechild's conceptionof space. London:Routledge& Kegan Paul. Piaget,J., Inhelder,B., & Szeminska,A. (1960). Thechild's conceptionof geometry.New York:Basic Books. Rogoff, B. (1990). Apprenticeshipin thinking: Cognitive development in social context. Oxford, England:OxfordUniversityPress. Saxe, G. (1991). Cultureand cognitivedevelopment:Studiesin mathematicalunderstanding.Hillsdale, NJ: Erlbaum. Shantz,C., & Smock,C. (1966). Developmentof distanceconservationandthe spatialcoordinatesystem. Child Development,37, 943-948. Sinclair, H. (1970). Number and Measurement. In M. Rosskopf, L. Steffe, & S. Taback (Eds.), Piagetian cognitive-developmentresearchand mathematicaleducation(pp. 135-149). Washington, DC: NationalCouncil of Teachersof Mathematics. Smedslund,J. (1961). The acquisitionof substanceand weight in children:II Externalreinforcement of conservationof weight and the operationof additionand subtraction.ScandinavianJournal of Psychology, 2, 71-84. Smedslund,J. (1963). Developmentof concretetransitivityof length in children.ChildDevelopment, 34, 389-405. Smedslund,J. (1965). The developmentof transitivityof length:A commenton Braine'sreply. Child Development,36, 577-580. Smith,I. (1968). The effects of trainingproceduresuponthe acquisitionof conservationof weight.Child Development,39, 515-526. Steffe, L., & Carey,R. (1972). Equivalenceand orderrelationsas interrelatedby four- and five-yearold children.Journalfor Research in MathematicsEducation,3, 77-88. Steffe, L., & Thompson,P. (2000). Interactionor intersubjectivity?A reply to Lerman.Journalfor Research in MathematicsEducation,31(2), 191-209.

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Stephan, M. L. (1998). Supportingthe developmentof one first-grade classroom's conceptions of measurement:Analyzing students' learning in social context. Unpublisheddoctoral dissertation, VanderbiltUniversity. Tomic, W., Kingma, J., & Tenvergert,E. (1993). Trainingin measurement.Journal of Educational Research, 86(6), 340-348. van Oers, B. (1996). Learningmathematicsas meaningfulactivity. In P. Nesher, L. Steffe, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 91-113). Hillsdale, NJ: Erlbaum. van Oers, B. (2000). The appropriationof mathematicalsymbols: A psychosemiotic approachto mathematicallearning.In E. Y. P. Cobb & K. McClain(Eds.), Symbolizingand communicatingin mathematicsclassrooms: Perspectiveson discourse, tools, and instructionaldesign (pp. 133-176). Mahwah,NJ: Erlbaum. von Glasersfeld,E. (1995). Radical constructivism:A way of knowingand learning.Bristol, PA: The FalmerPress. Vygotsky, L. (1978). Mind in society. Cambridge,MA: HarvardUniversityPress Wenger,E. (1998). Communitiesof practice. New York:CambridgeUniversityPress. Yackel, E., & Cobb, P. (1995). Classroomsociomathematicalnormsand intellectualautonomy.In L. Meira & D. Carraher(Eds.), Proceedings of the 19th InternationalConferencefor the Psychology of MathematicsEducation(vol. 3, pp. 264-272). Recife, Brazil:ProgramCommitteeof the 19thPME Conference.

Chapter 3

The Methodological Approach to Classroom-Based Research Michelle Stephan Purdue UniversityCalumet Paul Cobb VanderbiltUniversity

The previous chapterdetailed a considerablenumber of linear measurement studies and describedthree main themes that cut across the literature:analyzing learningin social context,accountingfor the roleof tools in students'development, anddeveloping instructionaldesign theoriesthatexaminethe meansof supporting learning.Stephanarguedthatnew investigationsmightbuildon these threethemes to supportstudents'developmentof linearmeasurementconceptions.To this end, we designedandimplementedan instructionalsequenceconcerninglinearmeasurement. Whereas the monographitself focuses on all three themes described in chapter2, this particularchapterfocuses on the thirdtheme, instructionaldesign, while integratingthe othertwo. More specifically, we expoundon an instructional design theory that integrates analyses of students' learning with the design of instructionalactivities. Both Cobb (chapter1 of this monograph)and Stephan(chapter2 of this monograph)describedDesign Researchas an iterativeprocessof integratingsociallysituated analyses of students'learningwithin the design of classroomenvironments, of which instructionalsequencesareone part.This approachdiffersfromprevious measurementstudies in that analyses of students' learningare describedboth in terms of the evolving taken-as-sharedmathematicaldevelopmentand the role of the instructionalactivities and tools in supportingthatdevelopment.This instructional design theory served as the basis for the constitutionof the measurement sequenceimplementedin the first-gradeclassroomthatis the subjectof this monograph.WhereasDesign Researchincorporatesboth analysesof students'learning and designing-refininginstruction,this chapterfocuses only on the formercharacteristic, building on the discussion from chapter 2 on the theoretical underpinnings of social constructivismby detailing the interpretiveframeworkthat guided the forthcominganalysesof students'measuringdevelopment(see chapter 5). We follow this discussionby describingthe methodused to analyzethe evolving taken-as-sharedlearning that occurredover 32 class periods. We conclude this chapterwith a descriptionof the varietyof sourcesused to collect dataduringclass-

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roomexperimentation.Althoughwe focus only on one aspectof Design Research, classroomanalyses,we save chapter4 for elaborationof the secondaspect,instructional design. ANALYSES OF STUDENTS' LEARNING As Cobb (chapter 1 of this monograph)noted, the purposeof a design experiment is not to implement an instructionalsequence and see whether it worked; rather,the purpose is to use ongoing and retrospective analyses of classroom events as fodderfor improvementsto the originaldesign. The interpretiveframework thatguides both ongoing and retrospectiveanalyses of students'individual andcollective learningis examinedin this chapter.Such a frameworkis necessary for understandingthe actuallearningthatemergesas studentsengage in theinstructional activities.The analyses, as guidedby the framework,then providefeedback to informthe designeras to how the originalinstructionalsequence supportedand constrainedthatlearningso thatmodificationscan be made. The theorywe drawon to makesense of students'learningwhile we arein a classroom or when we are conductinganalyses at the completionof an experimentis a versionof social constructivismcalled the emergentperspective.CobbandYackel (1996) andStephan(chapter2 of this monograph)describethis theorymorespecifically. Briefly, this theory draws from (a) constructivisttheories, which specify learningas an organic, autoregulatedseries of cognitive reorganizations(Steffe, von Glasersfeld,Richards,& Cobb, 1983; von Glasersfeld, 1995), and (b) interactionist theories, which emphasize learning as a social accomplishment (Bauersfeld, 1992; Blumer, 1969). These two perspectiveson how learningtakes place have been widely debatedover the years. The emergentperspectiveis one attemptto transcendthe individualversus social dichotomyby takinglearningto be both simultaneously.In otherwords, learningis characterizedas both an individual and a social process, with neithertaking primacyover the other. Students areviewed as reorganizingtheirlearningas they bothparticipatein, andcontribute to, the social andmathematicalcontext of which they are a part.Motivatedby this theoreticalperspective,Cobb and Yackel constructedan interpretiveframework useful for detailingthe learningof a classroomcommunityandits participants(see also Yackel, 1995). We subsequentlyelaboratehow this interpretiveframework guidesthe analyseswe conductbothon a dailybasisduringan experimentandretrospectively. An InterpretiveFramework Cobb and Yackel (1996) developed the frameworkthat shapes our analyses of students'learning.This interpretiveframeworkemergedfroman attemptto conduct analyses thatcoordinateindividualstudents'mathematicaldevelopmentwith the social context of the classroom (see Figure 3.1). The left side of the framework draws on an interactionistview of communal or collective classroom processes

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(Bauersfeld,Krummheuer,& Voigt, 1988;Blumer, 1969). The individualperspective draws on psychological constructivist views of students' activity as they participatein the development of these communal processes (von Glasersfeld, 1995). The relationshipbetween the two sides of the frameworkis said to be reflexive.

Social Perspective

IndividualPerspective

Classroomsocial norms

Beliefs about own role,others' roles, and the general natureof mathematicalactivityin school

Sociomathematicalnorms

Mathematicalbeliefs and values

Classroommathematical practices

Mathematicalconceptions

framework foranalyzingclassrooms. Figure3.1. An interpretive

The social perspective consists of three aspects: classroom social norms, sociomathematicalnorms, and classroom mathematicalpractices.In an effort to explainthis framework,we elaborateeach of these componentsby focusing on the reflexivity between the social and individualperspectives. Social Norms.The firstaspectof the interpretiveframeworkinvolves describing the classroom-participation structurethatwas establishedjointly by the teacherand the students. In the measurementexperiment,the social norms that were interactivelyconstitutedincludedexplainingandjustifyingsolutionmethods,attempting to make sense of otherstudents'solutionmethods,andaskingclarifyingquestions whenever a conflict in interpretationsarose. The teacher and the students we worked with had already negotiated these social norms before the classroom teaching experimentbegan, because we had worked with this classroom teacher on priorexperiments,and the teacherthus saw value in initiatingthe negotiation of these particularsocial normsin her classroom.We began the classroomteacher experimentin February,four to five months into the school year, so these norms hadalreadybeen relativelywell established.As the teachingexperimentprogressed and new situations arose, however, these social norms were renegotiated.For example, two days afterthe beginningof the measurementsequence,the students themselves initiateda discussion aboutaskingclarifyingquestions.In the episode thatfollows, a pairof studentshadjust measureda portionof the classroomfloor

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with a stripof papercontaininga tracedrecordof five of theirfeet-a footstripby placing it down end to end four times. In the middle of a lengthy whole-class discussion, Melanie explained to the class that 20 feet is the same length as four strips: T*: Ohhh,so 20 is thenumberof feet,butit's onlyfourfootstrips,it's not20 footstrips.., it's just4. HowmanypeopleagreewithwhatMelaniesays?[Some peopleraisetheirhands.]How manypeoplehavea questionfor her?If you disagree?[Nohandsareraised.]Alice? Alice: Well,I havea questionforthewholeclass.Umm,butnotforthepeoplewho, Melanieandwhatshe umm,raisedtheirhand.Well,if theydidn'tunderstand saidandagreedwithwhatshe said,thentheywouldall havequestions.[Perry raiseshis hand.] Perry: It seemsthatMs. Smithsaid,or you said,raiseyourhandif you agreewith Melanie,whatMelaniesaid.Somepeopleraisedtheirhandsandsomepeople didn't.Andthe,umm,somebodysaidraiseyourhandif youdon'tagree,and theydon'tanswer?[Heseemstomeanthatthosepeoplewhodidnotunderstand didnotindicateso.] T: Yeah,I wasjustwonderingaboutthatmyself.That'swhyI wasreallywanting to see whatpeopledo,becauseI thinkit's, I'.mreallyinterested andI thinkMs. Smithis reallyinterested to knowhowyou'rethinkingaboutthis.So if you're notsureaboutwhatMelaniesaid,oryoudisagreewithher,youneedto sayso. (*T refersthroughoutto the persontakingresponsibilityfor leadingthe discussion. At any time, this individualcould have been the classroom teacheror a member of the projectteam.) In this episode, Alice noticed that studentswho did not understandMelanie's explanationdid not raise their hands to ask a clarifying question. For her, these studentswereviolatinga social norm,andshe explicitlychallengedthemabouttheir obligations.Perry,for his part,reiteratedAlice's concernwhenhe askedwhy those people who "don'tagree.., don't answer"(i.e., they did not raisetheirhands).The teachercapitalizedon thisopportunityby repeatingthatit was importantfor students to vocalize their lack of understanding.As the classroom teaching experiment progressed,the social normof askingclarifyingquestionsbecametaken-as-shared. The individualcorrelateto social normsconcernsthe teacherandstudents'individual beliefs abouttheirown and others' roles. At the beginning of the teaching experiment,most studentsappearedto regardtheirrole as explicating their solution methods, attemptingto make sense of the solution methods of others, and askingclarifyingquestionswhenthey didnot understand.As illustratedby theforegoing episode, severalstudentsinitiatedconversationswhen they believed a social normhad been violated. SociomathematicalNorms. The second aspect of the interpretiveframeworkis describingthe sociomathematicalnorms that were interactivelyconstitutedover the course of the teachingexperiment.Cobb and Yackel (1996) discussed several sociomathematicalnorms including what counts as a different, an efficient, a sophisticated, and an acceptable mathematical solution. The most significant

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instancein which a sociomathematicalnormwas renegotiatedduringthe measurement teaching experimentconcerned the norm of what counts as an acceptable mathematicalsolution.Initially,the studentsexplainedtheir solution methodsby describing the calculational methods that led them to a result or answer. In Thompson, Philipp, Thompson, and Boyd's (1994) terms, studentswere giving explanationsthatwere consistentwith a calculationalorientation.Thompsonand his colleagues argue that solutions reflecting a calculational orientation focus mainly on the explanationof a procedure.An alternativeorientationwould be conceptualin natureif a studentexplainedhis or her interpretationof the problem situationthatled to a particularcalculation.Thompsonandhis coworkersemphasize thatexplanationsinvolving a conceptualorientationare more productivefor mathematicaldiscussions andfor learning.As a consequenceof thatresearch,the teacher attemptedto initiate a renegotiationof the criteriaby which an explanation was judged as acceptable.The teacherwantedstudents'explanationsto focus on their interpretationsof task situationsas well as on calculationalmethods (see Cobb, chapter1 of this monograph,for a similardiscussion). For example, prior to the measurementexperiment,the teachertried to initiate a conceptualdiscussion by askingstudentsto describethe ways in whichthey anticipatedsolving problems. In the following episode, the teacher had posed the following problem: "Lenahas 11 hearts.Dick has 2 hearts.How manymoreheartsdoes Lenahave than Dick?" T:

Whatwasit we wantedto knowabouttheproblem?

Sandra:

Whatthe answerwas.

T:

Whatwasit we wantedto knowaboutthehearts?

Sandra:

The answer.

T:

I thinkwe canall agreethatwe wantto knowtheanswerbecausethatis what yougetwhenyouaska question,butwhatkindof answerdoesthequestionwant youto tellus? Meagan: HowmanymoreheartsdoesLenahavethanDick. Mitch: We aretryingto findouthowmanymoredoesDickhave. T: Do otherpeopleunderstand whatMitchwas saying?Cansomeoneaskhima question? butI don'tknowwhatto ask.I thinkhe needsto explainit Meagan: I don'tunderstand, again. As can be seen in this excerpt,initially the studentshad a calculationalorientation (Thompsonet al., 1994) to solving problems.The purposeof their explanations was to explain their method for obtaining"the answer."The studentswere not familiarwith the type of explanationsthey were now expected to give. In fact, studentslike Meagandid not know how to ask questionsto make sense out of such explanations.Thus,the teacherattemptedto explicitlyrenegotiatewhatcountedas an acceptableexplanationjust priorto the measurementsequence.Two weeks after the foregoing example, explanationsthatfocused on conceptualinterpretationsof the task that led to the calculationalmethod ratherthan solely the method itself became taken-as-shared.

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In the following excerpt, Porterhad just explained his solution method to the following problem:"Tomhad 15 sandwiches.Len had 6 sandwiches.How many moredid Tom have thanLen?"The following excerptillustratesa shift in the types of explanationsthat were being given: withthesandwiches? T: Howdidyouthinkaboutwhathappened Hilary? Hilary: I thoughtaboutdrawinga picture.I wasthinkingaboutTom's15 sandwiches. I got6 outof it,andthenI circled [Shedraws15squaresto standforsandwiches.] 6 andthenI countedhowmanymorewerenotinthecirclewiththe6 andI knew howmany.Eight.[Sheaccidentallycircles7 insteadof 6]. The onesthatshecircledis 7. Melanie: Hilary: Thankyou,nowI got it better.It wouldequal9. Thisexcerptillustratesthatexplanationswerenow cast in termsof students'interpretationof the problemsituation,which led them to a calculation,ratherthanin termsof theirfocusing strictlyon the calculationalmethod.In fact, many students now drew pictures to describe how they reasoned about the problem. Drawing pictures not only helped students explain their reasoning but also served as a means for other studentsto understandtheir explanation.Thus, what constituted an acceptableexplanationinvolved describingboth the calculationalmethod and the underlyinginterpretationof the situation. The individualcorrelateof sociomathematicalnormsinvolves students'beliefs about what types of explanations are acceptable, efficient, sophisticated, and different.These beliefs arereflexively relatedto sociomathematicalnormsin that students play an active role in negotiating the criteria by which solutions are judged as acceptable. That is, as the teaching experiment progressed and new instructionalsituations arose, the teacher and studentscontinually modified the criteriaby which solutions werejudged as acceptable. ClassroomMathematicalPractices. The final aspect of the interpretiveframework involves documentingboth the collective mathematicaldevelopment of a classroom community and the learningof individual students. Such an analysis involves detailing the classroom mathematicalpractices that evolved over the course of the teachingexperimentwhile documentingindividualstudents'mathematicalgrowthas they participatedin, and contributedto, the emergenceof these mathematicalpractices.A mathematicalpracticecan be describedas a taken-assharedway of reasoning and arguingmathematically (Cobb, Stephan,McClain, & Gravemeijer,2001). Classroommathematicalpracticesevolve as the teacherand students discuss situations, problems, and solution methods and often include aspects of symbolizing and notating (Cobb, Gravemeijer,Yackel, McClain, & Whitenack, 1997). To be clear, the term mathematicalpractices has been used widely in the literatureto mean differentthings. Some researchersuse the termto meanthosepracticesfromthe mathematicalcommunitythatarealreadyestablished. The teachers' goal in this example is to help studentsappropriatea certainset of mathematicalpractices.In this monograph,we do not use the termin this manner. As we use the term, classroom mathematicalpractices are more localized to the

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classroomandareestablishedjointlyby the studentsandtheteacherthroughdiscussion; they emerge from the classroom ratherthan come in from the outside. We acknowledgethatin most situations,the local classroommathematicalpracticesare establishedas studentsbringin theirways of participatingin outsidemathematical practices.Hence, an interactionbetween outside and local mathematicalpractices is possible (cf. Whitenack,Knipping,andNovinger,2001). Classroommathematical practicesdiffer from social and sociomathematicalnormsin thatthe latterare constructsthat describethe participationstructureof the classroom;that is, they describewhatis normativein termsof how the teacherandthe studentswill communicatewith one another.Classroommathematicalpractices,therefore,can betterbe thatbecome normativethroughthese thoughtof as the mathematicalinterpretations interactions-practices thatare content specific (e.g., normativemeasuringinterpretationsand methods)and do not referto more generalsocial practices. Individualstudents'mathematicalinterpretationsandactionsconstitutethe indiand vidualcorrelatesof the classroommathematicalpractices.Theirinterpretations the mathematicalpracticesare reflexively relatedin that students'mathematical developmentoccursas they contributeto the constitutionof the mathematicalpractices. Conversely,the evolutionof mathematicalpracticesdoes not occurapartfrom students'reorganizationof theirindividualactivity (Cobb et. al., 1997). Here the priorcognitiveanalysesof Piagetbecomeimportantto us. Althoughwe have argued thatpriormeasurementstudieswereprimarilycast as individualisticandhave advocatedfor analysesthatdiscuss the role of social contextin learning,we find particular value in Piaget, Inhelder,and Szeminska's (1960) cognitive constructsfor analyzing students'interpretationsin the case studies that appearin chapter5 of this monograph.Specifically, we found Piaget and others' definitionof measurement as the mental activity of partitioningand iteratingto be more helpful for analysis than analyzing students' behaviors during measuring,in other words, learning whethera studentcan measure with a rulercorrectly.Furthermore,we incorporatedPiaget and others' mention that full measurementunderstanding means thatstudentsform inclusionrelationsamong subsequentiterations,and we renamedthis idea accumulationof distance (cf. Thompson& Thompson, 1996). In this way, we builton the cognitive analysesin the literaturereview to strengthen our analysis of students'developmentas it occurredin social context. The primaryfocus of chapter5 of this monographis to presenta sampleanalysis to illustratethe evolution of classroommathematicalpractices,because this is the leastdevelopedaspectof the interpretiveframework.Ourunitof analysiswill therefore be that of a classroom mathematicalpractice and students' diverse ways of participatingin, andcontributingto, its constitution(CobbandBowers, 1999;Cobb et al., 2001). In making reference to both communal practices and individual students'reasoning,this analyticunit capturesthe reflexive relationbetween the social and individualperspectiveson mathematicalactivity. As difficult as it is to documentthe evolution of students'learningfrom an individualisticperspective, an even more dauntingtask is to chroniclethe collective learning.Partof the difficulty lies in the fact that countless, complex interactionsoccur among a large

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numberof students.A logical question is, How does one justify the claim that a particularmathematicalinterpretationis taken-as-sharedin a classroom community?In the next section, we elaborateourmethodfor analyzingall datain general, and, more specifically, we establish the criteriaby which we documented the emergence of taken-as-shared,classroommathematicalpractices. METHODOF ANALYSIS The approachwe take follows Glaser and Strauss'(1967) constantcomparison method as adaptedto the needs of Design Research(Cobb & Whitenack, 1996). This method,like othersin the interpretivisttradition,treatsdataas text and aims to develop a coherent,trustworthyaccountof theirpossible meanings.The method can be organized into two distinct phases, initial conjectures and mathematical practices. Phase 1, Initial Conjectures Withregardto the actualprocessof analyzingdata,the firstphaseof the method involves working throughthe data chronologicallyepisode-by-episode.In doing so, we develop conjecturesaboutthe ways of reasoningand communicatingthat mightbe normativeat a particulartime andaboutthe natureof selected individual students'mathematicalreasoning.These conjectures,togetherwith the evidence that supportsthem, are explicitly documentedas part of a permanentlog of the analysis. This log also recordsthe process of testing and revising conjecturesas subsequentepisodes are analyzed.A useful approachhas been to focus on normative argumentationto develop these conjecturesaboutthe evolution of the collective mathematicalpractices(e.g., Bowers, Cobb, & McClain, 1999; Cobb, 1999; Cobb, in press; Gravemeijer,Cobb, Bowers, & Whitenack, 2000; Stephan & Rasmussen,in press).We have gainedinsightfrom Yackel (1997) who, following Krummheuer(1995), used Toulmin's (1969) model of argumentationto analyze the evolution of mathematicalpractices (see Figure 3.2). We have simplified Toulmin's model hereto drawout the aspectsmost relevantto documentingmathematical practices.For a more complete version that includes rebuttalsand qualifiers, see van Eemerenet al. (1996). Figure 3.2 illustratesthat for Toulmin (1969), an argumentconsists of at least threeparts,called the core of an argument(see dottedoval in Figure3.2): the data, the claim, andthe warrant.In any argument,the speakermakesa claim and,if challenged, can present evidence, or data, to supportthat claim. The data typically consist of facts thatlead to the conclusion. Even so, a listenermay not understand what the particulardatapresentedhave to do with the conclusion thatwas drawn. In fact, the listenermay challenge the presenterto clarify why the datalead to the conclusion.Whenthis type of challengeis made andthe presenterclarifiesthe role of the datain makingtheclaim,thepresenteris providinga warrant.Toulminargues that a warrantis always presentin an argument,whetherexplicitly articulated-

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MethodologicalApproachto Classroom-BasedResearch

DATA: Evidence

CLAIM: Conclusion

WARRANT: Explain how the data leads to the claim

BACKING: Explain why the warrant has authority

the Core

-the Validity

Figure3.2. A portionof Toulmin'smodelof argumentation.

usually as the result of a challenge-or implied by the presenter.As Toulmin's model shows, anothertype of challenge also can be made to an argument.The listenermay understandwhy the datasupportthe conclusionbutmay not agreewith the contentof the warrantused. In otherwords, the authorityof the warrantcan be challenged, and the presentermust provide a backing to justify why the warrant and, therefore,the core of the argumentis valid. Yackel (1997) and Cobb et al. (2001) have shown the usefulness of Toulmin's model for analyzing collective mathematicallearningby relating it to the documentation of mathematicalpractices. In the case of the sample measurement analysis thatwe presentin chapter5, the data in students'argumentsmay consist of the mannerin which a studentmeasuresthe physicalextensionof an object, and the claim is the numerical measure of that physical extension. An appropriate warrantthatserves to explain why the data supportthe conclusion might involve demonstratinghow and why a measurementtool was used to producethe numerical result.This warrantcan, of course, be questioned,thus obliging the studentto give a backing that indicates why the warrantor measurementprocedureshould be acceptedas havingauthority.A backingthatmightbe treatedas legitimatecould involve explaining how the use of the measurementtool structuresthe physical extension of the object into quantitiesof length. As partof the social normsthatthe teacherandclass established,studentsin the first-gradeclassroomwereexpectedto challengeone another's solutionswhen they did not understand.Often,challenges arosewhen new measuringtools were used. When challenges arose, studentswere obliged, with the teacher'shelp, to provide shouldbe acceptedas valid.This backingsfor why theirmathematicalinterpretation

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need for a backingindicatesthatthe way in which the use of a new tool structured the physicalextentof objectsinto quantitieswas nottaken-as-shared.In subsequent episodes, however, giving backingsto justify theirmathematicalinterpretationof a new measuringtool or measuringinterpretationwas no longer necessary for students.Shifts of this type are one indicationthata taken-as-sharedmeaning for the use of the tool has become established.In other words, when a backing is no longer necessaryto justify a particularmeasuringinterpretationor a studentcalls for a backingandthe class questionsthe need for one, we claim thata mathematical idea has become taken-as-shared. We thereforesearchfor instancesin thepublic in discourse which studentsno longer provide backings for a new measurement interpretation,and we recordthese occasions in the log for Phase 1. We test ourconjectures,for example,thata backinghas droppedout anda particular measurementidea is taken-as-shared,as we analyze subsequentvideotaped episodesby lookingfor occasionsin which a student's activityappearsto be at odds with a proposednormativemeaningand examine how membersof the classroom communitytreatthe student's contribution.If the communitydoes not accept the contribution,on the one hand,then we have furtherevidence thatan idea has been established.If, on the otherhand,the contributionis treatedas legitimate,we need to revise our conjecture. Phase 2, MathematicalPractices The resultof the first phase of the analysis is a chain of conjectures,refutations, andrevisions thatis groundedin the details of the specific episodes. In the second phase of the analysis, the log(s) of the first phase becomes data that are metaanalyzedto develop succinctyet empiricallygroundedchronologiesof the mathematicallearningof the classroomcommunityand of selected individualstudents. Duringthis phase of the analysis,the conjecturesdeveloped duringthe firstphase mathematicalideas arescrutinized aboutthe possibleemergenceof taken-as-shared froma relativelyglobal viewpointthatlooks acrossthe entireteachingexperiment. The resultsof the analysisarethenorganizedandcast in termsof the analyticunits that were alluded to previously-mathematical practices-and students' diverse ways of participatingin them. In summary,documentingclassroommathematicalpracticesinvolves working throughcontributionsto classroom discourse individually and using Toulmin's model of argumentationto develop conjecturesaboutthe mathematicalideas that emerge and eventuallybecome taken-as-sharedduringvariousdiscussions. Once thesenumerousideasarerecorded,they aretestedagainstone anotherandorganized aroundcommon mathematicalactivities to develop the classroom mathematical practices.In this sense, we say thatthe mathematicalpracticesare establishedby the community and not droppedin from outside, that is, predeterminedby the designer, fully formed. Rather, our documentation of mathematical practices involves analyzingstudent-teacherinteractionas it occurs andas the class decides whatbecomes legitimatemathematicsin theirclassroom.(StephanandRasmussen

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[in press]recentlyextendedthis analysis techniqueby analyzingthe mathematical practicesthatemergedin a class studyingdifferentialequations.) To analyze the communalpracticesand situatestudents'learningin those practices, we drawfrom a vast amountof data.The datacorpusfor an analysis of this type consists of videotaperecordingsof all classroom sessions conductedduring a teaching experiment, videotape recordings of student interviews conducted before andafterthe teachingexperiment,andaudiotaperecordingsof every project meeting held prior to, and during, the teaching experiment.Each of these data sources is describedsubsequently. DATA CORPUS The first-gradeclassroomthatwas the subjectof this studywas one of fourfirstgradeclassrooms at a privateschool in Nashville, Tennessee. The class consisted of 16 children:7 girls and9 boys. The majorityof the studentswere from middleclass backgrounds.The classroomteachingexperimenttook place over a 4-month periodfromFebruaryto May 1996.Justpriorto the teachingexperiment,the teacher and the students had been engaged in instructionon single-digit addition and subtraction.The teacherwas an active memberof the researchteam and continually workedat developinga teachingpracticeconsistentwiththe reformguidelines of ProfessionalStandardsfor TeachingMathematics(NCTM, 1991). She hadbeen involved with the project membersfor 3 years and originally sought help from professionals in the mathematicseducation field because she had become dissatisfiedwith the currentmathematicstextbooks.The researchteamhaddeveloped a close professionalrelationshipwith this teacher,such thatthe researchersoccasionally insertedcommentaryand questions duringwhole-class discussions. The studentsacceptedthe researchers'contributionsas if they were visiting teachers. In the sampleepisodesalreadypresentedin this paperandthroughouttheremainder of the monograph,we use the wordteachercollectively to referto both the teacher and the researcher.The classroom teacher,however, took primaryresponsibility for leadingall instruction.She clearlysharedthe researchgoals andthe importance of basinginstructionaldecisions on students'understanding.As such,we consider her an irreplaceablememberof the researchteam. Each of the data sourceselaboratedsubsequentlycontributedin its own way to the analysesthatwill be presentedin the next two chaptersof the monograph.Data were collectedfromthreedistinctsources:classroomsessions, formalandinformal meetings with the teacher,and two sets of interviews focusing on both measurementandthinkingstrategiesfor single- andtwo-digitadditionandsubtraction.Data collected from each of these sources were used to documentindividualstudents' learning(see case studiesin chapter5 of this monograph),the communallearning (see mathematicalpracticesin chapter5 of this monograph),andthe emergingtool use (see chapter6 of this monograph).

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ClassroomSessions Data from classroom sessions were collected in five modes: videotape recordings, field notes, clarifying questions, reflective journals, and studentwork. All mathematicslessons were recordedon videotape daily using two cameras. One camerawas placed at the back of the classroom and recordedthe teacherand all teacher-studentactivitytakingplace at the white boardat the frontof the room.The second camera was placed at the front of the room and recorded the students' activity duringwhole-class discussions. Two researchersdocumenteda total of five target students' individualdevelopmentby askingtheirtargetstudentsclarifyingquestionsduringpairor individual work on a daily basis. Each of these interactionswith targetstudentswas videotapedto infer shifts in theirindividualmathematicaldevelopment.Further,at the end of each day, both researchersreflectedon theirtargetstudents'activities and, in the formof reflectivejournals,madeinformalconjecturesconcerningtheirmathematicalactivity.All 16 of the students'writtenworkwas collecteddaily.The intent here was to infer shifts in notationaland thinkingstrategies. Researcherscollected threesets of field notes to documentinstructionalactivities andto recordwhole-classdiscussions.On completionof the classroomteaching experiment,copies of each set of field notes were made and were discussed for purposesof triangulation. Formal and InformalMeetings Aftereach classroomsession, the researchteam met with the classroomteacher for approximately10 to 15 minutesto planinstructionalactivitiesfor the following day. Often during these informal meetings, the team offered observations of students'reasoningand made conjecturesaboutindividualand collective mathematicalactivity.In addition,moreformalweekly meetings were held in which the pedagogical goals were revisited and the students' mathematicalactivity was discussed in terms of the conjecturedlearningtrajectory.This type of discussion includedhow the measurementsequencewas being realizedin interactionandways of furthersupportingthe students'development.Consequently,instructionalactivities were revised in light of the students'activity duringthatweek. These weekly meetings typically began by team members'sharingobservationsof the students' activity and makingconjecturesaboutthe communalmathematicaldevelopment. Both formaland informalmeetings were audiotapedto documentthe intentof the instructionalsequence as it evolved duringthe course of the teachingexperiment andto recordconversationsin which we conjecturedaboutstudents'development. Formal Interviews All 16 students were individually interviewed using two types of interview tasks.At theoutsetof theteachingexperiment,ourfocus was on supportingstudents' development of increasingly sophisticatedstrategies for mental estimation and

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computation. Therefore, one set of interviews focused on students' thinking strategiesfor single-digit and two-digit additionand subtraction.As noted previously, however,ourfocus becamethatof supportingstudents'increasinglysophisticated measuringconceptions.Therefore,a second set of interviewscenteredon students'measuringconceptions.The arithmetictasks were posed in both pretest andpostinterviewsituations,whereasthe measurementinterviewswere conducted as midinterviews and postinterviews.As soon as we learned how conceptually demandinglearningto measureis, we developed the measurementinterviewsand conductedthem 3 weeks afterinstructionon measurementhad begun and immediately following the experiment. The purposeof the pre-interviewsconcerningmentalestimationand computation was to assess students'mathematicalunderstandingso that the instructional sequencecould buildon theircurrentways of knowing.A second set of interviews was conductedon completionof the teachingexperimentandagainfocusedon children's thinkingstrategies.The intentof this postinterviewwas to assess students' problem-solving strategies to document growth over the 4-month period. The tasks from pre-interviewsand postinterviewsincludedsolving horizontalnumber relations, organizing unstructuredquantities, organizing structuredquantities, counting by l's, estimating,and solving story problems. The measurementinterviews were conducted, as mentioned,approximately3 weeks afterthe measurementsequencebegan.At thattime, the five targetstudents and three other studentswere interviewed. These interviews were conducted to assess initialinterpretations of theirmeasuringactivity.Afterthe completionof the teachingexperiment,we conducteda second set of postinterviewsthatfocused on all 16 students' conceptions of measurement.The interview tasks focused on students'measuringactivitywith tools used duringthe teachingexperimentas well as novel measurementtools. For all sets of interviews,at least two researcherswere present.One researcher conductedthe interviewby selecting problemsfrom an interviewprotocol sheet. A second researcher recorded the interview with videotaping equipment and recordedthe students'solution procedureson a protocol sheet. CONCLUSION In this chapter,we have detailed the interpretiveframeworkthat characterizes a student's learning as both an individual and a social endeavor.The framework enables researchersto account for individual learningby viewing it as an act of social participation.The goal of the analysis is to focus on how various students participatein, and consequently contributeto, the establishment of communal mathematicalpractices. To elaborate on how we identified these mathematical practices,we drew on Toulmin's (1969) model of argumentationto identify shifts in the mathematicalideas that became taken-as-sharedover the course of the teaching experiment. Finally, we elaborated each of the sources of data from which the subsequentanalyses were documented. In the next chapter,we focus

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on the second aspect of Design Research, designing testable classroom learning environments. REFERENCES Bauersfeld,H. (1992). Classroomculturesfroma social constructivistperspective.EducationalStudies in Mathematics,23, 467-481. Bauersfeld,H., Krummheuer, G., & Voigt,J. (1988). Interactionaltheoryof learningandteachingmathematicsandrelatedmicroethnographical studies.In H-G. Steiner& A. Vermandel(Eds.),Foundations and methodologyof the discipline of mathematicseducation (pp. 174-188). Antwerp, Belgium: Proceedingsof the TME Conference. Blumer,H. (1969). Symbolicinteractionism:Perspectivesand method.EnglewoodCliffs, NJ:PrenticeHall. Bowers, J., Cobb, P, & McClain, K. (1999). The evolution of mathematicalpractices:A case study. Cognitionand Instruction,17(1), 25-64. Cobb, P. (in press). Conductingclassroomteaching experimentsin collaborationwith teachers.In R. Lesh & E. Kelly (Eds.), New methodologiesin mathematicsand science education. (pp. 307-333). Dordrecht,Netherlands:Kluwer. Cobb, P. (1999). Individual and collective mathematicaldevelopment:The case of statistical data analysis.MathematicalThinkingand Learning, 1, 5-44. Cobb,P., & Bowers, J. (1999). Cognitive and situatedperspectivesin theoryandpractice.Educational Researcher,28(2), 4-15. Cobb, P., Gravemeijer,K., Yackel, E., McClain, K., & Whitenack,J. (1997). Symbolizing andmathematizing:The emergence of chains of significationin one first-gradeclassroom. In D. Kirshner& J. A. Whitson(Eds.),Situatedcognitiontheory:Social, semiotic,and neurologicalperspectives(pp. 151-233). Hillsdale, NJ: Erlbaum. Cobb, P., Stephan,M., McClain, K., & Gravemeijer,K. (2001). Participatingin mathematicalpractices. Journal of the LearningSciences, 10(1, 2), 113-163. Cobb,P., & Whitenack,J. (1996). A methodfor conductinglongitudinalanalyses of classroomvideorecordingsand transcripts.EducationalStudies in Mathematics,30, 213-228. Cobb, P., & Yackel, E. (1996). Constructivist,emergent,and socioculturalperspectivesin the context of developmentalresearch.EducationalPsychologist, 31, 175-190. Glaser, B. G., & Strauss,A. L. (1967). The discovery of grounded theory: Strategiesfor qualitative research. New York:Aldine. Gravemeijer,K., Cobb, P., Bowers, J., & Whitenack,J. (2000). Symbolizing, modeling, and instructionaldesign. In P. Cobb,E. Yackel, & K. McClain(Eds.),Symbolizingand communicatingin mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 225-273). Mahwah,NJ: Erlbaum. Krummheuer,G. (1995). The ethnographyof argumentation.In P. Cobb & H. Bauersfeld(Eds.), The emergence of mathematicalmeaning:Interactionin classroom cultures (pp. 229-269). Hillsdale, NJ: Erlbaum. NationalCouncil of Teachersof Mathematics(NCTM). (1991). Professional Standardsfor Teaching VA: NCTM. Mathematics. Reston, & J., Inhelder, Piaget, B., Szeminska,A. (1960). The child's conceptionof geometry.New York:Basic Books. Steffe, L. P., von Glasersfeld,E., Richards,J., & Cobb,P. (1983). Children'scountingtypes:Philosophy, theory,and application. New York:PraegerScientific. Stephan,M., & Rasmussen,C. (in press). Classroommathematicalpracticesin differentialequations. Journal of MathematicalBehavior. Thompson,A. G., Philipp,R. A., Thompson,P. W., & Boyd, B. (1994). Calculationaland conceptual orientationsin teaching mathematics.In Douglas B. Aichele (Ed.), Professional Developmentfor Teachers of Mathematics, 1994 Yearbook of the National Council of Teachers of Mathematics (NCTM) (pp. 79-92). Reston, VA: NCTM.

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Thompson,A., & Thompson,P. (1996). Talkingaboutratesconceptually,partII:Mathematicalknowledge for teaching.Journalfor Research in MathematicsEducation,27, 2-24. Toulmin, S. (1969.) The uses of argument.Cambridge,England: CambridgeUniversity Press. van Eemeren,F., Grootendorst,R., Henkemens,F., Blair,J., Johnson,R., Krabbe,E., et al. (Eds.)(1996). Fundamentalsof argumentation:A handbookof historical backgroundsand contemporarydevelopments(pp. 129-160). Mahwah,NJ: Erlbaum. von Glasersfeld,E. (1995). Radical constructivism:A way of knowingand learning. Bristol, PA: The FalmerPress. Whitenack,J., Knipping,N., & Novinger, S. (2001). Coordinatingtheoriesof learningto accountfor second-gradechildren's arithmeticalunderstandings.MathematicalThinkingand Learning, 3(1) 53-85. Yackel, E. (1995). The classroom teaching experiment.Unpublishedmanuscript,PurdueUniversity Calumet,Departmentof MathematicalSciences. Yackel, E. (1997, April). Explanationas an interactiveaccomplishment:A case studyof one secondgrade mathematicsclassroom. Paperpresentedat the annualmeeting of the AmericanEducational ResearchAssociation, Chicago.

Chapter 4

A Hypothetical Learning Trajectory on Measurement and Flexible Arithmetic Koeno Gravemeijer FreudenthalInstitute JanetBowers San Diego State University Michelle Stephan Purdue UniversityCalumet

In chapter 1, Cobb describedDesign Research as an approachthat integrates instructionaldesign and classroom-basedexperimentationin such a mannerthat the design effort shapes the researcheffort and the researcheffort shapes further design as well (see also Gravemeijer,1998). In chapter2, Stephanexaminedthe theoreticalperspectivethatunderliesthe researcheffort.And in chapter3, Stephan and Cobb exploredthe interpretiveframeworkand methodthat serve as the basis for analyzingthe ongoing events of a classroomenvironment.In this chapter,we first focus on the theoreticalunderpinningsthatguidedthe process of instructional design for the measurementexperiment.We then cast this perspectivein termsof an instructionaltheoryon measuringandflexible arithmeticthatguidedthe present teachingexperiment. DESIGN RESEARCH The type of Design Researchin which we engage falls underthe broadheadings of developmentalresearch (Cobb, Gravemeijer,Yackel, McClain,& Whitenack, 1997; Freudenthal,1991; Gravemeijer,2001; Gravemeijer,1998) and transformational research (NCTM ResearchAdvisory Committee,1996). Otherusers of the termDesign Research (Brown, 1992; Collins, 1999) define it in an interpretation similarto ours.At the heartof Design Researchis a cyclic processof designing instructionalsequences, testing and revising them in classroom settings, and then analyzingthe learningof the class so thatthe cycle of design, revision, andimplementationcan begin again. Therefore,testing and revising occur at both a formative level-that is, on an ongoing, daily basis duringclassroomintervention-and

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a summativelevel-that is, retrospectively.At the core of this cycle is the interplay between designing a potentialinstructionalsequence and using analyses of students' learningwhile they are engaged in the instructionalactivities to shape the redesignof the sequence.In what follows, we elaborateon the generalheuristics thatguide the developmentof an instructionalsequence in the design phase. Realistic MathematicsEducation The preliminarydesign andcontinualmodificationof an instructionalsequence is guidedby the domainspecific instructionaltheorycalled RealisticMathematics Education (RME), developed at the FreudenthalInstitute(Gravemeijer,1994a; Streefland, 1991; Treffers, 1987). RME is consistent with the emergentperspective in thatboth arebasedon a view of mathematicsas a humanactivityand a view of mathematicallearningas a processthatoccursas studentsdevelop ways to solve problemswithinthe social contextof a mathematicsclassroom(Cobb,2000). Given these assumptions,the main activity thatdesigners attemptto supportis progressive mathematization,or level-raising:Students'activityon one level is subjectto reflection and analysis on the next level. Although many importantideas are embodiedwithinthe idea of mathematization,we have chosento highlightthe three basic RME heuristicsthat, when taken together,inform designersin their efforts to supportstudents'reasoningwithin the cycle of Design Research. Heuristic 1. Sequences must be experientially real for students. One of the centralheuristicsof RMEis thatthe startingpointsof instructionalsequencesshould be experientiallyreal in that studentsmust be able to engage in personallymeaningful activity (Gravemeijer,1994b). Often, this requirementmeans grounding students'initial mathematicalactivity in experientiallyreal scenarios,which can includemathematicalsituations.To find such scenarios,the designermay carryout a didacticalphenomenologicalanalysis, within which he or she investigateswhat phenomenaareorganizedby the mathematicalconceptsor proceduresto be developed by students.For example, a scenario that was developed for the first-grade teaching experimentwas one in which a king needed to measurethe lengths of particularitems in his kingdom.Of course, by saying the startingpoints are experientially real, we are not suggesting that the studentshad to experience these startingpoints firsthand.Instead,we are saying simply thatthe studentsshouldbe able to imagine acting in the scenario,as king, for example. Heuristic 2: Guided Reinvention.A second heuristic of RME involves what Freudenthal(1991) termedguided reinvention.To startdeveloping a sequence of instructional activities, the designer first engages in a thought experiment to imagine a route the class might invent (Gravemeijer,1999). Here, knowledge of the historyof mathematicsas well as priorresearchconcerningstudents'invented mathematicalstrategiescan be used to develop what Simon (1995) has called a HypotheticalLearningTrajectory,or a possible taken-as-sharedlearningroutefor the classroom community. This construct is furtherelaboratedin a subsequent section of this chapter.

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Heuristic3. Emergenceof student-developedmodels.A final heuristicof Design Researchinvolves buildingon students'informalmodeling activityto supportthe reinventionprocess. As such, instructionalsequences should provide settings in which students can model their informal mathematicalactivity (Gravemeijer, 1994a).Often,therefore,the designerimagineswhattypes of practiceswill emerge as studentsreason with certaintools-physical tools, symbols, or notation-that mightsupporttaken-as-shared developmentof new mathematicalsolutions.Integral to this heuristic is the conjecturethat students' models of their informal mathematical activity evolve into modelsfor increasingly sophisticatedmathematical reasoning.Gravemeijer(1999) notes thatalthougha model may manifestitself in a variety of ways duringthe sequence, the model is a global notion that spreads over the instructionalsequence. The interplayamong the three heuristics:Anticipatinga transitionfrom model of to model for. In the RME approach,all three heuristicsare seen as workingin unison to supportthe development of increasingly sophisticated mathematical practicesas studentsparticipatein the sequence of activities. In what follows, we describethis unisonby firstelaboratingwhatis meantby a modelfromthe perspective of an RME developer.Next, we describethe model of to modelfor transition thatis the centerof this approach. The wordmodelcan be interpretedin differentways. Specifically, RME's interpretationdiffers from that of mainstreaminformation-processingapproachesor cultural-historicalapproaches,such as thatelaboratedby Gal'perin(1969). In the latter approaches,models come to the fore as didactic models that embody the formalmathematicsto be taught.In otherwords, the formalabstractmathematics is seen as concretizedto be madeaccessiblefor the studentsthroughdiscovery.This view is consistent with the socioculturalview of tools describedin chapter2. In contrastwith this view, RME models are not derived from the intended mathematics. These models are seen as student-generatedways of organizing their mathematicallygroundedactivity.Thus,the RMEmodelsareproductsof modeling; the startingpoint is in the experientiallyreal situationof the problemthat has to be solved, for example,the king scenario.The idea is thatactingwith these models will help the students(re)inventthe more formalmathematicsthatconstitutesthe goal of instruction. According to the theoryof RME, student-generatedmodels evolve as students collectivelyconstructnew mathematicalrealities.Therefore,a transitionin students' activity from model of to modelfor indicates a shift in the ways that the students areparticipatingin andhence constitutingthe mathematicalpracticesof the classroom community.In theory, the transitionin models occurs as follows. Initially, the models are to be understoodas context-specific models. The models referto concreteor paradigmaticsituationsthatareexperientiallyreal for the students.On this level, the model shouldallow for informalstrategiesthatcorrespondwith situated solution strategiesat the level of the situationthatis defined in the contextual problem.Fromthen on, the role of the model begins to change. As the classroom community gathers more experience with similar problems, a taken-as-shared

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model becomes more general in character.The process of acting with the model changes from one of constructingsolutions situatedin the context to one of using the model to communicatereasoningstrategies.In this sense, the model becomes an object in and of itself within the classroom community. By then, the model becomes more importantas a basis for mathematicalreasoningthan as a way to representa contextualproblem.As a consequence,the model can become a referential base for the formal mathematics.Or, in short, a model of informalmathematical activity becomes a modelfor more formalmathematicalreasoning. The shift from model of to modelfor can be seen as aligned with shifts in the collective mathematicalpractices.In brief,researcherslook for these shifts in practices by noting changes in students'justificationsthatreflect differencesbetween explanations of their solutions within the modeled context and explanationsof mathematicalrelations.As an aside, a helpful approachhas been to limit the use of the modelof-modelfor characterization to moreencompassingshifts.Otherwise, every symbolizing act could be predicatedas a model shift. Usually, somethingis symbolized (model of), and the symbolizationis used to reasonwith (model for). If we include all such shifts in the emergent-modelsheuristic,this heuristicwould lose its power. We thereforewant to delineate when we do or do not speak of a shift from a model of to a modelfor by specifying that such a shift coincides with the creationof some new mathematicalreality (Gravemeijer,1999). In summary,the threeRME heuristicscan be seen as integratedinto the process of supportingshiftsfromcontextualizedactivityto moreformalways of reasoning. In the following section,we describethe processby whichdesignersanticipatehow this transitionoccurs throughthe developmentof a hypotheticallearningtrajectory. After that, we illustratethe process by describingthe hypotheticallearning trajectorythatwas developedfor the measurementandnumber-relations sequences. DistinctionsBetweena HypotheticalLearningTrajectoryand TraditionalLesson Planning The RMEdesign heuristicsdescribedin the foregoingserveas tools for enabling instructionaldesigners to conceptualize how a teacher might build on students' models to supporttheirconstructionof increasinglysophisticatedreasoning.One question thatoften arises is the inherenttension between adaptingone')steaching to the ideas and suggestionsof the studentsand,at the same time, planninginstructional activities in advance.Ourresponse to this apparentparadoxof "on-the-fly planning"is to revisit Simon's (1995) study of the role of a constructivistteacher. Simon analyzedhis own role as a teacherwho tries to influence his class's argumentationby playing off their personallydeveloped mathematicalmeanings. He tries to envision ways in which his studentswill engage in the activities,then tries to anticipatehow his students'potentiallines of argumentationmight lead to the types of explanationsandjustificationshe wantsto become taken-as-sharedin the evolving classroom community.To describe this process, Simon introducesthe notion of a hypotheticallearning trajectory(HLT):

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The consideration of the learninggoal, the learningactivities,andthe thinkingand learningin whichthestudentsmightengagemakeupthehypothetical learningtrajectory.... (Simon,1995,p. 133) We have come to believe that this notion of an HLT addresses the apparent paradoxof "on-the-flyplanning"by taking into accountfour specific considerations of the process thatdistinguishit from the traditionalprocess of lesson planning: (1) the socially situatednatureof the learningtrajectory,(2) the view of planning as an iterativecycle ratherthan a single-shot methodology, (3) the focus on students'constructionsratherthanmathematicalcontent,and(4) the possibility of offeringthe teachera groundedtheorydescribinghow a certainset of instructional activities might play out in a given social setting. Each of these considerationsis furtherelaboratedsubsequently. Thefirstdifferencebetweenthe RMEapproachandthatof traditionallesson planning is that RME designers view learningas situatedwithin a classroom activity system. Two implicationsof this assumptionare that we do not assume that any given sequence will play out the same in any classroom, and we do not view a proposedtrajectoryas a series of conceptual stages along which each individual studentin the class will progress.Instead,when developing a learningtrajectory, we attemptto outline conjecturesaboutthe collective developmentof the mathematical communityby focusing on the practicesthat might emerge at the beginning of the sequence,thencreatingtools andactivitiesthatmight supportthe emergence of other practicesthat would be based on increasinglysophisticatedways of acting andjustifying mathematicalexplanations. The second aspect of the RME planningprocess thatdistinguishesit frommore traditionalteacher-only lesson planning is that it incorporatesfeedback from researchand, hence, is cyclic in nature.As the teacherand studentsengage in the activities, new insights into the types of mathematicalpracticesthatareemerging form the basis for the constitutionof a modified HLT for the subsequentlessons. Simon (1995) describesthis processas a mathematicsteachingcycle. Althoughthe teachingcycles thatSimon used as examples concernedthe teacher'spreparation for the next day's instructionalactivity,we haveextendedthis notionof cyclic planning to characterizethe process by which the teacher envisions a more encompassing learningroute. Thus, when enacting any activity, the teacheradjustsher or his time to build on what actually happensin the classroom. The overall plan provides a basis for thinkingthroughhow to get back on track.In relationto this flexibility, we may borrowSimon's journey metaphor.When making a journey, we may start out with a well-considered travel plan. Nevertheless, our actual journeywill differfromthis planbecauseof the circumstancesthatwe meet during ourjourney. Moreover,not every aspect of a journey can be plannedin advance. A preconceived,conjecturedlearningroute may be comparedwith a travel plan, in which what is actuallygoing on in the classroom constitutesthejourney. A thirdcharacteristicof the design process thatdistinguishesit from traditional lesson planningandalso resolvesthe apparent"on-the-flyplanning"paradoxis that the HLT takes students'cognitive developmentas the basis for design. This focus

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standsin starkcontrastwith traditionalinstructionaldesignprogramsthatorganize the goals of instructionin terms of mathematicalcontent. Therefore, Simon's notion of an HLTis moreconsistentwith reformrecommendationsthatplace high priorityon students'mathematicalreasoningandjustifications(NCTM, 2000). The final distinctionbetween Design Researchand traditionallesson planning is thatthe resultof a Design Researchexperiment,like the one thatis the topic of this monograph,can be cast in terms of a groundedtheory describinga learning route.The latteris also referredto as a conjecturedinstructionaltheory.The term instructionaltheoryis used to convey the intentof offeringa socially situatedrationale. That is, the objective of Design Research is not to offer an instructional sequence that "works"but to offer a groundedtheory that describes the tools, imagery, discourse, and mathematicalpractices that emerge as the instructional sequence is played out in the social setting of a mathematicsclassroom. This last criterion,offering a groundeddescriptionof how a sequence will play out, is consistentwith the message of a recenteditorialpublishedby the Research Advisory Committee(RAC) of the NationalCouncil of Teachersof Mathematics (NCTM)(1996). In its piece, the RAC observedthatfewer researcharticles,at least those published in the Journalfor Research in MathematicsEducation, simply reportfindingsthatshow thatinstructionalapproachA worksbetterthanapproach B. The rationalefor thistrendis thatresearchershaverealizedthatsimplycomparing two instructionalapproachesdoes not have much to offer to teacherswho are not successful with the approvedapproach.In other words, the only thing that those teachersknow is thatit workedelsewhere. In contrast,in the instanceof a justification in terms of a groundedtheory about how a new approachworks, teachers have a basis for thinkingthroughwhy something may not have worked in their classroomandbeginningto conjectureaboutwhatadaptationmightbe helpful.Note that,from this perspective,the enactmentof an instructionalsequencecan be seen as a continuationof the teachingexperiment,in thatteacherscan contributeto the research;the knowledge gained by teachersmay be used to enhance the conjecturedinstructionaltheory.Moreover,teachersbecome awareof the social nature of learning,in thatthe Design Researchperspectivetakes advantageof the diversity of classroom settings by offering a global learningrouteto be tailoredto the specific situationsby classroomteachers.In short,whenwe discussthe conjectured instructionaltheorythatis the yield of the teachingexperimenton linearmeasurement and flexible arithmetic,the readershould not expect a textbook-typeelaboration of an instructionalsequence. Instead, our presentationlooks more like a socially situated rationale for an instructional sequence (an example will be presentedin chapter6 of this monograph). DESIGNINGTHE HYPOTHETICALLEARNINGTRAJECTORY FOR THE FIRST-GRADESEQUENCE We begin by situatingthe classroomteachingexperimenton measurementand arithmeticin the context of the researchthatprecededit. One aspect that we need

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to elaboratehere is that originally, learningto measurewas a means to an enddeveloping a sequence for supportingstudents' constructionsof computational strategiesfor reasoningwith numbersup to 100-and not a goal in itself. Learning to measure,however, provedto be a sophisticatedprocess thatdeserved attention in its own right. Therefore, the actual teaching experiment encompassed two partialsequences consisting of linearmeasurementand mentalcomputation. Prior classroom-basedresearch One source that influenced the design of the currentactivities was our prior researchof the empty-number-lineinstructionalsequence.The goal of the emptynumber-linesequencewas to supportstudents'activitiesso thatthey woulddevelop flexible computationstrategiesfor additionand subtractionup to 100 (Whitney, 1988; Treffers, 1991). In priorresearchsettings, we found that studentsreasoned with this tool to describe the solution of two-digit additionand subtractionproblems by markingthe numbersinvolvedanddrawing"jumps"thatcorrespondedwith their partialcalculations. For example, solving 64 - 29 by subtracting4, 10, 10, and5, respectively,mighthave resultedin the inscriptionshownin Figure4.1. Such a solution method, which could be seen as a shortenedform of counting down, belongs to the most common informalsolution methodsthat studentsuse.

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Figure4.1. 64- 29 on theemptynumberline.

In relationto this categorization,we found thatwe were helped in our research by distinguishingbetween collection-type solution methodsand linear,countingtype solutionmethods.In the first type, studentspartitioneach numberinto a tens partand a ones part,then add or subtractthe numbersof those partsseparately.In contrast,when reasoningwith linear-basedmethods,studentsincrementor decrement directlyfrom one numberas in the example in Figure4.1. We foundthis distinctionuseful becauseresearchhas shownthatcollection-type solutionsmay cause problemswhen used with morecomplex subtractiontasks(see Beishuizen, 1993). Therefore, the empty-number-linesequence was especially designed to supportmore counting-typereasoning strategies.We also acknowledged, however, the fact thatcollections-basedreasoninghas its merits,in relation to both the written algorithms and estimation. Therefore, we also planned to

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provideopportunitiesin which studentscould develop well-understood,collectionbased strategiesfor reasoningwith tens andones. Note thatouraim was not to teach the students efficient solution strategies as such. Instead, our aim was to help studentsdevelop a frameworkof numberrelations,which would enable them to construeflexible solution methods. As partof the cyclingprocessof Design Research,ourpriorresearchon the empty number line led us to make revisions to the arithmeticinstructionalsequence (Cobbet al., 1997). The originalarithmeticsequencedevelopedin the Netherlands incorporateda bead stringwith red and white alternatingcolors every ten beads as the foundationalimagery for reasoning with the empty numberline. The indispensability of this kind of imagery proved itself in the teaching experiment conductedby Cobbandothers,in which the beadstringwas skipped.Withoutsome sort of linear-basedcontextualimagery,alternativeinterpretationsconcerningthe meaningof numeralsas specified on the numberline arose.Forexample, students interpreted"73" as either the 73rd instance of counting or as the result of having counted 73 times. In using the bead string,these interpretationsare equivalentto students' deciding whetherto put a clothespin, for example, directlyon the 73rd space or on the space after.Withoutthe imageryof, anddiscussionsabout,the bead string as a background,the studentscould not resolve these issues. Because the underlyingimageryhad been lacking with its introduction,the studentsneeded to createa semanticgroundingfor the empty numberline individuallyandprivately. They did so in a variety of ways, renderingimpossible their resolutionof interpretationissues; they had no common ground.Ratherthanfall back on the bead string,whichlacksin real-lifemotivation-no one wouldrealisticallyplace clothespins on a bead string-we designed the measurementsequence as a replacement for reasoningwith a bead string.Thus,this monographserves morebroadlyas one completeiterationof the Design ResearchCycle: using prioranalysesof the empty numberline to revise and subsequentlytest a new instructionalsequence and then analyzingand revising the enactmentof the resultinginstructionalsequence. As we beganplanninga new sequencethatwouldprecedethe emptynumberline, we focused on the ruler as the prime model. With this model in mind, we then focused on priorresearchrelatedto students'understandingsof measurement.As noted in chapter2, Piaget's priorresearchon children'slearningrevealed that to have studentsview measuringas an act of structuringspace,they mustfirstdevelop concepts of nested and iterableunits. To use Greeno's (1991) terms,the intentof the instructionalsequence would be that studentswould come to act in a spatial environmentin which the resultsof iteratingsignify an accumulationof distances. Our decision to use nonconventionalunits was influencedby the work of Piaget, Inhelder,& Szeminska(1960); Steffe, von Glasersfeld,Richards,& Cobb (1983); and Thompson& Thompson(1996), who maintainthatunderstandingmeasuring involves developingan image of accumulateddistance.Whereassome researchers agree with the approachof having studentsbuild a "ruler"out of iteratingnonconventionalandthenconventionalunits(Lehrer,in press),othersarguethatchildren preferto use a conventionalrulerfrom the beginning, and theirresearchfindings

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supportsuch an approach(Clements, 1999; Nunes, Light, & Mason, 1993). Our reasoning for the choice of nonconventionalunits is explained in the following section. TheHypotheticalLearningTrajectory We startedour design activity with the observationthat a strong similarityis evident between the empty numberline and a ruler.Speculatingon the genesis of the rulerin history, we reasonedthat we could take the view that the rulercame aboutas a recordof iteratinga measurementunit. So the rulercould be thoughtof as a modelof iteratingsome measurementunit. As describedpreviously, ourprior classroom-basedresearchindicatedthatthe empty numberline could function as a modelfor mathematicalreasoningin the context of mentalcomputationstrategies for reasoningwith numbersup to 100. The connectionbetweenthe two would be in the relationbetween iteratingmeasurementunits as accumulatingdistances and a cardinalinterpretationof positions on the numberline. In the context of measurement,we hoped that studentswould come to see distances on a ruleror numberline as magnitudes. By using the same tool as a means for supporting students'efforts to solve contextualproblemsthat do not involve measurement, we hoped studentswould engage in practicesthatinvolved reasoningaboutquantities moregenerally.Finally,even the directreferenceto specific quantitiesmight become less prominent,since the numberson the empty numberline would start to derive theirmeaningfrom a frameworkof numberrelationsfor the students. One of the overridingconcernsin this sequence was to bringthe decimal structure of our numbersystem into focus for whole-class discussions. In the context of measurement,we planned to do so by designing activities in which students would shortentheiriteratingactivity by means of a largermeasurementunit, that is, a unit of 10. Measuringwith tens and ones can form the basis for the development of students'view of the ruleras structuredin termsof tens and ones. At the same time, we hoped it would supportthe emergence of other strategiesthat use decuples as referencepoints. In summary,we envisioned the skeleton of a sequence for supportingstudents' arithmeticalreasoningwith numbersless than 100 that would unfold as follows: 1. Constructingcontext-basedmeaningfor measuring. a. Iteratingunits.The studentsstartmeasuringthe lengthsof variousobjectsby iteratingsome measurementunit. b. Organizingor structuringspace. The studentsdiscuss theirsolutionsin terms of structuredspace covered by iteratedunits. 2. Reasoning with a ruler.The studentsexplain theirefforts to shortentheiriterations of counting by using tens and ones. While doing so, the studentsstartto develop a frameworkof numberrelationswith decuples as referencepoints. 3. Extending the activity of measuring to incrementing, decrementing, and comparinglengths. Here the activity shifts from measuringto reasoningabout

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measures.This type of taskwould offer studentsthe opportunityto developarithmetical solution methods that might be supportedby referringto the decimal structureof the ruler. 4. Using a schematizedruleras a means of scaffolding and communicatingarithmetical solution methods for measurementproblems. The students develop increasinglysophisticatedmethodsfor arithmeticalreasoningbasedon an emergent frameworkof numberrelations. 5. Reasoning with a flexible ruler.The studentsdevelop new ways of communicating solution methodsfor all sorts of additionand subtractionproblems. As Cobb (chapter1 of this monograph)noted, the Design Researchernot only envisions a sequence of instructionbut also forms conjecturesaboutthe potential mathematicalargumentationand evolving tool use that might accompany the realizationof the sequence.As such,the global structureof the anticipatedlearning trajectorywould include conjecturesof possible discourseandtool use. Hence, in preparationfor the teaching experiment,this global structureof a hypothetical learningtrajectorywas workedout in the following manner. " The sequencewouldstartwith pacingto structurelineardistance.Measuringwith centimetersor inches would have been so familiarto the studentsthat the need for structuringof iterativeunitsmightnothave becomean explicittopicof discussion. Moreover,we imagined that measuringwith one's feet would create the opportunityto raise awarenessfor the need for a standardizedmeasurementunit; one such scenariointroducesthe king's foot as a standardizedmeasurementunit (cf. Lubinski& Thiessen, 1996;Lehrer,2003). We anticipatedthatthe discourse would focus on students'ways of structuringdistancesas they countedpaces in differentmanners. * Measuringwith the king's foot would be followed by measuringwith a different iterableunit:Unifix cubes in a smurfscenario.Accordingto the storythatwould be told to the students,the smurfs-little blue dwarfs-measure with theirfood cans, which happento have the size of Unifix cubes. An importantinstructional design considerationherewas ratherpractical,namely,thatthis smallermeasurement unit would allow studentsto deal with largernumbersmore easily. " Next studentswould be confrontedwith a dilemmain which the smurfsdecided to only use a small numberof food cans for measuringinstead of a bag of 50 cans. Studentswould offer suggestions of smallerquantities(e.g., 2, 5, 10, 20), and the teacherwould choose 10 to capitalize on the decimal structureof our numbersystem. As studentswould engage in measuringactivitieswith a rod of ten cans, we anticipatedthatthe discourse would focus on structuringdistance in tens and ones and coordinatingmeasuring with two different-sizedunits. Furthermore,the idea thata rodof ten would signify the resultof countingUnifix cubes might become taken-as-shared.Discussions concerningthe resultof iterating as an accumulationof distancemight emerge here in students'attemptsto clarify their measuringinterpretations.

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* As studentsreasonedwith these nonconventionalmeasuringtools, we envisioned that they would begin to stop counting every unit markand begin counting by tens andthen adjustingby addingor subtractingones. Ultimately,we hopedthat measuringwith tens andones would help the studentsreasonwith tens andones to structurethe numbersequence up to 100. * Measuringby tens and ones would lay the basis for introducinga measurement stripthatwas made of a paperstripon which 10 units of ten were iterated,each subdividedinto 10 units of one cube (Figure4.2). The idea was that measuring with the measurementstripwould become a taken-as-sharedway of signifying the result of iteratingwith the smurfbar and the individualcubes. Discussions mightcenteron the meaningthatcertainmarksor numeralshadfor students.We could imagine accumulationof distance becoming a taken-as-sharedinterpretation of the symbols on the measurementstrip.In other words, we hoped that eventually a numberon the measurementstrip would signify the distance that extends from the beginning of the measurementstrip to the line to which this numberbelongs.

10

20

30

40

50

60

70

80

90

Figure4.2. Measurement stripof 100.

* Afterthe studentsgrew accustomedto reasoningwith the measurementstrip,we envisionedthata shift mightbe madein taken-as-sharedmathematicalreasoning from measuring objects to using the measurement strip to reason about comparing, incrementing, and decrementing lengths of objects that are not physicallypresent.The focus of discoursewouldbe on identifyingandcomparing numericallythe lengths andheights of objects. In this way, the discoursewould shift from interpretingthe result of measuring to quantifying the difference among various heights. These tasks were designed to offer opportunitiesfor developing arithmeticalsolution strategiesas describedpreviously. * To furtherfoster a shift away from measuringto reasoningabout magnitudes, anothertool, the measurementstick, would be introducedalong with a new scenario. In this case, the stick was attachedto a wall and used to read off the height of the waterin a canal (see Figure4.3). The studentswould be invited to reasonwith the measurementstick to solve contextualproblemsaroundthe rise andfall of waterheights. The design of the measurementstick was basedon our priorresearchwith anothertool, the bead string.An importantfeatureof this tool was thatalthoughit has no numbers,values areeasy to readoff by countingtens

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and ones. The conjecture was that this feature might prompt discussions in which studentsstopcountingandinsteadrely on thedecimalstructureof the stick.

stick.(Herethestickis shownin a horizontal position;inthe Figure4.3. Measurement contextof thewaterheights,thestickis positionedvertically.)

" Next the empty numberline would be introduced.The goal of this tool was to

support students' efforts to reason with a tool that was not demarcatedwith specific units. The idea was that these line-based solution methods would still be groundedin measuringwith units of 10 and 1. We anticipatedthat students would drawon the practicesthatemergedduringmeasuringto develop andinterpret strategieswith the empty numberline. For example,jumps on the number line might symbolize distance quantitiesthat themselves signify accumulation of distances.For some students,thesejumps might also signify iteratingUnifix cubes or rods of ten. " As a final step, the students would be encouraged to use the same solution strategiesandexplanationsthatevolved when reasoningwith the empty number line to solve contextualproblems involving incrementing,decrementing,and comparing.Here we hoped thatjumps on the numberline would signify quantitative changes in the contextual situation under consideration. Eventually, however,we expectedthatthe students'thinkingstrategieswould become based on more generalizednumericalrelationships.The shift from numbersas referents to numbers as mathematicalentities is reflexively related to the model of-modelfor transition.Initially,the students'actionswiththe modelwouldfoster the constitutionof a frameworkof numberrelations.Afterthetransition,in which studentsdevelop a frameworkof numberrelations,the model would become a model for generalizedmathematicalreasoning. Cascade of Tools and Inscriptions The succession of varioustools constitutesan essential element of the instructional sequence. Althoughthe rulerservedas the overarchingmodel, it was manifested in the form of various tools or inscriptions throughoutthe hypothetical learningtrajectory(Gravemeijer,1999). This representationis characteristicof our view that reasoning with each new tool should supportthe students' efforts to shortenor stop theirpreviousactivitywhen reasoningwith earlierones. We hoped

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the class would participatein a series of emergentmathematicalpracticesin which each successive practicewas basedon students'increasinglysophisticatedtool use and discourse. Moreover, we believed a meaningfulbuildupof reasoning strategies wouldensurethatall studentshada way of participatingin thepracticesbecause anyone could go back to earlier,more familiarways of participatingif the need arose. Fromthis perspective,an essential image for the teacherto have was how the reasoningstrategiesthatwe envisionedfor each tool builton previousreasoning strategies. We find Latour's (1990) descriptionof a "cascadeof inscriptions"to be a useful way to describe how the proposed sequence of tools can be seen as reflecting RME's theoreticalreinvention process. We say theoretical because we need to acknowledge that we cannot expect studentsto spontaneouslyinvent these tools as needed atjust the righttime. Here, again, we rely on the teacherto resolve the apparent"on-the-fly planning"paradox. In this case, our solution was that the teacher would introducethe new tools as a way to do justice to the reinvention idea while also moving the class forward.The timing and process by which the teacher introducedeach new tool were crucial. First she had to make sure that each new tool was discussed in the whole-class setting as a solution to a problem or an encapsulationof some reasoning strategy.For example, the teacherwould tell a story about how the king did not want to do all the measuring by pacing each length personally. The teacher would then involve the students in this problemby asking them for their solutions (cf. Stephan,Cobb, Gravemeijer,& Estes, 2001). Next the solutions offered by the studentswould become the topic of a whole-class discussion in such a way that the students could explore the advantagesand disadvantagesof each suggested solution. Only then would the teacherinformthe studentsof the solutionthatwas chosen in the story.With luck, a similar solution had alreadybeen suggested by one of the students. If not, the preceding discussion would have at least have provided a basis on which the studentscould conclude that the teacher's proposed solution was sensible under the given conditions. Anotheraspect of the teacher's role in introducingeach new tool was that she needed to emphasize the tool's relationwith the precedingactivities. That is, the contextproblemfor which the new tool offereda solutioncould not be an arbitrary problem;it had to be one thatwas rootedin the precedingscenarioand that operated underthe precedingconstraints.In the instanceof the king who did not want to do all the measuring,the problemhad it roots in the earlieractivityof measuring with the king's foot, or measuringlike a king by pacing with students'feet. This aspect is importantbecause thathistory must providethe imageryunderlyingthe new tool. In the case of the unwilling smurfs,a barof 10 cubes was introducedto build on the activity of measuring with individual cubes. More precisely, the imageryunderlyingthe barof 10 was thatof a recordof measuringwith single cans. Studentswere expected to discuss advantagesof this tool, such as the fact that it could be used as a tool for a shortenedform of measuring,that is, it was easier to carry 10 cans than a whole bag, and counting by tens was simplerthan counting

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by ones. Takentogether,these threeelements-introducing each new tool as a solution to a problem,collectively discussing the advantagesanddisadvantagesof the new tool, and the importanceof a context-relevantscenario-were designed to engage the studentsin the experience of the reinventionprocess. CONCLUSION This chapterhas focused on the ways in which we were informedin our design process by the overall ideas underlying Realistic MathematicsEducation. We began the chapter by describing three heuristics: experientially real activities, guided reinvention,and emergentmodels. We then describedhow each of these heuristicsenabledus to addressthe "on-the-flyplanning"paradoxof how teachers can, on the one hand, adaptto the ideas and solutions of the studentsand, on the otherhand,move the class towarda mathematicalgoal. Following Simon (1995), we arguedthat the essential componentis anticipatingthe collective practicesin which the students might engage as they reason with the various tools in the context of the activity scenarios. In the second partof the chapter,we described the hypotheticallearningtrajectoryor conjecturedinstructionaltheorythat functioned as a frameworkof referenceas we enactedthe sequence. One of the majorthemes of this chapterhas been the importanceof considering how the sequence will supportthe taken-as-shareddevelopmentof increasingly sophisticatedthinking.The conjecturedinstructionaltheorywe offered addresses this concern by first describingthe general concept of modeling. We envisioned that,as studentsreasonedwith varioustools, the rulerwould shift from serving as a model of iteratingor measuringto being a model for arithmeticalreasoning.The modelof-modelfor shiftis the tie thatbindsthe measurementandarithmeticinstructional sequences. A second fundamentalthemethatwas presentedin this chapterwas our view of the role of tools in supportinglearning.In brief,the solutionsthatstudentsdevelop depend heavily on the fact that they are reasoning with the tools they are given. This view differs from a view of tools as devices thatenable one to transferone's mentalreasoningsto a physicaldevice. This view also differsfrom a view of a tool as a culturallyladen device whose use and meaningare predetermined. Teachers who want to adhereto the calls of reform mathematicseducationto build on students'own contributionsmay build on ourwork, but they will need to construe their own hypotheticallearningtrajectorieson a day-to-daybasis to be consistentwith the anticipatedpracticesin each social setting.Thus, teacherswill need to take into considerationtheirknowledge of theirown students,the instructional history of the class, and the end points they envision. In relationto these considerations,we can recall Simon's (1995) journeymetaphor.The instructional theoryoffersa travelplan-or maybea travelguide,on the basisof whichone could design a travel plan-but the actualjourney of the teacherand the studentswill always differ fromthe travelplan. Still, we would argue,the betterthe travelplan, the easier the journey.

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At this point, we hope that we have piqued the reader'sinterestto the point of wondering how the proposed sequence actually played out. The next chapter addressesthis question by presentingan analysis of how the mathematicalpractices actuallyemergedover the course of the enactedsequenceandhow this emergence compares with the hypotheticallearningtrajectorydescribedhere. In this chapter,we have presentedthe results of prior research and described how the sequence. analysesof thatresearchshapedthe designof the measurement-arithmetic Consistentwith the Design ResearchCycle, the classroom analysis found in the next chapteris used to generatepossible mathematicalpracticesthat accompany the proposedinstructionalsequence and to provide the rationalefor revisions to it. In the chapterfollowing the analysis, we discuss revisions and outline a generalized instructional theory on measurement and linear-based arithmetic, thus completingone iterationof the Design ResearchCycle. REFERENCES Beishuizen, M. (1993). Mental Strategiesand materialsor models for additionand subtractionup to 100 in Dutch second grades.Journalfor Research in MathematicsEducation,24(4), 294-323. Brown, A. L. (1992). Design experiments:Theoretical and methodological challenges in creating complex interventionsin classroom settings.Journal of the LearningSciences, 2, 141-178. Clements, D. (1999). Teaching length measurement: Research challenges. School Science and Mathematics,99(1), 5-11. Cobb,P. (2000). Conductingclassroomteachingexperimentsin collaborationwith teachers.In R. Lesh & E. Kelly (Eds.), New methodologies in mathematics and science education (pp. 307-333). Dordrecht,Netherlands:Kluwer. Cobb, P., Gravemeijer,K., Yackel, E., McClain, K., & Whitenack,J. (1997). Symbolizing and mathematizing:The emergenceof chains of significationin one first-gradeclassroom.In D. Kirshner& J. A. Whitson(Eds.), Situatedcognitiontheory:Social, semiotic,and neurologicalperspectives(pp. 151-233). Hillsdale, NJ: Erlbaum. Collins, A. (1999). The changing infrastructureof educationalresearch.In E. C. Langemann& L. S. Shulman(Eds.), Issues in education research (pp. 289-298). San Francisco:Jossey Bass. Freudenthal,H. (1991). Revisitingmathematicseducation.Dordrecht,Netherlands:Kluwer. Gal'perin,P. Y. (1969). Stages in the developmentof mentalacts. Tijdschriftvoornascholing en onderzoek van het reken-wiskundeonderwijs, 7(3, 4), 22-24. Gravemeijer,K. (1994a).Developingrealisticmathematicseducation.Utrecht,Netherlands:CD-PPress. Gravemeijer,K. (1994b). Educationaldevelopmentanddevelopmentalresearch.Journalfor Research in MathematicsEducation,25(5), 443-471. Gravemeijer,K. (1998). Developmentalresearchas a researchmethod.In J. Kilpatrick& A. Sierpinska (Eds.), Mathematicseducationas a research domain:A searchfor identity(an ICMI study) (book 2, pp. 277-295). Dordrecht,Netherlands:Kluwer. Gravemeijer,K. (1999). How emergent models may foster the constitutionof formal mathematics. MathematicalThinkingand Learning, 1(2), 155-177. Gravemeijer,K. (2001). Developmentalresearch:Fosteringa dialecticrelationbetweentheoryandpractice. In J. Anghileri (Ed.), Principles and practice in arithmeticteaching (pp. 147-16 1). London: Open UniversityPress. Greeno,J. (1991). Numbersense as situatedknowing in a conceptualdomain.Journalfor Research in MathematicsEducation,22, 170-218. Latour,B. (1990). Drawingthingstogether.In M. Lynch& S. Woolgar(Eds.),Representationsin scientific practice (pp. 19-68). Cambridge,MA: The MIT press. Lehrer,R. (2003). Developing understandingof measurement.In J. Kilpatrick,W. G. Martin,& D. Schifter (Eds.), A research companionto "Principlesand standardsfor school mathematics"(pp. 179-192). Reston, VA: NationalCouncil of Teachersof Mathematics.

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Lubinski,C., & Thiessen, D. (1996). Exploring measurementthroughliterature.Teaching Children Mathematics,2(5), 260-263. NCTM Research Advisory Committee (1996). Justification and reform. Journal for Research in MathematicsEducation,27(5), 516-520. National Council of Teachersof Mathematics(NCTM) (2000). Principles and standardsfor school mathematics.Reston, VA: NCTM. Nunes, T., Light, P., & Mason, J. (1993). Tools for thought:The measurementof length and area. Learningand Instruction,3, 39-54. Piaget,J., Inhelder,B., & Szeminska,A. (1960). Thechild's conceptionof geometry.New York:Basic Books. Simon, M. A. (1995). Reconstructingmathematicspedagogyfroma constructivistperspective.Journal for Research in MathematicsEducation,26, 114-145. Steffe, L. P., von Glasersfeld,E., Richards,J., & Cobb,P. (1983). Children'scountingtypes:Philosophy, theory,and application.New York:PraegerScientific. Stephan,M., Cobb, P., Gravemeijer,K., & Estes, B. (2001). The roles of tools in supportingstudents' developmentof measuringconceptions.In A. Cuoco(Ed.),Therolesof representationin school mathematics (2001 Yearbook)(pp. 63-76). Reston, VA: NationalCouncil of Teachersof Mathematics. Streefland, L. (1991). Fractions in realistic mathematicseducation. A paradigm of developmental research. Dordrecht,Netherlands:Kluwer. Thompson,A., & Thompson,P. (1996). Talkingaboutratesconceptually,partII:Mathematicalknowledge for teaching.Journalfor Research in MathematicsEducation,27, 2-24. Treffers,A. (1987). Threedimensions:A modelof goal and theorydescriptionin mathematicsinstruction-The WiskobasProject. Dordrecht,Netherlands:Reidel. Treffers,A. (1991). (Meeting)Innumeracyat primaryschool.EducationalStudiesin Mathematics,22(4), 309-332. Whitney, H. (1988). Mathematical reasoning, early grades. Unpublished manuscript, Princeton University, New Jersey.

Chapter 5

Coordinating Social and Individual Analyses: Learning as Participation in Mathematical Practices Michelle Stephan Purdue UniversityCalumet Paul Cobb VanderbiltUniversity Koeno Gravemeijer FreudenthalInstitute

Traditionally, analyses of students' learning has been cast as an individual endeavor(cf. Piaget,Inhelder,& Szeminska,1960),with socialaspects,if accounted for at all, seen as catalystsfor thoughtprocesses. Recently mathematicseducation researchhas seen a shift in emphasisfrom treatinglearningas solely an individual process toward analyzing learningin the social context of communities.As can happen in these kinds of profound shifts, a dichotomy was produced:that of subscribingto eitheran individualor a social perspective(see Cobb, 1994). Some, however, have attemptedto transcendthese two extreme positions by coordinatingthe two perspectives(Bauersfeld,1988;Cobb, 1994). In theirview, learning is simultaneouslya social and an individualprocess, with neithertaking primacy over the other. The primarygoal of this chapteris to use the results of the classroomteachingexperimenton linearmeasurementto detailan analysisof students' learningfrom a coordinatedperspective.We will show with this sample analysis how students'learningcan be characterizedas having both social and individual aspects simultaneously.To this end, we view the forthcominganalysisas situating students' learning in the social context of their classroom community. For us, students'learningis seen as participationin the local, emergingmathematicalpractices. This view will become cleareras the sample analysis is presented. The sample analysis found in this chapter gives an account of the collective learningaccompanyingthe measurementsequenceandthusservesas the backbone for revisions to the hypotheticallearningtrajectorylaid out in chapter4. Although the main objective of this particularchapteris to discuss the theoretical notion of coordinatingsocial and individualaspects of learning-the first of the threethemes of the monograph-the sampleanalysiswill illustrateeach of the

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other two monographthemes, as well. The analysis will show that the use of taken-as-sharedtools was integralto the evolutionof the measuringpractices.More theoretically,the analysisin this chapteralso takesits place in the Design Research Cycle becauseit is an instanceof classroom-basedresearchandbecauseit provides a rationalefor revisionsandcontinuationof the Cycle. We leave the tool-usetheme implicitfor now andtakeup its role in learningin chapter6. We visit thethirdtheme of this monograph-the Design Research Cycle-more specifically in the next chapter,as well. This chapteris organizedas follows. First,we presentananalysisthatcoordinates the evolution of both the collective learning and two students' personal mathematicaldevelopment.We documentthe emergenceof five classroommathematical practicesand situatetwo students'learningwithinthese practices.Second, we use the analysisas a contextwithinwhichto discusstheoreticalaspectsof analyzing learning in social context. Specially, we argue that the mathematicalpractice analysis and complementarycase studies serve as a more complete pictureof the learningof the first-gradeclassroom. Further,we contrastthis coordinationwith traditionalcognitive and socioculturalanalyses. Finally,we note the usefulnessof the classroom mathematicalpractice as a constructfor documentingcollective learningand supportinglearningin other socially situatedclassrooms. THE EVOLUTIONOF THE CLASSROOMMATHEMATICALPRACTICES This measurementsequence is divided into five phases, each correspondingto the emergence of a classroom mathematicalpractice.In this section, we discuss each phase of the measurementsequence, simultaneouslyelaboratingeach of the five mathematicalpractices.Concurrently,we discusstwo students'individualways of reasoningas they participatedin, and contributedto, the emergenceof each of the five mathematicalpractices.The two studentschosen for this analysishadbeen targetedat the outset of the teaching experimentand were followed almost daily duringthe entiretyof the project.These studentswere targetedon the basis of their distinct ways of reasoningat the outset. Phase 1-Measuring Items in the Kingdom(3 Days) At the beginningof the measurementsequence, the teachertold a story abouta king who lived in a kingdom and sometimes needed to measure for various purposes.The king decidedhe wantedto use his foot to measure.The teacherasked the studentsto imaginethatthey were servantsin the kingdomandto help the king determinehow he might use his feet to find how long various items were. They decidedthathe could paceby placinghis feet heel-to-toewith no spacesin between paces. During whole-class discussion, a studentpaced along the edge of a rug at the frontof the classroomto illustratehow the king might pace. As she paced, she counted each step to keep trackof how many steps had been taken.In subsequent instructionalactivities,studentspretendedto be the king andfoundhow long various

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items aroundthe classroom-the kingdom-were, usingtheirown feet. Organized in pairs,each studenttook turnsbeing the king andrecordedwhatthey foundeach time they measured. The classroom mathematicalpractice that became established as studentsparticipatedin these activities involved measuringby covering distance.(Unlike Piaget et al. (1960), we do not distinguishbetween distanceand length.Foreasierreadingwe use the worddistanceto referto boththe emptyspace between two points and the length of an object. We write the word space in our analysis when the studentsuse the termthemselves.) The emergenceof thefirst mathematicalpractice-Measuring by covering distance The first mathematicalpracticethat emerged involved a qualitativeshift from measuringby counting the numberof units used in the activity to measuringby covering distance. In addition,studentshad a variety of ways of participatingin this measuringpractice.Here we discuss the collective learningof the classroom communityby documentingthe establishmentof the firstmathematicalpracticeand the learningof two studentsas theyparticipatedin, andcontributedto, theemergence of the practice. First, we describe the negotiation process that led to the emergence andeventualestablishmentof the firstmeasuringpractice.Second,we detail the ways in which two students,Nancy and Meagan,participatedin this practice. Third, we provide another snapshot of classroom activity in which the first measuringpracticewas negotiated. Negotiating thepurpose of measuring.On the day the king's-foot scenariowas introduced,the teacherand the studentsdiscussed thatthe king could use his feet to pace the length of items in the kingdom. As students worked in pairs, we observedtwo distinctways of countingpaces. Some studentsplacedtheirfirstfoot down such thattheirheel was alignedwith the beginningof the carpetandcounted "one"with the placementof the next foot (see (a) in Figure 5.1). Otherstudents placed theirfirst foot such thatthe heel was aligned with the beginningof the rug and counted "two"with the placementof their next foot (see (b) in Figure 5.1).

"one" (a)

"one"

"two"

(b)

Figure5.1. Twomethodsof countingas studentspacedthelengthof a rug.

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We conjecturedthatthose studentswho countedpaces in the first manner(situation (a) in Figure5.1) did not see theirinitialpace as covering a distance.In other words, for these students,the goal of pacing was to count the numberof steps requiredto reachthe end of the rug ratherthan to cover a length.We thoughtthat if studentscontrastedthese two ways of countingpaces in the subsequentwholeclass discussion,thenthe mathematicallysignificantissue of measuringas covering distance might emerge as a topic of conversation.In the following whole-class discussion, the teacherand the studentsnegotiatedboth the method and purpose of pacing. The teacheraskedSandraandAlice to show the class how each of them would measure the rug at the front of the classroom so that the students could contrastthe two methodsof countingpaces. T: I was alsoreallywatchinghowa coupleof you weremeasuring. Whowantsto howyou'dthinkaboutit? showus howyou'dstartoff measuring, Sandra: Well, I started right here [places the heel of herfirstfoot at the beginning of the and went one [startscountingwiththeplacementof hersecondfoot as in rug] figure5.1(a)]two,three,four,five, six, seven,eight. T: Werepeoplelookingathowshedidit?Didyousee howshestarted? Whothinks theystarteda differentway?OrdideverybodystartlikeSandradid?Alice,did you starta differentwayorthewayshedidit? Alice:

Well, when I started,I countedright here [places the heel of herfirstfoot at the beginningof the rug and counts it as one as infigure 5.1(b)], "One,two, three."

T: Alice:

Whyis thatdifferentto whatshedid? Sheputherfootrighthere[placesit nextto rug]andwent,"One[countsoneas

T:

she places her secondfoot] , two, three, four, five." How manypeople understand?Alice says thatwhat she did andwhat Sandradid

wasdifferent.Howmanypeoplethinktheyunderstand? Do youthinkyouagree they'vegot differentways? As this episode progressed,the two students,with the teacher'said, articulated their initial methods and interpretationsof measuringthe length of the rug. In first explanation,she indicatedthatshe startedmeasuringby placing her Sandra'"s first foot down and counting the placement of her next foot as "one." Using Toulmin's (1969) model of argumentation,which is a useful analytictool because the studentsdid not view theircontributionsas constitutingpartsof this model, we argue that she was providinga claim-that is, that the rug to a certainpoint was eight paces long-and the data consisted of her method of counting-that is, counting "one"with her second step. In this bit of argumentation,Sandrahad not made her warrantexplicit to the class. As the argumentationcontinued,however, Alice presentedan alternativeexplanation,after which the teacherattemptedto make the warrantsand backings explicit. In contrastwith Sandra'sexplanation, Alice counted"one"the firsttime she placedherfoot downandcounted"two"with the placementof the second foot. Similarto Sandra,Alice did not explicitly articulate a warrant.As we see it, the question "Whyis that differentto [sic]what she did?"was an attemptto have Alice explain more explicitly how her interpretation of measuring led her to make the conclusion she did. In the next line, Alice contrasted the two solution methods by presenting Sandra's explanation. In

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Toulmin's terms,Alice was providinga warrant,or informationthatdescribedhow each of those students came to their alternativesolutions. Notice that Alice's warrantherecontainedexactly the samecontentas Sandra'sexplanation;however, we characterizeit as a warrant,not data,becauseit servesthe functionof answering a challenge (see chapter3 of this monograph).The teacher challenged Alice to justify why the two interpretationswere different,how each student'sdataled to the contrastingconclusions that were drawn.In response to the challenge, Alice offeredSandra's explanationimplicitlyas a contrastwith hers.In the next episode, we will see instances in which studentsprovide more detailed warrantsfor each of Sandraand Alice's solutions. Before we proceed to the next piece of dialogue, we note that, to this point, students' warrantsfor their measuring generally consisted of explaining their method of pacing, that is, which paces were counted. For students who paced Sandra's way, measuring initially appearedto signify going throughthe act of pacing without considering that their first foot covered a distance. To initiate a discussion in which the two different warrantswere made more explicit, the teacher asked Melanie to begin pacing the length of the rug so that a piece of masking tape could be placed at the beginning and end of each pace. Once this recordwas made,the studentswho countedtheirpaces Alice's way beganto argue thatSandra's methodof countingwould lead to a smallerresultbecauseshe missed the first foot. In the excerpt thatfollows, Melanie differentiatedbetween the two ways of counting paces, whereasother studentsjustified theirparticularmethod. Melanie:Sandradidn'tcountthisone [putsfoot infirst tapedspace];shejustputit down andthenshestarted "One,two."Shedidn'tcountthisone,though[points counting, to the space between thefirst two pieces of tape].

T:-

So she wouldcount,"One,two."[refersto thefirst threespaces,sincethefirst

space is not being countedby Sandra]. How would Alice count those? Melanie: Alice countedthem, "One,two, three."

T:

So forAlice,there'sone,two,threethere,andforSandra,there'sone,two.

Melanie: Because Alice countedthis one [pointsto thefirst tapedspace] andSandradidn't,

butif Sandrawouldhavecountedit, AlicewouldhavecountedthreeandSandra wouldhavetoo.ButSandradidn'tcountthisone,so Sandrahasonelessthanher.

T:

Whatdo you thinkaboutthosetwo differentways, Sandra,Alice, or anybodyelse?

Doesit matter?Orcanwe do it eitherway?Hilary? Hilary: Youcando it Alice'swayoryoucando it Sandra'sway. T:Andit won'tmakeanydifference? Hilary: Yeah,well, they'redifferent.Butit won'tmakeanydifferencebecausethey're s butjusta differentway,andthey'restillusingtheirfeet.Sandra' stillmeasuring, withthesecondone,butAlicedoesthesecond leavingthefirstoneoutandstarting oneandSandra's justcallingit thefirst. Phil: waywouldgo higherthanAlice's. ways.I thoughtSandra's They'rebothdifferent CauseAlicestartedby onesandgotthreeandSandraonlygottwo.Sandrawould go highercauseshewaslesserthanAlice.... Phil:

She's fifteen [refersto the total numberoffeet Sandracountedwhenshe paced]. Alice went to the end of the carpet[he means the beginningof the carpet]. Sandra

started afterthecarpet.Hersis lesser'causethere'slessermorecarpet.Alicestarted

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here,andthere'smorecarpet.It'sthesameway,butshe'sendingupwitha lesser numberthaneverybodyelse. Alex: She'smissingonerightthere.She'smissingthisonerighthere[pointsto thefirst tapedspace].She'sgoing"one,"butthisshouldbe "one"'causeyou'remissing a foot,so it wouldbe shorter. T: So he thinksthat'sreallyimportant. Whatdo otherpeoplethink? Alex: Sinceyouleavea spot,it's gonnabe a littlebitless carpet. Melanie's first and secondcontributionprovideda contrastbetweenSandraand Alice's measuringinterpretations.In other words, Melanie was explicitly articulating,accordingto Toulmin,the warrantsfor each of the students'arguments.She described how the data led each student to make the conclusion that they did: "Because Alice counted this one and Sandradidn't...."The argumentationtakes a turnwhen the teacherasks whetherit madea differencewhichway one measured. In our view, the teacherwas attemptingto elicit a backingfor the argumentations just presented.In otherwords,theteacherwas implicitlyaskingwhichof these interpretationsis mathematicallyacceptableto ourclassroomcommunity.In the ensuing argumentation,we saw a lengthynegotiationin whicha particularbackingbecomes taken-as-shared.By the end of the argumentation,Alice's interpretationwas treatedas legitimatein the publicdiscourse,andthe subsequentbackingsreflected the interpretationthatmeasuringwas aboutcovering distance (e.g., Alex: "Since you leave a spot, it's gonna be a little bit less carpet").If one measuredSandra's way, "you're missing a foot, so it would be shorter"[not as much carpet was measured]. This excerpt is significant because students' justifications suggest different interpretationsof measuring.Initially,in suchjustificationsas Melanie's, students were comparingtwo different sequences of paces instead of discussing that an amount of carpet was being measured. When the teacher asked the class if it matteredwhich way studentsbegancounting,the backingsbecame articulatedand involved measuringas covering distance. On the one hand, Hilary's explanation indicatedthatboth Sandraand Alice were measuringthe extent of the carpetbut were counting their paces differently. For Hilary, measuring was a matter of convention, thatis, studentscould count their paces either way even thoughthey were different.On the otherhand, Phil arguedthatthe difference in the counting methodswas indeed significantbecause Sandrawould obtaina smallernumberof feet: "there'smore carpet."Alex explainedthatbecause Sandrawas not counting her first foot, an amountof carpetwas not being measured. Alex's andPhil'sjustificationsappearedsignificantbecauseno one countedpaces using Sandra'smethodin the whole-classdiscussionthe followingday.In fact,from this point on, the goal of measuringas covering amounts of distance was now beyondjustification.In termsof Toulmin's (1969) model of argumentation,backings that involved measuringas covering distancedroppedout of students'arguments, which is to say that measuring as covering distance became taken-asshared.The foregoingepisode is an illustrationof the negotiationprocessinvolved in the constitution of a taken-as-sharedmathematical practice. Documenting

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instances in public discoursein which backingsemerge from the studentsto validate an argumentis thereforehelpful in delineatingthe emergenceof new mathematicalpractices.As we see it, mathematicalpracticesemerge duringargumentations in which the participantsprovide new backings that shift the mathematical interpretationsof the communityto a new level. When backings for a particular interpretation drop out of the discussions or when alternative backings are contributedby a studentand rejectedby the community,we say thata mathematical practicehas become established. Many students, interestingly, had an intuition that pacing with smaller feet would resultin morepaces. Also, studentswere very carefulnot to leave any space in betweentheirpaces. The students'explanationsof these two issues suggest that initiallymeasuringwas tied to the bodily actionof pacing.If a studentmissed some space between her feet as she paced, a taken-as-sharedexplanationwas that she hadtakenbigger steps, which resultin fewer paces. If two studentspacedthe same item and attaineddifferentresults,it was arguedthatthe personwith larger-sized feet measuredfasterthanthe otherperson.In otherwords, her feet were bigger so she would take fewer paces and do less countingratherthancover more distance by each pace.These types of explanationsfurthersupportthe argumentthatthe idea had become taken-as-sharedthatmeasuringwas tied to the bodily act of pacing. Meagan and Nancy. The episode discussed previouslyfocused on the developmentof a taken-as-sharedunderstandingof the purposeof pacing, thatis, to cover distance. Nancy's participationin the taken-as-sharedinterpretationof pacing involved covering amounts of distance. For example, the day after Alice and Sandra'smethodswere contrasted,we observedNancy measuringthe length of a rugby countingher paces as Alice had done. When she reachedthe end of the rug, she said, "Forty-threeand a little space."The fact thatNancy describedthe result of pacing as "Forty-threeand a little space" suggests that she interpretedher pacing activity as covering an amountof distance. Our interpretationof Nancy's activityin this exampleis consistentwith subsequentobservationsof hermeasuring activity. Such an interpretationwas made possible by her participationin the whole-class discussion describedin the previous section. In contrast,Meaganappearedto participatein this practiceby countingeach pace as she walked the length of an item, without regardingeach pace as covering distance or a part of the rug. For example, on the third day of the instructional sequence,Meaganmeasuredthe lengthof a wall as she workedwith Nancy during small-groupwork. She began by placing the heel of her foot at one end of the wall and countedthatfoot as "one."She continuedpacing heel-to-toe the entirelength of the wall as she counted her paces. When she reachedthe end, she counted her last pace as "fifteen"even thoughpartof the pace extendedpastthe end of the wall. In this instance,because she saw no conflict in having partof her last pace extend past the end of the wall, Meagan seemed to simply count the numberof paces she needed to take to reach the end of the wall. In otherwords, Meagandid not interprether paces as distance-coveringunits, as segmentingthe physical extent of the

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wall. Rather,her way of participatingin the practice of measuringby covering distanceinvolvedthe act of placingeach foot andcountingthe numberof stepsthat she took as she walked along the length of the wall. "Fifteen,"for her, seemed to signify the numberof paces she took her to reach the end of the wall ratherthan the amountof distance(wall) she had covered. Ourdescriptionof the second phase of the instructionalsequence shows clearly thatas Meaganparticipatedin the firstmathematicalpractice,she eventuallymade a conceptualreorganizationconcerningthe underlyingpurposeof measuring,that is, to cover distance. More learning in thefirst mathematicalpractice. Like Meagan,other students also had interestingways of countingtheirlast pace when partof it extendedpast the end of the item they were measuring.In fact, the teachercapitalizedparticularly on Perry'scontribution,and lateron Meagan's, in the following whole-class discussion. In the episode below, Sandrahad paced the extent of a rug at the front of the room and found that it was 18 of her feet long. Perryarguedthat because partof Sandra'slast pace extendedpastthe end of the rug, she could not know how long the rug was. Perry:

My question was, What could it be, 'cause her foot is right here? [He points to Sandra's lastfoot that extendspast the end of the rug.]

T:What'stheproblemwithherfootbeingrightthere? Perry: 'Causethenwe can'tfigureouthowmanyfeetit wouldbe. T:Whynot? Perry:

Well, because it's longer out.

[Note: An ellipsis such as this between lines of transcriptindicatesthatirrelevant conversationhas not been reported.] Perry:

Well you see, why would it be out here if that's not how long the rug is?

Max: Herfoot'slongerthanthecarpet. Perry: Thatstilldoesn'tmatter.'Causeit stilldoesn'tmakethefeetdifferent.Thecarpet can'tmove. What do youmeanthecarpetcan'tmove? T: I mean thecarpetisn'talive,so it can'tmove.So it hasto stayrightthere,so it Perry: doesn'tmakea differencewiththefeet... andthatdoesn'ttellus howmanyfeet thecarpetis... it doesn'tcount,they'restilloff therug,butthatdoesn'ttell us howlongtherugis andthat'swhatwe'retryingtofigureout.Sohowcanwe know it if people'sfeetaregoingoff therug? Is whatyoumeanthattheydon'tfit attheendexactly? T: Perry: [Agrees.] last foot did not fit exactly ForPerry,an item could not be measuredif a person'"s withinthe physicalboundariesof the item. Perryunderstoodthatwhen a student'"s last foot extended past the endpoint,the result of pacing did not determinehow manyfeet long the rugwas. His backingfor this interpretationconsistedof the idea thatthe "carpetcan't move." We interpretthat statementto mean that as students

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paced, they were, in effect, defining the distanceas they measured.For example, Samantha's last pace defined a distancethatextendedpast the end of the rug.This newly defined distancedid not fit with whatPerryhad anticipated.In otherwords, Perryanticipatedthatpaces would fit exactly withinthe physical endpointsof the rug. In his view, when the last pace was shorteror longer thanthe endpointof the rug, the rug could not be measured.To him, the rug itself seemed to take precedence over the measuringactivity,andhe could not conformhis measuringactivity to fit with the length of the rug;thatis, as he said, "thecarpetcan't move," andthe possibility of adjustinghis measuringactivity was not availablefor him. No one challengedPerry'sbacking,but they tried insteadto develop ways to measureso thatno partof theirpaces extendedpast the end of the item being measured.This lack of challenge indicatesto us thatPerry's interpretationwas taken-as-sharedin this mathematicalpractice. To correctfor their last pace extending past the endpoint of the carpet,many studentssimply turnedtheir feet sideways so that the last foot fit exactly within the end of the rug, and they counted the whole foot as "one."The teacherasked whethercountingthe last pace in this mannerwas a good way to measurethe extra distanceat the end of the rug. Some studentsarguedthatthey could not count the last pace as a whole foot because a whole pace was not needed. Thus, these studentsarguedthat the remainingdistance was about a half a foot. The teacher rememberedthatMeaganhadcountedthe turnedpace as "one"in smallgroupwork earlierin the class periodand asked studentswhetherthey could solve it thatway. That solution method was immediately rejected by the class, indicating that measuringas coveringdistancewas taken-as-shared.Finally,insteadof turningthe last foot and fitting it inside the remainingdistance, some studentsdid not place their foot at all and just visually estimated, for example, "Fourteenand a little space." To make a clarification,we see the contrastingreasoningof Meaganand others as acts of participationin the firstmathematicalpractice.Further,we considerthose acts of participationas simultaneouslycontributingto the emergenceof the mathematical practice.For example, when the teacheroffered Meagan's reasoningas a possible solutionmethod,the studentsrejectedit. In fact, the rejectionof Megan's solutionby her classmatesindicatesthather act of participationcontributedto the negotiationof the first mathematicalpractice.We view the coordinationof social andindividualperspectivesin thatsense: on the one hand,the firstmeasuringpractice emerged as Meagan and other studentsreasonedand contributedto it; on the otherhand,as we will see in the next phase of the instructionalsequence,Meagan and others reorganizedtheir evolving interpretationsas they participatedin the evolving mathematicalpractice. Describing the result of measuring.As studentscontinuedto engage in pacing activities,the idea thatthe resultof pacing signified a sequenceof steps thatcould be countedalso became taken-as-shared.Forexample,if studentspacedthe carpet and obtaineda result of 15, "13"was taken-as-sharedto be the distance covered

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by the 13thstepratherthanthe whole spanof therug.As Hilaryexplainedin wholeclass discussion,the 14th,the 13th,and so on, paces were "whatgot you up to 15." For us, this way of describingthe resultsof pacing could lead to a mathematically significantwhole-class discussion.Recall thatwe were tryingto supportstudents' coming to act in a spatialenvironmentin which measuringsignified an accumulation of distances. Hilary's explanation,togetherwith the fact that the community acceptedherexplanation,indicatedthatit hadbecome taken-as-sharedthatthe resultof pacing signified a chain or sequence of individualpaces. In otherwords, "fourteen"signified the distancecovered by the 14thpace in the sequence rather than an accumulationof the distancecovered by 14 paces. In summary, the first mathematicalpractice emerged as students compared differentways of countingtheirpaces. Thatmeasuringinvolveddefininga distance as one measuredappearedto be taken-as-shared.Defining this distanceseemed to be tied to the bodily activityof pacingsuchthatthe physicalplacementof each pace definedthe distancebeingmeasured.Also, the resultsof pacingsignifieda sequence of pacesthatcouldbe countedto findhow long an itemwas. Further,the lastnumber word that was spoken signified the distance covered by the last pace ratherthan the accumulationof distancescovered by all paces. Meaganand Nancy appeared to participatein this first mathematicalpracticein two differentways. Meagan's participationinvolvedinterpretingthe activityof measuringas countingthe number of paces she took to reachthe end of the object;pacing did not necessarilysignify covering distanceat this pointfor her.Alternatively,measuringsignified covering distancefor Nancy. Phase Two-Footstrip (7 days) The studentsengaged in pacing activitiesfor 3 days. On the 4th day, the teacher introduceda new scenario about smurfs who lived in a village and sometimes needed to know how long things were. The teacherexplainedthatto find out how long things were, the smurfs used their food cans (Unifix cubes) to measureby placingthemend to end andcountingthem.Ourintentwas for the collectiveactivity of measuringwith the Unifix cubes to serve as a basis for a more sophisticatedtool to be developedlaterin the sequence.As studentsengaged in initial activitieswith the Unifix cubes, however, we learnedthatthese new instructionalactivitieswere inappropriatefor them.Forexample,one activitythe teacherposed involved using food cans-Unifix cubes-to show the height of a wall, which was 41 cans tall, so thatthe smurfscould build a rope laddertall enough to climb over the wall. As studentsattemptedto solve this task, we soon realizedthatusing the cans as a unit with which to measurewas not taken-as-shared.Most studentsarguedthat it was impossible for the smurfsto find the height of the ladderunless they had enough cans to scale the wall on both sides. They imaginedthe smurfsactuallyclimbing a column of cans to get to the top of the wall. The smurfs, therefore, needed enough cans to climb up the wall andenoughextracans to get down the otherside. The differencebetween this new situationand the king's-foot scenariowas in the

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contrastbetween interpretingthe resultsof measuringandengaging in the activity of measuring.In the priorpacingactivities,studentscreatedthemeasureof anobject as they paced; students were engaged in the activity of measuringand actually carriedit out. In contrast,with the Unifix cubes, studentswere asked to take the resultsof measuringas given, thatis, theheightof the wall was given to theminstead of theirhaving to createit by actuallymeasuring.Ourconjecturewas thatstudents needed to have more experience shorteningtheir measuringactivity and needed to engage in discussions thatcenteredon what the results of measuringsignified. The emergenceof the second mathematicalpractice-Partitioning distance with a collection of units Creatinga recordofpacing. Given these reflections,the teacherreturnedto the king's-foot scenario the following day and told the students that the king was spendingall his time measuringinsteadof doing thingsthatwere necessaryto run a kingdom.She askedwhetherthey could thinkof anotherway thatthe king could measureso he would not always have to measurethingsfor otherpeople. Students madeseveralsuggestions,includingsendingone pairof the king's shoes outto those who needed them and distributingseveral pairs of the king's shoes aroundthe kingdom.The teacherthenrelatedthatone of the king's adviserssuggestedputting a pictureof the king's foot on a piece of paper.The studentsdecidedthattheywould need at least two feet so that a person who was using this picturewould not skip any distancebetween two feet. The teacherasked studentsto drawfive feet on a stripof paperand then asked how the king might use such a strip: T: Thekingsayshe wantsfive. "Iwantfive in a row."Howcouldyouuse somethinglikethis? Melanie:It wouldworkbecausehe wouldputthepaperin frontof itself. Hilary: Thiswouldbe faster.He'dhavelots morefeet,andif he hadlots morefeet,he couldjusttakea bigstepwithallthefeettogetherandtherewereten.Itwouldn't bejustten,twenty.Eachonewouldbethesamesize,andyoucouldjustaddthem all together. In this excerpt, Hilary drew on her priorparticipationin the first mathematical practiceto anticipatehow using the footstripwould be more efficient. Insteadof taking five little steps with their feet, studentscould now take one "big step"of five. More important,the studentsrealized that solving problemswould now be faster,indicatingthatsome of themanticipatedcountingtheirindividualpaces more efficiently, thatis, they would countfasterby five paces thanby single paces. This phenomenonof drawingon priorparticipationis exactly what we hope for when designing instructionfor students.The instructionin this teachingexperimentwas designed to build on students'priorunderstandingso that when studentsencounwould have a way to act. So, havingcome to this concluterednew problems,they sion, each pair of studentsthen createdtheir own stripsof five paces, which the class subsequentlynamedfootstrips. Meagan. Meagan's way of participating in the first mathematical practice appearedto involve countingthe numberof paces she took to reachthe end of an

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item, without necessarilyregardingher measuringactivity as covering the physical extension of an item. As she participatedin the whole-classdiscussion during the constitutionof the first practice, Meagan appearedto reorganizethe goal of measuringsuch thatpaces now became distance-coveringitems. Evidencefor this resultcomes as Meaganparticipatedin a whole-class discussionon day 5 in which two groups of studentswere comparingthe footstripsthey had made. One pair of studentshad constructedthe footstripseen in Figure5.2.

Figure5.2. Onepairof students'footstrip.

Meaganarguedthatif they used theirfootstrip"andyou were measuringthe rug, you would be missing some spaces."Paces, or recordsof paces, now seemed to be distance-coveringentities for Meagan because she argued that the paces were arrangedon the students'footstripin such a way that measuringwith that particularfootstripwould allow amountsof distanceto be missedwhen iteratingthe footstrip.In fact, she and Nancy had constructedtheirfootstripby drawingfive paces heel-to-toeandmarkinga line andnumeralaftereachpace (see Figure5.3). Meagan explainedthatwhen they were makingtheirfootstrip,they had placed numeralsat the end of each pace so they would not miss any distancein betweenpaces.

1

2

3

4

5

Figure5.3. NancyandMeagan'sfootstrip.

Nancy and Meagan-Measuring withthefootstrip.The 7th day markedthe first instancein which studentsmeasuredwith theirfootstrips.They were askedto work in pairsandmeasureitemson the playground.Observationsof NancyandMeagan's activity indicatedthat measuring,for them, was dependenton the act of placing

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down the footstrip.For example, consider how Meaganand Nancy measuredthe length of a small shed, which actuallymeasured24 1/2paces. They placed one end of the footstripat the beginningof the shed and counted"five."Next they iterated the footstrip end to end, counting by fives as they did so, "Five, ten, fifteen, twenty, twenty-five, twenty-five and a half [sic]" (see Figure5.4).

Shed

"5"

"...10" .

."15" .

"20"

".25"

a shed. Figure5.4. MeaganandNancy'smethodof measuring

Insteadof placingthe footstripa fifth time andcountingup by ones from20, that is, 21, 22, 23, 24 1/2,MeaganandNancy placedthe footstrip,counted25, andthen countedthe half a foot thatextendedbeyond the end of the shed. Whenthey began measuringthe shed, they seemingly intendedto cover the spatialextension of the shed with an exact number of footstrips. The final placement of the footstrip, however, extendedpast the end of the shed. This resultdid not fit with what they hadoriginallyintendedto do, which was to cover the spatialextent of the shed with a sequenceof footstrips.Thus,theyplacedthe footstripa fifth time, saying"twentyfive," where25 signifiedthe last partof the spatialextensionof the shed.They then realizedthatthe footstripextendedpast the end of the shed andaddedan extrahalf a pace. Measuringseemed to have become inseparablefrom the activity of iterating the footstrip.Because the distancethey had measured,the distancecovered by 25 paces, did not fit with what they had intendedto cover, they addedthe extra half pace. This interpretationof measuringwith the footstripwas typical of both Nancy's and Meagan's activity initially. As they participatedin the negotiation of a second mathematicalpractice described below, both eventually reorganizedtheir understanding. Constitution of the second mathematical practice. In the whole-class discussion

that follows, the taken-as-sharedinterpretationof measuring evolved and the second mathematicalpractice emerged. Whereas Nancy reorganizedher understanding in this episode, Meagan did not. A fundamentalissue reemerged as studentsparticipatedin the whole-class discussionfollowing the smallgroupwork

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describedin the foregoing.Recall thatthe previousmathematicalpracticeinvolved covering the spatialextensionof items with an exact numberof paces andthatthis activity was integrallytied to bodily action. As the second mathematicalpractice was constituted,the idea thatpartof a unitof measurecan extendpastthe endpoint of an item became an explicit focus of conversation. During the whole-class discussion thatfollows, Porterand Sandraused theirfootstripto find the length of a cabinet that was located at the side of the classroom and ended against a wall. After three iterations, Porter and Sandra found that they did not have enough room to place anotherfull footstrip.Insteadof sliding one end of the footstripup the wall, Porterand Sandraplaced the farthestend of the footstripagainstthe wall so thatit overlappedthe thirdplacementof the footstrip(see Figure5.5).

cabinet

"five"

'len"

17 16" ".181/4

Figure 5.5. Sandraand Porter'smethod of measuringthe cabinet.

SandraandPorterthencounted"Sixteen,seventeen,eighteenanda fourth"back along the footstripfrom the wall untilthey reachedthe end of the thirdplacement. Because many studentsseemedto be confusedby this solutionmethod,the teacher suggestedthatit mightbe easierfor othersto understandif PorterandSandramoved the footstripso thatthe excess ranup the wall. Lloyd arguedthatmoving the footstripup the wall would not work. Lloyd: Uh-uh.It doesn'tmeasurethewholecabinet.Butit won'twork. Whatdo othersthink? T:So Lloydsaysit won'twork.That'sinteresting. I thinkit wouldbe like, uh, I thinkthatwouldbe just 18 frommeasuringthe Pat: cabinets. fromdownthere[pointsto the Lloyd: I don'tthinkso. You'resupposedto be measuring beginning of the cabinet] to right here [points to where cabinet meets the wall].

I meanto thewall.Andthatis measuring upthewall.It'ssupposedto stayhere. the cabinet. it at the end of Pat: But,well, stopped Lloyd: Itdoesn'twork.'Causeshesaidfromthatbox[heispointingtoa boxthatsitsat the beginning of thecabinet]allthewaytothewall,andit'smeasuring upthewall.She saidfromtheboxto this wall righthere [pointsto wherethe wall and cabinetmeet].

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T:We'vegot twopeoplesayingdifferentthingshere. Sandra: I havea questionforPat.Shedidn'tsaygo allthewayupto thewall. Pat: Yeah,butif youjustleftit righthere[withfootstripextendingupthewall],then youwouldn'thavetheseextrafeetrighthere[pointingto the2 3/4feet extending upthewall]. T:So, Pat.Areyoukindof in yourmindseeingwherethewall,thewall...? I Thehalfandthetwofeet.Fifteen,sixteen,seventeen, Lloyd: see something[excitedly]. anda halffoot [pointsto eachfoot in thelastfootstripas he counts].It's about this.I thinkhe meantto, like,pretendthatthatpartwascutoff rightthere[points topartof stripgoingupthewall]. Pat: Yeah. So youjustimaginethatin yourmind,Lloyd?Doesthathelp,Edward? T: Edward:I agree. Max: Somepeoplecouldthinkthathe mightbe measuring up thewall,liketo get to twenty. Lloyd: No. That'snot whathe's doing.He'sjust cuttingit off. I thinkwe solvedthe problem. In this episode, Lloyd initially reasonedthatbecausethe footstripextendedpast the endpoint of the cabinet, then this method of measuringwould not work. In Toulmin's terms, Lloyd claimed that measuringin this mannerdid not work, the databeing thatthe footstripwent past the edge of the item. For Lloyd and others, placing the footstripdefined the distancebeing measured.When Lloyd placedthe footstripdown the fourthtime, the distancebeing measuredno longer stoppedat the end of the cabinet but ratherextended to the far end of the footstrip.Hence, measuringthis way "wouldnot work"becausethe distancethatwas being covered by the footstripextended past the physical extension of the cabinet. In this way, measuringwas tied to the activityof placing the footstrip.This measuringactivity grew out of students' participation in the first mathematical practice, where measuringby pacing was tied to bodily action;the footstripwas a recordof their physical actions. As a result of Pat's challenge, Lloyd provided a warrant,saying, "They're supposedto be measuring"from one end to the other. In our view, he is drawing on his participationin the priormathematicalpracticeof covering distance with an exact numberof units. This practicehad been taken-as-shared,and, therefore, Lloyd did not producea backing for his argument.In contrast,Pat arguedthathe could, in fact, place the footstripso thatpartof it extendedup the wall, andhe would still only measurethe cabinet. The data in this argumentinvolved counting only those paces thatwere needed and ignoringthe extrapaces. Studentsstill had difficulty with Pat's interpretation,so the teacher, for his part, attemptedto elicit a backing for Pat's argumentwhen he said, "Are you kind of in your mind seeing where the wall... ?" The teacherwas helping Pat articulatehow he mentallystructured the distance that was being measured. Lloyd reorganizedhis measuring understandingandrecastPat'sjustificationin termsof pretendingto cut off the footstripat a certainpoint. Significantly,mentallycuttingthe footstripalong any point became a taken-as-sharedbacking in this episode. As subsequent whole-class

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discussions proceeded, other students used mentally cutting the footstrip as a backingto justify why they extendedpartsof a footstrippast the end of an object and counted only those paces that were relevantto their measuring.This shift in the content of students'justifications-the evolution of the backing-marks the emergenceof a second mathematicalpracticeinvolving partitioningdistancewith collections of units, in this case, with five paces. When mentallycuttingthe footstripeventuallydroppedout of students'arguments,we claimthatthe secondmathematicalpracticewas relativelystable.The remainingmathematicalpracticesalso came under the same methodological scrutiny as the two practices we have described, althoughwe do not present a detailed analyses of the discourse with warrantsand backingsbecause we do not want to overwhelmthe reader. In this second mathematicalpractice,the distanceto be measuredwas independent of activity, independentof the placementof the footstrip.Now the distance to be measuredtook priorityover the measurementactivityinsteadof the footstripplacingactivity'9sdefiningwhatwas being measured.Measuringwas no longertied to the physical act of placing a footstrip;rather,the footstrip'sbeing mentally cut as the need arose became taken-as-shared. The initial interpretationof not extending a unit past the farthestendpointhad become taken-as-sharedin the first mathematicalpractice but was renegotiated duringthe establishmentof this second mathematicalpractice.The explicit attention given to this issue in this phase of the sequencecan be attributedto the takenas-shareduse of the footstrip.When measuringby pacing the length of an item, studentscould simply remove their last foot if it extended past the endpoint and refer to the extra distanceby pointing to it. Now, when using the footstrip,three or fourextrapaces mightextendpastthe endpoint,butthese paces were physically inseparablefrom one another.Therefore,opportunitiesto discuss what happened when paces extend past the endpointof an item were made possible. Meagan and Nancy: Reorganizingtheir measuringactivity.During the pivotal whole-class discussion described previously, Nancy, not Meagan, appearedto reorganize her prior measuring activity. For the remaining occasions in which Nancy measuredwith the footstrip,she coordinatedmeasuringwith a stripof five paces and counting single paces. In fact, Nancy continuallychallengedMeagan's measuringactivity duringsmall group and whole-class discussions in which she measuredwith her partner.For example, in the following episode, Meagan, for whom measuringdepended on the placement of the footstrip, and Nancy were measuringstringthatsignifiedthe lengthof fish thatthe king hadcaught.The string actually measured12 1/2 paces long, and Nancy explainedher result: Nancy: We didn'thaveenoughroomforfifteen,so I wentten,eleven,twelve,thirteen anda half [pointsto each of thefirst threepaces on the thirditerationof the footstrip].

For Nancy, measuringwith the footstripnow appearedto signify a more efficient form of pacing; "fifteen"was the amountof distance covered by 15 single paces. Thus, "We didn't have enough room for fifteen [paces]"indicatedthat,for

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her, the string was not long enough to be covered by exactly one more footstrip. Hence, "ten"signified the distance covered by 10 paces, and then she counted "eleven,twelve, thirteenanda half."This approachillustratesthatNancy now coordinatedmeasuringwith the footstripwith the resultshe would obtainif she actually paced. Measuringno longer dependedon the placementof the footstrip;the thirdfootstripdid not define the distanceshe had measuredthusfar.Rather,Nancy mentallyenvisionedcuttingthe last footstripon half of the 13thpace. This conception is a significant shift in her understandingfrom the beginning of the second phase of the measurementsequence. In contrast,Meaganhad difficulty coordinatingthe resultof measuringwith the footstripwith what she would have gottenif pacing.Meaganeventually,however, came to coordinate measuring with the footstrip and individual paces. She reorganizedher thinking during whole-class discussions in which students' interpretationsof the resultsof measuringwere madeexplicit andduringpairworkwith Nancy.AlthoughbothMeaganandNancy madethis coordinationwhile measuring with the footstrip, the result of measuring signified different things for them. Meagan,on the one hand,interpretedthe resultof measuringas a sequenceof individualpaces, whereasNancy understoodthe resultas an accumulationof distance. For example, when a researcherasked Meaganto show on the footstriphow long something3 1/2 feet would be, Meaganpointedto half of the fourthpace: T: Where'sthree? Meagan: There [points to the thirdpace].

T: So threeis thatonefootthere? Yeah. Threeanda halfof thefourthfoot. Meagan: T: Couldthreemeanall of thesefeet [pointsto thespacebetweenthebeginningof thefootstrip and the end of the thirdpace]? Meagan: That's numberone [points to the first pace], that's numbertwo [points to the second pace], and that's numberthree[pointsto the thirdpace].

From this exchange, Meagan appears to have interpreted the result of her measuring activity as a sequence of individual paces. "Three" signified the distance covered by the third pace ratherthan the distance covered by all three paces. Whenthe researcheraskedher to clarify if "three"could mean all threefeet together, Meagan indicated clearly that "one" meant the first pace; "two," the is significantin thatwe were second;and"three,"the thirdpace. This interpretation in a to act spatial environment in which trying to support students' coming measuringsignified the accumulationof distances.As Meaganparticipatedin the practice of partitioningdistance with a collection of units-for us, the cognitive or mentalactivity of segmentingspace with units-the resultof measuringsignified a sequence of individualpaces for her. As can be seen in the episodes related here, the results of partitioning were different for Nancy and Meagan. Even thoughshe did not interpretthe resultof measuringas an accumulationof distance, Megan'"sinterpretationis clearly more sophisticatedthanhow she initially participated in the practice.

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Nancy, in contrast,interpretedthe result of measuringas the accumulationof distance covered by paces. When asked to show how long something 8 1/2 feet would be, Nancy iteratedthe footstriptwice and replied: Nancy:

[Sweeps her hand over the spacefrom 0 to 8.] Eight and a half [points to half of the eighthfoot].

Nancy clearly indicatesby the sweeping motion of her hand that she interpreted the result of measuring as an accumulation of distance. "Eight"signified the distancecovered by 8 paces, as indicatedwhen Nancy demonstratedthat 8 would stretch from the end of the 8th pace to the beginning of the distance. In this episode, Nancy illustratedthatthe "half' from "eightand a half' is located on the half of the eighth foot. This answeris not in conflict with ourclaim thatmeasuring for her signified an accumulationof distance.Rather,whereto locate the half, for the teacher, was a matterof convention--either 8 and a half more or half of the eighth foot. The teacher,as a consequenceof observingsuch solutionsas Nancy's, establishedthatthe half is conventionallylocated on half of the next foot. This kind of interpretationremained consistent throughout the rest of the measuringactivitiesinvolvingthe footstrip.In fact, NancyandMeagansometimes measured during whole-class discussions. In this way, Nancy's and Meagan's participationin the practice of partitioningdistance with a collection of units contributedto the constitutionof this taken-as-sharedway of measuringwith the footstrip.In otherwords,as they andthe class discussedcontrastinginterpretations, the understandingthattheplacementof the footstripno longerdefinedthe distance measured became taken-as-sharedin the public discourse. Measuring became independentof the activityof iterating,andthe footstripcouldbe mentallycut along any point as the need arose. Conclusion of phase 2. In summary,the first mathematicalpractice involved measuringby covering distance,and the resultof measuringsignified a sequence of individualpaces thatextendedto the end of the item being measured;measuring was tied to the bodily act of pacing.The secondmathematicalpracticeevolved from the firstmathematicalpracticein thatthe footstripwas a recordof theirpaces;hence, that similar issues emerged as topics of conversationwas significant. Initially, measuringwith the footstripseemed to be integrallytied to the placementof the footstrip. As in the previous practice,the physical activity took precedenceover the physical extensionof an item. The idea becametaken-as-shared,however, that the footstripcould be mentallycut; the extension of the item now took precedence over the activity.Measuringhadto conformto the extensionof the item ratherthan the other way around.Therefore, the second mathematicalpractice involved a collective reorganizationof the first practice.In addition,a transitionin the takenas-sharedargumentationsabout the result of measuringbegan. As illustratedby Nancy's reasoning, she and several students began talking about the result of measuringas a whole measureddistance-an accumulationof distance-ratherthan a sequence of individualpaces. Further,they used their hands to show the accumulation of distances measured,for instance, from 0 to 8, and referredto these

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distances as "the whole eight."This way of talking and gesturingbecame takenas-sharedlaterin the instructionalsequence,but the transitionseemed to begin as studentsmeasuredwith the footstrip. Phase 3-Smurf Village (11 days) In this phase of the instructionalsequence, the teacherreintroducedthe smurfvillage scenario.Initially,studentswere given a bag of Unifix cubes and asked to find how long various items around the smurf village, the classroom, were. Subsequent instructionalactivities included giving students a piece of adding machinetape and telling them that it signified the length of an animalpen. They were also supplied several otherpieces of addingmachinetape that signified the lengths of different animals in the kingdom, such as a dog, a cat, or a horse. Studentswere then asked to determinewhethereach of the animals would fit in the pen. If the animaldid not fit exactly, the studentwas asked to find how much longer or shorterthan the pen the animalwas. The taken-as-sharedinterpretation of measuringitems in the classroom involved making a bar or rod of cubes that stretchedthe length of an item and thencountingthe cubes. The teachersuggested thatthe studentsmight find a moreefficient way of measuringwith the food cans and asked students whether they could think of a way that the smurfs could measure without carrying arounda whole bag of cubes. Students gave several suggestionsthatincludedcarryingonly a barof 10 cubes,whichtheynameda smurf bar becausethey hadpreviouslybeen told thatsmurfswere about10 cans, or cubes, tall. Instructionalactivities with the smurf bar included having studentsfind the lengths of various items aroundthe room and cutting pieces of paper signifying different-sizedwooden boardsfor buildinga smurfhouse. The instructionalactivities in this phase of the sequence differedfrom those in the first two phases in that now the studentsused a physical entity as a measuringtool insteadof using a part of their body or a record thereof. The mathematicalpractice that emerged as studentsparticipatedin the activitiesdescribedin the foregoinginvolvedmeasuring by accumulatingdistances. On the 14th day of the instructionalsequence, the class measuredwith a barof 10 cubes for the first time. The teacherasked the studentsto work in pairsand to measureitems in the classroom.Using a smurfbarfor the firsttime, Meaganbegan measuringthe height of an animal cage. She placed one end of the smurfbar at the bottom of the cage and said, "ten."She iteratedthe bar end to end along an edge of the cage and counted "ten, twenty." She placed the bar a thirdtime and counted "thirty"even thoughthe thirditerationextendedpast the top of the cage. Then, she counted the cubes, saying, "thirty-one,thirty-two, thirty-three"(see Figure 5.6). For Meagan, counting 30 as she placed the smurf bar for a thirdtime seemed to mean thatthe cubes within thatiterationshould be counted "thirty-one,thirtytwo, thirty-three.... "This way of reasoningindicatedthat Meagan was not coordinatingmeasuringusing the barof 10 with measuringwith the individualcubes

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"30"

"20"

"10"

theheightof a cage. Figure5.6. Meagan'smethodformeasuring

of which the bar was composed. In other words, iteratingwith the bar of 10 did not signify a recordof measuringwith individualcubes for her. Nancy indicatedthatshe disagreedwith Meagan'smeasurementandremeasured the height of the cage by counting as follows: [iteratesthe bar once] "ten,"[iterates the bara second time] "eleven,twelve, twenty," [iteratesthe bar thirteen,..... a thirdtime] "twenty-one,twenty-two,twenty-three." ForNancy, iteratingthe bar of 10 appearedto signify the distancecoveredby cubes thusfar,which can be seen by her counting by single cubes to justify her method of measuring.Although Meagan acceptedNancy's measurement,she continuedto measurein the manner she had before. The fact thatMeagandid not reorganizeher understandingin this episode might be due to the calculationalnatureof Nancy's explanation.Nancy described only how she counted cubes; she did not specify her interpretationof distancethatunderpinnedher measuringactivity. Otherstudentsinterpretedtheir measuringwith the smurfbar in the same manneras Meagan;hence, the teacher decided to ask studentsto compare and contrasttheir differing interpretationsin whole-class discussions.The teacherhopedthatan accumulation-of-distance interpretationwould become a topic for whole-class discussion. As we see in the next section, an accumulation-of-distanceinterpretationdid become an explicit focus in subsequentpublic discourse. The emergence of the thirdmathematicalpractice-Measuring by accumulating distance The episode that follows illustratesa pivotal whole-class discussion in which an accumulation-of-distance interpretationwas an explicit focus of conversation. This excerpt is significant because students' explanations indicate that an accumulation-of-distance becametaken-as-sharedduringthe discusinterpretation sion. Below, Alice andChriswere askedto cut a piece of addingmachinetape that signified wooden boardsto be used for a smurfhouse so thatit measured23 cans

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long. Alice iteratedthe smurfbartwice and uttered"ten,twenty"aftereach iteration.Next she placedthe smurfbardown once moreto finda lengthof 23 butpicked it up, brokeoff threecubes from the smurfbar,and laid the threecubes at the end of the second iterationto indicatea length of 23: Edward: I thinkit's thirty-three[pointsto wheretheyhave marked23 with the threecubes] because ten [iteratesthe smurfbar once], twenty [iteratesthe smurfbara second

[countsthefirst, second,andthird time],twenty-one,twenty-two,twenty-three

cube within the second iteration, thus measuringa length that was actually 13 cubes].

Let's be sureall the smurfscan understand, 'causewe have whatAlice had Weneedto be sureeverybodyundermeasured andwhatEdwardhadmeasured. standswhateachof themdid.So Edward, whydon'tyougo aheadandshowwhat cans. it is to measuretwenty-three Edward: Ten[iteratesthe smurfbar once], twenty [iteratesthe smurfbar again]. I changed my mind.She'sright. T." Whatdo youmean? T:

Edward: This would be twenty [pointsto the end of the second iteration].

T:

Whatwouldbe twenty?

Edward: This is twenty right here [places one hand at the beginning of the "plank"and the other at the end of the second iteration].This is the twenty.

Then,if I moveit upjustthreemore.There.[Breaksthebarto show3 cansand

places the 3 cans beyond20] That's twenty-three. Phil:

Whenyouput theseconddown, that'sthewholetwenty [pointsto the spacefrom the beginningof what was thefirst iterationto the end of the second iteration].

In the courseof his explanation,Edwardreconceptualizedhis measuringactivity. Initially,for Edward,placingthe smurfbardownfor a secondtime definedthecubes in the seconddecade,the twenties.In otherwords,if he were to countthe individual Thus, althoughiteratinginvolved cubes, he would count "twenty,twenty-one,. a shorteningof countingby individualcubes, it was based only on a number-word relation."Twenty"was the numberword associatedwith the second placementof the smurfbarratherthan with the amountof distancecovered by 20 cubes. After reflectingon Alice'9sexplanation,however,Edwardcame to view "twenty"as signifying the length covered by 20 cubes and realizedthat21, 22, and 23 must extend beyond the length whose measurewas 20. Similarto Alice' s reasoning,now if he were to count individualcubes in the second iteration,he would count "eleven, with the smurfbarwas a shorteningof covering twelve.....4"ForAlice, measuring distance with individual cubes and involved an accumulation of distance. To clarify the meaning of the number20, Phil explainedthat 20 signified the length from the beginningto the end of the second iteration,not the distancecovered by the last 10 cans. The main point of this excerptis not only thatEdwardandotherspossibly reorganizedtheirpriorunderstandingbutalso thatstudentsnegotiateda taken-as-shared

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interpretationof measuringwith the smurfbar.WhenAlice initiallyexplainedhow she measured23, she iteratedthe bartwice andbrokeoff threeextracubes to show 23. That she providedno backingfor why this solution was acceptableis significant. Edwardchallengedher explanation,offering a justificationin which he did not coordinate measuring with a bar and measuring with individual cans. As Edwardreorganizedhis understandingin the midst of a rejustification,Phil explicated a backing that involved an accumulationof distance. As the whole-class discussioncarriedon, no studentrejectedPhil's explanation.In fact, otherstudents used this way of reasoning as backing in subsequent conversations, with the backing eventually dropping out of students' arguments. Furthermore,when students offered an explanationthat did not reflect an accumulation-of-distance interpretation,the communityrejectedit. We claim thatin this way measuringby accumulatingdistancebecame taken-as-shared.Thatphysicallyiteratingalong an item createda partitioneddistance that could be structuredin collections of tens and ones also became taken-as-shared. MeaganandNancy.Ourobservationsof Nancy andMeagan'smeasuringactivity for the remainderof this phase indicatedthat both girls interpretedthe result of measuringas an accumulationof distance and coordinatedmeasuringby 10 cans and individualcans. Meagan, however, was better able to coordinatemeasuring with tens and ones when she symbolized the resultsof her measuringactivity. As she measuredan object, she kept trackof the numberof cans by recordingeach iteration of 10 with masking tape or numerals. As Meagan reasoned with her recordof iterating,two iterationssignified the accumulationof distance covered by 20 cubes. Anotherway to state this outcome is thatthe numberword "twenty" signified a numericalcomposite (Steffe, Cobb,& von Glasersfeld,1988), an entity or a distancepartitionedby 20 cubes ratherthanthe distancecovered by the 20th cube. This episode bringsto the fore the role of symbolizingin Meagan's activity. As she reasonedwith herrecordof iterating,Meaganinterpretedthe resultof each iterationas an accumulationof distancecoveredby cans. Further,when reasoning with such symbols, she coordinated measuring by iterating a bar of 10 and measuringby iteratingsingle cubes. Thus, as she participatedin the priorconversationsin which mathematicallysignificantissues arose, she developed relatively powerful ways of reasoning with symbols. When she did reason with symbols, Meagancoordinatedmeasuringwith a barof 10 andmeasuringby iteratingsingle cubes.WhenMeagandid not reasonwith symbols,however,she did not makesuch a coordination. AlthoughNancy symbolizedher activitywhen she workedwith Meagan,on her own, Nancy coordinatedmeasuringwith a bar of 10 and measuringwith single cubes withoutsymbolizing.Further,Meagan's reasoningwith symbols was an act of participationin the emergingmathematicalpracticeandconstitutednot only her learningbut also a contributionto the taken-as-sharedpracticesof the community. Conclusion of phase 3. In summary,the emergence of the thirdmathematical practice of measuringby accumulatingdistances can be seen to grow out of the

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second mathematicalpractice. As a result of participatingin the second mathematicalpracticeof partitioningdistancewith collections of units, takingthe result of measuringas a given had become taken-as-shared.As the thirdpractice was established,a numeral,such as 23, signified an object's measure,and immediately iteratingcollections of 10 cubes to specify its length was beyondjustification. Phase Four-Measuring with the MeasurementStrip(5 Days) In this phase of the instructionalsequence, the teacher built on the students' measuring activity to introduce a measurement strip that was 100 cans long. Studentshadengaged in instructionalactivities in which they cut pieces of adding machinetape that signified lengths of pieces of wood to be used for a smurfraft. Forexample,studentswere askedto cut pieces of wood thatwere 30 cans, 22 cans, and5 smurfbarslong. Whereasmost studentsiterateda smurfbarandcut thepaper at the end of the last iteration,Nancy and Meaganactuallyrecordedeach iteration of 10 on the pieces of paper(see Figure5.7).

10

20

ID

30 cans. Figure5.7. NancyandMeagan'swoodenboardmeasuring

The teacherbuilt on this innovationby suggesting to studentsthat Nancy and Meagan's papermight be easier to carryaroundthan a bag of cans. One student suggested they carryarounda stripthatwas 10 cans long. The teacherresponded that she liked their idea but that the smurfs decided they wanted to make a strip thatwas 50 cans long. The teacheraskedthe class to constructmeasurementstrips that were 50 cans long and thatdid not requirethem to carryany cans with them. As we observedpairsof students,we noticed thatmost studentsconstructedtheir strips as in Figure 5.8. An interestingfeatureof these strips was the fact that no individuallines were drawnto signify cubes withinthe decades.In fact, every pair of studentsconstructedtheir 50-strips by iteratinga smurfbar and markingonly the end of the barwith a numeraluntil they had made a stripthatwas 50 cans long. One interpretationof the students'constructionscould be thateach iterationof 10 that they marked on the strip did not signify composites of 10 but simply recordsof measuringwith the smurfbar. Anotherinterpretation,however, takes note of theirparticipationin the previousmathematicalpracticeinvolving iterating barsof 10. Considerhow studentsmeasuredthe lengthof a table43 cans long using

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10

20

30

40

50

of themeasurement strip Figure5.8. Initialconstructions

a smurfbar.Most studentswould iteratethe smurfbar"ten,twenty, thirty,forty," iteratethebarone moretime,andcountthefirstthreecubesin thatiterationto obtain 43. "Forty"signified what studentswould get if they countedthe cubes in the first four iterations by ones, and they needed only to count three individual cubes beyond "forty."Thus,in makingthe 50-strip,studentsmarkedonly the end of each iterationbecause they seemed to have no need to know wherethe individualcubes ended when measuringsomething, say, 50 cans long. Hence, the constructionof the 50-stripcan be tracedback to students'participationin the thirdmathematical practiceof measuringby accumulatingdistances.The marksthatstudentsdrew on the strip were a recordof their activity of iteratingthe smurfbar where they had no need to recordindividualcubes. The next day, the teacherasked studentsto make stripsthat were only 10 cans long instead of 50 cans. We thoughtthat to supportthe emergenceof a measurement strip 100 cans long, studentsmight build on theirconstructionsof 10-strips morenaturallythan50-strips.The intentof the 10-stripwas to buildon the previous practiceof measuringby accumulatingdistances;it fit with the activityof iterating to find a measure.At least two pairsof studentscut paper10 cans long andmarked individualcubes on the strip.In a subsequentwhole-class discussion,the need for individualmarkingsarose from the studentsand every pairrevised their strips. In the absence of symbolizing, both Meagan and Nancy had difficulty coordinatingmeasuringwith a 10-stripandmeasuringwith individualcans. Both of them, however, clearly understoodthe resultof iteratingas an accumulationof distance. The emergenceof thefourth mathematicalpractice-Measuring with a strip of 100 Subsequentinstructionalactivities with the 10-stripincluded measuringitems aroundthe classroom. In a whole-class discussion on the 24th day, two students came to the white boardat the frontof the classroomand showed how they would measureit with their 10-strip.They iteratedtheir strip,saying "ten,twenty,...." and recordedthe endpointof each iterationwith a markingpen (see Figure 5.9). They arguedthatcountingby tens wouldbe easierthancountingby ones each time. Then the teachertaped seven 10-stripsunderneaththeirdrawingand commented thateach stripmeant 10 cans they did not have to count by ones. In this way, the 100-can measurementstripemerged as a recordof the activity of measuring.The

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teacherasked studentssuch questionsas "Whereis twenty-five?"and "How long would somethingtwenty-sevenbe?"Duringthis discussion,one studentillustrated that49 cans would stretchfrom the beginningof the white boardto the line signifying the end of the 49th cube.Nancycountered,however,that49 actuallyextended from the beginningof the boardto the line signifyingthe end of the 39th. Tojustify her way of reasoning,Nancy pointedto each of the first four spaces and counted, "ten,twenty, thirty,forty,"and then, "forty-one,forty-two,..., forty-nine"within the space signifying the fourthiterationof the 10-strip.Because the new measurementstripintentionallyhadits rootsin iteratingthe 10-strip,a reasonableanddesirable outcome is that studentswould drawon theirparticipationin the priorpractice associatedwith iteratinga collection of ten to interprettheir activity with the new tool. Otherstudentscounteredher argumentby startingwith the numeral20 and counting by ones up to 39 to show Nancy that where she pointed actually marked39 cubes. Nancy acceptedthis justification and redescribedher solution methodby countingby ones to show thatwhat she originallythoughtwas 49 was only 39 cans long.

64 63 62

61 10

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1

60

thewhiteboard. Figure5.9. Onestudentpair'swayof symbolizingmeasuring

Thus, as students negotiated what became a taken-as-sharedway of interpretingmeasuringwith the stripof 100 in the foregoing example, their warrants consisted of countingindividualcubes, andbackingsof accumulationof distance re-emergedin the context of reasoning with a new tool. The result of measuring with the 10-strip quickly came to signify the accumulation of distance as the communityrejectedNancy's contribution.Fromthis point on, no studentduring subsequentwhole-class discussions had difficulty coordinatingmeasuring with a 10-stripwith measuring with individual cubes, indicating that it had become taken-as-shared.Such an interpretationwas supportedby the conceptual discussion illustratedpreviouslyin which studentsfolded back to countingby individual cubes. To clarify, we view Nancy's contributionhere as both her reasoning,and

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reorganization,and an act of contributingto the stabilityof the thirdmathematical practice. The community did not treat her act of participationas legitimate, indicating that the thirdmathematicalpracticehad indeed been established. Her contributionalso can be viewed as contributingto the negotiationof measuring with a new but similartool. Meagan and Nancy.As illustratedin the foregoingexample,Nancy reorganized her understandingduringthe whole-class discussion. From this point on, Nancy and Meagan no longer had difficulty coordinatingmeasuringwith a 10-stripand measuringwith individualcans. This reorganizationwas madeas they participated in, and contributedto, the constitutionof the fourthmathematicalpractice.To be sure,the reorganizationwas supportedby the teacher'srecordof iteratingthathad played such an importantrole in each child's mathematicallearning. The measurementstrip. After the discussion in which the teachertaped seven 10-stripson the whiteboard,she gave each pair of childrena papermeasurement stripthatmeasured100 cubes long (see Figure5.10). Studentsengaged in various activities, such as using the measurementstripto find the lengths of items in the classroom.Duringactivitiessuch as these, the mathematicalpracticeof measuring with a strip of 100 became established. Initially, many students measured by laying the stripdown alongside the item and countingby tens and ones until they reached the endpoint of the item. Significantly, many students measured by "building"the extent of an item by iteratingor countingby tens and ones instead of by finding where the farthest endpoint corresponded to a numeral on the measurementstrip. For example, when measuringthe length of a table, many studentslaid the measurementstripnext to the edge of the table andcounted each collection of 10 fromthe beginningof the table-"ten, twenty,thirty"-ratherthan simply read off the numeral30. Thus, measuringwith the measurementstripfor many studentsappearedto have evolved out of theirparticipationin priormathematicalpracticesin which they iteratedsingle or collections of units.

10

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50

60

70

80

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Figure5.10.Themeasurement strip.

Very quickly, studentsabbreviatedtheiractivityof countingup on the measurement strip,andthe fact became taken-as-sharedthatthe length of an item could be measuredby laying down the measurementstrip alongside the item and simply readingoff the numeralcorrespondingto the positionof the farthestendpoint.Also

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taken-as-sharedwas that when studentsmeasuredthe length of an item and read off a number,thatnumbersignified the distancethatextendedfrom the beginning of the measurementstripto the line signifying the end of the item. Studentsnow seemed to take-as-sharedthatthey were acting in a spatialenvironmentin which distance was alreadypartitioned.Distance no longer had to be partitionedin an activityby physically iterating.Rather,the fact thatlaying down the measurement stripsimplyspecifiedthe measureof an already-partitioned spatialextentwas takenas-shared.Studentsseemed to be acting in an environmentin which an item was alreadypartitioned-that is, it alreadyhad a measure-and the measurementstrip simply specified the measure. This taken-as-sharedinterpretationwas significantly more sophisticatedthan in prior mathematicalpractices and constitutes operationalmeasurementas defined by Piaget et al. (1960). Meaganand Nancy. Both Nancy andMeagancame to interpretmeasuringin the way just described.Initially, however, Meagan laid down the measurementstrip to find the length of a table and startedby counting individualcubes, "one, two, three......" Nancy interruptedMeagan's counting and continued from where Meagan had left off, counting by fives. Meagan also counted by fives until they had"iterated"the lengthof the table.Hence, initially,for Meaganthe measurement stripdid not signify the resultof iteratingstripsof 10. The very next day, however, she stoppedcountingby ones and, finally, no longer countedcubes at all. Rather, she simply locatedthe appropriateplace on the measurementstripandreadoff the measure of the item. This participationseemed to be supportedby the physical recordthatwas inherentin the measurementstrip.The stripitself became a record of the units that Meagan had initially established or counted along the strip. Further,the numeralssignifiedwhatshe wouldhave measuredhadshe actuallyiterated a 10-strip.In reasoningin this manner,she contributedto the establishment of this fourthmathematicalpractice.As before,symbolizingandtool use were integral to her reasoning. Otherthan in the foregoing example, Nancy never counted by tens or ones to establishthe lengthof an item when using the measurementstrip.The stripalready signified, for her,the resultof iteratinga 10-strip,or smurfbar.Whendetermining the measureof an item with the measurementstrip,the measuresignifiedthe result of accumulatingdistancesfor bothNancy andMeagan.The length they had determined signified a distance that could be partitionedinto tens and ones for them. Their reorganizationswere made as they participatedin the emergingmathematical practiceof measuringwith the measurementstrip.Theirreorganizationsalso constitutedtheircontributionto the ongoing evolution of the fourthmathematical practice. Conclusion of phase 4. In summary,the teacherbuilt on the priormathematical practiceof iteratingbarsof 10 by asking studentsto drawpaperstrips 10 cans long. Then studentsengaged in measuringactivities with these paper 10-strips. The teacher next taped the 10-strips together end to end to support students' construction of a measurementstrip 100 cans long. A taken-as-sharedway of

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measuringwith the measurementstripwas immediatelyconstituted.Laying down the measurementstrip seemed to partitionthe distance into cans, or cubes, that were themselves organizedinto composites of 10. That the spatial extension of an item was partitionedbefore the physical act of measuring took place was taken-as-shared,and when the measurementstripwas laid down, it simply specified the measure.Measuringwas now no longertied to physicallyiteratingto find an item's measure;rather,in a sense, the measurementstrip signified the prior activityof iteratingcollections of 10. Measuringseemed to be routineat this point. Thus, the fourthmathematicalpractice involving measuringwith a strip of 100 became established. Phase 5-Reasoning with the MeasurementStrip(5 Days) Because the foregoing measuringactivities became routinefor studentsfairly quickly, the teacherposed problemsthatinvolved makingcomparisonsof lengths on the measurementstrip.At thispointof the instructionalsequence,the fifth mathematical practice, which involved reasoning with a strip of 100, emerged. The teacherintroduceda scenarioin whichthe wizardsmurfwas testingsecretformulas thatcontrolledthe growthratesof sunflowers.The studentsweretoldthatthe wizard smurf was conducting an experiment in which he had created several secret formulasfor speedingthe growthof sunflowers.He noticedthatseeds treatedwith varioussecretformulasgrew into different-sizedsunflowers.Studentswere asked to help the wizardsmurfdecide whichformulawould makehis flowers grow taller. Wizard smurf needed to know how much taller each sunflower was than the others.One flower grew from a plain seed andwas 51 cans tall. Seeds treatedwith secretformulasA, B, C, andD grew to be 45, 35, 61, and70 cans tall, respectively. The students'task was to find out how much tallereach of the flowers grownwith secret formula were than the flower grown with a plain seed. This type of task differedfrompreviousactivitiesin thatstudentsno longermeasureda physicalitem with the measurementstrip.Previously,the stringsandaddingmachinetape signified the spatial extent of the dragons, fish, wooden boards, and so forth, and studentsactuallyhad to measurethe objects. In the sunfloweractivity, the spatial extents of the various sunflowers were given numerically,and studentshad to specify the length on the measurementstripratherthanmeasureit. The emergenceof mathematicalpractice 5-Reasoning with a strip of 100 Three days afterthe measurementstriphad been introduced,the teacherposed the sunflowersproblem.This instructionalactivity was the first one in which the studentsdid not measurean item that was physically present.As a consequence, a new mathematicallysignificant issue arose that continued to be the focus of discussion for the next two class periods.When the studentsspecified the spatial extent of two sunflowers' heights, say, 51 and 45, some counted the spaces between the two numbers,whereas others counted the lines. The students who

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countedthe spaces between the two specified heights obtained6 as the difference (see Figure 5.11).

70

-60 51

-50

45

-40

I

I 1-ftr Mo.-

I

Figure5.11.A portionof themeasurement strip.

The studentswho countedthe lines betweenthe two heightsstartedcountingwith the line beside 45 and stopped with the line beside 51, getting a result of 7. We thoughtthat for studentswho were counting lines, their use of the measurement stripwas divorcedfromtheirparticipationin priormathematicalpractices.The gap between 45 and 51 on the strip did not signify for them a distance that could be covered by cubes. Instead,they simply countedthe only thingsperceptuallyavailable-the lines. We thereforespeculatedthat these studentsmight be helped by engaging in a discussion in which theirdescriptionsfolded back to the context of measuringwith cubes. In the excerptthatfollows, the teacherasked studentshow much shorter the sunflower measuring 45 cans would be than the sunflower measuring51 cans.A measurementstripwas hungverticallyat thefrontof theclassroom, and Phil recordedwhere he thoughtthe two sunflowers' heights would be on the strip(see Figure5.11). Then he explainedthatthe sunflowermeasuring45 cans would be 7 shorterthanthe otherwould becausehe countedthe lines between the two numerals. Phil: Fromheredownto herewouldbe seven [pointsto the line beside51 and 45, Becauseone,two,three,four,five, six, seven[countsthelines]. respectively]. I havea question.You'resupposedto countspaces,notthelines,becauseit's ... Pat: Phil: It wouldbe the samebecauseone,two,.... six [seemsconfused].I'm counting thelinesbecausethey'rethesameasspaces.Here'sa line[pointstoa line],it goes down to here [moveshisfinger down to the line below it].

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... Evenif youcountthelines,it's stilllikethespacebecauseyou go downto here [pointsto a line and moves hisfinger down to the line below it].

... Pat:

Yeah,butthere'smorelinesthanonespace.I'm sayingthere'stwolinesin one spacebecause,see thefrontof one,theendof one.So there'stwo lines.

Phil and others seemed to be simply counting what was perceptuallyavailable without realizing they were counting an extra line. Our initial conjecture that studentswere not relatingcountinglines to the numberof cans thatpartitionedthe distance between the two heights had to be revised in light of Phil's explanation. In retrospect,Phil seemed to be relatinghis counting activity to counting spaces when he said, "I'mcountingthe lines becausethey'rethe same as spaces."In fact, he arguedthatcountingeitherlines or spaces would give the same number.Thus, Phil and others seemed to be indicatingthe numberof cans or the measureof the spatial extent between two heights by counting what was perceptuallyavailable withouttakingthe extraline into account.In the precedingexcerpt,the teachertried to encouragestudentsto justify theirparticularmethodin termsof what each line or space signified to them. ForPat, the lines signified the top andbottomof a cube and the space signified the distance covered by one whole cube. Thus, counting the lines gave an extra numberbecause "there'stwo lines in one space."During this and otherconversations,countingspaces to specify the measureof the spatial extension betweentwo lengthsbecametaken-as-shared.Students'explanationsof their activity indicatedthat the idea had become taken-as-sharedthat a measure, such as 35, signified the distancestartingfrom the bottomof the stripto the top of the 35th. Meagan and Nancy. Meaganinitially participatedin this practiceby first using a cube to markthe end of the spatial extensions of each of the two items being compared on the strip. Then she counted the number of cubes that fit exactly between the two cubes. For example, on the last day that the students used the measurementstrip (day 31), the teacher asked them to measure one another's heights andrecordthe results.The teacher,for her part,recordedon the boardthe resultsof each smallgroup,for example,Chris65 andHilary75; Lloyd67 andAlex 72. Then the studentswere asked to work in their small group to find the differences between each pair of students'heights. Alice and Meaganused a measurement strip to find the difference between Chris's (65) and Hilary's (75) heights. Meaganplaced a cube in the space signifying the 66th cube anda cube in the space signifying the 76th cube, as if she were marking the top of the two students' heights. Meagan then counted the numberof spaces on the measurementstrip between the two cubes, getting a total of 9 cubes (see Figure5.12). This example illustratesthatat the very least, Meagancould specify the spatial extensions of two items on the measurementstrip, that is, Hilary's and Chris's heights.These specifiedspatialextensionssignifiedcompositesunits,for example, the height stretchingfrom0 to 66. When it came to quantifyingthe difference(the

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50

60

70

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80

Figure5.12.Twocubesmarkingthe66thand76thspaces.

distance)between the two heights,however, she did not see Chris'sheight (65) as nested in, or as a part of, Hilary's height (75). Rather,Meagan seemed to be measuringthe gap between the two cubes, and that gap did not appearto signify an entity in and of itself. In other words, she measuredthe gap between the two cubes instead of reasoning about the difference between two quantities. This approachindicatesthatMeagandid not seem to be reasoningin part-wholeterms in this instance,that is, Chris's height was not nested in Hilary's height, with the remainingspace viewed as an quantityto be measured. Meagan'sreasoningwith symbols also contributedto the emergenceof the fifth mathematicalpractice.In the subsequentwhole-classdiscussion,the teacherasked Meaganto show how tall Lloyd (67) andAlex (72) were using a measurementstrip thathad been tapedverticallyat the frontof the classroom.Meaganpointedto the line beside the numeral67 and explainedthatLloyd would be thattall, indicating the lengthfromthe floor to whereshe has marked67. The teacherdrewa shorthorizontalline at the line indicatingLloyd'sheight.MeaganindicatedthatAlex's height began from the floor to the line beside the numeral72, and againthe teacherdrew a horizontal line to mark it. Note that the teacher's symbols served the same purposeas the cubes Meagan had used to markChris and Hilary's heights in the previousexample.Then,to find the differencebetweenLloyd's andAlex's heights, Meagan placed a smurf bar between the two horizontal lines the teacher had drawn,brokeoff six cubes, andplacedthe six cubes so thatthey fit exactly between the two lines. The space between the two marksdid, in fact, signify the difference between the two heights. This activity, however, is consistentwith her activity in the priorepisode,in thatshe did notreasonwith the two heightsin part-wholeterms. Rather, Meagan empirically measured to find the difference between the two marks. Several studentsobjected to the result and arguedthat the difference between the two heightswas only 5 cans long. Forexample,Alex reasonedthat3 morecubes from67 would be 70, and2 morecubes made5 cubes altogether.Hence, Alex took the gap between 67 and 72 as a measurabledistance,as well, but did not actually measureto find the result. Otherstudentssimply counted5 single cubes by ones. Phil argued,however, thatbecause Meaganhad measuredthe space and gotten 6, Alex musthave countedincorrectly.The teacherthenredrewthelines so they were straighterand more accurate,and when Meagan remeasuredin the same manner as before, she found that exactly 5 cans fit. The mathematicallysignificant issue

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thatwas the focus of thisconversationcenteredon how to quantifythe gapbetween the two heights-actually measuringversusreasoning.Meagan,for herpart,drew on herpriorparticipationin mathematicalpractice4 by actuallymeasuringthe space with cubes to solve the task at hand. Alex, for his part,could take that gap as a measurablespace andreasonwith it. Both Alex andMeagancould specify the two spatialextents very easily on the measurementstripby simplypointingto the lines correspondingto 67 and72. The spatialextensionthatsignifiedthe physicallength of an object seemed to be a partitioneddistancefor them;the distancesignified by the measurementstripwas alreadypartitioned,and when Meaganmeasuredwith the strip,she was simply structuringthatdistanceinto compositesof tens andones. When attemptingto reason with the spatial extensions that she had specified on the measurementstrip,however,those spatialextensionsdidnot appearto be nested in one another,thatis, the two spatialextents were not relatedin part-wholeterms and, therefore,she measuredthe space with cubes. In the conversationdescribed in the foregoing,Meagancontributedto the constitutionof a taken-as-sharedinterpretationthatthe measureof an item signified an objectivepropertyof thatitem65 signified Lloyd's height and was now beyondjustification. Nancy, in contrast, participatedin this practice by reasoning in part-whole terms.Forexample,on the 30th day of the instructionalsequence,the teachergave each pair of studentsa sheet of paperwith picturesof various pieces of furniture and theirrespectivemeasures,for example, box, 42 cans; television, 24 cans. The students were told that the smurfs wanted to stack the furnitureitems on top of one anotherso they could minimize the amountof storage space in a room that was 60 cans high. The following episode describeshow Nancy typicallyreasoned aboutthis problemsituation.In one instance,Nancy decidedto stacka box 42 cans high first. She identified the line beside the numeral40 and countedup two more spaces, "forty-one,forty-two."She explained, "That'sthe forty-two cans. Here, all the way to here [motionsfrom 0 to 42)]. It's the box."Clearly,the spatialextension that she had specified signified the height of the box. Then she counted by ones startingwith the 43rd space to see how muchroomremainedto stackanother item on top of the box, "one,two, eighteen."Afterconsultingthe activity three,..... she "We can the sheet, explained, put lamp [17 cans high]. Here's the one more that's left to the 60th space space]." When a researcheraskedher where [pointing the lamp was, Nancy replied, "fromright here [points to the line signifying the top of the 42nd can] to here [pointsto the line signifying the top of the 59th can]." FromNancy's activityin this example,she appearedto be ableto specify the spatial extension of two items on the measurementstrip, and, in turn,reason with these extensions. The space from 42 to 59 signified, for her, the spatialextension of the lamp. Furthermore,the two spatial extensions together-the distance from 0 to 59 as well as each of the two spatial extensions themselves-were nested in the height of the storageroom,60. She reasonedthatthe box andlamp wouldfit inside the room with only one can of distanceleft over. In fact, when Meagan suggested stacking the television (24 cans high) on top of the box, Nancy arguedthat the room would not be high enough. She seemed to be able to imagine the spatial

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extension signifying the height of the box (42) being nested in the spatialextension signifying the height of the room (60) with 18 cans of space remaining.She thereforeautomaticallyeliminatedfurniturethatmeasuredover 18 cans high. This realizationindicatedthat she was envisioning nesting two or more spatialextensions within the spatialextension signified by the height of the room and, in turn, could identifythe amountof space left over afterstackingthe furniture.The important point is that Nancy viewed the unknown difference as a quantity whose measureshe then found. Aform-functionshift. Stackingproblemssuch as those in the foregoing seemed to become relativelyroutine.Possibly, the idea thatthe measureof an item signified an objectivepropertyof thatitem hadbecome taken-as-shared.In otherwords, a reversalbetween the spatialextent and its measurehad takenplace. Before, the spatialextentof an item was physicallypresentandthe goal was to find its measure by layingthe measurementstripbeside it. Now, the measurewas given andthe goal was to use the measurementstrip to specify the spatial extent signified by its measure.This change constituteswhat Saxe (1991) termsaform-functionshift. In this way, the measureof an item signified an objective propertyof an item because the measurewas not being found but was being used, that is, taken as a given, to specify an item's spatialextent.The spatialextentnow signifiedthe heightof a stool or the height of a chair,and these heights took on a life of theirown, so to speakthey could be used to compareand combine with other heights. As can be seen, the taken-as-sharedway of reasoning with the measurementstrip went beyond simply specifying alreadypartitionedspatialextentson the strip.The practicethat was constitutedinvolved working from someone else's measureinstead of actually finding the measureitself. ANALYZING STUDENTS' LEARNINGIN SOCIALCONTEXT The first issue discussed in the literaturereview was that of locating students' development of measuring conceptions in the social context of the classroom. Previousresearch(e.g., Clements,Battista,& Sarama,1998; Inhelder,Sinclair,& Bovet, 1974; Piaget et al., 1960; Smedslund, 1963) typically analyzed students' solutionsto interviewtasks fromprimarilyan individualisticviewpoint.Thus, the maingoal of thischapterwas to analyzethe firstgraders'developmentof measuring conceptionsas it occurredin social context. To this end, we viewed learningas an act of participationin the local mathematicalpracticesof the classroom community. We detailed the emergence of five classroom mathematicalpractices that served to document the immediate social context of the individual students' learning.We presentedtwo case studiesin conjunctionwith the documentationof the collective learning.The purposeof presentingthe two simultaneouslywas to show that (1) students' learningoccurredas they participatedin these emerging practices,and (2) the mathematicalpracticesemergedas students,often the target students,contributedto them. A traditionalcognitive analysis would have treated

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the targetstudents'learningindependentlyof the local classroom communityby describing their development in terms of cognitive reorganizationswith little attentionto the social context, that is, to the collective argumentation.Although we describedthe cognitive developmentof each targetstudentby drawingon the cognitive constructsof Piaget and his colleagues and of Steffe et al. (1988)-for example, accumulationof distances and partitioning-we cast each instance of learning as an act of participationin the mathematicalpractice that was either emerging or was established.For instance,Meagan made certainreorganizations as she participatedin the whole-class discussion in which certainpracticeswere established. Furthermore, the mathematical practices were documented by analyzing the way in which structuringdistance evolved over time and became normativein the classroomdiscourse. When we talk aboutthe mathematicalactivity of a community,we refer to the taken-as-sharedactivityof the communityratherthanthe overlapof the meanings of all its individualmembers(Voigt, 1996). An analysisthatfocuses on the overlap in individualmeanings would constitutea cognitive analysis of the development of a plurality of individuals ratherthan the mathematicalactivity of a single community. Thus, when describing the classroom mathematicalpractices, we made claims aboutthe mathematicalpracticesof a communitybut not abouthow any one child was reasoning.Instead,we describedways of reasoningthatbecame normative over time as indicated by the taken-as-sharedmathematicalactivity observedin the public discourse.The documentationof the classroommathematical practicesreflectsthe way in which taken-as-sharedways of reasoning,arguing, symbolizing, and using tools evolved as the measurementsequence was enacted. When we focused on individual students' participationin these mathematical practices, we made claims about each student's personal way of reasoning, but always in the context of the taken-as-sharedpracticesof the community. One final issue thatshouldbe addressedhererelatesto the usefulnessof the classroom mathematicalpractice as a construct for documenting the learning of a community.As previouslymentioned,CobbandYackel (1996) firstdevelopedthe construct of the classroom mathematicalpractice as a way to account for the mathematicallearningof a classroomcommunityover long periodsof time. Thus thatconstructis useful for describingthe emerging social situationof the mathematical learning that is jointly constitutedby the teacher and the students. The analysis we have presentedshould clearly show that the classroommathematical practicesare not to be viewed as emergingindependentlyof the previouslyestablishedpractices.In otherwords,eachpracticegrewoutof practicespreviouslyestablished by the classroom community. Also, the mathematicalpractices did not suddenlyappearfully formed.The evolution of the taken-as-sharedmeaningsand involvedan analysisanddiscussionof eachof the mathematicalpracinterpretations tices with the intent of documentingthe process of their emergence. Thus the analysis of the classroom mathematicalpracticesserved not only as documentation of the evolving social situationin which the studentswere participatingbut also, retrospectively,as a summaryof the mathematicalcontentthatemergedover

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the course of the measurementsequence. Furthermore,the documentationof the collective mathematical practices, the evolving tool use, and the quality of whole-class discourse provide the necessary backdropfor teachers who wish to incorporateaspects of the instructionalsequence in their own socially situated classrooms. In contrastwith a purely cognitive analysis, some social theories might have involved analyzing the learningof the classroom independentlyof the variety of students' thought processes. In our view, the collective mathematicalpractices emergedonly as studentscontributedtheirpersonalmathematicalinterpretations. In socioculturaltraditions,the teacher guides students to fuller participationin broaderpracticesof the mathematicalcommunity.In this sense, mathematicalpractices arefixed a prioriandthe teacher'sgoal is to guide students'enculturationinto these practices.In our use of the term,mathematicalpractices arenot fixed prior to experimentationbut emerge as studentscontributeto them. Thus, classroom mathematicalpracticesarenormativefor the particularclassroomsocial structure in which they areestablishedand drawon the diversityof students'contributions. The role of the teacherin this type of classroomis to guide students'reasoningso thattheirpracticesbecome more consistentwith those of wider society. CONCLUSION In this chapter,we have used a sample analysis from a first-gradeclassroom to discuss the theoretical nature of coordinating social and individual aspects of learning.We presentedthe evolution of the collective mathematicalpracticesand complementarycase studies of two students'individualdevelopment.We argued thatalthoughfive practicesemergedin the collectivediscourse,individualstudents' developmentconstitutedacts of reasoningthatcontributedto the evolutionof these practices.Toulmin's (1969) model of argumentationwas used as a methodological tool to determine,from the observers' standpoint,when practicesappearedto be established. Finally, we reflected on one of the overarchingthemes of the monograph-analyzing students' learningin social context-by contrastingthis type of analysiswith both traditionalcognitive and socioculturalanalyses.During thatdiscussion,we arguedthatalthoughtraditionalanalysesareuseful,neithertraditional nor socioculturalanalyses alone gives a full explanationof the process of learning. Although the main focus of discussion has been on the theoreticalimplications of learning in social context, we have argued the practical merits of such an analysiselsewhere (Cobb,chapter1 of this monograph;Gravemeijeret al., chapter 4 of this monograph).The practicesandcase studiesprovidea backgroundagainst which to make revisions to the measurementsequence and to incorporatethe instructionalsequence within one's own classroom. We discuss these characteristics of mathematicalpracticesfurtherin the next chapter.

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REFERENCES

Bauersfeld,H. (1988). Interaction,construction,and knowledge: Alternativeperspectivesfor mathematics education.In T. Cooney & D. Grouws (Eds.), Effectivemathematicsteaching (pp. 27-46). Reston, VA: NationalCouncil of Teachersof Mathematics,and Hillsdale, NJ: Erlbaum. Clements, D., Battista,M., & Sarama,J. (1998). Development of geometric and measurementideas. In R. Lehrer& D. Chazan(Eds.), Designing learning environmentsfor developing understanding of geometryand space (pp. 201-226). Hillsdale, NJ: Erlbaum. Cobb, P. (1994). Whereis the mind?Constructivistand socioculturalistperspectiveson mathematical development.EducationalResearcher,23(7), 13-20. Cobb, P., & Yackel,E. (1996). Constructivist,emergent,and socioculturalperspectivesin the context of developmentalresearch.EducationalPsychologist, 31, 175-190. Inhelder,B., Sinclair,H., & Bovet, M. (1974). Learningand the developmentof cognition.Cambridge, MA: HarvardUniversityPress. Piaget,J., Inhelder,B., & Szeminska,A. (1960). Thechild's conceptionof geometry.New York:Basic Books. Saxe, G. (1991). Cultureand cognitivedevelopment:Studiesin mathematicalunderstanding.Hillsdale, NJ: Erlbaum. Smedslund,J. (1963). Developmentof concretetransitivityof length in children.ChildDevelopment, 34, 389-405. Steffe, L. P., Cobb,P., & von Glasersfeld,E. (1988). Constructionof arithmeticalmeaningsand strategies. New York: Springer-Verlag. Toulmin,S. (1969.) The uses of argument.Cambridge,England:CambridgeUniversityPress. Voigt, J. (1996). Negotiationof mathematicalmeaningin classroomprocesses.In P. Nesher,L. P. Steffe, P. Cobb, G. Goldin, & B. Greer(Eds.), Theoriesof mathematicallearning (pp. 21-50). Hillsdale, NJ: Erlbaum.

Chapter 6

Continuing the Design Research Cycle: A Revised Measurement and Arithmetic Sequence Koeno Gravemeijer FreudenthalInstitute JanetBowers San Diego State University Michelle Stephan Purdue UniversityCalumet

INITIATINGA NEW DESIGN RESEARCHCYCLE In chapter5, we analyzedthe developmentof the first graders'conceptions of measurementand,in so doing, completedthe secondphaseof the Design Research Cycle. In this chapter,we initiatea new cycle by reflectingon the revisedsequence. This discussion focuses on the threethemes thatwere discussed duringthe literaturereview.The firstthemeinvolvedanalyzingstudents'developmentof measuring conceptions as it occurs in social context. The second theme concernedthe need to delineatethe role of tools in supportingstudents'understanding.We addressthis issue by documentingthe evolutionof a chainof signification(Walkerdine,1988), which is based on our analyses of the taken-as-sharedmeanings that emerged as students used a variety of tools in the measurementsequence. The third theme centered on proactively supportingstudents' development through integrating researchand instructionaldesign. We addressthis issue by reflectingon our prior work to develop suggested revisions for ensuing phases of Design Research. Theme 1: Analyzing Students'Learningin Social Context In the previouschapter,we arguedthatidentifyingthe collective mathematical practicesand coordinatingthem with individualcase studies give a fuller account of the learning of the first-gradecommunity than traditionalanalyses that cast learning as primarily an individualistic endeavor. The main emphasis of this chapteris to demonstratethatthe mathematicalpracticeas a theoreticalconstruct has usefulness as it relates to instructionaldesign, as well. As Gravemeijeret al.

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(chapter4 of this monograph)noted, mathematicsresearchhas seen a shift in instructionalresearchinterestsfrom merely statingthatone instructionalapproach is betterthan anotheron the basis of some statisticalmeasures.Ourconcernwith these types of comparativestudiesis thatthey providelittle explanationas to how the instructionwas realizedin social context, and, therefore,teacherswould have difficulty knowing why the approachdid not work in their particularsituation. Furthermore,because classrooms are socially diverse entities, instructional approachesthatoffer recipes for implementationcannotdo so for every possible encounter.In contrast,we suggestthatthe productsof instructionaldesign research shouldpresentrichdescriptionsof the instructionaltasks,classroomactivitystructure,tools anddiscoursethatcan lead to the developmentof desirablemathematical practices.As Cobb and his colleagues have noted elsewhere (Cobb, Stephan, McClain, & Gravemeijer,2001), the pragmaticpower of the classroom mathematical practicelies in the fact that being able to anticipatethe learningroute of each of 25 individualstudentsis not feasible for a teacher.One can, however,think of guiding the discourseand tool use of a class in a way that allows for, and capitalizes on, the diversityof students'thinking.Therefore,the hypotheticallearning trajectoryin chapter4 andthe resultantinstructionaltheoryoutlinedin this chapter can be thoughtof in collective terms. The learningroute is that of a class, not of individual students.Therefore, we present the resulting instructionaltheory on measuringand arithmeticin this chapter,noting thatthese theorieswere designed using the mathematicalpracticesas the backdrop.In otherwords,the designeruses the analysisin chapter5 detailingthe mathematicalideas anddiscussiontopics that became taken-as-sharedmeasurementpractices to formulate the rationale and intentof the resultinginstructionaltheoryfoundin this chapter.To summarize,the power of the classroommathematicalpracticefrom the standpointof the designer is thatthe practicesserve as the basis for outliningthe potentialmathematicalideas andconversationtopics duringthe enactmentof the sequence.The usefulnessfrom a teacher'sstandpointis thatthese potentialpracticescan be used to lead a community, ratherthanindividualstudents,to sophisticatedmathematicalreasoning. Although we describe the sequence in terms of the collective mathematical reasoningthat can emerge as teachersand studentsengage in these instructional activities, we do not neglect individualstudents'reasoning.On the contrary,the case studies presented a rich account of how two students participatedin, and contributedto, the emergingpractices.Thus,any teacherimplementingthe instructional theory discussed in this monographcan use the theoryto anticipatepotential collective shifts in mathematicaldiscoursethataremadepossible by students' diverseways of reasoningandof contributingto the emergingtaken-as-shared practices. The case studies provideexamples of students'differentways of reasoning thata teachermight expect from her or his own studentsas the sequenceis implementedin the classroom.The teachercan use theseexamplesto anticipatethecontributions from studentsthat she or he wants to drawon in classroomdiscussions.

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Theme 2: Documentingthe Role of Tools with a Chainof Signification A second, but related,themefrom the literaturereview was thatof documenting the interplaybetweenthe developmentof meaningandthatof symbolizingandtool use. The rationalefor this aspect of the analysis reflectedthe growing realization in the mathematicseducationresearchcommunitythatlearningdoes not occurapart from reasoning with symbols and tools. Further,much of the prior literatureon students' development of measuring conceptions attributeda limited role to reasoningwith measurementtools. Given thatmeasuringinherentlyinvolves using bothconventionalandnonconventionalmeasurementdevices, this studyoffers an opportunityto investigatethe role of tool andsymboluse in students'mathematical development.Along with Meira (1998), we use the word tool to mean any physical device, such as a ruler, computermicroworld,or a calculator,and the word symbolto refer to semiotic systems, such as graphs,tables, icons, or drawings. A review of the emergenceof the classroommathematicalpracticesrevealsthat the use of tools and symbols was integral to the mathematicalactivity of the community.As figure 6.1 illustrates,the emergenceof the first four mathematical practicesoccurredas studentsacted with differenttools and symbols (see Figure 6.1). As indicatedby the transitionfrom the fourthto the fifth mathematicalpractice, however,a new mathematicalpracticecan emerge as studentsreasonwith the same tool. Recall from the analysis thatthe transitionfrom the fourthto the fifth practiceinvolved shiftingfrommeasuringwith the measurementstripto reasoning with specified quantities-spatial extensions-on the measurementstrip.Thus, as the taken-as-sharedfunctionof the measurementstripchanged,a new mathematical practicearose.The analysismakesclearthatthe communallearning,the emergence of mathematicalpractices,did not occur apartfrom reasoningwith tools. In the previousanalysisof classroommathematicalpractices,however,the focus was on coordinatingindividualandsocial perspectivesto give a full accountof learning, with explicit focus on the role of tools in this process fading into the background. In this section,we bringthe evolutionof tool use to thefore by elaboratingthetaken-

PRACTICES CLASSROOMMATHEMATICAL 1. Measuringby coveringdistance 2. Partitioning distance witha collectionof units 3. Measuringby accumulatingdistance 4. Measuringwitha stripof 100 5. Reasoningwitha stripof 100

Figure6.1. Theevolutionof theclassroommathematical practices.

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as-sharedmeaningthatdeveloped as studentsreasonedwith a varietyof tools. To do so, we detailboth the meaningand the taken-as-sharedgoals andpurposesthat evolved. Such an analysisis crucialfor documentingthe taken-as-sharedimagery and activity with tools that are part of the measurement instructional theory presentedat the end of this chapter. In this teaching experiment,the general notion of a ruler-that is, the general form and structureof a ruleralthoughwithoutthe inch markingsof a conventional ruler-served as the overarchingmodel (Gravemeijer, 1998). As described in chapter4, a model can takemanyformsthroughoutan instructionalsequence.One way to describe the variousmanifestationsof a model is to describethe evolving taken-as-sharedmeaningsassociatedwith the use of the tools and symbols. Cobb, Gravemeijer,Yackel, McClain& Whitenack(1997) andGravemeijer(1998) both suggest documentingthis evolutionaryprocess by describingthe emergence of a chain of signification.A chainof signification,as describedby Walkerdine(1988), captures the process by which taken-as-sharedactivity with symbols and tools comes to signify the use of another.When one symbol or tool comes to signify a previoussymbolor tool, Walkerdinestatesthatone servesas a signifierforthe other. This relationshipforms a signified-signifierpair and constitutesa link in a more global chain of signification.Forexample, in the measurementsequence, students initially paced the length of items and subsequentlymade recordsof this activity, for example, when the teacherplaced maskingtape to markeach pace. Thus, the first signified-signifier combinationthat became constitutedas the first mathematicalpracticeemergedinvolved pacing and recordsof pacing (see Figure6.2).

record of pacing

signifier)

pacing

signified)

sign1

Figure6.2. Thefirstlinkin a chainof signification.

More specifically,distance,as structuredby recordsof paces, servedas the signifier for pacing.As we learnedfromthe analysisof the classroommathematicalpractices, distancewas initially somethingto be filled by paces andthenby recordsof paces. Thus, as the first mathematicalpracticeemerged,recordsof paces came to signify pacing activity. Figure6.2 illustratesthatthe firstlink in the chainof significationis thatbetween recordsof pacing andpacing, designatedas {Irecordsof pacing/pacing}. As Cobb

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et al. (1997) note, one advantageof identifying a chain of signification is that a signifier-signifiedcombinationcan, in turn,be signified by other symbols as the interestsof the communitychange. Specifically, in the instance of the measurement sequence,the constitutionof the firstlink in the foregoing(Figure6.2) developed as the interestof the class involveddecidinghow one couldmeasurewith one's feet in accurateand spatiallylegitimateways. As studentsmeasuredwith the footstrip,the taken-as-sharedinterestsof the communitychangedand a secondmathematical practiceemerged that involved partitioningdistance with a collection of units.No longerwere studentsconcernedonly with measuringandcountingsingle units; the taken-as-sharedpurpose of the community changed to partitioning distancewith collections of units-in the footstriptool-that could not be physically separated.As this practicewas constituted,a second link in the chainof signification was establishedandcan be illustratedas shown in Figure6.3. In this case, the footstrip served as a signifier-signifier 2-for the first signified-signifier pair. Walkerdine(1988) uses the word sign to refer to a signifier-signifiedpair. Thus, {recordof pacing/pacing)Iconstitutesthe first sign, which is signifiedby the taken-as-sharedactivity and purposesfor measuringwith the footstrip.

footstrip

signifier2

record of pacing -

signifier)

pacing

signified1

sign2 sign) / signified2

Figure6.3. A secondlinkin thechainof signification.

As the first and second mathematicalpracticesemerged,the chain of signification exemplified in Figure 6.3 was constituted.These links were constitutedas studentsdiscussed theirinterpretationsof the resultof measuring.As this link was constituted,the relationshipbetween the signifierand the signified was not one in which one symbol merely replacedthe other. Rather,the mathematicalpractices became establishedas studentsreasonedwith subsequenttools and as the community's interestschanged;this outcome, in turn,changed the natureof the activity with previous signifieds; thus, the natureof distanceevolved for students.In this way, signified entities are said to slide undersucceeding signifiers ratherthan be masked or replacedby them (Cobb et al., 1997; Walkerdine,1988). As students

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measuredwith the footstrip, mentally structuringdistance into composites of 5 becametaken-as-shared. Measuringbecamedivorcedfromactivity,thatis, the idea was taken-as-sharedthat a numeral,such as 25 paces, signified the measureof a spatialextensionapartfromphysicallymeasuringto createthe measure.Therefore, as studentsmeasuredwith the footstrip, not only did the idea become taken-assharedthatthe footstripsignified the previoussign, but also the natureof distance again evolved. This example illustratesthe interplaybetween the taken-as-shared mathematicalactivity of the classroom communityand their evolving taken-assharedsymbolic meanings.In otherwords, as the mathematicalpracticesevolved and the interestsof the communitychanged, new links in the chain of signification emerged. A second chain of significationbegan to emerge when the teacherintroduced the smurf scenario. We say that a new chain began because the background scenario changed from measuringwith feet and collections of feet to measuring with single Unifix cubesfor the purposesof helpingsmurfsperformtheirmeasuring tasks. Also, the initial tasks in the new scenario recapitulatedthe first chain; the new scenariobegan with iteratingsingle units and moved to iteratingcollections of units. At the beginning of this scenario, studentsused Unifix cubes as substitutes for food cans. The understandingthat a measure,say, 27 cans, signified the length of an item became taken-as-sharedfairly quickly. Thus, the first link in the second chain consisted of food cans signified by Unifix cubes (see Figure6.4).

Unifixcubes------------signifier food cans-------------signified

Figure6.4. Thebeginningof a secondchainof signification.

The fact thatthis link was constitutedfairly quickly can be explainedby taking note of the students'participationin priormathematicalpracticesduringwhich the first chain of significationwas constituted.Recall thatwe had attemptedto introduce the smurfsscenarioearlier,priorto the students'constructionof the footstrip. When we did so, it became apparentthat most students had difficulty taking a measure, such as 41 cans, as a given for the height of a wall in the smurfvillage. We conjecturedthatthe inabilityof the class to take the resultof measuring--41 cans-as a given for the length of an item was taken-as-shared;rather,measuring was groundedin the activity of iterating,thatis, iteratingto createthe measure41

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cans. As the studentsengaged in activities where they iteratedthe footstrip and participatedin discussions where interpretingthe result of measuring was an explicit topic of conversation, the second mathematicalpractice emerged. As discussedpreviously,distancewas no longersomethingto be coveredwith an exact numberof footstrips.Rather,as the second mathematicalpracticewas established, the fact thatthe resultsof measuringcould be divorcedfromthe activityof counting becametaken-as-shared.Twenty-fivesignified the numberof times thatone would pace to measurethe lengthwithoutactuallycarryingout the action.In otherwords, 25 specifiedan amountof distancethatcould be measured,or a potentiallymeasurable distance.Therefore,as the link between the footstripand its signifiedbecame established,the natureof distanceevolved such thattakinga measure,such as 41 cans, as a given for the lengthof an object was now taken-as-shared.The firsttime we introducedthe smurfscenario,only the first link of the chain (see Figure6.2) thatdistancesignihadbeen established.At thatpoint,the idea was taken-as-shared fied something to be covered in activity, something to be createdby physically measuring.Forthatreason,the introductionof the smurfscenariowas inappropriate at thatpoint, and the studentsengaged in activities with the footstrip.In thatway, the constitutionof the first chain of significationwas integralto the emergenceof the second chain (see Figure6.5). As the instructionalactivitieschangedfrom measuringwith individualcubes to iteratinga smurfbar, measuringby accumulatingdistances emerged as the third mathematicalpractice.Further,issues thatemergedas topics of conversationnow focused on the students' interpretationof distance ratherthan their method of measuringper se. Forexample,the studentswere askedto describehow they interpreted a measure,such as 20. Many studentsexplained that 20 was the distance accumulatedby iteratinga bartwice and saying, "tenandten is twenty."Thus, the interests and activity of the community had changed from finding methods of

footstriprecord of pacing- -pacing -

Unifix cubes

signifier2 signifier)

sign2

signified)

signifier) signs

food cans -

CHAIN 1

signs / signified2

CHAIN 2

signified)

chainsof signification. Figure6.5. Theemergenceof twointerrelated

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iteratingandcountingto solve the measuringdilemmasof the smurfsto interpreting and organizingdistancein measuresof tens and ones. As this thirdmathematical practicewas establishedand the interestof the class changedto describingparticularways of organizingdistancemathematically,a furtherlinkin the chainof signification was formed. Now the food can-Unifix cube combination,designatedas {food cans/Unifix cubes }, became signified by the smurfbar (see Figure6.6).

smurf bar Unifix cube food cans

signifier2 signifier)

sign2 sign1 / signified2

signified,

Figure6.6. A secondlinkin thesecondchainof signification.

The firstchainof signification,involvingthe footstripandpacing,was also paramount to both the food can-Unifix cube link and the constitution of the link between the smurfbar and the food cans or Unifix cubes. As studentsmeasured with the footstrip,distancewas structuredin collections of 5 and single paces. As a result of discussing their engagement in these activities, measuring became divorced from pacing activity and this way of reasoning about and structuring distance became taken-as-sharedin the classroom community. Because of this historical constructionof an understandingof distance, measuring with single Unifix cubes became taken-as-sharedalmost immediately. Further,iterating a collection of 10 cans was, in some sense, similarto measuringwith a collection of 5 paces. Although the numberin the collection was different-10 insteadof 5the underlyingimage of distance as divorced from physically measuringwas the same in the context of the smurfs.Thus, the first chain of significationwas paramount to the emergenceof both the first and second links of the second chain of signification (see Figure6.7). The link between the smurfbar and the food cans or Unifix cubes was constituted as the third mathematicalpractice, measuringby accumulatingdistances, emerged. The emergence of the thirdmathematicalpracticesignals a shift in the interests of the classroom community. The interest had changed from simply discussing how to count paces to structuringdistanceinto collections of tens and ones, for example, 20 is two tens. At the beginningof the instructionalsequence, the interestof the communitywas on how to physicallymeasurean item. Now the

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Chain1

{foodcans.-----------.Unifixcubes)}----------.

smurfbar

(Chain2)

Figure6.7. An emergingchainof signification.

interestsandconversationsfocused on organizingand structuringdistancemathematically.Thus, the meaningof the previoussign combinationsevolved once the second chain was established.Further,the natureof distanceevolved, in thatnow the resultof measuringsignified an accumulationof distance. As the fourthpracticeof measuringwith a stripof 100 emerged,two new signifying links were formed. First, in the fourth practice, the recognition became taken-as-shared thatmeasuringwith the 10-stripservedas a symbolicrecordof iteratinga smurfbar.Recall thatthis meaningemergedas studentsdiscussedthe utility and meaningof markingeach successive food can on their strips.The interestsof the studentsagainhadchangedfrommeasuringby iteratinga barto findingan efficient anduseful way of markingthe 10-stripso thatit could be used as a measuring device without the presence of any Unifix cubes. Then the purpose of their measuringand discussions was to find ways of both structuringdistancewith the 10-stripandcoordinatingmeasuringwith tens andones by accumulatingdistances. Although their practicalgoals were to find the measure of objects in the smurf village, their mathematicalgoals had changed to interpretingand refining their methodsof countingtheiriterationsof the 10-strip.A linkwas thusformedin which measuringwith a 10-stripcame to signify the meanings associatedwith iterating a smurfbar (Figure6.8).

1O-strip---------------------signifier3

signifier2

smurfbar.......----------------------.

sign3 sign2I signifier3

Unifixcubes.---------.signifier sign1 / signified2

foodcans--------signifiedJ

Figure6.8. A newlinkin thesecondchainof signification.

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Measuringwith the 10-stripsoon gave way to the more efficient 100-stripand came to be signified by the resultsof measuringwith the 100-strip.This new link again involved a shift in the purposesof measuringfor the community.Insteadof finding methodsof iteratinga tool to measure,now the agreementbecame takenas-sharedthat the purposeof their measuringwas to develop a way to interpret measuringwith a stripthat they did not need, for the most part,to iterate.Recall fromchapter4 thatstudentsoften looked back (Pirie& Kieren,1994) to measuring with smurf bars to interpretmeasuring with the 100-strip.Thus, the history of evolving links is strong,in that new links may be formedby drawingon mathematical practicesformed in several previous links, not just the one immediately preceding it. As the fourth practice was established, the understandingbecame taken-as-sharedthatdistancesignified a partitionableunitthatcould be structured in collections of tens andones. Further,the idea was taken-as-sharedthatdistance had the characteristicor propertyof a measurethatwas independentof physically iteration.Layingdownthe measurementstripsimply specifiedthe item's measure. Thus,the globalchainof significationthatemergedover the courseof the measurement sequence can be picturedin its entiretyas in Figure6.9. The transitionfromthe fourthto the fifth mathematicalpracticeinvolved a shift from measuringwith the measurementstrip to reasoning with the measurement strip.This transitionin practicesandinterestssignaleda shift in the taken-as-shared understandingof distance. Now studentsspecified spatialextensions of items on the stripandreasonedaboutthe differencesbetweenlengthsandheights.Following Saxe (1991), this transitioncan be seen to involve a form-functionshift,in thatnow the function of the measurementstriphas changed from measuringto reasoning. This form-functionshift is also consistent with the "modelof-model for" transition describedby Gravemeijer(1994).

measurementstrip 10 strip

footstrip record of pacing

pacing

smurf bar Unifix cubes

foodcans

Both dotted and full lines indicate the constitution of a signifying relationship. The dotted lines indicate an instance of one signified sliding under a new signifier. The two arrows indicate the places where the chain associated with the king scenario influenced the second chain (recall Figure 6.7).

Figure6.9. Theglobalchainof signification.

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Again, an importantpoint to note is that the shift from measuringto reasoning with the measurementstripcoincidedwith a changein the interestof the classroom community.The taken-as-sharedgoals shifted from measuringan item to finding ways to interpretand use the measurementstrip to make comparisons among spatialextensions. This outcome is indicativeof a generalpattern:As the interests of the communitychange,thatis, becomemoremathematical,new linksin the chain of significationevolve. The realizationis imperativethat Figure6.9 incorporates not only the change in physical tools but also the evolution of classroom mathematicalpracticesand taken-as-sharedinterestsandpurposesthat accompanynew symbolizations. Although the focus of the analysis in chapter5 ended with students'construction of, and reasoningwith, the measurementstrip, two more tools were created and used by the students:the measurementstick and the empty numberline. We have not includedthese two tools in the global chain of significationbecause we have not conductedformal analyses of the mathematicalpractices and taken-assharedgoals associatedwith using these two tools. We have, however, conducted an informal analysis to anticipate how or whether we can include these tools within a hypotheticallearningtrajectory.We discuss the importanceof these two tools in two subsequentsections of this chapter:"Generalization"and "Revisions to the Sequence." A semiotic analysisof this type bringsto light the role thattools play in supporting students' learningalong with the mathematicalpracticesthat are constituted duringthe evolving semiosis. Furthermore,the analysis emphasizesthe evolving purposes and goals of the community that accompany the transitionsinvolving new tools. Such an analysis bringsto the fore the taken-as-sharedsymbolizations andmeaningsthataccompanythe mathematicalpracticesandthe taken-as-shared intereststhatevolve as the practicesareconstituted,bothimportantaspectsof classroom mathematicalpractices(Cobb, 1999). An additionalbenefit of the chain-ofsignification analysis is that the analysis can shed light on why certaintools can be included or omitted in an instructional sequence. For instance, when we designed the initial measurementsequence, we did not intend to have students reason with a footstrip. The chain of signification indicated, however, that the taken-as-sharedlearningthatemergedas studentsactedwith thefootstripwas paramount in the sequence of the use of two tools, the 10-stripand the measurement strip,thatis, the 100-strip.Thus, the measurementsequence has now been modified to includethe footstripandotherrevisionsas describedin a subsequentsection of this chapter. Theme 3: Design Research A thirdissue from the literaturereview concernsthe relationbetween research and instructionaldesign. Chapter2 indicatedthat a numberof the priormeasurement studiesused trainingmethodsto supportstudents'developmentof measuring conceptions. Typically, researchers who engaged in training studies drew on

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Piaget's cognitive theory to derive trainingtasks. Instructionon these tasks was thenimplemented,andthe studentswere assessedto determinewhetherthe training instructionhad been effective (see Figure6.10).

theory-(Piaget'sdevelopmental)

trainingprocedure--

measuredoutcomes

Figure 6.10. Trainingprocedures.

The linearityin Figure6.10 is basedon a one-wayprocessin whichexperimenters list a set of hypotheses prior to the study and then test their conjectures with pretestsand posttests.The accountof students'learninggiven in trainingstudies provides evidence only as to whether the hypotheses were proved. In contrast, althoughdeveloperswho areengaged in the Design Researchparadigmalso make conjecturesthat are tested and revised, their intent in researchis to constructthe means of supportingdevelopmentratherthan to determinewhethera conjecture was provedtrueor false. The iterativecycles of Design Researchcan be viewed at two levels. At a global level, the first phase of a cycle begins when designersanticipatea global learning trajectory-ratherthanspecificinstructionalactivities---onthe basisof resultsfrom previous research.Then the designers carryout an initial thoughtexperimentin which they envision what studentsmight learn as they participatein instructional activities,the qualityof discourseandmathematicalpracticesthatsupportslearning, andthe evolving cascadeof inscriptions(Latour,1990) or potentialchainof signification (Walkerdine, 1988). Gravemeijer(1999) notes that in conducting this thought experiment,the developers formulateconjecturesboth about students' mathematicaldevelopment and about the means of supportingit. In the second phase, the researchteam carriesout a classroomteachingexperimentin which the initial conjecturesconcerningthe realizationof the proposedsequence are tested andrevised in action. Once the classroomteachingexperimentconcludes, a retrospective analysis, such as the one describedin chapter5, is conductedto inform the designersof any unanticipatedinterpretationsthatarosein the social settingof the classroom. One interestingaspect of Design Research is that it is inherentlyrecursive in nature.At a microlevel,daily minicycles areenactedduringthe researchphase.For example, as the measurementteaching experimenttook place, researchersused informal analyses of students' mathematicalreasoning to make decisions about daily revisions of the instructionalactivities. Consequently,the resultinginstructional sequence was shaped by our daily analyses of students' mathematical

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activity, which were made againstthe backdropof a conjecturedlearningtrajectory. Further,revisions to the sequenceshapedstudents'subsequentmathematical activity. Therefore,we can now augmentthe initial Design Researchfigure introduced in chapter2 by addingthe minicycles shown in Figure6.11.

informalanalyses

revisions a) Minicyclesconsistingof informalanalyses and revisions to the instructionalsequence.

Instructionaldesign

Classroom-basedresearch informalanalyses

rXXXsXX

b) Revisionof the graphicwithminicyclesconsistingof informalanalyses and revisionsto the instructional sequence.

Figure6.11. Minicyclesin DesignResearch.

In summary, the documentation of the mathematicalpractices that became taken-as-sharedin thecommunityprovidedan accountof theinstructionalsequence that emerged as a consequence of these daily modifications.As we documented the emergenceof the mathematicalpractices,we took care to describeour role in the enactment of the instructionalsequence. Where appropriate,we provided informationconcerning our conjectures about students' understandingand our subsequentdecisions concerningfurtherinstructionalactivities to clarify our role in supportingstudents'learning.The analysis of the mathematicalpracticesgave a situatedaccount of the learningthat was tied to students'participationin practices as they reasonedwith tools. Thus the accountof students'learningpresented in thismonographwas integrallytied to the meansof supportingthatlearning.These analyses have led us to initiatea new cycle of Design Researchby describingour revised sequence.

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REVISIONSTO THE SEQUENCE In this section, we firstdescribethreerevisionsto the sequenceandtheiraccompanyingrationales.We thenpresenta tablethatincorporatesourinstructionaldesign theory by listing the imagery, taken-as-sharedinterests of the community, and potential topics for discourse for each tool in the revised sequence. Our goal in constructingthis table is to develop a heuristicfor othersattemptingto envision a hypotheticallearningtrajectory. Revision 1: The Introductionof the Footstrip The genesis for this revision occurredthe first time we introducedthe smurfs scenario and asked studentsto show the height of a wall with Unifix cubes. Our intention was to use pacing activities only as a setting to introducethe idea of measuring.We conjecturedthatstudentswould move easily from pacingto activities with the Unifix cubes, but, as documentedin the analysis, they did not. We foundthattakingtheresultof measuringwiththe food cansas a given was not takenas-shared,that is, the quantityof 41 cans did not signify the result of measuring but ratherwas createdby physicallypacing.As a consequence,we conjecturedthat if studentsengagedin activitiesin which they iteratedcollectionsof paces symbolized on paper-the footstrip--insteadof measuredwithunitsthatwerepartsof their bodies, takingthe resultof measuringas a given might become taken-as-shared. Revision 2: The Constructionof the MeasurementStrip Duringone activityin the teachingexperiment,two studentsrecordedeach iteration of 10 on a piece of addingmachine tape that signified a piece of wood that had to be cut 30 cans long (see Figure 6.12). The teacherused this incident to suggest thata piece of paperwould be much easier to carryaroundthanthe smurf bar,which was 10 connected Unifix cubes, and asked the studentsto make a strip 50 cans long as a new measurementtool. The studentsmade a stripof 50, but they markedonly the decades, as describedin Nancy and Meagan's case studies. The reasonthese studentsmay not have felt the need to makemarksfor individualcubes may have been thatthey were makinga recordof the activityof iteratingthe entire smurf bar. So when they were measuringwith the stripof 50, the studentscould have intendedto use individualcubes to measurethe ones, just as they had done with the smurfbar.

10

20

Figure6.12.Student'srecordof iteratingtens.

30

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The teacherlateraskedthe studentsto makea strip10 canslong thatcould replace the smurfbar,in contrastwith the student-initiated tool thatarosefromthe students' efforts to stop iteratingthe smurfbar.At this point, some of the studentsdid draw marksfor the individualcubes because, when they were measuringwith the smurf bar,they could measurewith eitherthe whole baror the individualcubes on an asneeded basis. Hence they reasonedthatif they were going to replacethe bar,they wouldneedmarksfor individualcubes.This reasoningled naturallyto the construction of the measurementstripof 100, which was introducedby taping 10 ten strips togetherin the context of a discussion on measuringwith ten strips. Revision 3: The DiminishedRole for the MeasurementStick As partof our hypothesizedlearningtrajectoryoutlined in chapter4, we anticipateda transitionin measuringactivity from reasoningwith a measuringstripto reasoningwith a measuringstick (see Figure 6.13). An analysis of the teaching experiment,however, led us to diminishthe role of the measurementstick in the revised teaching experiment. We learned that the measurementstick was not neededto help supportthe transitionfromreasoningwith the stripto reasoningwith the emptynumberline. One reasonwas thatthe measurementstickwas not needed to introducetasksconcerningincrementing,decrementing,andcomparing.As the studentsengaged in such activities, they reasonedwith the measurementstripin a naturalway. In fact, when the measurementstick was introduced,severalstudents had difficultyreadingthe unit measures.For these reasons,we suggest thatits use be limited or eliminated altogether as teachers and researchersinitiate a new Design ResearchCycle.

Stick Revision3: TheDiminishedRolefortheMeasurement

Conclusions Takentogether,these three revisions serve as a paradigmcase of the feedback mechanismfoundin the Design ResearchCycle. In otherwords,the daily andretrospective analyses of students' learning in situ provided feedback to inform the designerof revisionsto theinstructionalsequence.Afterreflectingon theserevisions, we were able to createtable 6.1, which serves as the outline of our measurementinstructiontheory.This table was createdusing the analyses of classroom mathematical practices and chains of signification as a basis. Specifically, the cascade

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of tool-use and taken-as-shared purposes of the community as they acted with these tools formed the basis for the first three columns, whereas the analysis of mathematical practice shaped all columns, but mainly the last.

Table 6.1 Overviewof the ProposedRole of Tools in the InstructionalSequence

Tool Feet (heelto toe)

Imagery

Activity/Taken-assharedinterests

Potentialmathematical discoursetopics

Measuring

Recordof activity Reasoningabout Focuson coveringdistance of pacing activityof pacing Recordof pacing Measuring witha as divorcedfrom Measuring (buildson masking "bigstep"of five = activityof measuring; tape) measuring by iterating a collectionof paces structuring distancein collectionsof fivesandones (form-function a record shift:using of pacingas a tool formeasuring) Smurfcans Stackof Unifix divorced Measuring by creating Buildsonmeasuring cubessignifies a stackof Unifixcubes fromactivityof iterating resultof iterating Smurfbar Signifiesresultof Measuring of distances; by iterating Accumulation a collectionof 10 iterating Unifixcubes; coordinating measuring by tenswithmeasuring by ones distance structuring intomeasuresof tens andones of distances 10-strip Signifiesmeasur- Measuring by iterating Accumulation ingtensandones the 10-strip,andusing withthesmurfbar thestripas a rulerfor theones Measure- Signifiesmeasuring(1) measuring: Distanceseenas already mentstrip with10-strip; extension stripalongsideitem; partitioned; countingby tensand alreadyhasa measure startsto signify ones resultof measuring Part-whole reasoning; thegapsbeReadingof endpoint quantifying shift tweentwoormorelengths (form-function devel- (2) Reasoning inscription opedformeasuringaboutspatialextensions Shiftin focus:focuson is usedforscaffold (resultsof measuring number relations; developing ing andcommuni- havebecomeentitiesin andusingemergentframeandof themselves) workof numberrelations cating) Empty SignifiesreasoningMeansof scaffolding Numbersas mathematical numberline withmeasurementandmeansof communi-entities(numbers derivetheir strip catingaboutreasoning meaningforma framework aboutnumberrelations of numberrelations) Masking tape Footstrip

Various arithmetical strategies

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GENERALIZATION At the end of a Design ResearchCycle, what can be takenfrom the experience? In otherwords,what,if anything,fromthe measurementclassroomteachingexperiment generalizesbeyond this one enactmentof an instructionalsequence?In our view, at least two aspects of this study generalizebeyond Ms. Smith's first-grade classroom:(1) an instructionaltheoryon linearmeasurement,and (2) theoretical innovationsregardingDesign Researchingandanalyzinglearningin socialcontext. MeasurementInstructionalTheory One of the most, if not the most, importantproductsof a series of design experiments is the generationof an instructionaltheory for a specific mathematical content area that can be used by classroom teachers. After all, one of the main commitmentsof Design Researchersis to test and revise an instructionaltheory with the goal of producing a viable theory that can be adapted by classroom teachers.We suggest that the MeasurementInstructionalTheory outlined in this monographand summarizedin Table 6.1 can be generalized.We are very deliberatewhen we claim thatthe instructionaltheory,versusan instructionalsequence, generalizes.One of the main problemswith claiming that a sequence generalizes is thatresearchersexpect thatif teachersfollow theirinstructionalrecipes,they will obtain the same learningresults in their classrooms. We believe that no instructionalsequenceor curriculummaterialscan be writtenso thatif the teacherfollows directionsprecisely,the teacherwill obtainthe sameresultsattainedin the measurementexperiment.The reasonfor this discrepancyis thatclassroomsare,by nature, idiosyncraticentities with a wide rangeof studentswith diverse backgroundsand abilities.Therefore,we would neverexpect thatthe measurementsequencewould generalizeandproduceexactly the same learningresultsin a varietyof classrooms. Rather,we proposethatwhat can be generalizedis the measurementinstructional theory-the ideasembodiedin table6.1-that expressthepotentialtools, discourse, mathematicalactivity,andimagerythatcanbe integratedto supportstudents'understandingof linearmeasurement.Individualteachers,who arefamiliarwiththediversity of their particularclassrooms, should use the results of our measurement experiment as a guide to develop their own hypothetical learning trajectories specific to the needs of theirstudents.More specifically, we see threeaspectsthat teachersshouldconsiderwhendesigninghypotheticallearningtrajectoriesfor their own classrooms: 1. How the studentsare expected to act and reason with the tools 2. How this activity relatesto the precedingactivities 3. The relevantmathematicalconcepts and the students'conceptualdevelopment in relationto them Ouroverall goal is thatas differentteachersdevelop theirown ways of adapting the MeasurementInstructionalTheoryto theirclassrooms,the results-their own

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personalinstructionaltheorieson measurement-can be sharedwith othersso that they contributeto the overall strengtheningof the instructionaltheorydeveloped in this monograph.In otherwords, the creating,testing, and revising of measurement instructionaltheories by other teachers and researchersconstitutesanother phase of the Design ResearchCycle and contributesto the researchthatbegan in this measurementteachingexperiment. TheoreticalInnovations A second aspectof this measurementexperimentthatwe contendcan be generalized is certain theoretical and practical ideas about designing instructionand analyzinglearningin situ.Ideas,such as the methodologicalapproachto analyzing students' learningin context, involving using Toulmin's model of argumentation (chapter 3) and Walkerdine's chains of signification can be used by other researchers who are interested in conducting similar types of analyses. Furthermore,the processes involved in designing instruction,such as developing hypotheticallearningtrajectories,can be furtherdeveloped by other researchers who wish to conduct design experiments.Already, several of the authorsof this monographhave used the ideas developed in this analysis to design and conduct experiments in differential equations and algebra (Stephan & Rasmussen, in press;Presmeget al., in press).These new studies show thatnot only can the ideas in this monographbe used to conductnew researchbutthe new researchhas served to modify the original theoreticalideas here presented.For example, as a result of our experiences analyzing the mathematicalpracticesfrom the measurement experiment, the methodology for documenting classroom mathematicalpractices has now been expanded(see Stephan& Rasmussen,in press).Therefore,the theoreticalandpracticalinnovationsdescribedin this monographarealreadybeing generalized and used by ourselves and other researcherswho are engaged in similar research. CONCLUSION In this chapter,we have revisitedthe threethemes of the monographin termsof the Design Researchmethodology.First,we made the pointthatclassroom-based analyses, such as those foundin chapter5, thatdocumentlearningin social context provideteacherswith the detailsfor implementingan instructionalsequencein their own classrooms.Teachersmay drawon boththe classroommathematicalpractices and the diversityof students'participationin these practicesto designhypothetical we arguedthat learningtrajectoriesthatfit theirclassroomsituations.Furthermore, being able to conjectureabout25 individuallearningtrajectoriesis not feasible for a teacherand thatthe classroommathematicalpracticeprovidesthe teachera way to supportthe learningof a community.These types of analyses satisfy two of the threecriteriathatCobboutlinedin chapter1, thatan analyticapproachshouldmake possible the documentationof both the collective mathematicallearningand the

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mathematicalreasoningof individualstudentsas they participatein the practices of the community. Second, we used the notionof a chainof significationto analyzethe tool use that coincided with the evolving interestsand mathematicalpracticesof the first-grade classroom. We stressed that the chain provides an account of the history of the evolving meaningassociatedwith the formationof signifying links. The chain and associated practices were pragmaticallyuseful in that they provided a backdrop againstwhich to make revisions to the instructionalsequence and to elaboratethe MeasurementInstructionalTheory(Table6.1). We have thereforesatisfiedCobb's (chapter1 of this monograph)criterionthatan analyticapproachshouldfeed back to informresearchersin the improvementof instructionaldesigns. We presented the revisions and resultinginstructionalsequence on measuringand arithmetical reasoningin table form, which includeda descriptionof each tool along with the imagery,activity,andpotentialmathematicaltopics thatcould possibly ariseas the sequences are enacted. Finally, we arguedthatthe Design ResearchCycle, which consisted of successive roundsof design, classroom-basedresearch,andsocially situatedanalysesand revisions, differs from traditionalmeasurementstudies.The differencelies in the fact thatprioranalysestypicallyfocused on pretestandposttestresultsand seldom tied the processes of learning to the training techniques that were employed. Anotherway to state this point is that the learningthat occurredin the measurement trainingsessions did not account for the socially situatednatureof the experimentation.Thusteachershave few resourcesor guides for makingadjustments to the instructionso that it fits their particularclassroom community.One of the strengthsof Design Research is that the decisions, whether good or bad, of the research team and the rationale for daily revisions were articulatedduring the analysis. This reflective process of using students' learning in social context to informpractitionersin theirdaily instructionaldecisionsandretrospectiverevisions is the hallmarkof this approach. REFERENCES

Cobb, P. (1999). Individualandcollective mathematicallearning:The case of statisticaldataanalysis. MathematicalThinkingand Learning, 1, 5-44. Cobb, P., Gravemeijer,K., Yackel, E., McClain, K., & Whitenack,J. (1997). Symbolizingand mathematizing:The emergence of chains of signification in one first-gradeclassroom. In D. Kirshner& Situatedcognition theory: J. A. Whitson semiotic,and neurologicalperspectives(pp. Social, (Eds.), 151-233). Hillsdale, NJ: Erlbaum. Cobb, P., Stephan,M., McClain, K., & Gravemeijer,K. (2001). Participatingin mathematicalpractices. Journal of the LearningSciences, 10(1, 2), 113-163. Gravemeijer.K. (1994). Developingrealisticmathematicseducation.Utrecht,Netherlands:CD-PPress. Gravemeijer, K. (1998). Developmental Research as a Research Method. In J. Kilpatrick & A. Sierpinska(Eds.), Mathematicseducation as a research domain:A searchfor identity(An ICMI study) (book 2, pp. 277-295). Dordrecht,Netherlands:Kluwer. Gravemeijer,K. (1999). How emergent models may foster the constitutionof formal mathematics. MathematicalThinkingand Learning, 1(2), 155-177. Latour,B. (1990). Drawingthingstogether.In M. Lynch& S. Woolgar(Eds.),Representationsin scientific practice. Cambridge,MA: The MIT Press.

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Meira,L. (1998). Makingsense of instructionaldevices:The emergenceof transparencyin mathematical activity.Journalfor Research in MathematicsEducation,29(2), 121-142. Pirie, S., & Kieren,T. (1994). Growthin mathematicalunderstanding:How can we characterizeit and how can we representit? EducationalStudies in Mathematics,26, 61-86. Presmeg, N., Ddrfler,W., Elbers, E., van Amerom, B., Yackel, E., Underwood,D., et al. (in press). Realistic mathematicseducationresearch:Leen Streefland'swork continues. EducationalStudies in Mathematics. Saxe, G. (1991). Cultureand cognitivedevelopment:Studiesin mathematicalunderstanding.Hillsdale, NJ: Erlbaum. Stephan,M., & Rasmussen,C. (in press). Classroommathematicalpracticesin differentialequations. Journal of MathematicalBehavior. Walkerdine,V. (1988). Themasteryof reason: Cognitivedevelopmentand theproductionof rationality. London:Routledge.

Chapter 7

Conclusion Michelle Stephan,Purdue UniversityCalumet

The goals of this monographwere both theoretical and pragmaticin nature. Theoretically,we examined three main researchquestions:(1) How do we characterizelearning,in particular,learningto measure,in termsof bothsocial andindividual processes? (2) What role do tools play in supportingstudents'measuring development?(3) How do we accountfor our presence and influence as instructionaldesignersin the learningprocess?Eachof theseresearchquestionswas recast as a themeandexploredthroughoutthe monograph'schapters.Stephan,in chapter 2, used priorresearchon students'developmentof measuringconceptionsboth to reveal the motivationfor furtherinvestigationof each theme and to develop our theoreticalposition on each. Throughoutthis monograph,the emergentperspective was describedas a view thatcoordinatesboth social and individualaspects of learningandtakeslearningwith tools as centralto mathematicaldevelopment.The emergent perspective was cast as a view that could expand prior measurement studies that did not take social context or tool use into account duringanalyses. Further,the heuristicsof Design Researchwere outlined to describe the instructionaldesigntheoryunderlyingthe researchdocumentedin the monograph.Design Researchwas posed as an alternativeto design theoriesused duringpriormeasurement studies. In the thirdchapter,we describedthe methodology that guided the analysispresentedin the fifth chapter.The analysisconsistedof documentingboth the collective andindividuallearningof the classroomcommunityandthe evolving tool use thatemerged duringthe course of the measurementsequence. Pragmatically,we hopedto extendpriormeasurementinvestigationsby outlining a potentiallyviableinstructionaltheorythatcouldbe adaptedby classroomteachers. In the fourthand sixth chapters,Gravemeijerand his colleagues summarizedthe processof conjecturinga possiblelearningtrajectoryandusinganalysesof students' learningto revisethe measurementsequencebothin actionandretrospectively.This processof experimentationyieldeda potentiallyviableinstructionaltheoryon linear measurementand arithmeticthat consisted of the tools that might be used along with the conceptual developmentthat evolves simultaneously.Gravemeijerand others went on to describe how teachers might adaptthe sequence to their own students. In the sectionsthatfollow, I concludethe monographby revisitingandclarifying some importantaspects of each of the three themes. I discuss each theme in turn and close the monographby pointingto futureresearchableissues.

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THEME 1: LEARNINGAS A SOCIALAND INDIVIDUALPROCESS The first main theme discussed in the literaturereview concerneddocumenting students'learning,in particular,of measurementconceptions,as it developsin social context. From the literaturereview, I argued that current research advocates building on cognitive analyses of students'measurementdevelopmentto include a social perspective on learning (cf. Bauersfeld, Krummheuer,& Voigt, 1988; Krummheuer,2000; Vygotsky, 1978). Rather than swing from one side of the social-versus-individualdichotomy to the other, we advocateda coordinationof these two perspectives. In this regard,we were indebted to the pioneers of coordinatingsocial andcognitiveperspectives(e.g., Bauersfeldet al., 1988;Cobband Yackel, 1996). Cobb, Stephan,McClain, & Gravemeijer(2001) describethis coordinationbest by taking the relationbetween social and individualperspectives as reflexive. The relationshipbetween these two perspectivesis extremely strong in thatone does not exist withoutthe other. The analysis presentedin the fifth chapterof this monographwas one example of such a coordinationof perspectives.On the one hand,we took a social perspective to documentthe mathematicalinterpretationsthatbecame taken-as-sharedin the public discourse. The teacher's and students' interpretationswere seen as constitutingthatwhich becamenormativein the publicdomain-a social perspective. On the other hand, the case studies presentedtwo students'developmentan individualperspective-yet their learningwas cast as acts of participationin, andcontributionto, the evolving normativeactivity. Cobbet al. (2001) offer more detail concerning aspects of coordinating these two perspectives. The results, however, of the complementaryanalysis served as documentationof the evolving classroommathematicalpracticesanda diversityof ways of participatingin them. Coordinatingthese two perspectivesgave a fuller pictureof learningthan either one of the perspectives could do alone. For example, if we presentedonly the analysisof the classroommathematicalpractices,one mightwonderhow individual studentsparticipatedin these practices.Althougha mathematicalpracticeis clearly a social construct,we emphasizedthatstudents'reasoningswere consideredas acts of participationin the practiceandconstitutedthe reasoningfromwhich the practices arose. Therefore,an analysis of mathematicalpracticesis complementedby individual students' reasoning.If we had documentedonly individualstudents' reasoning,the analysiswould not have consideredthe social environmentin which these students'learningoccurred.As the analysisshowed,the reorganizationsthat these two studentsmade did not occur apartfrom the classroomconversationsin which they participated.In the next sections, I addresssome remainingconcerns aboutcoordinatingthe two perspectives. MathematicalPractices I wish to clarifytwo mainpointsconcerninganalysisof mathematicalpractices. The first concernsthe methodologicalcriteriathatone adoptswhen documenting the emergence of mathematicalpractices. As we detailed in the third and fifth

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chaptersof the monograph,the absence of a backingin students'argumentations was one indicationthata practicehadbecome established.This statementdoes not imply that any absence of backings suggests that a practice is taken-as-shared. Mathematicaljustification,or backings,mustbe exploredpubliclyfirstbeforethey can dropout of the discourse and indicatea shift in practices.In other words, the absence of discussion aboutan interpretationdoes not necessarilyimply taken-assharedunderstanding.When backingsthathave been debated,however, publicly thenwe can agreethatthisabsencehas signifidropout of students'argumentations, cance. A second indicationthata practiceis relativelystableis thatwhen a student offers a contributionthat appearsto violate a practice,the other membersof the classroomtreatthe proposalas a breach.Methodologicallyspeaking,this statement implies thatthe social and sociomathematicalnormsmust also be relatively stable for the researcherto form such conclusions as those mentioned.If studentsarenot obliged to explain theirthinkingor to challenge one anotherwhen a differencein interpretationoccurs, then the researchercannotassume thatthe absence of backings indicatesa shift in taken-as-sharedlearning. A second point I wish to clarify concernsthe use of a commonly adoptedcriterion forjudging the establishmentof a mathematicalpractice.Underthis criterion, a questionsuch as "Whendo we decide thata mathematicalpracticeis established, when 90 percentof the studentssharethe same view?" is asked. We would argue that such a question imposes an individualperspectiveon a social phenomenon. To answerthe question, one must assess each student's currentcognitive understanding and determine whether 90 percent of the students share a sufficiently similar understanding.Our perspective is that a mathematicalpracticedoes not assumethateveryone"shares"an interpretation.In fact, to claim the establishment of practices makes no claims about how any individual student is reasoning. Rather,the establishmentof a mathematicalpracticemeansthatthe communityhas for thepurposesof continuedcommuacceptedas taken-as-sharedan interpretation nication and progression of its members' reasoning. Therefore, we would not consider whethera certainpercentageof studentssharedthe same interpretation (see also Bowers and Nickerson, 2001). Instead, we analyze the mathematical reasoningthat becomes normativein the public discourse, using the two criteria discussedin the precedingparagraphto determinethe collective interpretationthat is taken-as-shared. Case Studies The two case studies presentedin the fifth chapterdocumentdifferentways of participatingin mathematicalpractices.By documentingthese differentways, we hoped to account for the diversity of students'reasoning as they participatedin, and contributedto, the local classroom community. A traditionalcognitive case study would have documentedthe shifts in Meagan's and Nancy's reasoning,as well, but this evolution would have been cast purelyin termsof theircognitive reorganizations.In other words, traditionalcase studies would have done well at

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documentingthe shifts thatoccurredin students'mindsbut would not necessarily have accountedfor the students' developmentin terms of social interaction.We arguethatcognitive analyses can be a useful but incompleteaccountof students' development.We presentedstrongevidence in the fifth chapterto suggest thatnot only did Meagan and Nancy make substantialprogressin theirthinking,but they did so as theyparticipatedin theirclassroomcommunity.Oftenduringthe analysis of the mathematicalpractices,we saw thatNancy andMeaganmadeimportantreorganizationsduringthe course of a whole-class discussion or small-groupinteraction. Therefore,an analysis that documents only students'cognitive reorganizations stops short of explaining the context within which these reorganizations occurred.In this spirit,the case studies were presentedalongside, and sometimes within, the analysis of classroom mathematicalpractices. The practices of the communityandindividualstudents'reasoningco-evolved, andpresentingthe two analyses togetherwas an attemptto illustratethe simultaneityof social and individual learningprocesses. The Pragmatics of the Coordination We have arguedtheoreticallythatcoordinatingsocial andindividualperspectives of learning is essential for understandingstudents' conceptual development. Pragmatically,such coordinationhas importantconsequences.The analysis of the mathematicalpracticesin the fifth chapternot only presentedan account of the learningof the classroomcommunitybut also outlinedthe emergenceof a potentially powerful instructionaltheory on linearmeasurement.The learningdetailed in the measuring practice analysis summarizedthe mathematicalcontent that emerged duringthe enactmentof the sequence of instruction.This analysis was helpful for Gravemeijerand his colleagues, who, in chapter6, revised the hypotheticallearningtrajectoryto presentan instructionaltheoryon linearmeasurement. Gravemeijerand otherswere able to drawon the analysisto anticipateboth major shiftsin collectivelearningandmathematicaltopicsof conversationthatcanbe initiated by the teacherduringthe enactmentof the measurementsequence. The case studies are helpful for teacherswho want to enact a similar measurement sequence,becausethe studiesdetailthe contributionsmadeby studentsas the sequenceis enacted.As I havearguedbefore,diversityin students'thinkingis crucial for the growthof the communitybecauseteachersneed to drawfrom a wide range of contributionsfor productivediscussions.Thusthe case studiesprovideteachers with differingaccountsof students'developmentandhelpteachersbetteranticipate students'reasoningthatwill contributeto theirpedagogicalagenda. THEME2: TOOLUSE AS CENTRALTO MATHEMATIZATION The literature review in chapter 2 first documented, then challenged prior researchon students'developmentof measuringconceptionsandthe role of tools in supportinglearning. The measurementexperimentthat is the subject of this

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monograph was an excellent domain to explore this issue because measuring necessarilyincorporatesthe use of physical devices. Had we startedthe measurement instructionalsequence by presentingstudentswith a conventionalrulerand taughtthem how to use it, the studentswould have experiencedthe introduction of this tool in a top-downmanner.The tool would have been something separate from their current mathematicalexperiences, and students would have had to create meaning as they decoded the device. Rather,the approachwe took used informaltools to build on students'existing mathematicalunderstandingand led them to progressivelyconstructmeaningwith severaldevices priorto the "ruler." In this way, the studentsdevelopedfoundationalimagerythatgrew out of theirexperiencein a bottom-upmanner.At the heartof this approachwas the conviction that tool use and mathematicalmeaning develop simultaneously;tool use and meaningmakingwere viewed as reflexively related. Even thoughthe designerintendsfor the tools to be experiencedin a bottom-up manner,this situationdoes not always occur. For example, recall from chapter5 thatthe first introductionof Unifix cubes as measuringdevices was problematic; the task called for reasoning with the new tool in a mannerinconsistentwith the existing taken-as-sharedreasoning.Thus, the analysis of mathematicalpractices in the fifth chapterandthe ensuingchain of significationin chapter6 were significantin thattheydescribedthe learningthatoccurredas studentsreasonedwithtools. We have found an analysis that details the emerging chain of significationto be morehelpfulthanapproachesthatneglectedthe role of culturaltools, bothfor determining the progressionof tool use and for documentingthe changing interestsof the classroom. The chain of signification elaborated in chapter 6 detailed the evolving tool use more explicitly thanpreviousstudies were able to do and documentedthe importanceof includingseveraltools thatwere not originallyintended for the instructionalsequence. The chain of signification is importantfor a second reason. In their chapter, Gravemeijer and his colleagues described the emergent-models heuristic that undergirdsthe design of an instructionalsequence. The backboneof a sequence relies on the idea that studentslearnas they model theirmathematicalactivity. A sequence is typically organizedarounda global "model of-model for" shift. As Gravemeijerandothersnoted,however,the model can takeon differentmanifestations during the realizationof an instructionalsequence. In the example of the took several such measurementsequence,the "ruler,"the overarching model, forms, the smurf the measurement as the masking-taperecord,the footstrip, bar, strip,and so on. The chain of significationdetailednot only the manifestationsandprogression of the model during the experimentbut also the evolving taken-as-shared meaningand purposesof the classroomcommunity.Therefore,a chain of significationcanbe a helpfuldesignheuristic,in thatthe skeletonof a hypotheticallearning trajectorycan be organizedas an anticipatedchain of signification.As Table 6.1 indicated,a sequence can be organizedaccordingto the anticipatedtools thatwill be used and the imagery and the mathematicalactivities, interests,and practices thatcorrespondto them.

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The analysisof the classroommathematicalpracticesin the fifth chaptershowed thatstudents'learningdid not occur apartfrom reasoningwith tools. The chain of significationprovideda detailed accountof thatlearningfrom the perspectiveof the role thatvarioustools played in students'developmentof measuringconceptions. This accountaddedto priormeasurementstudiesthatdid not analyzethe role thatreasoningwith tools plays in supportinglearning. THEME3: DESIGN RESEARCH We have argued throughoutthe monographthat mathematicalpractices and tools are inseparableaspects of learning. We have also argued that an instructional designer capitalizes both on mathematicalpracticesto form a conjectured, or realized, learning trajectoryand chains of signification to anticipate how a model might take on life duringinstruction.The instructionaltheoryon measurement as detailed by Gravemeijerand his colleagues in the sixth chapter is the productof such analyses and experimentation.The chaptersin this monograph are the next iterationin the long cycle of experimentationguided by the Design Research Cycle described in the second chapterby Stephan.The measurement experiment encompasses both phases of the cycle. In the Design Phase, the measurementsequence was promptedby and designed to enhance priorinstructional sequenceson arithmeticalreasoning.A hypotheticallearningtrajectorywas conjectured and subsequently tested in a classroom teaching experiment. The ResearchPhasewas the measurementclassroomteachingexperiment.The experiment consisted of testing and revising the instructionalsequence with a class of students and analyzing the learning that occurred as students reasoned with tools. This analysis then led Gravemeijerandhis coresearchersto make revisions to the sequence, the results of which they presented in the sixth chapter, thus completing anotheriterationof the Design Research Cycle. That Design Researchersare more than participantobservers during experimentation should be clear by now. In fact, we view our role as proactive in shaping the evolving learningthatoccurs in classrooms.As such, the analyses in this monographtied students'developmentto the decisions the researchersmade duringexperimentation.In other words, our accounts of students' learningwere tied to the means by which we supportedthat learning. This type of account provided a clearer picture as to the reasons instructionwas changed during and subsequent to experimentation.Thus Design Researchers,one of whom is the classroom teacher,have an importantrole in shapingthe mathematicalpractices of the classroom community,and this role, from a design perspective,was highlighted and accounted for during the analysis. This type of analysis stands in contrastwith priormeasurementstudies thatfocused on whetherstudentsscored betteron tests aftermeasurementtrainingwas conducted(see Stephan,chapter2 of this monograph). Finally, if we take seriously our contentionthat learningis inherentlysocial in nature,then we must emphasize thatthe measurementsequence is not replicable

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in its purestsense. Social perspectiveson learningwarnus thateach classroomis a different social setting and that an instructionalsequence can be realized in a varietyof ways. Ratherthanview this warningas a negative, Gravemeijerandhis colleagues arguedin chapter6 not to expect the instructionalsequenceto be replicable.Instead,they detaileda potentialinstructionaltheorythatteachersshouldadapt to theirparticularsetting by formulatingtheirown hypotheticallearningtrajectories. To aid in this endeavor,Gravemeijerandothersdetailthe mathematicalideas thatcan arisefor studentsas theyengagein the sequenceratherthanprovideteachers with a list of activitiesor tools thatthey can use duringinstruction.The importance of such a description is that it organized instructionon the basis of students' thinkingratherthan mathematicalcontent and it viewed the teacheras a professional decision makerratherthan an invariantof the social context. UNEXPLOREDTERRITORY Throughoutthis monographwe have examined a variety of researchsubjects, includinginvestigatingthe role that social situations,tools, and designersplay in supportingstudents'mathematicaldevelopment.In particular,we exploredthese issues in the context of an investigationin a first-gradeclassroomthatengaged in a measurementinstructionalsequence. Consequently,we were able to not only examine theoreticalideas but also develop an instructionaltheory in the domain of linear measurement.For all our ambitiousefforts, we are awarethat we have underemphasizedimportantaspects of classroom life. For example, althoughwe detailed the decision-making process of the researchers-who included the teacher-during experimentation,the role of the teacherwas not examinedto its full potential.Whereas powerful mathematicaltasks and tools are importantfor supportinglearning,a teacherwho understandsthe mathematicalintentof these componentsis more important.As seen duringthe analysesin this monograph,the teacherwas very skilledat initiatingandguidingclassroomdiscourseto bringmathematicallysophisticatedideas to the forefront.An analysis of this teacher'spedagogical practicesis vital to understandingthe way in which the teachermakes onthe-spotdecisionsthatcanpromotesophisticateddiscourseand,therefore,learning. Anotherareathatshouldbe furtherexploredfollowing this studyis the role that students' participationin outside cultural practices played in supportingtheir measuringdevelopment.Forexample, studentsparticipatein measuringpractices we definedsocial at homeor in otheroutside-of-schoolenvironments.Furthermore, at the outset to mean social interactionsat the level of the classroom,not as one'9s social status in the world. An interesting exploration would investigate how studentswith diversesocietalbackgroundsparticipateddifferentlyin the classroom mathematicalpractices. Although these lines of investigations were beyond the scope of this study, we believe they are extremelyimportantaspectsof classroom life andthatunderstandingthemwouldfostera strongermeasurementinstructional theory.

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REFERENCES

Bauersfeld,H., Krummheuer,G., & Voigt, J. (1988). Interactionaltheoryof learningandteachingmathstudies.In H-G. Steiner& A. Vermandel(Eds.),Foundations ematicsandrelatedmicroethnographical and methodologyof the discipline of mathematicseducation (pp. 174-188). Antwerp:Proceedings of the TME Conference. Bowers, J. S., & Nickerson,S. (2001). Identifyingcyclic patternsof interactionto studyindividualand collective learning.MathematicalThinkingand Learning,3, 1-28. Cobb, P., Stephan,M., McClain, K., & Gravemeijer,K. (2001). Participatingin mathematicalpractices. Journal of the LearningSciences, 10(1&2), 113-163. Cobb, P., & Yackel, E. (1996). Constructivist,emergent,and socioculturalperspectivesin the context of developmentalresearch.EducationalPsychologist, 31, 175-190. Krummheuer,G. (2000). Mathematicslearningin narrativeclassroomcultures:Studies of argumentation in primarymathematicseducation.For the Learningof Mathematics,20(1), 22-32. Vygotsky, L. (1978). Mind in society. Cambridge,MA: HarvardUniversityPress.